Operator assignment problem in aircraft assembly lines: a new planning approach taking into account economic and ergonomic constraints

Transcription

1 Operator assignment problem in aircraft assembly lines: a new planning approach taking into account economic and ergonomic constraints Dmitry Arkhipov, Olga Battaïa, Julien Cegarra, Alexander Lazarev May 12, 2018 D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 1/30

2 Overview 1 Context & Motivation 2 Operator assignment problem 3 Mathematical model 4 Numerical experiments 5 Conclusion D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 2/30

3 Overview 1 Context & Motivation 2 Operator assignment problem 3 Mathematical model 4 Numerical experiments 5 Conclusion D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 3/30

4 Considered problem A brief problem formulation There is an aircraft assembly line. How to schedule assembly tasks and assign them to operators optimally? How to satisfy precedence, resource, time and ergonomic constraints? D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 4/30

5 RCPSP Resource Constrained Project Scheduling Problem (RCPSP) Considers resources of limited availability and activities of known durations and resource utilization, linked by precedence relations. The problem consists of finding a schedule of minimal duration by assigning a start time to each activity such that the precedence relations and the resource availabilities are respected. The objective is to minimize the project makespan. Complexity The problem is NP-complete in a strong sense (Garey, Johnson 1975). D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 5/30

6 Industrial motivation Aircraft companies to reduce takt time; to minimize the number of human errors; to improve working conditions. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 6/30

7 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 the repetitive environment for other assembly lines e.g. automotive); consideration of cognitive and physical ergonomic factors. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 7/30

8 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. Battaïa, J. Cegarra, A. Lazarev 8/30

9 Ergonomics Physical ergonomics evaluation methods D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 9/30

10 Ergonomics Cognitive ergonomics Type of actions: motor vs cognitive. Worker s personal factors: skill, age, sex. Learning, fatigue and motivation effects. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 10/30

11 Operator assignment problem D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 11/30

12 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. Battaïa, J. Cegarra, A. Lazarev 12/30

13 Operator assignment problem Tasks r j release time; p j processing time; a jx amount of resource x R required to process task j; b js number of operators with skill s S required to process task j. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 13/30

14 Operator assignment problem Physical ergonomic risks M set of ergonomic risk evaluation methods; erg mj ergonomic score evaluated by method m M for one time unit of task j N; U mo an upper bound on total ergonomic impact for operator o evaluated by method m; D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 14/30

15 Overview 1 Context & Motivation 2 Operator assignment problem 3 Mathematical model 4 Numerical experiments 5 Conclusion D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 15/30

16 Constraint programming model for the aggregated demand Decision variables interval j interval variable associated to the execution of task j N, i.e. interval j = [S j, C j ); Objective function The objective is to find a schedule π with the minimal makespan i.e. min max (S π j(π) + p j ). (1) j N D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 16/30

17 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 : interval j = p j. (2) Task processing intervals must satisfy the precedence relations with time lags, i.e. e ji E : S j (π) + l ji S i (π). (3) D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 17/30

18 Constraint programming model for the aggregated demand Resource capacity constraints Resource capacity constraint: F (x, t) = a jx f (interval j, t), (4) j N where f (interval j, t) = 1 if t interval j and f (interval j, t) = 0 otherwise. Then resource capacity constraint can be formulated as x R, t : c x F (x, t). (5) D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 18/30

19 MIP model for Operator assignment problem Decision variables assign oj binary variable equals to 1 if operator o O assigned on task j N, otherwise assign oj = 0. Objective function The objective function is to minimize the highest ergonomic impact calculated for each pair (m M, o O). min max m M,o O j N assign oj erg mjso (6) D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 19/30

20 MIP model for Operator assignment problem Constraints For each task j N, the number of operators with skill s S has to be equal to b js j N, s S : o O:s o=s assign oj = b js. (7) The total ergonomic impact of the tasks assigned to the same operator o O measured by method m M has to be less than the defined critical level U mo, i.e. m M, o O : U mo j N erg mjso assign oj. (8) D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 20/30

21 MIP model for Operator assignment problem Incompatibility constraints Since the schedule of the tasks is known, the incompatible sets E of tasks can be defined, i.e. the sets of the tasks e that cannot be performed by the same operator. e E, o O : j e assign oj 1. (9) D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 21/30

22 Overview 1 Context & Motivation 2 Operator assignment problem 3 Mathematical model 4 Numerical experiments 5 Conclusion D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 22/30

23 Numerical experiments Implementation Software: IBM ILOG CPLEX 12.6 Processor: Intel(R) Core(TM) i GHz RAM: 16 GB D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 23/30

24 Numerical experiments Instance tasks; 7 operators with 3 skills; 3 ergonomic evaluation methods. Optimal solution found in 18 minutes. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 24/30

25 Numerical experiments Instance tasks; 5 operators with 2 skills; 3 ergonomic evaluation methods. Optimal solution found in 20 minutes. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 25/30

26 Numerical experiments Gantt chart for the optimal solution for instance 2 D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 26/30

27 Overview 1 Context & Motivation 2 Operator assignment problem 3 Mathematical model 4 Numerical experiments 5 Conclusion D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 27/30

28 Conclusion Obtained results Operator assignment problem for aircraft assembly line subject to ergonomic constraints was considered; Constraints programming and Integer linear programming models were developed; Optimal solutions were found for two industrial instances in reasonable time. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 28/30

29 Conclusion Future perspectives Consideration of cognitive and physical ergonomic factors together. Evaluation of impacts of sequences of tasks. D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 29/30

30 Thanks! Questions? O. Battaïa D. Arkhipov, O. Battaïa, J. Cegarra, A. Lazarev 30/30