# Empirical working time distribution-based line balancing with integrated simulated annealing and dynamic programming

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## Abstract

According to the Industry 4.0 paradigms, the balancing of stochastic production lines requires easily implementable, flexible and robust tools for task to workstations assignment. An algorithm that calculates the performance indicators of the production line based on the convolution of the empirical density distribution functions of the working times and applies dynamic programming to assign tasks to the workstations is proposed. The sequence of tasks is optimised by an outer simulated annealing loop that operates on the set of interchangeable task-pairs extracted from the precedence graph of the task-ordering constraints. Eight line-balancing problems were studied and the results by Monte Carlo simulations were validated to demonstrate the applicability of the algorithm. The results confirm that our methodology does not just provide optimal solutions, but it is an excellent tool in terms of the sensitivity analysis of stochastic production lines.

## Keywords

Line balancing Simulated annealing Dynamic programming## List of symbols

- SALBP
Simple assembly line balancing problem

Probability distribution function

- CDF
Cumulative density function

- DP
Dynamic programming

- SA
Simulated annealing

## Notations

- \(V = \{1, 2, \ldots , n\}\)
Set of tasks

- \(k = 1, 2, \ldots , m\)
Workstations

- \(T_j\)
Deterministic task time of task

*j*, \(j\in V\)- \(P=(p_1, p_2, \ldots , p_n), p_i \in V\)
Partially ordered sequence of tasks

- \(P_{(i:j)}=(p_i, p_{i+1}, \ldots , p_j)\)
Sub-sequence of

*P*, \(i<j\)- \(S_k\)
Sub-sequence of tasks assigned to workstation

*k*- \(T(S_k) = \sum _{j \in S_k}T_j\)
Deterministic station time

- \(C = \max _k T(S_k)\)
Cycle time

- \(IT(S_k) = C - T(S_k)\)
Idle time of workstation

*k*- \(T_{sum} = \sum _{j}T_j\)
Lead time

- \(T_{max} = \max _{j}T_j\)
Longest task time

- \(E(S_k) = \frac{T(S_k)}{C}\)
Efficiency of workstation

*k*- \(E = \frac{T_{sum}}{mC}\)
Efficiency of assembly line

- \(SX=\sqrt{\sum _{k=1}^m(C-T(S_k))^2}\)
Smoothness index (Moodie 1965)

*M*(*i*,*j*)Precedence matrix, 1 if task

*i*precedes*j*- \(M^*\)
Transitive closure of precedence matrix

- \(f_j(t), f_{P_{(i:j)}}(t), f_{S_k}(t)\)
Discrete PDF of task

*j*, sequence \(P_{(i:j)}\) and workstation*k*- \(F_j(t), F_{P_{(i:j)}}(t), F_{S_k}(t)\)
Discrete CDF of task

*j*, sequence \(P_{(i:j)}\) and workstation*k*- \(\alpha \)
Confidence level, e.g. 0.9 is equal to 90%

- \(T_\alpha (j), T_\alpha (P_{(i:j)}), T_\alpha (S_k)\)
Expected duration of task

*j*, sequence \(P_{(i:j)}\) and workstation*k*with confidence level \(\alpha \)- \(E^s(S_k)\)
Stochastic efficiency of workstation

*k*- \(E^s\)
Stochastic efficiency of the line

- \(IT^s(S_k)\)
Stochastic idle time of workstation

*k*- \(SX^s(S_k)\)
Stochastic smoothness index of workstation

*k*- \(SX^s\)
Stochastic smoothness index of the line

- \(S_j(t), S_{P_{(i:j)}}(t), S_{S_k}(t)\)
In-progress function of task

*j*, sequence \(P_{(i:j)}\) and workstation*k*.- \(S_j'(t), S_{P_{(i:j)}}'(t), S_{S_k}'(t)\)
Real in-progress function of task

*j*, sequence \(P_{(i:j)}\) and workstation*k*.- \(cost(k,p_j)\)
Element of cost matrix that belongs to workstation

*k*and task \(p_j\)*B*1First lower bound of cycle time, expressed in Eq. 20

*B*2Second lower bound of cycle time, expressed in Eq. 21

- \(\varDelta _1\)
Function calculates the expected cycle time, if a new workstation is used for the new task, expressed in Eq. 22

- \(\varDelta _2\)
Function calculates the expected cycle time, if an old workstation is used for the new task, expressed in Eq. 23

- \(W = (w_1, w_2 \ldots , w_e)\)
Task-workstation assignment path

- \(P^i\)
Sequence in iteration

*i*- \(P^p\)
Previous sequence

- \(C_i\)
Cycle time in the case of \(P^i\)

- \(C_{sub}\)
Sub-solution for cycle time

- \(C_{Best}\)
Best solution for cycle time

*Temp*Temperature of simulated annealing

- \(\tau \)
Temperature reduction rate

*I*Number of iterations in simulated annealing

- \(I_{sub}\)
Number of sub-iterations in simulated annealing

- \(\varDelta \)
Difference between \(C_i\) and \(C_{sub}\)

- \(tr_{MC}\)
Number of Monte Carlo simulations

## Mathematics Subject Classification

06A06 11B99 44A35 60G25 62E17 62G30 62N02 90C27 90C39## Notes

### Acknowledgements

This research was supported by the National Research, Development and Innovation Office NKFIH, through the project OTKA-116674 (Process mining and deep learning in the natural sciences and process development). Daniel Leitold was supported by the ÚNKP-17-3 New National Excellence Program of the Ministry of Human Capacities.

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