# An ant colony optimisation algorithm for balancing two-sided U-type assembly lines with sequence-dependent set-up times

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## Abstract

Some practical arrangements in assembly lines necessitate set-up times between consecutive tasks. To create more realistic models of operations, set-up times must be considered. In this study, a sequence-dependent set-up times approach for two-sided u-type assembly line (TUAL) structures is proposed for the first time. Previous studies on TUAL have not included set-up times in their analyses. Furthermore, an algorithm based on the Ant Colony Optimization (ACO) algorithm, which is using a heuristic priority rule based procedure has been proposed in order to solve this new approach. In this paper, we look at the sequence-dependent set-up times between consecutive tasks and consecutive cycles, called the “forward set-up time” and the “backward set-up time”, respectively. Additionally, we examine the “crossover set-up time”, which arises from a new sequence of tasks in a crossover station. In order to model more realistic assembly line configurations, it is necessary to include sequence-dependent set-up times when computing all of the operational times such as task starting times and finishing times as well as the total workstation time. In this study, the proposed approach aims to minimize the number of mated-stations as the primary objective and to minimize the number of total workstations as a secondary objective. In order to evaluate the efficiency of the proposed algorithm, a computational study is performed. As can be seen from the experimental results the proposed approach finds promising results for all literature-test problems.

## Keywords

Assembly line balancing U-type assembly lines two-sided assembly lines sequence-dependent set-up times ant colony optimization priority rules## Notations

*IS*Iteration size (number of iterations)

*iter*Iteration index \( \left( {1 \le iter \le IS} \right) \)

*CS*Colony size

*a*Colony index \( \left( {1 \le a \le CS} \right) \)

*n*Number of tasks

*i,j*Task index \( \left( {1 \le i,j \le n} \right) \)

*CL*A list composed of candidate tasks

- \( PM_{i,j} \)
Precedence matrix which keeps the precedence relations between all tasks

- \( TM_{i,m } \)
Task matrix which keeps the required values of each task

- \( \tau_{t,i} \)
Pheromone matrix saves real numbers which indicate the pheromone trail intensity of the task

*i*stored in the*t*^{th}task assignment process- \( \eta_{a,t} \)
Heuristic information matrix saves one of the six different heuristic information which is required to calculate the selection probability (

*P*) of*t*^{th}task assignment process for the ant*a*- \( S_{a,i,k} \)
Solution matrix saves detailed solutions for each task (

*i*) of each ant- \( SR_{a,l} \)
Solution Result matrix saves objective function values for each ant

*NP*Position index

*NS*Station index

*loc*Assignment locations, (

*loc*= 1,2, 3, 4)*pos*The selected position for assignment, (

*pos*= 1, 2, …, pos^{max})*C*Cycle time

- \( t_{i} \)
Task time of each task, \( i \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( fs_{i,j} \)
Forward set-up time for all \( i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( bs_{i,j} \)
Backward set-up time for all \( i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( cs_{i,j} \)
Crossover set-up time for all \( i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( FSM_{i,j} \)
Forward set-up matrix consists of the setup values between each task, for all \( i,j \;\;{\text{where}}\; i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( BSM_{i,j} \)
Backward set-up matrix consists of the setup values between each task, for all \( i,j \;\;{\text{where}}\; i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( CSM_{i,j} \)
Crossover set-up matrix consists of the setup values between each task, for all \( i,j \;\;{\text{where}}\; i,j \in \left\{ {1,2, \ldots ,n} \right\} \)

- \( {\text{F}}_{pos,loc} \)
The feasibility value of the current assignment operation at position

*pos*and location*loc*- \( RT_{pos,loc} \)
Remainder time of the current assignment operation at position

*pos*and location*loc*- \( X_{r,i} \)
It is used to save all of the priority rule values, which are determined in the initialization step, for all tasks

*i*in*CL*- \( pr_{i} \)
It is used to save all of the calculated relative priority rule value of each candidate task

- \( P_{i} \)
The selection probability value of task

*i*. It is calculated using the ant’s pheromone value and the selected priority rule- \( SP_{l} \)
Cumulative selection probability matrix

## Notes

### Acknowledgement

This research was supported by Scientific Research Fund of Erciyes University under the contract no: FBA-2017-7349.

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