Mathematical models and migrating birds optimization for robotic U-shaped assembly line balancing problem

Abstract

Modern assembly line systems utilize robotics to replace human resources to achieve higher level of automation and flexibility. This work studies the task assignment and robot allocation in a robotic U-shaped assembly line. Two new mixed-integer programming linear models are developed to minimize the cycle time when the number of workstations is fixed. Recently developed migrating birds optimization algorithm is employed and improved to solve large-sized problems. Problem-specific improvements are also developed to enhance the proposed algorithm including modified consecutive assignment procedure for robot allocation, iterative mechanism for cycle time update, new population update mechanism and diversity controlling mechanism. An extensive comparative study is carried out to test the performance of the proposed algorithm, where seven high-performing algorithms recently reported in the literature are re-implemented to tackle the considered problem. The computational results demonstrate that the developed models are capable to achieve the optimal solutions for small-sized problems, and the proposed algorithm with these proposed improvements achieves excellent performance and outperforms the compared ones.

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Acknowledgements

The first author acknowledges the financial support from the National Natural Science Foundation of China under Grant 61803287 and China Postdoctoral Science Foundation under grant 2018M642928.

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Correspondence to Mukund Nilakantan Janardhanan.

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Appendix: Pseudocodes of the tested algorithms and utilized operators

Appendix: Pseudocodes of the tested algorithms and utilized operators

The implemented and compared algorithms in this paper are: simulated annealing algorithm (SA), particle swarm optimization (PSO1), particle swarm optimization (PSO2), genetic algorithm (GA), teaching–learning-based optimization algorithm (TLBO), artificial bee colony algorithm (ABC), discrete cuckoo search (DCS). The psuedocodes of the implemented algorithms are listed in sequence as follows.

SA algorithm

The main procedure of SA algorithm is presented in Algorithm 1.

figured

PSO1 algorithm

PSO1 utilizes the random-key method to tackle the considered discrete problem. Suppose that there are nine tasks and the original \( x_{i,0} \) is [0.42, 0.68, 0.35, 0.01, 0.70, 0.25, 0.79, 0.59, 0.63], the floating-point vector is transferred into the task permutation in the following method. Here, the tasks with lowest value have the highest priority and thus task 4 with the lowest value 0.01 is allocated to the first position in the task permutation; the task 6 with the second lowest value 0.25 is allocated to the second position in the task permutation. Subsequently, the floating-point vector is transferred into the task permutation [19]. In short, by employing the random-key method, the original evolution techniques in PSO can be applied directly, where the task permutation is applied in decoding to obtain the fitness values.

figuree

PSO2 algorithm

PSO2 is a discrete PSO utilizing neighbor operator and crossover operator to update the population. The main procedure of PSO2 is presented in Algorithm 3.

figuref

GA algorithm

The main procedure of GA algorithm is presented in Algorithm 4.

figureg

TLBO algorithm

TLBO utilizes the random-key method to tackle the considered discrete problem. Suppose that there are nine tasks, nine floating-point numbers between 0 and 1 are generated for one individual. For a vector \( \psi \) = (0.42, 0.68, 0.35, 0.01, 0.70, 0.25, 0.79, 0.59, 0.63), the floating-point vector is transferred into the task permutation in the following method. Here, the tasks with lowest value have the highest priority and thus task 4 with the lowest value 0.01 is allocated to the first position in the task permutation; the task 6 with the second lowest value 0.25 is allocated to the second position in the task permutation. Subsequently, the floating-point vector is transferred into the task permutation [19]. In short, by employing the random-key method, the original evolution techniques in TLBO can be applied directly, where the task permutation is applied in decoding to obtain the fitness values.

figureh

DCS algorithm

The main procedure of DCS algorithm is presented in Algorithm 5.

figurei

ABC algorithm

The main procedure of ABC algorithm is presented in Algorithm 6.

figurej

Neighbor operator

The implemented algorithms employ the insert operator and swap operator as the neighbor operator, and the two-point crossover operator to combine two individuals. Specifically, the insert operator or swap operator is randomly selected and applied to obtain one new task permutation. Figure 5 depicts an example of insert operator and swap operator with nine tasks, and Fig. 6 depicts an example of utilizing two-point crossover operator.

Fig. 5
figure5

Insert operator and swap operator

Fig. 6
figure6

Two-point crossover operator

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Li, Z., Janardhanan, M.N., Ashour, A.S. et al. Mathematical models and migrating birds optimization for robotic U-shaped assembly line balancing problem. Neural Comput & Applic 31, 9095–9111 (2019). https://doi.org/10.1007/s00521-018-3957-4

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Keywords

  • Robotic U-shaped assembly line
  • Integer programming
  • Migrating birds optimization
  • Artificial intelligence