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Bacterial Foraging Optimization Algorithm for assembly line balancing

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Abstract

Assembly line balancing is the problem of assigning tasks to workstations by optimizing a performance measure while satisfying precedence relations between tasks and cycle time restrictions. Many exact, heuristic and metaheuristic approaches have been proposed for solving simple straight and U-shaped assembly line balancing problems. In this study, a relatively new optimization algorithm, Bacterial Foraging Optimization Algorithm (BFOA), based heuristic approach is proposed for solving simple straight and U-shaped assembly line balancing problems. The performance of the proposed algorithm is evaluated using a well-known data set taken from the literature in which the number of tasks varies between 7 and 111, and results are also compared with both an ant-colony-optimization-based heuristic approach and a genetic-algorithm-based heuristic approach. The proposed algorithm provided optimal solutions for 123 out of 128 (96.1 %) test problems in seconds and is proven to be promising.

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Correspondence to Yakup Atasagun.

Appendices

Appendix 1: Mathematical formulation of SALB

Notation

i, r, s :

Task

j :

Station

C :

Cycle time

M max :

Maximum number of stations

n :

Number of tasks

t i :

Processing time of task i

S :

Set of precedence relations

(r,s) ∈ S :

A precedence relation; task r is an immediate predecessor of task s

x ij :

1, if task i is assigned to station j; 0, otherwise

The mathematical model that has been presented by Talbot and Patterson [33] for SALB is as follows:

$${\text{Minimize}} \mathop \sum \limits_{j = 1}^{{M_{ \hbox{max} } }} jx_{nj}$$
(6)
$$\mathop \sum \limits_{j = 1}^{{M_{ \hbox{max} } }} x_{ij} = 1 \quad \forall i$$
(7)
$$\mathop \sum \limits_{i = 1}^{n} t_{i} x_{ij} \le C \quad \forall j$$
(8)
$$\mathop \sum \limits_{j = 1}^{{M_{\hbox{max} } }} j (x_{rj} - x_{sj} )\le 0\quad \forall \left( {r,s} \right) \in S$$
(9)
$$x_{ij} \in \left\{ {0,1} \right\}\quad \forall i; \forall j$$
(10)

Equation (6) is the objective function of the mathematical model. The objective function minimizes the number of stations by assigning the last task to the earliest possible station. Equation (7) ensures that a task is assigned at least and at most one station. Equation (8) guarantees that total processing times of tasks assigned to a station do not exceed the cycle time. Equation (9) ensures that assignments do not violate the precedence relations by assigning a predecessor task to an earlier or the same station with its successor. Finally, (10) indicates that all x ij variables in the model are binary variables.

Appendix 2: Mathematical formulation of SULB

Notation

i, r, s :

Task

j :

Station

C :

Cycle time

M min :

Theoretical minimum number of stations

M max :

Maximum number of stations

n :

Number of tasks

t i :

Processing time of task i

S :

Set of precedence relations

(r,s) ∈ S :

A precedence relation; task r is an immediate predecessor of task s

x ij :

1, if task i is assigned to station j and to the front side of the line; 0, otherwise

y ij :

1, if task i is assigned to station j and to the back side of the line; 0, otherwise

z j :

1, if station j is utilized; 0, otherwise

The mathematical model that has been presented by Urban [10] for SULB is as follows:

$${\text{Minimize}} \mathop \sum \limits_{{j = \left[ {M_{\hbox{min} } } \right] + 1}}^{{M_{\hbox{max} } }} z_{j}$$
(11)
$$\mathop \sum \limits_{j = 1}^{{M_{\hbox{max} } }} (x_{ij} + y_{ij} ) = 1\quad \forall i$$
(12)
$$\mathop \sum \limits_{i = 1}^{n} t_{i} (x_{ij} + y_{ij} ) \le C \quad j = 1,2, \ldots ,[M_{\hbox{min} } ]$$
(13)
$$\mathop \sum \limits_{i = 1}^{n} t_{i} (x_{ij} + y_{ij} )\le Cz_{j} \quad j = [M_{\hbox{min} } ]+ 1, \ldots , [M_{\hbox{max} } ]$$
(14)
$$\mathop \sum \limits_{j = 1}^{{M_{\hbox{max} } }} (M_{\hbox{max} } - j + 1 ) (x_{rj} - x_{sj} )\ge 0 \quad \forall \left( {r,s} \right) \in S$$
(15)
$$\mathop \sum \limits_{j = 1}^{{M_{\hbox{max} } }} (M_{\hbox{max} } - j + 1 ) (y_{sj} - y_{rj} )\ge 0\quad \forall \left( {r,s} \right) \in S$$
(16)
$$x_{ij} ,y_{ij} ,z_{j} \in \left\{ {0,1} \right\}\quad \forall i; \forall j$$
(17)

Equation (11) is the objective function of the mathematical model minimizes the number of utilized stations. Equation (12) ensures that all tasks are assigned and a task is assigned at least and at most one side of one station. Equations (13) and (14) guarantee that total processing times of tasks assigned to a station do not exceed the cycle time. Equations (15) and (16) ensure that assignments do not violate the precedence relations. Finally, (17) indicates that all x ij , y ij and z j variables in the model are binary variables.

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Atasagun, Y., Kara, Y. Bacterial Foraging Optimization Algorithm for assembly line balancing. Neural Comput & Applic 25, 237–250 (2014). https://doi.org/10.1007/s00521-013-1477-9

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