Balancing two-sided U-type assembly lines using modified particle swarm optimization algorithm

Abstract

In this paper, a new two-sided U-type assembly line balancing (TUALB) procedure and a new algorithm based on the particle swarm optimization algorithm to solve the TUALB problem are proposed. The proposed approach minimizes the number of stations for a given cycle time as the primary objective and it minimizes the number of positions as a secondary objective. The proposed approach is illustrated with an example problem. In order to evaluate the efficiency of the proposed algorithm, the test problems available in the literature are used. The experimental results show that the proposed approach performs well.

Appendix: Notations used in the paper

n :

Number of tasks

i, j, h :

Index for task $(1\le i, j, h \le n)$

Maxiter :

Number of iterations

m :

Index for iteration $(1\le m\le Maxiter)$

np :

Number of particles

p :

Index for particle $(1\le p \le np)$

sd :

Index for side of the assembly line (0 -> left, 1 -> right)

CLF :

Consists of tasks whose predecessors have already been assigned or tasks which have no predecessor

CLB :

Consists of tasks whose successors have already been assigned or tasks which have no successor

$c_{1}, c_{2}$ :

Two positive constants indicating cognition and social learning factors

$r_{1}, r_{2}$ :

Two random real numbers drawn from uniform distribution U[0–1]

w :

Inertia weight, controls the impact of previous velocity value on the new one

$X_{p,i,j,sd}$ :

The relative priority matrix, which consists of the values which show the selection probability of task j immediately after task i in the solution string p, for all $i,j\,where\,i\,and\,j\in \left\{ {1,2,\ldots ,n}\right\} \,and\,sd\,\in \left\{ {0,1} \right\}$

$V_{p,i,j,sd}$ :

The velocity matrix, which consists of the values which show the rate of the change at $X_{p,i,j,sd}$, for all $i,j\,where\,i\,and\,j\in \left\{ {1,2,\ldots ,n} \right\} \,and\,sd\,\in \,\left\{ {0,1} \right\}$

$PBX_{p,i,j,sd}$ :

The best relative priority vector found by each particle so far

$GBX_{i,j,sd}$ :

The best relative priority vector found by the swarm so far

$S_{p,i,k}$ :

Solution matrix is used to save detail solution vector for each task (i) of each particle (p)

$SR_{p,l}$ :

Solution result matrix is used to save each objective function value for each particle (p)

$PBS_{p,i,k}$ :

The best solution vector found by each particle (p) so far

$PBSR_{p,l}$ :

The best solution result vector (objective function values) found by each particle (p) so far

$GBS_{i,k}$ :

The best solution vector found by the swarm so far

$\hbox {G}BSR_{l}$ :

The best solution result vector (objective function values) found by the swarm so far

NP :

Index for position

NS :

Index for station

loc :

The selected location for assignment, (1 $=$ front_left, 2 $=$ front_right, 3 $=$ back_left, 4 $=$ back_right)

N(loc) :

Station indexes for each location, respectively

ST(loc) :

The station load including unavoidable idle times of each location

TS(i):

Side value of task i, (0 -> left, 1 -> right, 2 -> either)

AT :

Available time of each location

opposite :

opposite side of the currently used location

C :

Cycle time

$tt_{i}$ :

Task/operation time of each task, $i\in \{1, 2, {\ldots },n\}$

$tf_{i}$ :

Finishing time of task $i, i\in \{1, 2, {\ldots }, n\}$

P(i):

Set of immediate predecessors of task $i, i\in \{1, 2, {\ldots }, n\}$

LB :

Lower bound

LTotal :

The total task times of the L directional tasks

RTotal :

The total task times of the R directional tasks

ETotal :

The total task times of the E directional tasks