A modified particle swarm optimization algorithm to mixed-model two-sided assembly line balancing

Abstract

In this paper, a new modified particle swarm optimization algorithm with negative knowledge is proposed to solve the mixed-model two-sided assembly line balancing problem. The proposed approach includes new procedures such as generation procedure which is based on combined selection mechanism and decoding procedure. These new procedures enhance the solution capability of the algorithm while enabling it to search at different points of the solution space, efficiently. Performance of the proposed approach is tested on a set of test problem. The experimental results show that the proposed approach can be acquired distinguished results than the existing solution approaches.

Appendices

Appendix 1: Notations used in model formulations

$i,j,h,p$ :

Index for task

Maxiter :

Number of iterations

iter :

Index for iteration

ns :

Number of swarms in the population

$x$ :

Index for swarm

np :

Number of particles in the swarms

$y$ :

Index for particle

$S_{x,y}$ :

Solution and side string number $y$ in the swarm $x$

$M$ :

Number of product models

$m$ :

Product model index

List :

List of candidate tasks

${\textit{RPW}[i]}$ :

Ranked positional weight for each task $i = 1, 2,\ldots , n$

${\textit{Sn}[i]}$ :

Number of successors for each task $i = 1, 2,\ldots , n$

${\textit{Pl}[i]}$ :

Priority list for each task $i = 1, 2,\ldots , n$

${\textit{Tt}[m]}$ :

Cumulative task times for each product model $m = 1, 2,\ldots , M$

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

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

$L_{best}$ :

Particle with the best solution value in the current swarm

$L_{worst}$ :

Particle with the worst solution value in the current swarm

$G_{best}$ :

Particle with the best solution value in the current population

$G_{worst}$ :

Particle with the worst solution value in the current population

$\textit{FWPM}_{[i]}$ :

The first walk probability matrix, which consists of the values which show the selection probability of a task as the first task in the solution string, $\hbox {where }i\in \left\{ {1,2,\ldots ,n} \right\}$

$\textit{JPPM}_{[i,j]}$ :

The joint probability position matrix, which consists of the values which show the selection probability of task $j$ immediately after task $i$ in the solution string, for all $i\ne j \hbox { where }i\hbox { and }j\in \left\{ {1,2,\ldots ,n} \right\}$

$\textit{JPVM}_{[i,j]}$ :

The joint probability velocity matrix, which consists of the values which show the rate of the change at $\textit{JPPM}[i,j]$, for all $i\ne j\hbox { where }i\hbox { and }j \in \left\{ {1,2,\ldots ,n} \right\}$

Elit_List :

List for the best solution of algorithm

NM :

Index for mated station

NL, NR :

Indexes for left and right stations, respectively

$\textit{mWL}_{NM}^1$ :

The station load including unavoidable idle times of the left-side station of the current mated-station for all $m\in \{1,2,\ldots ,M\}$

$\textit{mWL}_{NM}^2$ :

The station load including unavoidable idle times of the right-side station of the current mated-station for all $m\in \{1,2,\ldots ,M\}$

SL, SR :

Starting times of the task for the left and the right-side stations, respectively

$t_{im}$ :

Completion time of task $i$ for model $m$

$C$ :

Cycle time

$t_{im}^f$ :

Finish time of task $i$ for model $m$

$P(i)$ :

Set of immediate predecessors of task $i$

$\textit{TL}_{NM}^1$ :

Set of tasks which are assigned to the left-side station of the current mated-station

$\textit{TL}_{NM}^2$ :

Set of tasks which are assigned to the right-side station of the current mated-station

$\textit{WL}_{max}$ :

Maximum of the cumulative station load including unavoidable idle times of the last stations for all $m\in \{1,2,\ldots ,M\}$

WLL :

The station load including unavoidable idle times of the left-side station of the current mated-station for all $m\in \{1,2,\ldots ,M\}$

WLR :

The station load including unavoidable idle times of the right-side station of the current mated-station for all $m\in \{1,2,\ldots ,M\}$

Appendix 2: Task times for each product model of P205 problem

See Table 7.

Table 7 Problem P205