Designing an Assembly Line for Reliability

Figure 1 represents an assembly line balancing problem. This is a famous problem studied in Ray Wild. We have adopted it for the purpose of explaining how the proposed model works. The numerical figure within a circle represents the task number.

Figure 1: Precedence diagram of workelements.

In Table 1, the above mentioned problem is summarizedintermsof workelements, immediate predecessor(s), expected task durations and their variances. We assume independent normality for each task duration. Using the above table, we can easily get the minimum number of workstation, $\mathrm{N}_{\min }$ as 5. So, minimum trial cycle time, $C_{\min }$, comes out as $C_{\min }=\left[\sum_{i=1}^{K} \frac{\mu_{i}}{N_{\min }}+1\right]$, i.e. $C_{\min }=29$ time units.

Table 1: Precedence relation and task times of work elements.

Table 2: Trial Configurations

Table 3: Final Optimum Configuration

Now, we consider in the final solution the Cycle time $\mathrm{C}$ as 35 time units. So, the trial cycle time starts from 29 time units and goes upto 35 time units. For the given problem, we get no feasible solution for the trial cycle times as 29 and 30 time units. For the rest of the cycle times we get feasible solutions. These trial configurations are presented in Table 2.

The final solution based on optimization criterion is presented in Table 3 for trial cycle time as 31 time units.

In the optimum configuration it is found that the optimum value of $R_{A L}$ is $0.873450476$ having the configuration of 5 workstations with work elements $2,3,7,8,11$ assigned to workstation 1 , work elements $1,4,5,6,10,12$ assigned to workstation 2, work elements $9, 13, 14, 15$ assigned to workstation 3 , work elements $16,19 , 17,20$ assigned to workstation 4 and work elements $18,21$ assigned to workstation 5 .