Designing an Assembly Line for Reliability

Under the deterministic setup, the uneven allocation of works to different workstations results in loss of efficiency. The efficiency of an assembly line is therefore measured in terms of balancing loss, $B=\left(N C-\sum t_{i}\right) / N C$. Under the stochastic setup since $\mathrm{t}_{\mathrm{i}}$ 's are random variables this balancing loss itself becomes a random variable. So, one may like to minimize the expected value of the same, i.e.,

$E(B)=\frac{\left(N C-\sum \mu_{i}\right)}{N C}$

But this measure alone is not sufficient to ensure efficiency of the production system. For example, for a perfectly balanced situation with $E(B)=0$, the chance of failure of an assembly line under symmetric distribution of each workstation time is $\left(\frac{1}{2}\right)^{N}$ which tends to zero as the number of workstations becomes large. So, there must be some other consideration for ensuring high chance of meeting the cycle time requirements in each workstation. Drawing analogy with the concept of product reliability in terms of meeting the mission requirement, we may define the reliability of a workstation in terms of idle time meeting the non-negativity restriction. Thus, reliability of $j$th workstation, $R_{j}$ can be defined as

$R_{j}=\operatorname{Pr} .\left[L_{j} \geq 0\right]$

Then the assembly line can be viewed as an arrangement of $\mathrm{N}$ workstations in series in the sense if one workstation fails to meet the cycle time requirement the entire assembly line faces operational failure. This observation translated in terms of reliability indicates $R_{A L}=\prod_{j=1}^{N} R_{j}$, where $R_{A L}$ is the reliability of the assembly line. We propose to consider the reliability of the assembly line along with the expected balancing loss as dual measure of system efficiency. Thus, the efficiency of the total system will be maximum when both the expected balancing loss will be minimum and the system reliability will be maximum. Therefore, the objective of our proposed method can be equivalently expressed as the minimization of the number of workstations ($N$) and maximization of the system reliability, $R_{AL}$, subject to precedence constraints.

Let us consider the binary variable, $a(i,j)$ such that

$a(i, j)= \begin{cases}1, & \text { if } i \in W_{j}, i \text { th task is assigned to } W j \\ 0, & \text { if } i \notin W_{j}, i \text { th task is not assigned to } W j\end{cases}$

and is true for $\mathrm{i}=1,2, \ldots ., \mathrm{K}, \mathrm{j}=1,2, \ldots . ., \mathrm{N}$.

The following condition must hold for each $\mathrm{i}=1,2, \ldots ., \mathrm{K}$, under the restriction that the $\mathrm{i}^{\text {th }}$ task can be assigned to only one workstation:

$\sum_{j=1}^{N} a(i, j)=1$

Further, according to precedence constraints if task $i^{\prime}$ is to be assigned before assigning task $\mathrm{i}$, that is $i^{\prime} < i$, then

$a(i, j) \leq \sum_{r=1}^{j} a\left(i^{\prime}, r\right) \quad \forall i^{\prime} < i \quad \forall j$

Since the task times are random variables, the condition for completion of tasks in a workstation within the assigned cycle time can be described in terms of reliability measure,

$\begin{aligned}&R_{j}=\operatorname{Pr} \cdot\left[L_{j} \geq 0\right]=\operatorname{Pr} \cdot\left[\frac{L_{j}-E\left(L_{j}\right)}{\sqrt{\operatorname{var}\left(L_{j}\right)}} \geq-\frac{E\left(L_{j}\right)}{\sqrt{\operatorname{var}\left(L_{j}\right)}}\right] \\&=1-\Phi\left(-\frac{E\left(L_{j}\right)}{\sqrt{\operatorname{var}\left(L_{j}\right)}}\right),\end{aligned}$

under normality of the each elemental times.

Thus,

$R_{j}=\Phi\left(\frac{E\left(L_{j}\right)}{\sqrt{\operatorname{var}\left(L_{j}\right)}}\right)$

The reliability of the assembly line, $R_{A L}$, can be expressed as

$R_{A L}=\prod_{j=1}^{N} \Phi\left(\frac{E\left(L_{j}\right)}{\sqrt{\operatorname{var}\left(L_{j}\right)}}\right)$

following the properties of the series system and the fact that workstations are arranged in series.

Thus, the optimization framework of the line balancing problem can be expressed in terms ofthe following objectives:

Minimize $\quad \mathrm{N}$

Maximize $\quad R_{A L}=\prod_{j=1}^{N} R_{j}$

subject to the following constraints,:

(i) $\sum_{j=1}^{N} a(i, j)=1 \quad \forall i$

(ii) $a(i, j) \leq \sum_{r=1}^{j} a\left(i^{\prime}, r\right) \quad \forall \quad i^{\prime} < i$

(iii) $a(i, j)=0,1 \quad \forall i, j$

We prefer to address the above optimization problem in two stages. First, we undertake the task of minimization of $$\mathrm{E}(\mathrm{B})$ and thereby generate, in the first instant, feasible solutions with minimum value of $\mathrm{E}(\mathrm{B})$. Then we obtain the final solution of the problem by imposing the second objective of maximization of $R_{A L}$. Even for generating the set of feasible solutions, we consider a sequential approach of assigning trial cycle time that results in slack time. This slack time is to be assigned to each workstation meeting the optimality condition arising out of the first objective of the above formulation. In this approach the trial cycle time starts from some lowest value and gets increased step by step some lowest value and gets increased step by step so as to reach the maximum limit C. Determination of the lowest value for trial cycl time depends on the following consideration.

Given a choice of $\mathrm{C}$, it may be noted that the theoretical minimum number of workstations, $\mathrm{N}_{\min }$, must satisfy the following constraints:

$\displaystyle\sum_{i=1}^{K} T_{i} / C \leq N_{\min } \leq \displaystyle\sum_{i=1}^{K} T_{i} / C+1 \text {, }$

from where we arrive at $\mathrm{C}_{\min }$, the minimum value of $\mathrm{C}$, as

$C_{\min }=\left[\displaystyle\sum_{i=1}^{K} T_{i} / N_{\min }+1\right]$

Thus, given a cycle time, C, one may conceptually consider a trial cycle time, $C_{t}$, satisfying the condition $\mathrm{C}_{\min } \leq \mathrm{C}_{\mathrm{t}} \leq \mathrm{C}$, to arrive at the set of feasible workstation configurations and maintain the same cycle time $\mathrm{C}$ by uniformly adding to each workstation a slackness $S_{t}$ to $C_{t}$, where $S_{t}=C-C_{t}$. This will help to increase the system stability.

Our proposed two-stage procedure with sequential generation of feasible solution and selection of final solution can be best described by the following algorithm.

1. Calculate the theoretical minimum number of workstations, $N_{\min }$, following the formula $\sum_{i=1}^{K} \frac{\mu_{i}}{C} \leq N_{\min } \leq \sum_{i=1}^{K} \frac{\mu_{i}}{C}+1$. Calculate the minimum cycle time, $C_{\text {min }}$, using the relationship, $C_{\min }=\left[\sum_{i=1}^{K} \frac{\mu_{i}}{N_{\min }}+1\right]$.

2. Set the trial cycle time $C_{t}$ at $C_{\min }$.

3. Prepare the list of all unvisited tasks - call it List $U$.

4. Prepare List $R$ from the tasks of List $U$ with no immediate predecessor or whose immediate predecessors have been visited. The tasks of $R$ are ready for selection.

5. Prepare List $A$ from the task of List $R$ having assembly time less than that of cycle time and is allowable for inclusion.

6. Randomly select a task from List $A$ and reset the cycle time as {$C_{t}$ assembly time}.

7. If cycle time is less than the assembly time, then open a new workstation. Reinitialize cycle time to its original value and repeat the above steps until all nodes are visited.

8. After getting the complete distribution of tasks to workstations, calculate $R_{A L}$, the reliability of the assembly line.

9. After each run, the new reliability value $R_{A L}$ is compared with the previous $R_{A L}$ value. If the new $R_{A L}$ value is greater than the previous value, the new solution is stored as the basis for next comparison.

10. Increase the cycle time by one unit until it crosses $C$ value. If $C$ value is crossed, go to step 12.

11. Repeat step 2 to 10.

12. Check whether all the work elements have been assigned to specified number of work stations. If not, increase the value of $N_{min}$ by 1 and go to step 2.

13. Print the best solution in terms of overall maximum 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 .