BUS606: Operations and Supply Chain Management

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.