BUS606: Operations and Supply Chain Management

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.