# A New Self-Balancing Assembly Line Based on Collaborative Ant Behavior
## 3 The Efficiency of BB-Based Assembly Cell  
We now examine efficiency losses in the BB-based system caused by hand-offs and worker blocking in this section on assembly lines under uncertainties. Let us first examine the effect of the uncertain events on the assembly time distribution. The uncertain events in assembly systems include assembly tool breakdown, defected parts, workers' job skills, and utility problems, to name a few. Even though the net assembly task time is deterministic, the uncertain events make effective assembly times stochastic. Suppose net assembly time for a task is deterministic with \(t_{\text{s}}\) and an assembly tool is broken in the middle of an assembly activity. Then, the assembly time is expanded to the net assembly time plus an independent tool replacement time. Suppose the tool breakage takes place with probability \(p\) and it requires \(t_{r}\) minutes on average to fix the problem. The effective task time will be \(t_{s}\) with probability 1-\(p\) and \(t_{s}+t_{r}\) with probability \(p\). On average, the effective assembly time, \(\tau_{s}\), will be \(\tau_{s}=(1-p)t_{s}+p(t_{s}+t_{r})=t_{s}+p\cdot t_{r}\). Now, suppose the tool breakage time is stochastic with a coefficient of variation, \(v_{r}\). The coefficient of variation (CV) is defined as the standard deviation over the average of a variable. Then, the CV of the effective assembly time, \(v_{s}\), is calculated by using the model presented in [35] as follows:  
\[v_{s}\ =\ \sqrt{\frac{t_{s}^{2}\ +\ p_{s}\big{(}v_{r}^{2}\ t_{r}^{2}\ +\ 2t_{r}t_{s}\ +\ t_{r}^{2}\big{)}}{\tau_{s}^{2}}}-1 \tag{1}\]  
To understand the effect of uncertain events on the effective assembly time, let us take a simple example. Suppose the net assembly time is constant with 3.0 min, and it undergoes a tool breakdown with a probability of 0.02 following an exponential distribution of average 9.0 min. Then, the average assembly time is prolonged from 3.0 min to 3.18 min (= 3.0 + 0.02 cdot 9.0) and the CV of the effective assembly time is 0.56, calculated from Equation (1). Hence, even with a constant net assembly time, it is reasonable to assume that the effective assembly time is not deterministic but stochastic in the system under uncertainties. Figure 2 shows a situation where a hand-off loss happens. Suppose the upstream worker is performing an assembly task starting at time \(ta\), whose expected assembly time is \((tc-ta)\) time units. If the downstream worker becomes available and ready to take over a new assembly job at time \(tb\) from the upstream worker who is performing the assembly task, he should wait until the upstream worker finishes his current task. Since the current task is expected to be finished at time \(tc\), the expected waiting time of the downstream worker for the hand-off is \((tc-tb)\) time units. We now estimate the performance of workers when the hand-off times are present. We will adopt the result of [36] where it is proved that the expected time per each hand-off, \(h_{o}\), is \(h_{o}\ =\ \frac{E\big{[}t^{2}\big{]}}{2E\big{[}t\big{]}}\), where \(t\) is a variable representing assembly time. Since \(E[t^{2}]=V(t)+E[t]^{2}=(cv(t)^{2}+1)\ E[t]^{2}\) and \(E[t]=T/n\), we have  
\[h_{o}\ =\ \frac{E\big{[}t^{2}\big{]}}{2E\big{[}t\big{]}}\ =\ \frac{\big{(}cv(t)^{2}+1 \big{)}T}{2n} \tag{2}\]  
where \(V(t)\) is the variance of \(t\), \(cv(t)\) is the CV of \(t\), and \(T\) is the total assembly time for a final product. It is seen from Equation (2) that there is a hand-off time even with no assembly time variability with \(cv(t)=0\). Consider an \(m\)-worker assembly line dedicated to a product type where the final product requires \(n\) assembly tasks. When all the workers are fully utilized, then the maximum production rate per unit  
Figure 2: Hand-off loss in BB-based assembly cell. The expected hand-off time is \(tc-tb\). time of the assembly line is \(m/T\). In the BB-based cell, each product experiences \((m-1)\) hand-offs in the \(m\)-worker line.