## 5 Discussion on Unbalanced Lines Performance  
So far, the assumption of balanced line has been considered, involving that machine processing times are taken from a distribution with the same mean value. However, in practice, this assumption is infrequently verified, as the case in which there is not a process more critical than the others is uncommon to find (i.e., there is often a process for which increasing the processing rate is costly than the others). As a result, it is of scientific and practical interest to try relaxing the balanced line assumption. To do this, additional input parameters need to be provided to the performance estimation model. We have already shown that, from the performance estimation point of view and referring to the balanced line case, the bottleneck rate acts as a simple scale factor. Thus, we wonder if it is possible to reduce the amount of information requested to describe the unbalancing of the production line, thereby reducing the data collection and the training effort required when, as an example, a machine learning algorithm is considered. Starting from the [PERSON] and [PERSON]'s modeling approach, the production line can be generally identified by two distinct factors: the processing time of the bottleneck machine (\(1/r_{b}\)) and the average crossing time \(T_{0}\) when queuing on machines does not occur. Letting \(n\) be the number of machines in the line, it is possible to state that the average processing time \(T_{0}/n\) and the bottleneck processing time \(1/r_{b}\) are unquestionably the first two mutually independent parameters to be used to identify an unbalanced production line. However, it has to be noted that they alone are not enough to express the degree of unbalancing of the line. As an example, let us consider the case of two distinct production lines with five machines that have the following distribution of average processing times:  
\[\text{Line }1:12;10;10;10;8\text{ [min]}\] \[\text{Line }2:12;12;11;8;7\text{ [min]}\]  
Both of the two production lines have the same raw processing time (\(T_{0}=50\) min) and the same average processing time for the bottleneck machine (\(r_{b}=12\) min). However, they perform significantly different in function of the WIP released. In fact, it is reasonable to expect Line 1 performing better than Line 2, owing to the fact that the first is less unbalanced than the second. This highlights the insufficiency of using only the raw processing time and the bottleneck machine processing time to distinguish between the performances of production lines with different degrees of unbalancing. In order to identify possible parameters that can uniquely distinguish production lines with the same unbalancing magnitude among stations in the production line, it is worthwhile to consider the [PERSON] [25] iterative model summarized in Equations (1)-(4), which is exact in the case of a production line that is CONWIP-controlled, and the processing times are drawn from an exponential distribution. Therefore, by applying the mentioned model, it is possible to develop the iterative calculation for the throughput value (the TH formula is directly shown for reasons of space, as these can be obtained through simple mathematical steps by applying the equations iteratively):  
\[TH(1)=\frac{1}{\sum(t)}=\frac{1}{T_{0}}\] \[TH(2)=\frac{2\cdot T_{0}}{\sum(t^{2})+T_{0}  
become constant, collapsing into terms that are always constant except for the value of \(T_{0}\). In fact, if the production line is balanced, the average processing time of different workstations is always equal to \(1/r_{b}\), meaning that:  
\[\sum(t) =T_{0}\] \[\sum(t^{2}) =\frac{1}{r_{b}^{2}}+\frac{1}{r_{b}^{2}}+...=n\cdot\frac{1}{r_{b}^ {2}}=W_{0}\cdot\frac{1}{r_{b}^{2}}=\frac{T_{0}}{r_{b}}\] \[\sum(t^{3}) =\frac{1}{r_{b}^{3}}+\frac{1}{r_{b}^{3}}+...=n\cdot\frac{1}{r_{b}^ {3}}=W_{0}\cdot\frac{1}{r_{b}^{3}}=\frac{T_{0}}{r_{b}^{2}}\] \[...\]  
where \(n\) is the machine number of the production line, which is equal to \(W_{0}\) for the balanced production line.