# Design Model of Flow Lines to Include Switch-Off Policies Reducing Energy Consumption
## 2 Flow Line Design Model  
The problem deals with the designing of a flow line composed by \(M\) stations that manufactures one product type. The product consists of \(N\) operations to process; these operations should be assigned to the M workstations, following the precedence constrains. The variables of the model are the assignments of the operations to the workstations of the flow line. The operation assignment binary variable \(x_{ij}\) is defined as follows (Equation (1)):  
\[x_{ij}= \begin{cases}1&if\;\;the\;i-th\;operation\;is\;assigned\;to\;the\;j-th\; station\\ 0&otherwise\end{cases}  \tag{1}\]  
The precedence constrains binary variable ensures that the \(i\)-\(th\) operation must be completed before the \(k\)-\(th\) operation, and it is computed as (Equation (2)):  
\[v_{ik}= \begin{cases}1&if\;\;the\;i-th\;operation\;prec\;the\;k-th\;station\\ 0&otherwise\end{cases}  \tag{2}\]  
The processing time \(T_{j}\) of the \(j\)-\(th\) station is the sum of the processing time of the operations assigned to \(j\)-\(th\) station, as follows (Equation (3)):  
\[T_{j}=\sum\
olimits_{i}^{N}t_{i}x_{ij} \tag{3}\]  
The cycle time \(C\) of the production line is equal to the maximum of the stations' processing times, as shown in Equation (4)  
\[C=MAX_{j}\big{[}T_{j}\big{]} \tag{4}\]  
The \(j\)-\(th\) station idle time is defined as follows (Equation (5)):  
\[T_{d,j}=C-T_{j} \tag{5}\]The idle time for each cycle is given by Equation (6):  
\[TT_{d}=\sum_{j=1}^{M}\left(T_{d,j}\right) \tag{6}\]  
The difference of processing time between nearby stations is called distance \(d_{j}\) and it is defined as follows (Equation (7)):  
\[d_{j}=T_{j+1}-T_{j} \tag{7}\]  
A positive value of the distance \(d_{j}\) forms a couple of workstations when the first has a higher velocity than the second workstation. Then, the first workstation can fill the downstream buffer and goes into the off state, reducing energy consumption. Therefore, the flow line consists of a couple of workstations to facilitate the off state of the first workstation of the couple. Figure 1 shows the concept of the distance of a couple of workstations. In the following figure, the term WSj has been used to identify the j-th workstation; the distance between two stations has been obtained as the difference of the processing time of the second workstation (with higher working time) and the first workstation of the couple. In Figure 1, the processing time is linked to the stations with, respectively, lower and higher productivity per couple, as represented in blue and green. To achieve an unbalanced flow line, the objective function is achieved by maximizing the sum of the distance between stations (Equation (8)):  
\[MAX\left(\sum_{j}d_{j}\right)\ \ with\ j=1,3,5,\ldots \tag{8}\]  
This is subject to the following constrains (Equations (9)-(11))  
\[\sum_{j=1}^{M}x_{ij}=1\qquad\ with\ i=1,\ldots,N \tag{9}\]  
\[\sum_{j=1}^{M}j\ \ x_{ij}\leq\sum_{j=1}^{M}j\ x_{kj}\qquad\ \forall\ v_{ik}=1 \tag{10}\]  
\[\sum_{i=1}^{N}t_{i}\ \ x_{ij}\leq C_{max}\qquad\ with\ j=1,\ldots,M \tag{11}\]  
Equation (9) ensures that each operation is only assigned to one machine. Equation (10) ensures that the constraints on the precedence of operations are respected. Finally, Equation (11) ensures that the station processing times are below the maximum fixed cycle time. By setting the cycle time,  
Figure 1: Distance between stations. maximizing the objective function achieves an unbalanced line that respects the targeted productivity. Equation (8) results in an unbalanced flow line with high idle times between sequential stations. In order to achieve more downtime, and therefore higher energy saving, this research proposes a different objective function from past literature, where the focus instead is on obtaining the minimum cycle time [17], the minimum number of stations, and the minimum idle time [18]. Figure 2 shows the framework used for the design of unbalanced flow lines.