## 3 The GLSP with Rework and Lifetime Constraint for Defective Items  
The GLSP-RP was introduced by [1] to model process environments where conflicting items are required to avoid contact as they are subject to contamination, for example, the food processing industry where an-allergenic food items are produced on the same machine as regular food items. The problem considers an imperfect production system where a fraction of the planned lot size is not of serviceable quality and goes to a separate rework inventory where it is stored till it is reworked and shipped to customers or disposed of, which is illustrated in Figure 1.  
Note that, in this paper, no distinction will be made between the quality of good quality items and reworked items. Furthermore, we assume that the rework process always succeeds. For scheduling rework lots, a joint setup variable is considered, i.e., if the machine is set up for the original production in micro-period \(m\) (i.e., \(x_{j,m}=1\)) and followed by a rework lot for product \(j\) in \(m+1\) and vice versa, no additional setup is needed and, therefore, no additional setup costs occur. To address this in the model, the binary setup variable \(x_{j,m}\) of the GLSP formulation is redefined as the binary joint setup-variable. In addition, the defectives have a fixed lifetime while being stored in the rework inventory, i.e., if the defective items are not reworked in a fixed time frame expressed by a number of consecutive micro-periods (\(\Omega_{j}\)) they become scrap and have to be disposed, resulting in disposal costs (\(\lambda_{j}\)). Table 2 shows the additional variables and parameters of the GLSP-RP. \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameters** & \\ \hline \(h_{j}^{R}\) & Inventory holding cost factor for defective items of product \(j\) \\ \(\lambda_{j}\) & Disposal cost for one perished item of product \(j\) \\ \(\Omega_{j}\) & Lifetime of product \(j\) in micro-periods \\ \(\Theta_{j,t}\) & Proportion of defective items \(0\leq\Theta_{j,t}<1\) of product \(j\) in \(t\) \\ \(tp_{j}^{R}\) & Process time for reworking a defective item per unit of product \(j\) \\ \hline
**Variables** & \\ \hline \(R_{j,m}\) & Number of defective items of product \(j\) produced in micro-period \(m\) \\ \(s_{j,m}^{D}\) & Number of perished items of rework inventory of product \(j\) \\  & in micro-period \(m\) \\ \(s_{j,m}^{R}\) & Number of defective items stored in rework inventory of product \(j\) \\  & in micro-period \(m\) \\ \(x_{j,m}\) & Binary joint-setup variable, 1 if product \(j\) is set up for production and/or rework \\  & in micro-period \(m,0\) otherwise \\ \(y_{j,m}^{R}\) & Quantity of defective items of product \(j\) reworked in micro-period \(m\) \\ \(y_{j,m}^{S}\) & Production amount of good-quality items of product \(j\) in micro-period \(m\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Additional variables and parameters for the general lot sizing and scheduling problem with defective items and lifetime constraints (GLSP-RP). Figure 1: Illustration of the defective production process (see [49]). items. It is composed of the rework inventory in \(m-1\), the number of defective items of micro-period \(m\) minus the reworked items in \(m\) and the number of items that passed their lifetime while being stored (\(s_{j,m}^{D}\)). The interrelation between storage time and lifetime of the defective items is shown in Equation (16). It states that the amount of spoiled defectives for a given micro-period \(m\) is calculated by the defectives produced by the original production process from the beginning of the planning horizon until micro-period \(m\) deduced by the lifetime length \(\Omega_{j}\) of the defectives, less those items that are already reworked until micro-period \(m-1\) and defectives that were already disposed in previous micro-periods. Constraint (17) ensures that defectives of micro-period \(m-1\) can be reworked in \(m\) the earliest.