## 2 Literature Review
### The General Lot Sizing and Scheduling Problem
\begin{table}
\begin{tabular}{c l} \hline \hline
**Indices** & \\ \hline \(i,j\) & Products with \(i,j=1,\ldots,J\) \\ \(L_{t}\) & Denoting the last micro-period of a macro-period \(t\) \\ \(m\) & Micro-periods \(m=1,\ldots,L_{T}\) with \(L_{T}=\sum_{t=1}^{T}|M_{t}|\) \\ \(M_{t}\) & Set of micro-periods \(m\) within macro-period \(t\) \\ \(t\) & Macro-periods with \(t=1,\ldots,T\) \\ \hline \hline \multicolumn{2}{l}{**Parameters**} \\ \hline \(b_{t}\) & Available capacity in macro-period \(t\) in time units \\ \(d_{j,t}\) & Demand of product \(j\) in period \(t\) \\ \(f_{i,j}\) & Sequence-dependent setup costs for a change over from product \(i\) to \(j\) \\ \(h_{j}\) & Inventory holding cost factor for product \(j\) \\ \(\kappa_{j}\) & Minimum lot size amount of product \(j\) \\ \(|M_{t}|\) & Fixed number of micro-periods within a macro-period \(t\) \\ \(st_{i,j}\) & setup times for a change over from product \(i\) to \(j\) \\ \(tp_{j}\) & Process time per unit of product \(j\) \\ \(M\) & big number, e.g., \(max(b)_{t\in T}\) \\ \hline \hline \multicolumn{2}{l}{**Variables**} \\ \hline \(s_{j,t}\) & Inventory of product \(j\) in period \(t\) \\ \(x_{j,m}\) & Binary setup variable, 1 if product \(j\) is set up for production \\  & in micro-period \(m\), 0 otherwise \\ \(y_{j,m}\) & Production amount of product \(j\) in micro-period \(m\) \\ \(z_{i,j,m}\) & Changeover variable, 1 if production is changed from product \(i\) to \(j\) \\  & in micro-period \(m\), 0 otherwise \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables, parameters and indices used in the model. \[\sum_{j=1}^{l}x_{j,m}=1\qquad\forall m \tag{5}\]  
\[y_{j,m}\geq\kappa_{j}(x_{j,m}-x_{j,m-1})\qquad\forall j,t,m\
eq L_{t} \tag{6}\]  
\[y_{j,m}+y_{j,m+1}\geq\kappa_{j}(x_{j,m}-x_{j,m-1})\qquad\forall j,t,m=L_{t} \tag{7}\]  
\[z_{i,j,m}\geq(x_{i,m-1}+x_{j,m})-1\qquad\forall i,j,m,i\
eq j \tag{8}\]  
\[y_{j,m},s_{j,t},z_{i,j,m}\geq 0\qquad\forall j,m,t \tag{9}\]  
\[x_{j,m}\in\{0,1\}\qquad\forall j,m \tag{10}\]  
The objective Function (1) minimizes the overall costs consisting of the holding costs for produced items and the sequence-dependent setup costs. Equation (2) depicts the inventory balancing constraint. It states that the inventory in macro-period \(t\) is composed of the inventory of the previous macro-period \(t-1\), the production amount of all micro-periods belonging to \(t\) subtracted by the demand of period \(t\). Inequality (3) states that the maximum available capacity (\(b_{t}\)) of macro-period \(t\) reduced by the sum of sequence-dependent setup times incurred by setup changes from product \(i\) to product \(j\) with \(m\in M_{t}\) must be greater or equal to the sum of processing times in all micro-periods belonging to macro-period \(t\). Constraints (4) and (5) ensure that product \(j\) can only be produced in micro-period \(m\), if a setup for product \(j\) is performed in \(m\), i.e., if the binary setup variable (\(x_{j,m}\)) is set to one and that only a single product can be produced at a time in a micro-period \(m\). The authors of [12] pointed out that minimum lot sizes have to be included, otherwise wrong calculations of setup costs could occur as a result of setup state changes without product changes in cases where the setup cost matrix does not satisfy the triangle inequality (i.e., \(f_{i,k}+f_{k,j}\geq f_{ij}\) with \(i,j,k\in 1,\ldots,l\)). As shown by [14], this originally introduced minimum lot size constraint was strictly limited as a production state between two consecutive periods; it is only conserved if the available capacity exceeds the minimum production quantity. Therefore, the authors adjusted Constraint (6) and introduced Constraint (7) to enforce the production continuity across macro-periods. Constraint (8) presents the changeover logic and Constraints (9) and (10) enforce non-negative and binary variables, respectively.