## 4 Operations Research Model
### Mathematical Model  
In this section, the sets and indices, parameters, and decision variables for the proactive model are defined in Table 1. In addition, the objective function and constraints are presented. The model is based on the model by [PERSON] et al. (2016). Adjustments to the model were made based on the industrial use case and the resulting requirements. In the industrial use case, a distinction must be made in the connections between the warehouses. The newly introduced Set \(L\) contains all possible connections. However, some connections force an integrated return transport. A Set \(G\) has been introduced for these connections. The model formulation had to be adapted due to the introduction of the sets. In addition, Constraints (8) are introduced. \begin{table}
\begin{tabular}{c l} \hline \hline  & \multicolumn{2}{c}{Sets and indices} \\ \hline \(i\), \(j\), \(k\;\in N\) & Set of all locations \\ \(N^{\text{Warehouses}}\) & Set of all warehouses, with \(N^{\text{Warehouses}}\subseteq N\) \\ \(N^{\text{Customers}}\) & Set of all customers, with \(N^{\text{Customers}}\subseteq N\) \\ \(L\) & Set of all tuples of warehouses and customers that are subject to a direct \\  & precedence relationship \((k,\;j)\in L\) \\ \(G\) & Set of all tuples of warehouses that are subject to a direct precedence \\  & relationship for transports with obligatory return transport \((k,\;j)\in G\subseteq L\) \\ \(x\in P\) & Set of parts \\ \hline  & \multicolumn{2}{c}{Parameters} \\ \hline \(B_{xk}\) & Stock of part \(x\) in warehouse \(k\), parameterized in \(\mathbb{B}\) \\ \(D_{xk}\) & Demand of part \(x\) by customer \(k\), parameterized in \(\mathbb{B}\) \\ \(c_{k}^{\text{Storing}}\) & Capacity of warehouse \(k\) for storing parts, parameterized in \(\mathbb{N}_{0}\) \\ Handling & Capacity of warehouse \(k\) for handling parts, parameterized in \(\mathbb{N}_{0}\) \\ \(c_{k}^{\text{Vehicles}}\) & Capacity of vehicles for transporting parts, parameterized in \(\mathbb{N}_{0}\) \\ \(r_{kj}^{\text{T}}\) & Cost rate for traveling locations \((k,\;j)\), parameterized in \(\mathbb{R}_{0}^{+}\) \\ \(r_{xi}^{\text{F}}\) & Approximated future transshipment costs for delivery of part \(x\) if stored in warehouse \(i\), parameterized in \(\mathbb{R}_{0}^{+}\) \\ \(\epsilon\) & A sufficiently small number, parameterized in (0, 1] \\ \hline  & \multicolumn{2}{c}{Decision variables} \\ \hline \(z_{xkj}\in\mathbb{B}\) & \(\begin{cases}1,&\text{if part $x$ is transshipped between locations $(k,j)$}\\ 0,&\text{otherwise}\\ \text{Number of vehicles traveling $(k,\;j)$}\\ R^{\text{T}}\in\mathbb{R}_{0}^{+}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{ }\text{}\text{}\text{}\text{}\text{}\text{The objective function and the constraints are given below:  
\[\min R^{\text{T}}+\varepsilon\ \cdot\ R^{\text{F}} \tag{2}\]  
\[\sum_{(k,\ j)\ \in\ L\ |\ k=i}z_{xkj}\ \ \leq\ B_{xi}+\sum_{(j,\ k)\ \in\ L\ |\ k=i}z_{xjk}\ \ \ \forall x\in P;\ i\in N^{\text{ Warehouses}} \tag{3}\]  
\[D_{xi}=\sum_{(j,\ k)\ \in\ L\ |\ k=i}z_{xjk}\ \ \ \forall x\in P;\ i\in N^{\text{ Customers}} \tag{4}\]  
\[c_{i}^{\text{Storing}}\ \geq\ \sum_{x\ \in\ P}B_{xi}+\sum_{x\ \in\ P}\sum_{(j,\ k)\ \in\ L\ |\ k=i}z_{xjk}-\sum_{x\ \in\ P}\sum_{(k,\ j)\ \in\ L\ |\ k=i}z_{xkj}\ \ \ \forall i\in N^{\text{ Warehouses}} \tag{5}\]  
\[c_{i}^{\text{Handling}}\ \geq\ \sum_{x\ \in\ P}\sum_{(j,\ k)\ \in\ L\ |\ k=i}z_{xjk}+\sum_{x\ \in\ P}\sum_{(j,\ k)\ \in\ L\ |\ k=i}z_{xkj}\ \ \ \forall i\in N^{\text{ Warehouses}} \tag{6}\]  
\[c^{\text{Vehicles}}\ \geq\ \frac{1}{w_{kj}}\ \sum_{x\ \in\ P}z_{xkj}\ \ \ \forall(k,\ j)\ \in L \tag{7}\]  
$x$0  
The objective function (2) minimizes the realized costs for transshipments and the approximated future transshipment costs. The parameter \(\varepsilon\) is chosen to be sufficiently small in order not to influence the primary objective of minimizing transport costs.