# Exploring Decentralized Warehouse Management Using Large Language Models: A Proof of Concept
## 3 System Architecture
### Physical Layer  
The physical layer in the decentralized warehousing system represents the real-world, tangible components that form the foundation of the system's operations. This includes packages, warehouse slots, and transport processes. The physical layer manages the movement, storage, and spatial arrangement of goods, ensuring that every entity is properly accounted for in terms of its physical position and availability. In this layer, each package is characterized by its location, as shown in Equation (12):  
\[P_{i}=(x_{i},y_{i},z_{i})\quad\forall i\in\{1,2,\ldots,N_{P}\} \tag{12}\]  
where \(P_{i}\) represents the location of the \(i\)-th package, \((x_{i},y_{i},z_{i})\) denotes the 3D coordinates of the package within the system, and \(N_{P}\) is the total number of packages. Similarly, each warehouse slot \(W_{j}\) is characterized by both its location and its occupancy status, as shown in Equation (13):  
\[W_{j}=(x_{j},y_{j},z_{j},S_{j})\quad\forall j\in\{1,2,\ldots,N_{S}\},\quad S_{j }=\{0,1\} \tag{13}\]  
where \((x_{j},y_{j},z_{j})\) are the 3D coordinates of the \(j\)-th warehouse slot, \(S_{j}\) is the occupancy status, where \(S_{j}=1\) if the slot is occupied and \(S_{j}=0\) if it is vacant, and \(N_{S}\) represents the total number of warehouse slots. Transport processes are responsible for changing the states of packages and warehouse slots. They facilitate the movement of packages from one location to another, changing their location. The process operates between two points in time: the initial time \(t\), when transport begins, and the final time \(t+t_{\text{transport}}\), when transport is completed. Let \(P_{i}(t)\)  
Figure 1: System overview of the decentralized warehousing network. represent the location of package \(i\) at the starting time \(t\), and \(P_{i}(t+t_{\text{transport}})\) represent its location after the transport process is completed. Initially, the package's location is defined by Equation (14):  
\[P_{i}(t)=(x_{i}(t),y_{i}(t),z_{i}(t)), \tag{14}\]  
And, after the transport process (Equation (15), the new location is  
\[(P_{i}(t+t_{\text{transport}})=(x_{i}(t+t_{\text{transport}}),y_{i}(t+t_{\text{ transport}}),z_{i}(t+t_{\text{transport}})). \tag{15}\]  
The transition can be represented as shown in Equation (16):  
\[P_{i}(t+t_{transport})=P_{i}(t)+(\Delta x_{i},\Delta y_{i},\Delta z_{i}) \tag{16}\]  
where \((\Delta x_{i},\Delta y_{i},\Delta z_{i})\) represents the change in the position of the coordinates of the package caused by the transport process. The changes in coordinates \((\Delta x_{i},\Delta y_{i},\Delta z_{i})\) are defined as \((x_{i}(t+t_{transport})-x_{i}(t),y_{i}(t+t_{transport})-y_{i}(t),z_{i}(t+t_{ transport})-z_{i}(t))\). The state of the warehouse slots also changes during the transport process, reflecting the movement of packages between locations. As a package is transported from one warehouse to another, the origin slot becomes vacant, and the destination slot becomes occupied. Let \(W_{j}(t)\) (Equation (17)) represent the state of the origin warehouse slot at time \(t\):  
\[W_{j}(t)=(x_{j}(t),y_{j}(t),z_{j}(t),1) \tag{17}\]  
And let \(W_{k}(t)\) represent the state of the destination warehouse slot at time \(t\) (Equation (18)):  
\[W_{k}(t)=(x_{k}(t),y_{k}(t),z_{k}(t),0) \tag{18}\]  
\(S_{j}(t)=1\) indicates that the origin slot is occupied, and \(S_{k}(t)=0\) indicates that the destination slot is vacant. After the transport process, the occupancy of the slots changes as shown in Equation (19):  
\[ \begin{split} W_{j}(t+t_{\text{transport}})&=(x_{j} (t+t_{\text{transport}}),y_{j}(t+t_{\text{transport}}),z_{j}(t+t_{\text{transport }}),0)\\ W_{k}(t+t_{\text{transport}})&=(x_{k}(t+t_{\text{ transport}}),y_{k}(t+t_{\text{transport}}),z_{k}(t+t_{\text{transport}}),1)\end{split}  \tag{19}\]  
This means that, after the transport process, the origin slot becomes vacant (\(S_{j}(t+t_{\text{transport}})=0\)), and the destination slot becomes occupied (\(S_{k}(t+t_{\text{transport}})=1\)).