# A Facility Layout Algorithm for Logistics Scenarios Driven by Transport Lines
## 2 Problem Description and Mathematical Model  
### Problem Description  
The logistics facility layout is a hybrid layout problem, which involves considering both the layout of logistics facilities and the routing of transport lines. However, logistics facilities and transport lines have inherently different characteristics in the actual production process, it is impractical to use a single representation structure to define both types of facilities, and a correlation exists between the two in the layout problem. Based on the functional characteristics and geometric features of different types of logistics facilities in industrial logistics scenarios, this paper posits the following assumptions regarding research content related to facility layout:  
1. Workshop areas and all working facilities are considered as rectangular structures.
2. The orientation of the working facilities is classified into four types: up and down, left and right, which cannot be placed at an inclined angle.
3. All working facilities are of a fixed size, and the sum of their total area is less than the area of the workshop.
4. Each working facility has designated entrance and exit points for material transportation by transport lines.
5. Transport lines are constrained to extend only in horizontal or vertical directions.
6. Transport efficiency and unit costs are available for material transport routes and each individual transport line.  
The realistic logistics scene mainly includes storage facilities, processing facilities, and transport lines. Within the facilities category, processing facilities can be further divided into three types: manufacturing facilities, turning facilities, and sorting facilities. Figure 1 gives the distribution of entrance and exits of various types of processing facilities. The logistics workshop exists within a two-dimensional plane coordinate system, the position of each facility within the workshop is not predetermined, and it is crucial to ensure that no overlap occurs between any of the facilities in the layout. The size of the workshop and all of the facilities are known, along with the relative locations of the entrance and exits. Transport lines exist between the exits and entrances of different facilities, with line lengths indicating the actual distance of material transport. Figure 2 shows the facility layout of the logistics scenario. The sizes of the workshop are represented by \(L\) for length and \(W\) for width. The workshop entrance and exit are designated as \(S_{in}\) and \(S_{out}\). Facilities \(i\) and \(j\) are classified as processing facilities, while facility \(k\) is categorized as a storage facility. The center coordinates of facility \(i\) in the workshop are \((x_{i},y_{i})\). The transport lines are represented by a set of directed line segments between the facilities, with \(h_{min}\) representing the minimum horizontal spacing and \(v_{min}\) representing the minimum vertical spacing required between the facilities.  
Figure 1: Examples of three types of processing facilities. (**a**) Manufacturing facility; (**b**) turning facility; and (**c**) sorting facility.  
Figure 2: Logistics facility layout diagram.