# A Facility Layout Algorithm for Logistics Scenarios Driven by Transport Lines
## 2 Problem Description and Mathematical Model
### Mathematical Model  
To assess the impact of transport lines on operating costs in industrial logistics scenarios, two evaluation metrics have been established: material handling cost and transport facility cost. The effectiveness of layout solution \(X\) is directly proportional to its ability to minimize the target cost for both evaluation metrics. The MHC is calculated as shown in Equation (1), which represents the total cost associated with transportation materials along a specified route. Due to the presence of multiple material entrances and exits within logistics facilities, the routing of transport lines linked to each entrance and exit may vary, so Equation (1) is formulated based on the perspective of the transport lines. It is assumed that the facility needs to process, sort, and store multiple materials, and the total number of transport lines required for material transportation is denoted as \(n\). Any given transport line \(i\) in the facility has an entrance \(q\) and an exit \(p\) on either end. The set of exits connected to entrance \(q\) is denoted as \(Q_{i}\), while the set of entrances connected to exit \(p\) is represented by \(P_{i}\). Additionally, the quantity of materials being handled by line \(i\) is expressed as \(g_{i}\). The transport cost per unit distance \(c_{pq}\) for material handling depends on the efficiency of each working facility, and the actual distance of line \(i\) is denoted as \(d_{pq}\). \[f_{MHC}(X)=\sum_{i=1}^{n}\sum_{p\in Q_{i}}\sum_{q\in P_{i}}g_{i}c_{pq}d_{pq} \tag{1}\]  
The TFC is calculated as shown in Equation (2), which represents the total cost required for the layout of the transport facilities in the workshop. The objective is based on the actual layout of logistics facilities as the transport facilities in logistics workshop are also part of the cost of the company. The total cost of the transport facilities in the logistics workshop comprises two components, which are the total cost of linear conveyors and the total cost of turning conveyors. The cost per unit distance of linear conveyors is denoted as \(c_{s}\), the cost of each individual turning conveyor is represented by \(c_{t}\), and the total number of turning conveyors in the transport line is represented as \(n_{t}\). \[f_{TFC}(X)=\sum_{i=1}^{n}\sum_{p\in Q_{i}}\sum_{q\in P_{i}}c_{s}d_{pq}+c_{t}n_ {t} \tag{2}\]  
The constraints involved in the logistics facility layout problem are divided into three categories, which are the constraints in the layout of the working facilities, the constraints in the routing of the transport facilities, and the constraints existing between the transport facilities and the working facilities, which are expressed in the following equations. The working facilities must adhere to two specific constraints, namely, the in-boundary constraint and the non-overlapping constraint. The in-boundary constraint requires that all logistics facilities be situated within the workshop during layout and that the coverage of these facilities does not surpass the area of the workshop. This is depicted mathematically in Equations (3)-(6). The non-overlapping constraint mandates that the logistics facilities cannot overlap with one another and must maintain either horizontal or vertical spacing. The fulfillment of this constraint can be achieved by satisfying one of the conditions specified in Equations (7) and (8). Assuming n denotes the number of working facilities located within the workshop, \(l_{i}\) and \(w_{i}\) are the length and width of the facility \(i\), respectively. The coordinates \((x_{i},y_{i})\) represent the center coordinates of the rectangular area for each facility.