# A Facility Layout Algorithm for Logistics Scenarios Driven by Transport Lines
## 3 Force-Directed Logistics Layout Algorithm
### Objective Optimization Mechanism  
In this paper, the two objectives of MHC and TFC are defined to evaluate the quality of feasible solutions for industrial logistics scenarios. After obtaining the target solution metrics through layout and wiring, the MOSA-FD algorithm evaluates the likelihood of accepting the target solution by utilizing both its dominance value and the present temperature. This assessment determines whether the new solution will replace the current one and whether it will be added to the archive. The concept of dominance value is defined in the literature [40], which is represented by Equation (25) and indicates whether solution b is dominated by solution a. In the scenario of dual objectives, the dominance value signifies the proportion of the rectangular region where the diagonal is formed by solutions a and b in relation to the entire objective rectangular area. Figure 8 illustrates that a smaller dominance value implies that the objective values of both solutions are more similar. \[\Delta dom_{a,b}=\frac{\left|f_{MHC}(a)-f_{MHC}(b)\right|}{R_{MHC}}\times \frac{\left|f_{TFC}(a)-f_{TFC}(b)\right|}{R_{TFC}} \tag{25}\]  
Figure 8: Dominance value and crowding degree. In Equation (25), the function \(f_{MHC}(a)\) represents the MHC objective value of solution a, whereas \(f_{TFC}(a)\) denotes the TFC objective value of solution a, and the range of values for the MHC and TFC objective functions in the new solution and archive are represented by \(R_{MHC}\) and \(R_{TFC}\), respectively. To compare the current solution with the target solution, Pareto dominance relationship is employed. The three possible cases are: the current solution dominates the target solution, the target solution dominates the current solution, or the two solutions do not dominate each other. The likelihood of accepting the target solution can be determined by analyzing the dominant value and the current temperature, which are discussed in detail below:  
1. If the current solution dominates the target solution, it means that the current solution outperforms the target solution on all objective values. If the target solution is dominated by \(k\) solutions in the archive, then the acceptance probability can be calculated using Equations (26) and (27) to replace the current solution, but the target solution will not be added to the archive. \[\Delta dom_{avg}=\frac{\sum_{i=1}^{k}\Delta dom_{i,new}+\Delta dom_{current, new}}{k+1}\] (26) \[prob=e^{-(\Delta dom_{avg}/T)}\] (27)
2. If the target solution and the current solution are non-dominated, it means that both have advantages and disadvantages on the objective values. If the target solution is not dominated in the archive, it will be set as the current solution and added to the archive, while all feasible solutions dominated by the target solution will be removed from the archive. If the target solution is dominated by \(k\) solutions in the archive, the acceptance probability can be calculated using Equations (27) and (28) to replace the current solution, but the target solution will not be added to the archive. \[\Delta dom_{avg}=\frac{\sum_{i=1}^{k}\Delta dom_{i,new}}{k}\] (28)
3. If the target solution dominates the current solution, it means that the target solution is superior to the current solution on all objective values. If the target solution is not dominated in the archive, set it as the current solution and add it to the archive data while removing all feasible solutions dominated by the target solution in the archive. If the target solution is dominated by \(k\) solutions in the archive, use the non-dominated solution corresponding to \(\Delta dom_{min}\) to calculate the acceptance probability for replacing the current solution; otherwise, directly replace the current solution with the target solution. The formula for calculating the probability is detailed in Equation (29), where \(\Delta dom_{min}\) represents the smallest domination value distance from the target solution in the archive.