## 2 Optimization Model
### Solution Algorithm
#### 2.4.1 Solution Procedure for Multi-Objective Optimization
\(\bullet\) Multi-Objective evolutionary algorithm based on Decomposition (MOEA/D)  
The Multi-Objective evolutionary algorithm based on Decomposition (MOEA/D) uses an aggregation function to generate a set of uniform weight vectors, through which the multi-objective optimization problem is divided into a set of single-objective sub-problems, and then updates the \"bad solutions\" in the neighborhood according to the value of the aggregation function so as to ultimately find the optimal Pareto solution set [23]. The steps are as follows:  
Step 1: Initialization. Initialize the population and calculate the objective function value. Generate a set of weight vectors and define neighborhoods (e.g., based on weight distances or solution spaces) for each subproblem. Determine the ideal point (minimum set of objective values). Step 2: Subproblem co-optimization. For each subproblem (corresponding to a weight vector), randomly select two solutions from the neighborhood and use DE or SBX+ variant to generate child solutions. Evaluate and update the optimal solutions and ideal points in the neighborhood according to aggregation functions (e.g., Tchebycheff, weighted sum) as needed. Step 3: Population update. Update the solutions in the neighborhood of all subproblems after each generation. Step 4: Judgment termination. Repeat steps 2-3 until the maximum number of generations or function evaluations is satisfied. \(\bullet\) Multi-Objective Brown Bear Optimization (MOBBO)  
MOBBO is a novel multi-objective optimization algorithm based on the foraging, territory marking, and sniffing behavior of brown bears. The algorithm simulates the natural strategy of \"exploration-exploitation\" balance in the foraging process of brown bears and has the ability of both global search and local exploitation. In order to effectively deal with the multi-objective optimization problem, MOBBO introduces the non-dominated sorting and external archive strategies to dynamically maintain the optimal non-dominated solution set and improve the diversity and balance of the solution set. With the help of the \(\varepsilon\)-domination mechanism, the algorithm can effectively prevent the aggregation of solutions while ensuring the quality of the solution set, which is suitable for problems with complex constraints and multi-objective engineering optimization. Step 1: Initialization. Randomly generate the population and initialize the tread markers, i.e., decision variables. Set the objective function, population size, range of design variables, and termination criteria. Step 2: Non-dominated sorting and profile initialization. Perform Pareto sorting to identify non-dominated solutions and initialize the external archive. Filter the set of solutions through the \(\varepsilon\)-domination mechanism to ensure that the solutions retained in the external archive are distributed equally in the target space. Step 3: Iterative evolution. In each iterative generation, sequentially perform tread-marking behavior (alternating exploration and exploitation), sniffing behavior (local fine search), and updating the nondominated ordering and external archive. After each step, the better solution is retained by greedy selection. (1) Footprint Scent Marking Behavior  
Characteristic Gait Walking (First One-Third of Iterations)  
\[P_{i,j,k}^{new}=P_{i,j,k}^{old}-\left(\theta_{k}\cdot\alpha_{i,j,k}\cdot P_{i, j,k}^{old}\right) \tag{41}\]  
\[\theta_{k}=\frac{C_{t}}{N_{t}} \tag{42}\]  
where \(\alpha_{i,j,k}\) is a random number within the range; \(C_{t}\) is the current iteration number; and \(N_{t}\) is total number of iterations.