## 2 Optimization Model
### Solution Algorithm
#### 2.4.1 Solution Procedure for Multi-Objective Optimization
Cautious Stepping Feature (From One-Third to Two-Thirds of Iterations)  
\[P_{i,j,k}^{new}=P_{i,j,k}^{old}+F_{k}\cdot\left(P_{j,k}^{best}-L_{k}\cdot P_{ j,k}^{worst}\right) \tag{43}\]  
\[F_{k}=\beta_{1,k}\cdot\theta_{k} \tag{44}\]  
\[L_{k}=\text{round}(1+\beta_{2,k}) \tag{45}\]  
where \(\beta_{1,k}\) and \(\beta_{2,k}\) are random numbers within the range [0, 1]; \(F_{k}\) is the step factor at the \(k\)-th iteration; and \(L_{k}\) is the step size at the \(k\)-th iteration. Twisting Footsteps Feature (Final One-Third of Iterations)  
\[P_{i,j,k}^{new}=P_{i,j,k}^{old}+\omega_{i,j,k}\cdot\left(P_{j,k}^{best}-\left| P_{i,j,k}^{old}\right|\right)-\omega_{i,k}\cdot\left(P_{j,k}^{worst}-\left|P_{i,j,k}^{ old}\right|\right) \tag{46}\]  
\[\omega_{i,j,k}=2\pi\theta_{k}\gamma_{i,k} \tag{47}\]  
where \(\gamma_{i,k}\) is a random number within the range [0, 1], and \(\omega_{i,k}\) is the twisting angular velocity of the \(i\)-th individual at iteration \(k\). (2) Sniffing Behavior  
The mathematical model of brown bear sniffing behavior is as follows:  
\[P_{m,j,k}^{new}= \begin{cases}P_{m,j,k}^{old}+\lambda_{j,k}\cdot\left(P_{m,j,k }^{old}-P_{n,j,k}^{old}\right),f\left(P_{m,k}^{old}\right)<f\left(P_{n,k}^{old }\right)\\ P_{m,j,k}^{new}=P_{m,j,k}^{old}+\lambda_{j,k}\cdot\left(P_{n,j,k}^{old}-P_{m,j,k}^{old}\right),f\left(P_{m,k}^{old}\right)\geq f\left(P_{n,k}^{old}\right) \end{cases}  \tag{48}\]  
where \(\lambda_{j,k}\) is a random number within the range [0, 1]. Step 4: Termination and Output. Terminate when the maximum number of iterations is reached or there is no improvement, and output the optimal Pareto solution set. To comprehensively evaluate the performance of the multi-objective optimization algorithms used in this study, three widely recognized performance metrics were selected:computational time, Hypervolume (HV), and Inverted Generational Distance (IGD) [46; 47]. These metrics, respectively, assess the computational efficiency, the diversity and convergence of the obtained solution sets, and the proximity of the solutions to the true Pareto front. The Hypervolume (HV) metric measures the volume of the objective space covered by the obtained solutions and reflects both convergence and diversity. Its formula is defined as follows:  
\[HV=\text{Lebesgue measure}\left(\bigcup_{x\in P}[f_{1}(x),r_{1}]\times[f_{2} (x),r_{2}]\times\ldots\times[f_{m}(x),r_{m}]\right) \tag{49}\]  
where \(P\) is the current set of non-dominated solutions (Pareto front); \(f_{i}(x)\) is the value of solution x in the i-th objective; \(r=(r_{1},r_{2},\ldots,r_{m})\) is the reference point, usually chosen as a worst-case point in the objective space; and \(m\) is the number of objectives. A larger HV value indicates a better-performing solution set. The Inverted Generational Distance (IGD) metric evaluates the proximity and diversity of the obtained solutions with respect to the true Pareto front. Its formula is given by:  
\[IGD(P,P^{*})=\frac{1}{\left|P^{*}\right|}\sum_{w\in P^{*}}\min_{u\in P}d(u,v) \tag{50}\]  
where \(P\) is the obtained non-dominated solution set; \(P^{*}\) is the true or reference Pareto-optimal solution set; \(d(u,v)\) denotes the Euclidean distance between solutions \(u\) and \(v\); and \(\left|P^{*}\right|\) is the number of solutions in the reference set. A smaller IGD value indicates better convergence and distribution of the solution set.