## 2 Preliminaries
### Expression of Solution and Objective Function  
The proposed Multi-objective Five-element Cycle Optimization algorithm based on Multi-strategy Fusion (MOFECO-MS) is utilized in this study to tackle the BITTP. In the context of MOFECO-MS applied to BITTP, each solution corresponds to an element within the FECM framework. The population, comprising \(N\) individuals of MOFECO-MS, is segmented into \(q\) cycles, each containing \(L\) elements. They are denoted as \(\mathbf{x}_{ij}(k)\), an element representing the \(i\)th element in the \(j\)th cycle at the \(k\)th iteration, constituting a feasible solution for BITTP. The decision variables of BITTP encompass two components, the route plan \(\mathbf{\pi}_{ij}(k)\) and the item plan \(\mathbf{z}_{ij}(k)\). The expression \(\mathbf{x}_{ij}(k)=[\mathbf{\pi}_{ij}(k),\mathbf{z}_{ij}(k)]\) succinctly captures this duality.  
The route plan \(\mathbf{\pi}_{ij}(k)\) comprises \(n\) decision variables, corresponding to the number of cities, denoted as:  
\[\mathbf{\pi}_{ij}(k)=[\pi_{1,ij}(k),\ \pi_{2,ij}(k),\ \cdots,\ \pi_{n,ij}(k)] \tag{8}\]  
The item plan \(\mathbf{z}_{ij}(k)\) encompasses \(n^{\prime}\) decision variables, representing the number of \(n^{\prime}\) items, denoted as:  
\[\mathbf{z}_{ij}(k)=[z_{1,ij}(k),\ z_{2,ij}(k),\ \cdots,\ z_{n^{\prime},ij}(k)] \tag{9}\]  
In MOFECO, the number of masses associated with each element is equivalent to the number of objective functions. Here, \(m_{ij,r}(k)\) denotes the \(r\)th objective function value of the element \(\mathbf{x}_{ij}(k)\):  
\[m_{ij,r}(k)=f_{r}(\mathbf{x}_{ij}(k)) \tag{10}\]  
For BITTP, \(r\) takes values of 1 and 2.  
Based on the relationship among FECM, MOFECO-MS, and BITTP as shown in Table 2, MOFECO-MS is designed to solve the BITTP. Table 2 illustrates the relationship between the Five-element Cycle Model (FECM), the Multi-objective Five-element Cycle Optimization algorithm (MOFECO-MS), and the Bi-objective Traveling Thief Problem (BITTP). In FECM, elements \(\mathbf{x}_{i}(k)\) represent entities within the cycle, with mass \(m_{i}(k)\) indicating their state, and force \(F_{i}(k)\) reflecting the interactions driving system evolution.  
In MOFECO-MS, these elements correspond to potential solutions within the algorithm's population, where each element's mass \(m_{ij,r}(k)\) is linked to the objective function values in BITTP, indicating solution quality. The force \(F_{ij,r}(k)\) applied to these elements represents the evolutionary pressure guiding the optimization process. The BITTP context translates these elements into concrete solutions (tour plans and picking plans), with their mass reflecting objective function values like travel time and total profit.  
This table clarifies how MOFECO-MS leverages the principles of FECM, with each component in the optimization algorithm corresponding directly to the theoretical constructs in FECM, thereby providing a strong foundation for solving BITTP.  
\begin{table}
\begin{tabular}{c c c} \hline
**FECM** & **MOFECO-MS** & **BITTP** \\ \hline Elements \(\mathbf{x}_{i}(k)(i=1,2,\cdots,L)\) & Elements \(\mathbf{x}_{ij}(k)\)\((i=1,2,\cdots,L;j=1,2,\cdots,q)\) & Solution (Tour plan, picking plan) \(\mathbf{\pi}_{ij}(k)\), \(\mathbf{\pi}_{ij}^{k}\) \\ Mass of elements \(m_{i}(k)\) & Mass of elements & Objective functions (Travel time \(f_{1}\) and Total profit \(f_{2}\)) \\ Force exerted on elements \(F_{ij}(k)\) & Force exerted on elements \(F_{ij,r}(k)\) & Variables estimating the quality of solutions \\ \hline \end{tabular}
\end{table}
Table 2: Relationship among FECM, MOFECO-MS and BITTP.