## 2 Preliminaries  
### Bi-Objective Traveling Thief Problem  
The TTP is a novel combinatorial optimization challenge that amalgamates two classical problems, the TSP and the KP. The BITTP, as adopted in this paper following the definition from [23], is a variant of TTP with dual objective functions. Specifically, it entails a scenario where a traveling thief traverses \(n\) cities while selectively acquiring items from a pool of \(n^{\prime}\) items and placing them in their knapsack, ultimately returning to the departure city. Notably, the departure city does not contain any items, while other cities may contain one or more items. The objective is to devise an optimal route plan and item acquisition strategy to minimize the thief's travel time while maximizing the total profit from the items. It is crucial to note the interdependence between the two objectives, as the selection of heavier items may impede the thief's speed, thereby elongating travel time. The formulation of objective functions and constraints for BITTP is outlined below.  
The route plan in the BITTP is represented by the vector \(\mathbf{\pi}=[\pi_{1},\pi_{2},\cdots,\pi_{i},\cdots,\pi_{n},\pi_{1}]\), where \(\pi_{i}\) is a permutation vector denoting the sequence of cities visited by the thief. Here, \(\pi_{i}\) represents the \(i\)th city visited by the thief, with \(i\) ranging from 1 to \(n\).  
The item plan represented as \(\mathbf{z}\) is a vector denoted by \(\mathbf{z}=[z_{1},z_{2},\cdots,z_{j},\cdots,z_{n^{\prime}}]\), where \(z_{j}\) corresponds to the \(j\)th item. Each element \(z_{j}\) in \(\mathbf{z}\) is a binary value: \(z_{j}=1\) signifies that the \(j\)th item is selected, while \(z_{j}=0\) indicates its exclusion. The index \(j\) ranges from 1 to \(n^{\prime}\).  
The speed of the traveling thief diminishes linearly with the augmentation of the cumulative weight of the selected items [21]. Hence, the velocity at which the traveling thief departs from city \(\pi_{i}\) can be expressed as follows:  
\[v(W_{\pi_{i}})=v_{max}-(v_{max}-v_{min})\frac{W_{\pi_{i}}}{Q} \tag{1}\]  
where \(v_{min}\) and \(v_{max}\) represent the thief's speed when the knapsack is empty and when it is full, respectively. In this study, we set \(v_{max}=1\) and \(v_{min}=0.1\). \(W_{\pi_{i}}\) denotes the total weight of items carried as the thief departs from the city \(\pi_{i}\), while \(Q\) signifies the knapsack capacity.  
\[W_{\pi_{i}}(i,\mathbf{\pi},\mathbf{z})=\sum_{k=1}^{i}\sum_{j=1}^{n^{\prime}}a_{j}( \pi_{k})w_{j}z_{j} \tag{2}\]  
where \(a_{j}(\pi_{k})=1\) indicates that item \(j\) located at city \(\pi_{k}\) is stolen, and conversely, if it has not been stolen, \(a_{j}(\pi_{k})=0\). The weight of item \(j\) is denoted by \(w_{j}\).  
The distance matrix \(\mathbf{d}_{n\times n}\) between \(n\) cities is known and symmetric, implying that \(d_{\pi_{i},\pi_{i+1}}=d_{\pi_{i+1},\pi_{i}}\). Therefore, the traveling time, which represents the first objective function of BITTP, is computed as shown in (3).  
\[min\ f_{1}(\mathbf{\pi},\mathbf{z})=\sum_{i=1}^{n-1}\frac{d_{\pi_{i},\pi_{i+1}}}{v( W_{\pi_{i}}(i,\mathbf{\pi},\mathbf{z}))}+\frac{d_{\pi_{n},\pi_{1}}}{v(W_{\pi_{n}}(n, \mathbf{\pi},\mathbf{z}))} \tag{3}\]  
The second objective function of the BITTP, representing the thief's profit, is expressed by (4). For convenience and consistency with profit minimization, profit maximization is also transformed into a minimization solution.  
\[min\ f_{2}(\mathbf{z})=-\sum_{j=1}^{n^{\prime}}p_{j}z_{j} \tag{4}\]  
where \(p_{j}\) denotes the value of the \(j\)th item.  
Subject to:  
\[\pi_{1}=1 \tag{5}\]  
\[\sum_{j=1}^{n^{\prime}}w_{j}z_{j}\leq Q \tag{6}\]  
Equation (5) signifies that the thief always initiates the journey from city 1, while (6) dictates that the cumulative weight of the stolen items cannot surpass the knapsack's capacity.