## 3 Problem Formulation
### Problem Description
The transportation cost of one ship of type \(s\) completing a full round trip on route \(r\) includes fuel costs (\(F_{sr}\)), berthing costs (\(B_{sr}\)), and any extra penalties specifically incurred by China-built ships when deployed on routes calling at U.S. ports (\(B_{sr}^{\text{extra}}\)). The company's owned fleet includes \(A_{s}\) ships of type \(s\). To reduce operating costs, such as avoiding penalty fees associated with deploying China-built ships on routes that call at U.S. ports, the company may choose to charter in additional ships of alternative types and charter out idle ships that are not required. The weekly cost of chartering in a ship of type \(s\) is given by \(c_{s}^{\text{in}}\), and the revenue obtained from chartering out an idle ship of type \(s\) is denoted by \(c_{s}^{\text{out}}\). A route represents a complete, cyclical sequence of port calls that a ship follows. A leg is the segment of a route connecting two adjacent ports. An OD pair represents a specific transport demand from a port of origin to a port of destination, both of which lie on the same route. The path for fulfilling an OD pair's demand consists of one or more sequential legs along the route. To visually illustrate these relationships, Figure 1 presents an illustrative diagram. Figure 1 shows an example circle route \(r\) that consists of four ports. The segments connecting adjacent ports form the four legs of the route: \(l_{1}\), \(l_{2}\), \(l_{3}\), and \(l_{4}\). The figure also illustrates two distinct OD pairs. The first is a transport demand from origin port \(o_{1}\) to  
Figure 1: OD pairs and legs on a route \(r\). destination port \(d_{1}\), with its path covering legs \(l_{1}\) and \(l_{2}\). The second is a transport demand from origin port \(o_{2}\) to destination port \(d_{2}\), which is fulfilled by traversing legs \(l_{2}\) and \(l_{3}\). This structure highlights how multiple OD demands can share capacity on common legs within a single shipping route. The objective of this study is formulated as a bi-objective optimization that simultaneously maximizes the weekly profit and maximizes the total weekly transport volume across all OD pairs. The weekly profit consists of the total revenue from accepted transport demand, minus the total transportation costs of deployed ships, minus the cost of chartering in ships, plus the revenue generated from chartering out idle ships. To this end, we need to decide which candidate routes to operate. This decision is represented by the binary variable \(x_{r}\), which equals 1 if route \(r\) is operated, and 0 otherwise. We also determine the fleet composition on each operated route by assigning specific ship types. This is described by the integer decision variable \(n_{sr}\), which indicates the number of ships of type \(s\) assigned to route \(r\). In addition, we consider whether ships are rented in or chartered out. The integer decision variable \(n_{s}^{\text{in}}\) denotes the number of ships of type \(s\) chartered in during the planning horizon, while \(n_{s}^{\text{out}}\) represents the number of ships of type \(s\) chartered out. For each OD pair \((o,d)\) and each route \(r\in R_{od}\), the transport volume is determined by the continuous decision variable \(z_{odr}\), which specifies the amount of accepted demand (TEUs/week) allocated to route \(r\).