## 3 Methodology
### Profitability Results
#### 4.2.2 The Effect of Revenue-Related and Cost-Related Parameters  
\(c_{in},c_{re}\) and \(c_{p}\) are parameters related to the operating revenue and the operating cost. As shown in Figure 7a,b, \(Y(r_{\text{max}})\) increases with the increase of \(r_{\text{max}}\) under the influence of \(c_{in}\) and \(c_{re}\). \(Y(r_{\text{max}})\) increases with the decrease of \(r_{\text{max}}\) under the influence of \(c_{p}\) (Figure 7c). It can be explained by the different meanings of these parameters. The relocation strategy can reset the number of bikes at each node to an initial state, which make the bike-sharing system able to supply the customers with enough bikes after each relocation. Due to the definition of \(c_{in}\), the high price means operators can generate more revenue by satisfying the customer demand, which can be promised by frequent relocations. Therefore, with the increase of \(c_{in}\), both the optimal relocation frequency and the maximum profit increase. According to the change of \(r_{\text{max}}\) and \(Y(r_{\text{max}})\), as shown in Figure 7a, \(r_{\text{max}}\) is insensitive to the increase of \(c_{in}\) and \(Y(r_{\text{max}})\) is sensitive to the increase of \(c_{in}\). In reality, operators can make more profits by raising the unit price. However, they have to consider the unit price given by their competitors, who have the ability to pull regular customers from them. Figure 6: Effect of relocation-related parameters. (**a**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(c\); (**b**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(m\). Although acquiring a large revenue from customers can cover the expense on re-locations to some extent, the increase of the optimal relocation frequency is slight. For operators, pricing decisions have a significant influence on the total profit but have a slight effect on the decision-making about relocating bikes. As shown in Figure 7b, with the increase of \(c_{re}\), both \(r_{\max}\) and \(Y(r_{\max})\) decrease. It is because that the high cost of relocating bikes is supposed to decrease the relocation frequency to save cost for operators. With the decrease of relocation frequency, the total profit decreases due to the huge expense on relocations and the decreased revenue caused by the occasional shortage of bikes at some nodes. The slope of the curve in Figure 7b indicates that \(r_{\max}\) is more sensitive to the change of \(c_{re}\) than \(Y(r_{\max})\), especially when \(c_{re}\) has a small value. It is suggested that operators should consider how to reduce the expense on relocating bikes in the system, which can be very beneficial to solve the imbalance problem of bike distribution. In Figure 7c, \(r_{\max}\) increases and \(Y(r_{\max})\) decreases with the increase of \(c_{p}\). The large value of \(c_{p}\) means a large penalty of losing customers, which happens when customers arrive at the node where there is no bike available. The increase of relocation frequency can minimize the losses by relocating bikes to the nodes with high demands. With the increase of the penalty and the increased cost derived from frequent relocations, the total profit decreases. In the real world, the unit cost of penalty from unmet demands is difficult to measure. In the numerical experiments, the value of \(c_{p}\) is set far larger than the values of \(c_{in}\) and \(c_{re}\). It is because unmet demands may have a significant impact on the loyalty of the regular customers. If the customers often cannot find any bikes to ride, they are likely to feel dissatisfied with the dockless bike-sharing system and turn to alternatives, which may bring about great losses (e.g., an obvious decrease in the potential revenue) for the bike-sharing company.