# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 5 Numerical Experiments
### Results and Analysis
#### 5.2.3 Managerial Insight  
This section discusses the importance of considering shipowner satisfaction in CT seaside operations from a managerial perspective and validates the value of our proposed multi-objective optimization model in practice. First, we analyze the relationship between terminal operating costs and shipowner dissatisfaction to explore whether there is a conflict. With the objective of minimizing the terminal operating cost, we solved instance 20-1 and recorded the trends of terminal operating costs and shipowner dissatisfaction during the optimization process. Figure 9 shows the convergence curve of the optimization process, where the horizontal axis represents the number of iterations and the vertical axis represents the terminal operation cost and shipowner dissatisfaction, respectively.  
From Figure 9, it can be observed that the terminal operation cost gradually decreases and stabilizes as the number of iterations increases. Meanwhile, shipowner dissatisfaction shows an upward trend and eventually reaches a high level, thus demonstrating that the two objectives are not positively correlated, but are in some conflict (not always). This result is explained by the fact that reducing operating costs usually relies on reducing the number of QCs assigned to each vessel and arranging vessels in positions that result in high berth utilization, which can lead to longer waiting times and delays. This conflict provides a theoretical basis for the use of multi-objective optimization and highlights the need to balance the two objectives in actual operations.  
Based on the clarification of the conflict between the two objectives--in order to further evaluate the advantages of the bi-objective optimization method, this section compares it with the traditional weighted single-objective optimization method. The weighted single-objective optimization method is performed by constructing the objective function \(Obj=w_{a}.Obj_{1}+(1-w_{a}).Obj_{2}\) and testing the results with different values of weight \(w_{a}\) in order to analyze the performance difference between the two methods. Each instance is run independently 10 times in the same environment and the average objective function value is taken.  
The results of the comparison are shown in Table 9, where the third through eighth columns show the average shipowner dissatisfaction (Obj1) and the total cost of the marina (Obj2) for the single-objective optimization with different weights, \(w_{a}\), and the ninth  
Figure 9: Convergence curve of the optimization process.  
through twelfth columns show the two endpoints (Min obj1 and Min obj2) of the Pareto frontier solution for the bi-objective optimization.  
In the Pareto frontier, the left endpoint value (Min obj1) decreases the average costs by 7.1% compared to a single-objective optimization (\(w_{a}~{}=~{}1\)) that aims to minimize Obj1, while customer dissatisfaction increases by only 5.4%. The right endpoint value (Min obj2) decreases customer dissatisfaction by 8.7% compared to a single-objective optimization (\(w_{a}~{}=~{}0\)) that aims to minimize Obj2, but the terminal operating cost only increases by about 5.6%. These results indicate that the bi-objective optimization method provides a flexible and diverse set of solutions covering a wide range of trade-offs between minimizing Obj1 to minimizing Obj2 (as shown in Figure 7). This approach provides terminal managers with an expanded choice space to select the most appropriate solution according to the actual needs (e.g., prioritizing shipowner satisfaction in the peak season scenario and prioritizing cost in the off-season scenario). In contrast, the weighted single-objective method can only generate a single solution for each optimization due to the fixed weight limitation, and relies on problem-specific knowledge and a priori preferences of decision makers.