# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 5 Numerical Experiments
### Experiment Settings  
#### 5.1.1 Instances Generation and Parameter Setting  
To test the performance of the algorithm, some instances are generated following the criteria applied by [PERSON] [18], [PERSON] et al. [43], and others [4,44]. These instances are extended in this paper in order to consider various types of cost- and shipowner satisfaction-related parameters in this study. The generated set \(|V|\ =\ \{10,15,20,30,40,50\}\) contains a total of 30 instances of vessels. These instances can be further categorized into two groups: small-scale instances, which include instances with a number of vessels ranging from 10 to 20, with 5 instances for each number of vessels; and large-scale instances, which include instances with a number of vessels ranging from 30 to 50, with 5 instances for each number of vessels. According to the previously described criteria, for all instances in this set, we consider a terminal of length \(L\ =\ 1000\) m and equipped with \(Q\ =\ 10\) QCs, with a planning time period of \(T\ =\ 210\) h, discretized in units of 10 m and 1 h, respectively. There are three types of vessels: feeder, medium, and jumbo, each having different length and QC requirements at the terminal, accounting for 60%, 30%, and 10% of the total vessels in each instance. The length of the vessel, \(l_{i}\), the QC requirements in crane hours, \(m_{i}\), the minimum number of QCs, \(q_{i}^{\min}\), and the maximum number of QCs, \(q_{i}^{\max}\), are generated, as seen in Table 3. For the expected arrival time of the vessel \(a_{i}\), small-scale instances are generated using uniform distribution, \(U[0,T/2]\), while large-scale instances are generated using uniform distribution, \(U[0,168]\), corresponding to a one-week period to avoid the generation of solutions that are out of the planning horizon. The desired position of the vessel, \(x_{i}^{*}\), is generated by the uniform distribution, \(U[0,L-l_{i}]\). For each vessel, its loading and unloading times, \(t_{i}\), are calculated according to Equation (14), and its expected departure time, \(d_{i}\), is \(a_{i}+1.5m_{i}/q_{i}^{\max}\). The processing time and estimated departure time are rounded to the next integer. The maximum waiting time, \(rt_{1i}\), and the maximum delay time, \(rt_{2i}\), tolerated by the vessel are \(d_{i}-a_{i}\) and \(T-d_{i}\), respectively. For all vessels, the cost coefficients and the parameters of shipowner satisfaction are the same, namely, \(\omega_{1}\ =\ 1.2\), \(\omega_{2}\ =\ 1.2\), \(\omega_{3}\ =\ 1.2\), \(\omega_{4}\ =\ 1.2\), \(\omega_{5}\ =\ 1.2\), \(\omega_{6}\ =\ 1.2\), \(\omega_{7}\ =\ 1.2\), \(\omega_{8}\ =\ 1.2\), \(\omega_{9}\ =\ 1.2\), \(\omega_{10}\ =\ 1.2\), \(\omega_{11}\ =\ 1.2\), \(\omega_{12}\ =\ 1.2\), \(\omega_{13}\ =\ 1.2\), \(\omega_{14}\ =\ 1.2\\(c_{1}\ =\ 0.38\), \(c_{2}\ =\ 0.2\), and \(c_{3}\ =\ 0.1\). According to [PERSON] and [PERSON] [45], the value function coefficients for satisfaction are generally set to \(\lambda\ =\ 2.25\), \(\eta\ =\ 0.88\). Before using the algorithm, the necessary parameters need to be determined. Some main parameters need to be set for the IMOCS algorithm, including the population size, \(N\), the maximum number of iterations, \(Iter\), and the abandonment probability, \(p_{a}\). Figure 5 illustrates the results of the sensitivity analysis experiments for the main parameters of the IMOCS. Using solving instance 50-1 as an example, the main focus is on the optimization objective of the terminal cost. According to Reference [41], the settings of the parameters involved in the test include \(N\ =\ \{60,80,100,120,140\}\), the abandonment probability \(p_{a}\ =\ \{0.15,0.2,0.25\}\), and \(Iter\ =\ \{300,400,500\}\). As shown in Figure 5 and Table 4, the IMOCS algorithm obtains high-quality solutions in a relatively short time when \(p_{a}\ =\ 0.25\) and \(N\ =\ 100\). In addition, the algorithm usually converges within 300 iterations.