## 6 Numerical Experiments
### Input Data Generation  
The parameter values for the **BSPSR** mathematical model were selected based on the available BSP literature and relevant online sources [7; 8; 9; 10; 11; 12; 14; 15; 16; 17; 18; 19; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. The adopted parameter values are presented in Table 2. It was assumed that each berthing position was available at the beginning of the planning horizon (i.e., time \"0\"): \(T_{j}^{B}=0\ \forall j\in J\) (h). The arrival pattern of vessels at the MCT was assumed to follow an exponential distribution with the average inter-arrival time (\(dT\)) of 2 h, i.e.,\(dT=exp[2]\) (h). The latter statistical distribution has been frequently used in the BSP literature for modeling vessel arrivals [8; 11; 17]. The handling time of vessels at their \"preferred berthing positions\" was generated using the following relationship: \(T_{i}^{HT*}=U[8;20]\)\(\forall i\in I\) (h), where \(U[Val_{1};Val_{2}]\) is a notation used for uniformly distributed pseudorandom numbers that vary between \(Val_{1}\) and \(Val_{2}\). The \"preferred berthing position\" for each vessel was assigned randomly from the available berthing positions as follows: \(j_{i}^{s}=U[1;n]\)\(\forall i\in I\) (berthing position), where \(j_{i}^{s}\) is a notation used for \"preferred berthing position\" of vessel \(i\). The handling time of a given vessel was assumed to increase, if it was not assigned for service at its \"preferred berthing position\" based on the following relationship: \(T_{ij}^{HT}=T_{i}^{HT*}. (1+0.03\cdot|j-j_{i}^{s}|)\)\(\forall i\in I,j\in J\) (h). The negotiated vessel departure times were generated based on their arrival times and handling times at \"preferred berthing positions\" as follows: \(T_{i}^{D}=T_{i}^{A}+T_{i}^{HT*}.U[1.20;1.50]\)\(\forall i\in I\) (h). It was assumed that the MCT operator was expected to provide service for the following types of vessels: (1) Panamax (length: 820.2 ft., draft: 41.0 ft.); (2) Panamax Max (length: 951.4 ft., draft: 41.0 ft.); (3) Post Panamax (length: 935.0 ft., draft: 42.7 ft.); (4) Post Panamax Plus (length: 984.3 ft., draft: 47.6 ft.); (5) New Panamax (length: 1200.8 ft., draft: 49.9 ft.); and (6) Triple E (length: 1312.3 ft., draft: 50.9 ft.). The vessel dimensions were adopted based on the available literature [36]. The length of berthing positions was set based on the minimum and maximum length values of the aforementioned vessel types as follows: \(L_{j}^{B}=1.20\cdot\min_{i}(L_{i}^{V})+U[0;1.20\cdot\max_{i}(L_{i}^{V})-\min_{ i}(L_{i}^{V})]\)\(\forall j\in J\) (ft.). Similarly, depth of berthing positions was set based on the minimum and maximum draft values of the aforementioned vessel types as follows: \(H_{j}^{B}=1.20\cdot\min_{i}(H_{i}^{V})+U[0;1.20\cdot\max_{i}(H_{i}^{V})-\min_{ i}(H_{i}^{V})]\)\(\forall j\in J\) (ft.). The horizontal clearance requirement for vessels was assigned using the following relationship: \(D_{i}^{H}=U[50;100]\)\(\forall i\in I\) (ft.). The vertical clearance requirement for vessels was set as follows: \(D_{i}^{V}=U[4;8]\)\(\forall i\in I\) (ft.). The vessel weights were generated using the following relationship: \(W_{i}=U[0.10;1.00]\)\(\forall i\in I\). The large positive number was set to \(\Gamma=10,000\). Using the generated data, a total of 60 problem instances were developed by changing the number of arriving vessels and the number of available berthing positions. The generated problem instances can be classified in the following two groups: (1) small size problem instances (1-30), where the number of arriving vessels was altered from 5 to  
\begin{table}
\begin{tabular}{l l} \hline \hline  
14 with an increment of one vessel, while the number of available berthing positions was altered from 2 to 4 with an increment of one berthing position; and (2) large size problem instances (31-60), where the number of arriving vessels was altered from 55 to 100 with an increment of five vessels, while the number of available berthing positions was altered from 4 to 8 with an increment of two berthing positions.