## 2 MCT Operations Description  
This study focuses on modeling the seaside operations at a multi-user MCT, where vessels from different liner shipping companies deliver and pick up containers. Various MCT berthing  
Figure 1: EA parameter selection approaches. layouts have been reported in the BSP literature (including discrete, continuous, hybrid, indented, and channel) [6; 17], and the commonly used discrete berthing layout will be adopted in this study for the considered MCT. Based on the discrete berthing layout, the MCT wharf is partitioned in a set of berthing positions, which will be referred to as \(J=\{1,\ldots,n\}\), and one vessel can be moored at a given berthing position at the time. The MCT layout, which will be modeled in this study, is presented in Figure 2. Let \(I=\{1,\ldots,m\}\) be a set of arriving vessels, which must be served at the MCT. There are two common types of vessel arrivals, which have been modeled in the BSP literature, including the following [6]: (a) static vessel arrivals--all vessels have already arrived at the MCT, and the berth schedule should be designed based on the specific objectives; and (b) dynamic vessel arrivals--vessels have not arrived at the MCT yet, but the MCT operator has the information regarding the expected vessel arrival time (which is negotiated with the corresponding liner shipping company). The dynamic vessel arrivals will be considered in this study. The vessels are assumed to arrive at the MCT at the scheduled time (\(T_{i}^{A}\), \(i\in I\)--measured in h). The scope of this study does not include modeling uncertainty in vessel arrivals due to different factors, including adverse weather, reduction in the vessel sailing speed as a result of the main engine malfunctioning, potential delays in vessel service at preceding ports, human errors, and others. Upon arrival at the MCT, each vessel will be towed by tug boats to the assigned MCT berthing position. In some cases, the assigned berthing position may not be available at the moment (as the other vessel is being served), and the vessel will be towed to the dedicated waiting area, which is located near the MCT. Note that increasing waiting time (\(\mathbf{T}_{i}^{WT},\ i\in I\)--measured in h) of vessels may negatively affect the MCT operations, as increasing number of waiting vessels may cause congestion in the access channel and result in navigational difficulties for the vessels, entering and leaving the MCT. The MCT operator should consider the length (\(L_{i}^{V}\), \(i\in I\)--measured in ft.) and draft (\(H_{i}^{V}\), \(i\in I\)--measured in ft.) of the arriving vessels in the design of berth schedules. Each berthing position has two spatial attributes,  
Figure 2: Layout of the considered MCT. including length (\(L_{j}^{B},\ j\in J\)--measured in ft.) and depth (\(H_{j}^{B},\ j\in J\)--measured in ft.). If vessel \(i\) does not satisfy certain horizontal and vertical clearance requirements (which will be denoted as \(D_{i}^{H},\ i\in I\) and \(D_{i}^{V},\ i\in I\), respectively, and are both measured in ft.), it cannot be assigned for service at berthing position \(j\). Moreover, service of a given vessel cannot start before the time, when the assigned berthing position becomes available in the planning horizon at the first time (\(T_{j}^{B},\ j\in J\)--measured in h). Once the assigned berthing position becomes available, the vessel is moored, and the on-shore quay cranes start loading and unloading containers (see Figure 2). There are several other important factors, which should be considered by the MCT operator in the design of berth schedules, including the following: (1) vessel handling rates; (2) allocation of the available handling equipment; and (3) allocation of the available storage space. Vessel handling rates, measured in the number of twenty-foot equivalent units (TEUs) handled per hour, are negotiated between the MCT operator and liner shipping companies.