## 2 Literature Review
### The Berth Allocation Problem (BAP)  
From the point of view of port management, one of the problems to address is to reduce congestion and delays in vessel loading and unloading as much as possible. This is the Berth Allocation Problem (BAP). The objective of berth allocation is to service a set of vessels on a set of piers for a given period. The objectives addressed more frequently in the literature are (i) minimizing total vessel operating and waiting time; (ii) minimizing early or late departures with respect to scheduled times; and (iii) minimizing fuel consumption and emissions. There are several spatial and temporal constraints involved in BAP problems, leading to a multitude of formulations. Time restrictions are related to the vessel arrival process, the start of the service, and vessel handling times. Spatial restrictions are based on the design of the docks and their use (shared or not). According to [PERSON] and [PERSON] [17], a vessel's arrival process can be considered static or dynamic. In static arrivals, all ships are already in port. In dynamic arrivals, only a number of the scheduled ships are in port and the rest are assigned arrival times. These arrival times can be considered deterministic, with fixed values, or stochastic, in which a distribution of arrival times can be given to reflect the uncertainty of arrivals. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
\begin{tabular}{c} Shipping companies, \\ terminal operators, \\ port authorities \\ \end{tabular} & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Commercial products for port operations. Spatial restrictions limit the feasible docking positions of vessels according to a pre-established division of the quay into alignments. Based on berth design, the BAP can be classified as discrete, continuous, or hybrid [17]. In the discrete case, the dock is divided into a set of sections and only one ship can be serviced per section at any given time [18; 19; 20]. In the continuous case, there is no dock division and a ship can occupy any arbitrary position along the dock [21; 22; 23]. This leads to better utilization of dock space; however, it is computationally more complicated. In the hybrid case, the dock is divided into a set of sections but a vessel can occupy more than one section at a time, and several vessels are also allowed to share the same alignment at the same time [24; 25]. The study of the berth allocation problem in liquid bulk terminals (liquid bulk-BAP) has received little attention in recent years, mainly due to the uncertainties and more demanding restrictions, such as specialized pipelines or conveyers, involved in these types of operations. Table 2 shows recent papers about liquid bulk-BAP problems. For each reference, the type of algorithms used, the purpose of their objective function, and the characteristics considered, based on vessel arrival times and spatial constraints, are indicated. Some authors [26; 27; 28] proposed a mixed integer programming (MIP) model to minimize total vessel service time, considering a dynamic vessel arrival time and hybrid [26; 27] or discrete spatial constraints [28]. Moreover, [29] study and solve the problem of recovering a baseline vessel berthing schedule in a port in real time as disruptions occur. The uncertainty of vessel arrival and handling times is modeled on probability distributions derived from past data. Various metaheuristic models are used to solve bulk-BAP problems. These algorithms allow several problems to be worked on together, such as bulk-BAP and yard assignment problems [27]. Other algorithms minimize waiting time, ship operating time after berthing, and ship priority deviation based on decision support systems [30; 31]. Recently, machine learning techniques have been applied to minimize the cost associated with vessel handling operations [32]. In this case of study, the BAP is based on dynamic vessel arrivals since this arrival time is a variable decision and discrete from spatial restrictions.