# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 3 Description and Formulation of BACASP  
### Problem Description  
BACASP involves a set of vessels \(V~{}=~{}\{1,2,\ldots,|V|\}\), which load and unload cargo during the planning horizon \(T~{}=~{}\{1,2,\ldots,|T|\}\) at a continuous quay. The quay has a length of \(L\) and is equipped with \(Q\) quay cranes mounted on the rail (as shown in Figure 1). A contract is usually signed between the shipowners and the terminal managers prior to the planning horizon [4]. This contract is used to provide information--in advance--about the relevant information of the vessel, such as length, expected arrival time, desired departure time, and desired position. The desired position usually refers to the position closest to the vessel's loading and unloading yard. If the actual berthing position deviates from the desired position, additional transportation costs are incurred. Terminal managers then determine the berthing position, berthing time, and specific QC assignments for each vessel based on the relevant information to formulate a berthing plan that is both efficient and responsive to the shipowners' needs. Each vessel \(i\) arrives at the expected arrival time, \(a_{i}\), at which point, it is necessary to wait at the anchorage before berthing at the terminal if there are no available berths and enough seaside QCs to service it. Otherwise, the vessel is scheduled to berth. After vessel \(i\) berths, the CT assigns specific QCs to serve it. Once assigned, the number of QCs assigned to a vessel remains constant during its processing (time-invariant). Each vessel \(i\) occupies the length, \(l_{i}\), of the terminal at position \(y_{i}\) without interruption until its loading and unloading processes are complete. The time for processing is related to the number of QCs assigned, the interference among the QCs, and the workload of horizontal transportation [18]. The vessel then departs from the terminal after the completion of its loading and unloading processing. Typically, terminal managers seek to minimize the operational costs on the seaside of the terminal, which include the cost of berth occupancy before vessel berths, the cost of loading and unloading operations on the QCs, and the additional transportation costs associated with vessel deviations from desired positions. At the same time, human behavior in the system can have a significant impact on the scheduling system [37]. As the customers and ultimate recipients of the CT's production scheduling process, the level of dissatisfaction of shipowners can be a measure of a terminal's level of service, which in turn affects the attractiveness and competitiveness of the terminal. [PERSON] and [PERSON] [38] and [PERSON] and [PERSON] [39] illustrated that waiting times and delays in the queuing process cause customer dissatisfaction and that there is some negative correlation between longer waiting times and customer satisfaction. Therefore, the delayed departure of vessels and waiting before berthing may lead to shipowner dissatisfaction, which in turn reduces satisfaction with CTs. As stated in Section 2.3, prospect theory is used to measure shipowner dissatisfaction. The value function in prospect theory represents human behavior in dealing with risk in an  
Figure 1: Layout of a container terminal. uncertain environment [37; 40]. The gist of prospect theory is that the actor focuses only on the relative change in returns when making decisions, and the value function can be expressed in Equation (1):  
\[v(x)\ =\ \begin{cases}x^{\varphi}&x\geq 0\\ -\lambda(-x)^{\eta}&x<0\end{cases}  \tag{1}\]  
where \(x\) denotes the value of gain or loss; \(\varphi\) is used to describe the shape of the gain function; \(\lambda\) denotes the loss aversion coefficient; and \(\eta\) denotes the losses risk attitudes coefficient. In BACASP, shipowner dissatisfaction mainly depends on the time of delayed departure of the vessel from the terminal and the waiting time before berthing.