# Bi-Objective Optimization for Joint Time-Invariant Allocation of Berths and Quay Cranes
## 2 Literature Review
### Joint Allocation of Berths and Quay Cranes  
Many studies have focused on time-variant BACAP and BACASP. [PERSON] and [PERSON] [18] examined the possibility of speeding up the arrival of each vessel. They also considered that as the number of cranes serving a vessel increases, the marginal crane productivity decreases due to interference among them. [PERSON] et al. [19] proposed a mixed-integer linear scheme for optimizing berth position, berth time, and crane schedule. [PERSON] et al. [20] proposed a set partitioning model for the BACAP and introduced column reduction techniques to improve the solution process. [PERSON] et al. [21] developed an integer linear programming model considering tidal effects and proposed three heuristic solution algorithms. [PERSON] et al. [22] developed a recoverable robust optimization approach for the weekly berth and quay crane planning problem, focusing on uncertainties in vessel arrival times and quay crane handling rates. They introduced a proactive baseline schedule with reactive recovery costs to balance efficiency, robustness, and recoverability. [PERSON] et al. [23] proposed an integrated mathematical model and a fast local exploitation heuristic algorithm aimed at minimizing the container transit distance. [PERSON] et al. [24] aimed to minimize the total turnaround time of the vessel and the total penalty cost of the maintenance time of the terminal QC tardiness. [PERSON] et al. [25] proposed an optimization scheme, integrating continuous BACAP and shore power allocation to achieve service efficiency improvement and carbon emission reduction.  
Recently, time-invariant BACAP and BACASP have begun to receive attention from scholars. [PERSON] et al. [26] aimed to minimize the total service time and shipowner dissatisfaction level of all vessels. [PERSON] et al. [27] aimed to minimize the total terminal turnaround time and the number of QC moves and proposed a multi-objective hybrid genetic algorithm to solve the problem. [PERSON] et al. [28] introduced a binary integer linear scheme, BACAP, and extended it to BACASP by combining it with a specific time-invariant QC assignment problem. [PERSON] et al. [29] proposed a combined laycan and time-invariant BACASP methodology in a tidal port with multiple terminals. [PERSON] et al. [4] proposed a synthesis method for solving time-invariant BACASP. [PERSON] et al. [30] aimed to minimize the expected total dwell time of a vessel at the terminal. [PERSON] et al. [31] proposed a new approach to incorporate QC trips and setup times into BACASP. They developed a new mixed-integer linear programming model in which berth position and time are considered continuous variables, and they used a genetic algorithm to solve the problem.  
After reviewing the problems related to berth allocation and QC assignment, we conclude that the reality of BACAP and specific QC assignments has only been partially considered in the existing research. The research work on BACASP is mainly single-objective, and its time-invariant version has received less attention compared to the time-variant version. Only a few of the existing multi-objective BACASP studies have considered shipowner satisfaction, where shipowner satisfaction is usually considered as a direct, linear metric, such as measuring shipowner satisfaction directly in terms of delay or waiting time [26; 32]. However, shipowner satisfaction is a complex psychological process that is influenced by many factors, including cognitive and affective factors, and its responses are usually nonlinear [33; 34]. For example, with respect to delay time, shipowners may react minimally to short delays but strongly to long delays. Such nonlinear responses can be captured by prospect theory, which emphasizes the asymmetric response of people to losses and gains. In other fields, many studies based on prospect theory have been used to measure shipowner satisfaction, achieving good results [35; 36]. Therefore, prospect theory is used to calculate shipowner satisfaction in the proposed bi-objective time-invariant BACASP.