Optimizing Berth Allocation for Maritime Autonomous Surface Ships (MASSs) in the Context of Mixed Operation Scenarios

Abstract

This study deals with berth allocation for Maritime Autonomous Surface Ships (MASSs) in the context of the mixed operation of MASSs and manned vessels from the perspective of port-shipping companies’ collaboration. Two berth allocation strategies, namely the separated-type and the mixed-type, are proposed in this article. Two mixed integer nonlinear programming models aimed at minimizing the total docking cost of the vessels in the port and the waiting time for berths are developed and solved using Gurobi, respectively. A large-scale simulation of the mixed-type berth allocation model is carried out using an improved simulated annealing algorithm. Several experiments are conducted to test the effectiveness of the model and to draw insights for commercializing autonomous vessels. The presented results show that multi-objective modeling and optimization should be conducted from the collaboration of port-shipping companies, which is more efficient from the perspective of shipping companies or ports, respectively. When berth resources are limited or there is a high requirement for operational safety, the separated-type berth allocation strategy is more efficient. When the number of MASS-dedicated berths reaches a certain proportion, the total docking cost of the vessel no longer changes, indicating that more dedicated berths are not better.

1. Introduction

In 2018, the International Maritime Organization (IMO) introduced the concept of Maritime Autonomous Surface Ships (MASSs) at the 100th Maritime Safety Committee meeting. The IMO defines MASSs as ships that can operate independently of manned vessels to varying degrees and categorizes them into four levels of autonomy: ships that need to provide automated processes and decision support, remotely operated ships (ROVs) that need to carry seafarers, ROVs that do not need to provide seafarers, and fully autonomous ships [1]. This article discusses autonomous ships considered to be large, unmanned container commercial vessels, and unmanned at least during part of their voyages, hereafter referred to as MASSs.

MASSs can be applied to maritime traffic, waterway transportation intelligence, and marine environmental protection and have become a trending topic in the field of maritime traffic engineering and an important trend in the future development of high-tech ships [2]. According to a “Global Autonomous Ship Industry” report released by ReportLinker, a famous French market research company, the global autonomous ship market, will reach USD 10.1 billion by 2026 [1], although only a few companies have attempted to actually operate autonomous ships. For example, the Norwegian ship operating company Massterly has developed an autonomous ship remote operation center and is attempting to operate MASSs [3] (Figure 1). China’s Wisdom Navigation (Qingdao) Technology Co., Ltd., has developed an autonomous ship named “Zhi Fei”, which was officially put into operation on the Qingdao Port–Dongjiakou route on 22 April 2022, [4] (Figure 2). The “Zhifei” is China’s first full-container type vessel with autonomous navigation capabilities which is designed to carry a cargo volume of less than 2999TEU and a maximum speed of 12 knots [5]. It can be foreseen that the mixed operation of manned vessels and MASSs will obviously become the main development trend for a long time in the future (Kim T et al., 2022) [6]. In addition, the South Korean “Prism Courage” large LNG carrier has completed an autonomous transoceanic navigation test; Japan is continuously promoting the “MAEGURI2040” project, aiming to achieve autonomous navigation for 50% of its domestic vessels by 2040. So far, it has completed the long-distance verification of key technologies such as autonomous collision avoidance and automatic berthing for six types of vessels including small sightseeing boats, ro-ro ships, and container ships; Netherlands has proposed that 25% of its inland cargo ships reach the R2 (remote control with no crew on board) level by 2030 and has formed an autonomous ferry fleet; Russia has provided a legal framework for autonomous vessel operations by amending the “Merchant Shipping Act” and promulgating MASS regulatory regulations. It has verified autonomous navigation systems on three test vessels, including icebreakers on the Arctic route and bulk carriers on the Black Sea route [5].

Regarding the gradual commercialization of MASSs, navigation safety is considered the most crucial element. Therefore, path planning, collision avoidance, and risk prevention have become the hot topics of extensive research today [7]. This mainly includes studies on various adaptive obstacle-avoidance and path planning algorithms [8], and the development of autonomous navigation decision making models [9]; research on the real time identification of ship positions, speeds, and directions; and dynamically adjusting the distance between MASSs and obstacles to achieve collision avoidance [10,11,12,13]. In particular, Liu et al. [14] paid special attention to the collision risk between MASSs and manned vessels and proposed an effective collision-avoidance decision making model based on reinforcement learning methods. There are also scholars who consider the risks of MASSs in practical operations from the perspective of cyber-attacks [15] and provided solutions. Although MASS technology has matured and been widely applied, there has been relatively little discussion on the key determinants of its commercial application in specific ports, and it has not been deeply analyzed in combination with the actual operational characteristics of the ports [16]. The motivation of this study lies in one of the major problems in the commercialization of MASSs, that is, the scheduling problem when manned vessels and MASSs operate in a mixed manner within a region. Ship sailing arrangements are influenced by many factors, and one of the most important factors is the availability of berths. Evidently, not all berths are suitable for MASSs. Port operators should do berth allocation for autonomous vessels [17]. The Berth Allocation Problem (BAP) is a key issue in resource scheduling during port operations and a crucial link in port management [18]. As for the path optimization of autonomous ships, the congestion or collision between vessels when sailing in crowded or restricted waters, as well as issues, such as how to achieve navigation, turning, berthing, and unberthing in narrow waters, are not the research objectives of this article.

Actually, the berth allocation plan of port operators affects the time a vessel spends in port and its safety directly, and the efficiency and profitability of both the port-shipping companies’ collaboration. Therefore, as customers of the terminal, shipping companies’ interests and costs should be taken into account during the optimization of port berth allocation. From the perspective of port-shipping companies’ collaboration, this article conducts research on berth allocation in the context of the scenario of the mixed operation of MASSs and manned vessels. According to different application scenarios, two berth allocation strategies, namely the separated-type and the mixed-type, are proposed. Specifically, this article explores the estimation method of the total docking cost of MASSs, and analyzes the time discount factor of manned vessels during loading/unloading at the berths designated for MASSs. Two mixed integer programming models aiming to minimize the total docking cost of the vessels in port and the waiting time for berths are established and solved using Gurobi, respectively. An improved simulated annealing algorithm is designed to solve large scale examples of the mixed-type berth allocation model. Finally, a series of experiments are carried out to compare and analyze the effectiveness of different strategies. The research of this article provides scientific support for decision making in the promotion and application of MASSs and the updating planning of port berths.

The rest of this article is organized as follows: Section 2 reviews the relevant literature. Section 3 describes the research problem, clarifies the basic assumptions of the problem, and provides notation explanations. Section 4 discusses the analysis and calculation of the total docking cost of MASSs, establishing a mixed integer nonlinear programming model for separated-type berth allocation and solved using Gurobi. Section 5 develops and solves a model for mixed-type berth allocation, and then, conducts a comparative analysis of the two strategies. Section 6 designs an improved simulated annealing algorithm to verify the model for mixed-type berth allocation through large scale examples. Section 7 presents a comparative study for the two strategies through multiple experiments.

2. Literature Review

Exploring how to improve port operational efficiency and reduce costs, the study of BAP can be reviewed from two perspectives as follows: (i) BAP optimization based on the port’s own resources; (ii) BAP optimization research based on port-shipping companies’ collaboration.

2.1. BAP Study Based on the Port’s Own Resources

BAP research based on the port’s own resources usually involves integration berths and the port’s own resources under uncertainty, such as yard space, vessels’ arrival times, quay-crane allocation, and so on. They are as follows: Zhen et al. [19] addresses the integrated allocation problem of berths and yard space under uncertainty and proposed an integrated planning method. They combined a two-stage stochastic integer programming model and solved it using a column-generation-based decomposition algorithm, thus improving port operational efficiency. Scholars, such as Ji et al. [20] and Chargui et al. [21], took into account factors, such as the uncertainty of arrival times, and proposed optimized berth allocation methods suitable for uncertain arrival times. The former adopted the NSGA-II algorithm, while the latter designed an algorithm based on exact decomposition for solving. Dai et al. [22] focused on the optimization of berth allocation in maritime transportation considering the quay-crane setup time. They used a bi-level programming model to deal with the uncertainties in berth allocation and optimized the berth allocation through an adaptive cross-entropy algorithm and heuristic adjustment. Wang et al. [23] proposed a column-generation method for the integrated problem of berth allocation, quay-crane allocation, and yard allocation. Liu et al. [24] considered the ship route of the port optimization into the BAP model and proposed an improved branch-and-price heuristic algorithm. Yu et al. [25] proposed a multi-objective optimization model aiming to reduce emissions and balance economic demands, providing management suggestions for the green development and sustainable operation of the port.

2.2. BAP Research Based on Port-Shipping Companies’ Collaboration

Unlike the BAP research that is centered on the resources of individual ports, the BAP research focusing on port-shipping companies’ collaboration emphasizes cooperative optimization among ports, as well as between ports and shipping companies, and among shipping companies themselves. Msrtin-Iradi et al. [26] studied the berth allocation problem of collaboration among multiple ports based on a cooperative game through sharing port resources via mathematical analysis. Guo et al. [27] and Zheng et al. [28] explored the optimal number of berths and collaboration strategies when multiple liner companies cooperate to lease dedicated berths from a tactical level. Zheng et al. [29] addressed the co-existence of general berths and dedicated berths in ports and proposed joint optimization strategies for the lease of the two types of berths. Only Zhang et al. [17], when analyzing the vessel scheduling and berth allocation problems of MASSs in the shipping network, proposed a mixed integer nonlinear programming model. By minimizing the total docking cost of fuel consumption and delay penalties, they optimized the arrival time of MASSs and berth allocation. This research not only demonstrates the potential of port-shipping companies’ collaboration in the application scenarios of emerging technologies but also optimizes the overall scheduling efficiency and reduces fuel consumption and environmental impacts.

Altogether, the literature review above reveals that the BAP study in the context of the mixed operation of manned vessels and MASSs is still in the primary stage. Only Zhang et al. [17] analyzed the autonomous vessel scheduling and berth allocation problem in a shipping network under the single operation scenario of MASS. Moreover, only a small-scale case validation has been carried out for the proposed model. Details are presented in

Table 1. Literature review.

Author(s)	Year	BAP Optimization	Type of Vessel	Mixed Operation Scenario of Manned Vessel and MASSs
Zhen et al. [19]	2022	Y	Manned vessels	N
Ji and Huang [20]	2022	Y	Manned vessels	N
Dai et al. [22]	2023	Y	Manned vessels	N
Wang et al. [23]	2018	Y	Manned vessels	N
Yu et al. [25]	2023	Y	Manned vessels	N
Zheng and Wang [29]	2023	Y	Manned vessels	N
Zhang and Wang [17]	2020	Y	MASS	N
This study	2024	Y	Manned vessels + MASS	Y

.

3. Problem Description

For the berth allocation problem, it can be geometrically represented as a two-dimensional bin packing problem, as presented in Figure 3. A two-dimensional space–time diagram of time–shoreline is established with the shoreline $L$ of the berth wharf as the X-axis and the time $t$ of berth allocation as the Y-axis [30].

In Figure 3, taking vessel $i$ as an example, $t_i^f$ represents the start docking time of vessel $i$, $t_i^e$ represents the departure time of vessel $i$, and $x_i$ is the berth position of vessel $i$ along the shoreline. The value of $x_i$ indicates the docking position of the bow of vessel $i$ on the shoreline $L$.

The requirements for MASSs’ berths are obviously different from those of ordinary vessel berths, and not all berths are suitable for MASS operations [17]. First, MASSs require precise positioning systems and advanced sensors to support their autonomous docking operations, and the berths need to be equipped with additional navigation aids and sensor interfaces. Additionally, owing to the operational characteristics of MASSs, berths need to improve collision prevention facilities, install emergency shutdown buttons, and implement remote monitoring systems [31]. Although ordinary port berths can theoretically accommodate MASSs, upgrading and modifying ordinary berths is necessary to ensure operational safety and efficiency (Zhang et al., 2022) [32]. In order to clearly describe this, this article divides port berths into two types: Class A and Class B. Class A berths refer to ordinary berths which are suitable only for manned vessels, whereas Class B berths are mixed berths that can accommodate both MASSs and manned vessels. The total number of berths in a port is the sum of Class A berths and Class B berths.

This article discusses two berth allocation strategies in the following two scenarios.

In the first scenario, we consider that the manned vessels’ number will still far exceed the number of MASSs for a long period in the future [6], and that in the initial phase of mixed operations between MASSs and manned vessels, safety will still be the most critical issue due to the MASSs’ technology and experience. Therefore, when the number of Class A berths of a port is sufficient to meet the docking requirements for manned vessels, according to the analysis based on the historical docking records of the ports, the separated-type berth allocation strategy should be recommended. That is, manned vessels will dock at Class A berths, while MASSs will dock at Class B berths. Class A and Class B berth allocation will be optimized, respectively.

In the second scenario, as the scale of the MASSs’ operations increases, the number of Class B berths gradually increases, and a mixed-type berth allocation strategy for MASSs and manned vessels should be adopted where all berths participate in a unified mixed scheduling optimization.

3.1. Model Assumptions

Through a literature analysis and joint discussion with practitioners from port-shipping companies’ collaboration (co-authors), the following hypotheses were derived. Hypotheses (1), (3), and (9) are commonly found in port practices (Dalian Port, etc.). The remaining hypotheses are derived from the literature Refs. [17,29].

(1)

Communication between the MASSs and manned vessels and with the port management system is reliable, with no packet loss, error codes, or delays;

(2)

The port equipment and berth structure in Class B berths are capable of adapting to the docking and loading/unloading needs of manned vessels [17];

(3)

The model assumes that all vessels at sea, whether manned vessels or MASSs, strictly comply with all rules and regulations related to navigation and docking, such as rules of the road and safety distances;

(4)

In this article, we study discrete BAPs, where the arrival time and loading/unloading times of all vessels are known [29];

(5)

The physical elements of both types of berths can accommodate and serve all arriving vessels of suitable capacity [29];

(6)

Berth preference is reflected between Class A and Class B berths [29];

(7)

All vessels loading/unloading operations are carried out immediately after the vessels dock [29];

(8)

Each vessels leaves the port immediately after loading/unloading the cargo [17];

(9)

Environmental factors, such as the wind speed, current speed, and other oceanographic conditions, are assumed to be known and relatively stable during berth allocation.

3.2. Notation

The notation is defined as in

Table 2. Notations of the model.

Sets:
$B$	A collection of port berths $B$$=\{B\_1$$, B\_2$$, \dots B\_m$$\left\}\right), \mathrm{where} B\_m$ is the total number of berths in the port.
$I$	Arriving vessels muster $I$$=\{I\_1$$, I\_2$$, \dots I\_m$$\left\}\right), \mathrm{where} I\_m$ is the total number of vessels arriving at the port.
$I_l$	Arrive at the MASS assembly $(I_l=$$\{I_l\_1$$, I_l\_2 \dots I_l\_n$$\left\}\right), \mathrm{where} I_l\_n$ is the total number of MASS arrivals.
Parameters:
$C$	The total docking cost of the vessels in the port.
$C_{\mathrm{i}}^{\left(op\right)}$	Manned vessels $i$ operating cost per unit time in port (USD).
$C_{\mathrm{i}}^{\left(run\right)}$	$MAS{S}_i$ operating cost per unit time in port (USD).
$C_i^{\left(handling\right)}$	Manned vessels $i$ cost per unit time for loading/unloading operations at ordinary berths (in millions of dollars).
$C_{\mathrm{i}}^{\left(load\right)}$	$MAS{S}_i$ Cost per unit time for loading/unloading operations at Class B berths (in millions of dollars).
$T_{ai}$	Vessels $i$ the projected time of arrival at the port.
$h_i$	Vessels $i$ loading/unloading time at Class A berths (hours).
$D$	Number of days in the planning period.
$M$	A sufficiently large positive number.
$\beta$	The degree of efficiency improvement of Class B berths compared to Class A berths.
$T_{MASS}$	It represents the loading/unloading time of MASSs at Class B berths, and also indicates the basic loading/unloading time brought by automated equipment.
$T_{prep}$	Additional preparation time for manned vessels in Class B berths.
$T_{man}$	Loading/unloading time at Class A berths.
$S_{MASS}$	The operating speed of port loading/unloading equipment.
$D_{MASS}$	The demand for MASS loading/unloading services.
$E_{MASS}$	Scheduling optimization device count.
$S_{man}$	The loading/unloading speed of manned vessels.
$T_{man}^A$	The adjustment time of crew operation.
$D_{man}$	The demand for manned vessel loading/unloading services.
$n$	Number of Class B berths.
$n^{(\max )}$	Maximum number of Class B berths. The maximum number of Class B berths is $n^{(\max )}$$=\left[\left|B\right|\right]$.
Decision variables
$b_{ir}$	Vessels $i$ at berth $r$ the time of docking.
${\delta }_i$	Manned vessels $i$ availability of berths in Class B berths.
$y_{ir}$	0–1 variable, if vessel $i$ at berthr If you berth, then $y_{ir}=1$; otherwise, it is 0.
$Z_{ijr}$	0–1 variable; if vessel $i,j$ at berth $r$ docking, and $i$ earlier than $j$, if you berth, then $Z_{ijr}=1$; otherwise, it is 0.
$K_r$	0–1 variable; if berth is set $r$ as a Class B berths, then $K_r=1$; otherwise, it is 0.

:

4. Separated-Type Berth Allocation Strategy

As for the separated-type berth allocation strategy, which is presented in Figure 4, manned vessels are required to choose Class A berths (blue region), while MASSs are expected to dock only in the Class B berth (purple region). In this chapter, a multi-objective optimization model is constructed and solved for this scenario.

4.1. Mathematical Model

In this section, we construct a mathematical model for the scenario of separated-type berth allocation from the perspective of port-shipping companies’ collaboration. The optimization objectives include minimizing both the cost associated with vessel docking in the port and the waiting time for berths. Specifically, cost minimization serves as the primary goal for shipping companies, while minimizing the waiting time for berths is aimed at benefiting ports. The berth docking cost mainly includes the operating cost of the ship in the port time and the cost related to cargo loading/unloading [29] in terms of $C_1$ and $C_2$, respectively, and is optimized as the main objective. Due to the higher degree of automation and intelligence of Class B berths, the loading/unloading time for vessels in the port is reduced. Therefore, in the model of this chapter, a discount factor $\beta$ is added to represent the efficiency improvement of Class B berths compared to Class A berths. The specific mathematical model is as follows.

$$\begin{array}{c}{C}_1={{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I-I_l}{C}_{\mathrm{i}}^{\left(op\right)}{y}_{ir}\left(b_{ir}+h_i-T_{ai}\right)+{{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I_l}{C}_{\mathrm{i}}^{\left(run\right)}{y}_{ir}\left[b_{ir}+\left(1-\beta \right)h_i-T_{ai}\right]\end{array}$$

(1)

$$\begin{array}{c}{C}_2={{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I}{C}_i^{\left(handling\right)}{y}_{ir}{h}_i+{{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I_l}{C}_{\mathrm{i}}^{\left(load\right)}{y}_{ir}\left(1-\beta \right)h_i\end{array}$$

(2)

Upon analyzing the aforementioned berth allocation model, it was found that the utilization efficiency of berths within this framework is suboptimal. As demonstrated in the provided example (Figure 4), when a Class A berth becomes available at the port, Ship No. 5 remains in a state of waiting and does not promptly occupy the vacant berth, resulting in resource wastage. Consequently, an objective optimization function ${\mathrm{T}}_3$ for minimizing the waiting time for berths has been incorporated into the overall objective function (refer to Equation (3)).

$$\begin{array}{c}{T}_3={\displaystyle {\displaystyle \sum }_{i\in I}{\displaystyle \sum }_{j\in I,j\ne i}{\displaystyle \sum }_{r\in B}}\left(b_{jr}-\left(b_{ir}+h_i\right)\right)\cdot z_{ijr}\end{array}$$

(3)

The multi-objective optimization model based on the separated-type berth allocation strategy can be expressed as the following mathematical model (Equation (4)):

$$\begin{array}{l}\begin{array}{c}Mi{n}_{s}C=\end{array}{C}_1+C_2\\ \begin{array}{c}Mi{n}_{s}T=\end{array}{T}_3\end{array}$$

(4)

Subject to:

$$\begin{array}{c}{{\displaystyle \sum }}_{r\in B}{y}_{ir}=1, \forall i\in I\end{array}$$

(5)

$$\begin{array}{c}{y}_{ir}\le M{K}_r, \forall i\in I, \forall r\in B\end{array}$$

(6)

$$\begin{array}{c}{y}_{ir}\le M\left(1-K_r\right), \forall i\in I, \forall r\in B\end{array}$$

(7)

$$\begin{array}{c}{K}_r\le {{\displaystyle \sum }}_{i\in I}{y}_{ir}, \forall r\in B\end{array}$$

(8)

$$\begin{array}{c}n={{\displaystyle \sum }}_{r\in B}{K}_r\end{array}$$

(9)

$$\begin{array}{c}n\le n^{\max }\end{array}$$

(10)

$$\begin{array}{c}{b}_{ir}\le M{y}_{ir}, \forall r\in B, \forall i\in I\end{array}$$

(11)

$$\begin{array}{c}{{\displaystyle \sum }}_{r\in B}{b}_{ir}\ge T_{ai},\forall i\in I\end{array}$$

(12)

$$\begin{array}{c}{b}_{ir}+h_i\times \left(1-k_r\right)\le b_{jr}+M\left(1-z_{ijr}\right),\forall i, j\in I,i\ne j,\forall r\in B\end{array}$$

(13)

$$\begin{array}{c}{Z}_{ijr}+Z_{jir}\le y_{ir},\forall i, j\in I,i\ne j,\forall r\in B\end{array}$$

(14)

$$\begin{array}{c}{Z}_{ijr}+Z_{jir}\ge y_{ir}+y_{jr}-1,\forall i, j\in I,i\ne j,\forall r\in B\end{array}$$

(15)

$$\begin{array}{c}{b}_{ir}\ge 0,\forall i\in I,\forall r\in B\end{array}$$

(16)

$$\begin{array}{c}n\ge 0\end{array}$$

(17)

$$\begin{array}{c}{y}_{ir}\in \left\{0,1\right\},\forall i\in I,\forall r\in B\end{array}$$

(18)

$$\begin{array}{c}{K}_r\in \left\{0,1\right\},\forall r\in B\end{array}$$

(19)

$$\begin{array}{c}{Z}_{ijr}\in \left\{0,1\right\},\forall i,j\in I,i\ne j,\forall r\in B\end{array}$$

(20)

The constraints of the model are specifically described as follows:

Constraint (5) ensures that each vessel is docked at a certain berth.

Constraint (6) ensures that MASSs cannot dock at other Class A berths but only at Class B berths.

Constraint (7) ensures that manned vessels cannot dock at Class B berths but only at Class A berths.

Constraint (8) ensures that if a port sets up a Class B berth, it will definitely dock at this berth to avoid the waste of Class B berth resources.

Constraint (9) is used to calculate the number of Class B berths to be renovated at the port.

Constraint (10) ensures that the number of Class B berths at the port does not exceed the upper limit.

Constraint (11) defines the relationship between $y_{ir}$ and $b_{ir}$.

Constraint (12) ensures that the docking time of a vessel is not earlier than its arrival time at the port.

Constraint (13) ensures that for two consecutive vessels served at the same berth, the docking time of the latter is not earlier than the completion time of loading/unloading of the former.

Constraints (14) and (15) express the relationship between $y_{ir}$ and $Z_{ijr}$, ensuring that two vessels docked at the same berth do not have a time conflict.

Constraints (16)–(20) give the range of values for the decision variables.

4.2. Model Solving

The small-scale examples examined in this article are based on the analysis of a port featuring six berths, comprising four Class A berths and two Class B berths, where a total of twelve vessels are docked. The incoming vessels are categorized into four groups, with three ships assigned to each group.

This section begins by calculating the total docking costs for both manned vessels and MASSs. Subsequently, it identifies the model parameters and ultimately resolves the separated-type berth allocation model. For detailed explanations of specific symbols, please refer to Section 3.2.

4.2.1. Total Docking Cost Analysis

(1)

The time of the vessel’s arrival at the port and loading/unloading.

This article conducts a case study based on the information of vessels arriving at a certain port within a week in September 2022 collected from the website of ShipVision (https://market.myvessel.cn/) and randomly selects the arrival times of 12 vessels. The small-scale case studies in this article are the same as follow: six berths (four Class A berths and two Class B berths), the arriving vessels are divided into four groups of three vessels each. As analyzed in Section 4.2.1, the loading/unloading times were randomly generated with a uniform distribution with the expectation of the average loading/unloading time [29], as presented in

Table 3. Vessel loading/unloading time information.

Arrival Vessels Group Number	Vessel Loading/Unloading Time $\overline{\mathit{h}}$
No1	11.51
No2	16.24
No3	11.88
No4	10.98

.

(2)

Analysis and calculation of total docking cost for manned vessels

The total docking cost of vessels consists of vessel loading/unloading cost $C_2$ and vessel operating cost $C_1$ [29], and their calculation formulas are (21) and (22), respectively.

$$\begin{array}{c}Vessel operating cost=Unit operating cost\times Time of vessel in port\end{array}$$

(21)

$$\begin{array}{c}Vessel loading/unloading cost=Unit Loading/unloading cost\times Loading/unloading time\end{array}$$

(22)

Among them, the per unit time loading/unloading cost of the vessel $C_{\mathrm{i}}^{\left(handling\right)}$ can be calculated based on Formulas (23) and (24) [33,34]; that is $C_{\mathrm{i}}^{\left(handling\right)}$=144.88 ($\forall i\in I$) USD/hr; operating costs per unit time for manned vessels $C_{\mathrm{i}}^{\left(op\right)}=289.74$ ($\forall i\in I$) USD/hr.

$$\begin{array}{c}{C}_i^{\left(handling\right)}={\displaystyle \frac{\left(\begin{array}{c}loading/unloading equipment cost+loading/unloading worker wages\\ +Other loading/unloading cost\end{array}\right)/USD}{Total operating time/hour}}\end{array}$$

(23)

$$\begin{array}{c}{C}_i^{\left(op\right)}={\displaystyle \frac{\left(\begin{array}{c}Crew wages+Fuel cost+Maintenance cost+Insurance cost\\ +Other operating cost\end{array}\right)/USD}{(Total loading/unloading time)/hour}}\end{array}$$

(24)

(3)

Calculation of the total docking cost for MASS

The total docking cost for MASSs comprises vessel operating costs as well as loading/unloading costs, with the respective calculation formulas being (21) and (22). However, there are significant differences between MASSs and manned vessels in terms of operational processes, equipment requirements, and human resource needs. These disparities result in varying operating costs and loading/unloading costs in practical applications. A detailed analysis is provided below.

(i)

Calculation of MASS Operating Costs $C_{\mathrm{i}}^{\left(run\right)}$

MASS operations involve a significantly reduced number of crew members, primarily relying on shore-based control centers [35]. Additionally, the insurance costs associated with MASSs are higher than those for manned vessels [36]. Consequently, following a comprehensive analysis and discussions with practitioners from port-shipping companies’ collaboration, this article incorporates both insurance costs and operational expenses related to shore-based control centers into the overall cost structure of MASS operations. In contrast, it eliminates most crew salary expenses. The MASS unit time operating cost $C_{\mathrm{i}}^{\left(run\right)}$ can be calculated by Equation (25). The annual maintenance cost is approximately 135,000 USD per vessel; the annual human resource cost for the onshore control center is approximately 33,000 USD per vessel; and other operating costs (communication, port call, etc.) are approximately 10% higher than those of manned vessels [35]. Moreover, due to the risk uncertainty of MASSs, the insurance cost is higher than that of manned vessels. Because the insurance cost of MASSs accounts for approximately 1% of the total operating cost of manned vessels, the insurance cost of the MASS per hour is estimated to be approximately 2.77 USD [36]. Substituting the above parameters into Equation (25) yields the following result: the operating cost of the MASS per unit time is $C_{\mathrm{i}}^{\left(run\right)}=245.93$ ($\forall i\in I$) USD/hr.

$$\begin{array}{c}{C}_{\mathrm{i}}^{\left(run\right)}={\displaystyle \frac{\left(\begin{array}{c}Maintenance cost+Fuel cost+Insurance cost\\ +Shore Control Center operating cost+Other operating cost\end{array}\right)/USD}{Total operating time/hour}}\end{array}$$

(25)

(ii)

Calculation of MASS loading/unloading cost $C_{\mathrm{i}}^{\left(load\right)}$

Compared with the loading/unloading costs of manned vessels, the MASS has greater automation and intelligence characteristics, and the cost of loading/unloading equipment is different from the cost of labor support [37], so it is adjusted to the cost of labor support after research demonstration and analysis. Therefore, the MASS unit time loading/unloading cost can be calculated according to Formula (26).

$$\begin{array}{c}{C}_{\mathrm{i}}^{\left(load\right)}={\displaystyle \frac{\left(\begin{array}{c}loading/unloading equipment cost+Auxiliary labor cost\\ +Other loading/unloading cost\end{array}\right)/USD}{(Total loading/unloading time)/hour}}\end{array}$$

(26)

After joint research and analysis with industry experts, the cost of loading/unloading the equipment of Class B berths is usually 1.5 to 2 times that of Class A berths (depending on the level of automation and the complexity of the equipment). This article assumes that the cost of loading/unloading the equipment of Class B berths is 1.8 times that of Class A berths.

$$\begin{array}{c}{C}_{\mathrm{i}}^{\left(load\right)}=\mu \cdot C_{\mathrm{i}}^{\left(handling\right)}\end{array}$$

(27)

where $\mu$ is the ratio of the cost of a Class B berth to the cost of a berth for a manned vessel, i.e., $\mu =1.8$. Accordingly, the cost of loading/unloading equipment per unit time at a Class B berth is approximately 260.78 USD/hr.

At the same time, according to the actual situation at Chinese ports, the average hourly wage for positions such as loading/unloading operations usually ranges from 20 to 40 USD/hr (depending on the region and skill requirements of the job type) [37]. The actual hourly rate may be higher if overtime, night shift allowances, and technical support needs are accounted for, especially for the special skills required to assist in the operation of autonomous equipment. Therefore, three staff members are set up to assist in the example in this article, and the labor support cost is extrapolated based on a rate of 45 USD/hr for each person, which results in a total labor support cost of approximately 120 USD/hr. Substituting the above parameters into formula (26) yields the following results: MASS unit time loading/unloading costs $C_{\mathrm{i}}^{\left(load\right)}=$380.78 ($\forall i\in I$) USD/hr.

(4)

Calculation of $\beta$

The more automated equipment at Class B berths is more efficient in the operation of MASSs but also requires additional time $T_{prep}$ to accommodate manned vessels. Therefore, $T_{MASS}+T_{prep}$ represents the loading/unloading time of manned vessels at Class B berths, and $T_{man}$ represents the loading/unloading time of manned vessels at conventional Class A berths. Thus, by calculating ${\displaystyle \frac{T_{MASS}+T_{prep}}{T_{man}}}$, the ratio of the loading/unloading time of Class B berths to that of Class A berths can be obtained. $1-{\displaystyle \frac{T_{MASS}+T_{prep}}{T_{man}}}$ denotes the proportion of loading/unloading time saved, i.e., the degree of efficiency of improvement of Class B berths compared to Class A berths, where the discount factor, $\beta$, is as follows.

$$\begin{array}{c}\beta =1-{\displaystyle \frac{T_{MASS}+T_{prep}}{T_{man}}}\end{array}$$

(28)

(i) $T_{MASS}$ in Equation (29): This time is related mainly to $R_{MASS}$, $D_{MASS}$, and $E_{MASS}$ correlation [38]. Therefore, $T_{MASS}$ can be extrapolated by the following formula.

$$\begin{array}{c}{T}_{MASS}={\displaystyle \frac{D_{MASS}}{S_{MASS}\times E_{MASS}}}\end{array}$$

(29)

(ii) $T_{prep}$: This time mainly includes the time required for commissioning equipment, coordinating personnel, etc., when manned vessels call at Class B berths (Zhang, S et al., 2022) [32].

(iii) $T_{man}$: It is mainly related to $S_{man}$, $T_{man}^A$, and $D_{man}$ [39]. Therefore, $T_{man}$ can be calculated by the following formula.

$$\begin{array}{c}{T}_{man}={\displaystyle \frac{D_{man}}{S_{man}}}+T_{man}^A\end{array}$$

(30)

Liao et al. [40] obtained $S_{man}=66$ in standard containers/hour. Each manual operation takes about 1 h to adjust the time ($T_{man}^A=1$). MASS is highly automated, and according to port investigation (Chinese ports), $S_{MASS}=80$ standard containers/hour; the adjustment time for manual operation is very short and negligible; and $T_{prep}$ is approximately 1 h.

Based on the scale of the example in this article, $D_{MASS}=120$ and $D_{man}=120$. The number of available devices for each MASS ($E_{MASS}=2$) is assumed.

$$\begin{array}{c}{T}_{MASS}={\displaystyle \frac{120}{80\times 2}}=0.75 h\end{array}$$

(31)

$$\begin{array}{c}{T}_{man}={\displaystyle \frac{120}{66}}+1=2.82 h\end{array}$$

(32)

Equation (28) is used to calculate $\beta =0.38$ (retain two decimal places).

4.2.2. Model Validation

To test the effectiveness of the proposed models in this article, we conduct several experiments. Experiments are performed using a PC with Intel^® Core™ i7-3770 CPU (2.60 GHz speed) and 8 GB RAM. The algorithm program is compiled by Python 3.9, and the model was solved using Gurobi 10.0.3. The solution results are presented in

Table 4. Validation results of the separated-type berth allocation model.

Berths	Vessel Name	Actual Docking Time (Time Point)
Class B berth 2	MASS 2’	5.22
Class A berth 4	Manned vessel 3	7.87
Class B berth 1	MASS 3’	9.89
Class A berth 4	Manned vessel 1	16.11
Class A berth 4	Manned vessel 5	25.91
Class B berth 1	MASS 1’	30.02
Class A berth 4	Manned vessel 4	34.88
Class A berth 3	Manned vessel 6	40.00
Class A berth 4	Manned vessel 2	46.02
Class A berth 4	Manned vessel 7	58.32
Class A berth 4	Manned vessel 8	66.96
Class A berth 3	Manned vessel 9	71.86

and Figure 5. In Figure 5, the point where the red dotted line intersects with the red circle represents the total time all vessels spent in the port.

5. Mixed-Type Berth Allocation Strategy

Obviously, as the number of MASSs increases and the mixed operation of MASSs and manned vessels expands, the mixed-type berth allocation strategy offers higher operational efficiency. Therefore, the following content of this article studies the mixed-type berth allocation strategy. The basic principle of mixed-type berth allocation is that when all Class A berths are occupied, manned vessels can choose to dock at Class B berths, whereas MASSs are not allowed to choose Class A berths. As presented in Figure 6, if a manned vessel was originally scheduled to dock at a Class A berth but the Class A berths are full, the vessel must change its plan and dock at a Class B berth instead. In this chapter, a multi-objective optimization model is constructed and solved for this scenario.

5.1. Mathematical Model

In the scenario of mixed-type berth allocation, when all Class A berths are occupied, manned vessels can choose to dock at the vacant Class B berths. Based on this practical consideration, compared with the mathematical model in Chapter Four, the model constructed in this chapter incorporates the above situation (including (33)–(35)).

(1)

Mathematical modeling of a Mixed-type berth allocation strategy

$$\begin{array}{c}{C}_1={{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I-I_l}{C}_{\mathrm{i}}^{\left(op\right)}{y}_{ir}\left[b_{ir}+h_i\left(1-\beta K_r\right)-T_{ai}\right]+{{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I_l}{C}_{\mathrm{i}}^{\left(run\right)}{y}_{ir}\left[b_{ir}+\left(1-\beta \right)h_i-T_{ai}\right]\end{array}$$

(33)

$$\begin{array}{c}{C}_2={{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I}{C}_i^{\left(handling\right)}{y}_{ir}{h}_i\times \left[\left(1-\beta \right)K_r+\left(1-K_r\right)\right]+{{\displaystyle \sum }}_{r\in B}{{\displaystyle \sum }}_{i\in I_l}{C}_{\mathrm{i}}^{\left(load\right)}{y}_{ir}\left(1-\beta \right)h_i\end{array}$$

(34)

$$\begin{array}{c}{T}_3={\displaystyle {\displaystyle \sum }_{i\in I}{\displaystyle \sum }_{j\in I,j\ne i}{\displaystyle \sum }_{r\in B}}\left[b_{jr}-\left(b_{ir}+h_i\right)\right]\cdot z_{ijr}\end{array}$$

(35)

$$\begin{array}{l}\begin{array}{c}Mi{n}_{m}C=\end{array}{C}_1+C_2\\ \begin{array}{c}Mi{n}_{m}T=\end{array}{T}_3\end{array}$$

(36)

Subject to the following:

$$\begin{array}{c}{{\displaystyle \sum }}_{r\in B}{y}_{ir}=1, \forall i\in I\end{array}$$

(37)

$$\begin{array}{c}{y}_{ir}\le M{K}_r, \forall i\in I,\forall r\in B\end{array}$$

(38)

$$\begin{array}{c}{K}_r\le {{\displaystyle \sum }}_{i\in I}{y}_{ir},\forall r\in B\end{array}$$

(39)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{j\in A}}{y}_{i,j}=0\Rightarrow {\displaystyle {\displaystyle \sum }_{j\in B}}{y}_{i,j}\le {\delta }_i\end{array}$$

(40)

$$\begin{array}{c}{\displaystyle {\displaystyle \sum }_{j\in A}}{y}_{i,j}\ge 1-{\delta }_i\end{array}$$

(41)

$$\begin{array}{c}n={{\displaystyle \sum }}_{r\in B}{K}_r\end{array}$$

(42)

$$\begin{array}{c}n\le n^{\max }\end{array}$$

(43)

$$\begin{array}{c}{b}_{ir}\le M{y}_{ir},\forall r\in B, \forall i\in I\end{array}$$

(44)

$$\begin{array}{c}{{\displaystyle \sum }}_{r\in B}{b}_{ir}\ge T_{ai}, \forall i\in I\end{array}$$

(45)

$$\begin{array}{c}{b}_{ir}+h_i\times \left(1-\beta K_r\right)\le b_{jr}+M\left(1-z_{ijr}\right),\forall i, j\in I, i\ne j, \forall r\in B\end{array}$$

(46)

$$\begin{array}{c}{Z}_{ijr}+Z_{jir}\le y_{ir}, \forall i, j\in I, i\ne j, \forall r\in B\end{array}$$

(47)

$$\begin{array}{c}{Z}_{ijr}+Z_{jir}\ge y_{ir}+y_{jr}- 1, \forall i, j\in I, i\ne j, \forall r\in B\end{array}$$

(48)

$$\begin{array}{c}{b}_{ir}\ge 0, \forall i\in I, \forall r\in B\end{array}$$

(49)

$$\begin{array}{c}n\ge 0\end{array}$$

(50)

$$\begin{array}{c}{y}_{ir}\in \left\{0,1\right\}, \forall i\in I, \forall r\in B\end{array}$$

(51)

$$\begin{array}{c}{K}_r\in \left\{0,1\right\}, \forall r\in B\end{array}$$

(52)

$$\begin{array}{c}{Z}_{ijr}\in \left\{0,1\right\}, \forall i, j\in I, i\ne j, \forall r\in B\end{array}$$

(53)

(2)

Constraints

Constraint (37) ensures that every vessel is docked at a particular berth.

Constraint (38) ensures that MASSs cannot dock at any ordinary berths but only at Class B berths.

Constraint (39) ensures that if the port has set up Class B berths, docking operations will definitely be carried out at this berth to avoid the waste of Class B berth resources.

Constraint (40) ensures that ${\delta }_i$ allows the vessel to choose a Class B berth only when all Class A berths are occupied.

Constraint (41) indicates that if a manned vessel $i$ fails to find a position in a Class A berth, then ${\delta }_i=1$, meaning it is allowed to dock at a Class B berth.

Constraint (42) is used to calculate the number of Class B berths for port renovation.

Constraint (43) ensures that the number of Class B berths in the port does not exceed the upper limit.

Constraint (44) defines the relationship between $y_{ir}$ and $b_{ir}$.

Constraint (45) ensures that the docking time of a vessel is not earlier than its arrival time at the port.

Constraint (46) ensures that for two consecutive vessels served at the same berth, the docking time of the latter is not earlier than the completion time of the loading/unloading of the former.

Constraints (47) and (48) express the relationship between $y_{ir}$ and $Z_{ijr}$, ensuring that two vessels docked at the same berth do not have a time conflict.

Constraints (49)–(53) give the range of values for the decision variables.

5.2. Model Solving

5.2.1. Model Linearization

The objective function of the mixed-type berth allocation strategy model consists of Equations (33)–(35), where both Equations (33) and (34) contain nonlinear components $K_{\mathrm{r}}{\mathrm{y}}_{\mathrm{ir}}$. Therefore, first, this article transforms the mixed-type berth allocation strategy model into a linear model ($\left[Mode{l}_2\right]minC$) for solving.

This article introduces 0–1 auxiliary variables such that $u_{ir}=K_{\mathrm{r}}{y}_{ir}$. If a vessel $i$ docks at a Class B berth $r$, then $u_{ir}=1$; otherwise, $u_{ir}=0$. In addition, when vessel $i$ is not at berth $r$, $y_{lr}=0$, the docking time $b_{ir}$ of vessel $i$ at the berth is also 0. In this case, the objective function can be expressed as follows.

$$\begin{array}{c}\left[Mode{l}_2\right]minC={\displaystyle {\displaystyle \sum }_{r\in B}{\displaystyle \sum }_{i\in I-I_l}}{C}_{\mathrm{i}}^{\left(op\right)}\left[y_{ir}\left(b_{ir}+h_i-T_{ai}\right)-\beta h_i{u}_{ir}\right]+{\displaystyle {\displaystyle \sum }_{r\in B}{\displaystyle \sum }_{i\in I_l}}{C}_{\mathrm{i}}^{\left(run\right)}{y}_{ir}\left[b_{ir}+\left(1-\beta \right)h_i-T_{ai}\right]\\ +{\displaystyle {\displaystyle \sum }_{r\in B}{\displaystyle \sum }_{i\in I}}{C}_i^{\left(handling\right)}\left[y_{ir}{h}_i-\beta u_{ir}{h}_i\right]+{\displaystyle {\displaystyle \sum }_{r\in B}{\displaystyle \sum }_{i\in I}}{C}_{\mathrm{i}}^{\left(load\right)}{y}_{ir}\left(1-\beta \right)h_i\end{array}$$

(54)

Moreover, the following constraints need to be introduced.

$$\begin{array}{c}2u_{ir}\le K_r+y_{ir},\forall i\in I, \forall r\in B\end{array}$$

(55)

$$\begin{array}{c}{u}_{ir}\ge K_r+y_{ir}-1, \forall i\in I, \forall r\in B\end{array}$$

(56)

$$\begin{array}{c}{u}_{ir}\in \left\{0,1\right\}, \forall i\in I, \forall r\in B\end{array}$$

(57)

5.2.2. Model Validation

(1)

Model parameter determination

The calculation of the values for the vessel’s arrival time at the port, vessel loading/unloading time, total docking cost for manned vessels, and total docking cost for MASSs in this section’s model is the same as that in Section 4.2.2.

(2)

Validation of the effectiveness of the Mixed-type berth allocation model

Using the same example in Chapter 4, the berth allocation scheme is obtained via the Gurobi solution, which proves the validity of the model, as presented in

Table 5. Validation results of the mixed-type berth allocation model.

Berths	Assigned Vessel Numbers	Actual Docking Time (Time Point)
Class B berth 2	MASS 1’	18.60
Class B berth 1	MASS 2’	46.87
Class B berth 2	MASS 3’	54.95
Class B berth 2	Manned vessel 1	28.96
Class B berth 1	Manned vessel 2	39.91
Class B berth 1	Manned vessel 3	15.04
Class B berth 1	Manned vessel 4	57.88
Class A berth 3	Manned vessel 5	3.23
Class B berth 2	Manned vessel 6	40.02
Class A berth 3	Manned vessel 7	26.67
Class B berth 2	Manned vessel 8	44.56
Class B berth 2	Manned vessel 9	66.83

and Figure 7. In Figure 7, the point where the red dotted line intersects with the red circle represents the total time all vessels spent in the port.

A comparison of the separated-type berth allocation strategy and the mixed-type berth allocation strategy (Table 4 and Table 5, Figure 5 and Figure 7) reveals that under the same conditions, the mixed-type berth allocation strategy can save at least 7% of the vessel’s time in the port compared with the separated-type berth allocation strategy (see the red marked parts in Figure 5 and Figure 7). This indicates that the mixed-type berth allocation strategy is more efficient. With increasing MASSs operating in scale, the number of Class B berths gradually increases, and the mixed-type berth allocation strategy has the advantages of greater practical demand and theoretical berth allocation efficiency. Therefore, it is necessary to verify the effectiveness of the mixed-type berth allocation model with large-scale examples.

6. Improved Simulated Annealing Algorithm

As seen from Section 5.2.2, the mixed-type berth allocation problem proposed in this article involves complex constraints such as vessel docking time, berth type restrictions, and loading/unloading cost differences, and is a typical NP-hard combinatorial optimization problem. With increasing problem scale, the ordinary exact algorithm has difficulty solving the problem within a reasonable time under large-scale cases; heuristic algorithms are used to try to solve the problem in this article, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Simulated Annealing (SA).

For this problem, SA presents unique advantages. SA has a strong global search capability, can effectively jump out of the local optimal solution through the mechanism of the probabilistic acceptance of inferior solutions, and is suitable for solving highly nonlinear and high-dimensional optimization problems. However, the algorithm also has the limitation that it will stop early when the temperature drops too fast in the late stage of the search, which affects the quality and the stability of the solution. An improved SA is addressed in this article; the adaptive cooling strategy is used to dynamically adjust the temperature drop rate to avoid the problem of early stopping, enhance the global search capability, and improve the quality and stability of the solution. The experimental results are presented in that the improved SA algorithm is not only close to the exact solution of Gurobi in small-scale problems but also has superior computational efficiency and stability in large-scale problems.

6.1. Algorithm Design

6.1.1. Preprocessing for Solution

The solution algorithms for multi-objective optimization models can be roughly divided into two categories. One category is to transform the multi-objective optimization problem into a single-objective optimization problem; that is, to first unify the dimensions of each objective through dimensionless processing and other means and then process the objectives through methods such as linear weighting, converting them into a single objective. The advantage of this approach is that it can obtain the optimal solution, which is convenient for intuitive analysis and comparison. However, this type of method has a high sensitivity requirement for parameters. The other category of algorithms is to seek the Pareto solution set, that is, to find all nondominated solutions through heuristic algorithms. The advantage of this type of method is that it eliminates the influence of dimensions and parameters, and each solution’s objective can be reflected in the coordinate system. The disadvantage is that all solutions on the Pareto frontier are equally important, and it cannot provide the best solution for managers, making it inconvenient for decision-making and analysis [41].

Since resource allocation does not require the retention of multiple nondominated solutions [41], this article adopts the linear weighting method to transform the multi-objective problem into a single-objective one, that is, to a dimensionless process for the total docking cost of the vessels in the port and the waiting time for berths, and then convert it into a new single objective through the linear weighting method. Considering the complexity of the model constraints, this article uses SA to solve it, conducts extensive exploration in the complex solution space, fully considers various possible berth allocation schemes, and thus, finds a better solution.

(1)

Dimensionless processing

The total docking cost of the vessels in the port ($C$ ($C=C1+C2$)), and the waiting time for berths ${\mathrm{T}}_3$ are, respectively, dimensionless:

$$C^{\prime }={\displaystyle \frac{C-C_{\min }}{C_{\max }-C_{\min }}},\hspace{1em}{T}^{\prime }={\displaystyle \frac{T-T_{\min }}{T_{\max }-T_{\min }}}$$

(58)

Of which,

$C_{\min }$ and $C_{\max }$: Minimum and maximum values of the total docking cost of a vessel in the port; they will be calculated on the basis of actual example data.

$T_{\min }$ and $T_{\max }$: Minimum and maximum values of the waiting time for berths, which are based on the scheduling scheme of berths in the example.

(2)

Linear weighting

The dimensionless objective function is combined into the fitness function by linear weighting:

$$F={\omega }_1{C}^{\prime }+{\omega }_2{T}^{\prime }$$

(59)

where ${\omega }_1$ and ${\omega }_2$ denote the weights of the two objectives and satisfy ${\omega }_1+{\omega }_2=1$.

6.1.2. Specific Steps of Algorithm Design

An adaptive cooling strategy is adopted to dynamically adjust the temperature according to the current iteration number (Equation (61)), which improves the diversity of the solution and the global search capability, and solves the local optimal problem.

Step 1: Parameter initialization

(1)

Setting the initial temperature, $T_0$, minimum temperature, $T_{\min }$, maximum number of iterations $N_{\max }$, and adjustment factors $\beta$.

(2)

Randomly generate an initial solution such as $X_{\mathrm{current}}$, ensuring that they satisfy model constraints (e.g., berth type limitations, time constraints, etc.).

(3)

Calculate the value of the objective function of the initial solution, $C_{\mathrm{current}}=f(X_{\mathrm{current}})$.

(4)

Setting the initial number of iterations to $k=0$.

Step 2: Generating a new solution

(1)

Candidate solutions are generated by randomly adjusting vessel berth assignments $X_{\mathrm{new}}$.

(2)

Inspection of $X_{\mathrm{new}}$. Check if the constraints are satisfied or not, and if not, then regenerate.

Step 3: Objective function calculation

The value of the objective function of the candidate solution is calculated as $C_{\mathrm{new}}=f(X_{\mathrm{new}})$.

Step 4: Principles of acceptance

If $C_{\mathrm{new}}<C_{\mathrm{current}}$, which is accepted directly $X_{\mathrm{new}}$:$X_{\mathrm{current}}=X_{\mathrm{new}}$,$C_{\mathrm{current}}=C_{\mathrm{new}}$.

If $C_{\mathrm{new}}>C_{\mathrm{current}}$, then $X_{\mathrm{new}}$ is accepted as probability $P$:

$$P=\exp \left(-{\displaystyle \frac{C_{\mathrm{new}}-C_{\mathrm{current}}}{T_k}}\right)$$

(60)

If the random number $r\in [0,1]$ satisfies $r<P$ and accepts $X_{\mathrm{new}}$ for the current solution.

Step 5: Dynamic temperature adjustment

Dynamic temperature adjustment according to an adaptive cooling strategy.

$$T_{k+1}=T_k\cdot {\displaystyle \frac{1}{1+\alpha \cdot k}}$$

(61)

where $\alpha$ is the adjustment factor and $k$ is the current iteration count.

If the updated temperature is below the minimum temperature, then the algorithm will be terminated.

Step 6: Stop condition

(1)

If the temperature $T_k\le T_{\min }$ or the maximum number of iterations $N_{\max }$ is reached, then the algorithm stops and returns the current optimal solution.

(2)

Otherwise, the number of iterations of the update, $k=k+1$, returns to Step 2.

6.2. Arithmetic Analysis

This section uses Dalian Container Terminal Co., Ltd. (DCT) in China as a case to present the berth allocation strategy in the context of the mixed operation of MASS and manned vessels. DCT is the largest container terminal in Northeast China. The terminal has a coastline of 5759 m and currently operates 14 container berths, with a maximum water depth of −17.8 m. Among these, five deep-water berths can accommodate container vessels with a capacity of up to 200,000 tons. The terminal covers an area of 2.351 million square meters and is equipped with 35 quay cranes, 68 rubber-tired gantry cranes, and 24 rail-mounted gantry cranes. The port area has a total of 18 railway tracks, directly extending to the terminal, achieving seamless sea–rail intermodal transportation. Given the current tonnage of the MASS [17] (the “Zhifei” has a deadweight of approximately 5353 tons), there is no need to consider upgrading berths for very large tonnage MASSs in the short term. Therefore, this study conducts a planning analysis around the existing 10 berths of Dalian Port with a deadweight of about 10,000–50,000 tons. In this case, it is assumed that there are four Class B berths and six Class A berths, and 40 vessels are simulated to dock in the poet (Vessels 1–10 are MASS, and vessels 11–40 are manned vessels). The specific data for the vessels are obtained from the Ship View website (https://market.myvessel.cn/).

6.2.1. Parameters

After several experimental adjustments, the parameter settings of the algorithm are shown in

Table 6. Algorithm parameter setting.

Parameters	Symbol	Values
Initial temperature	$T_0$	1200
Minimum temperature	$T_{\min }$	0.01
The maximum number of iterations	$N_{\max }$	300
Cooling factor	$\alpha$	0.90

.

6.2.2. Algorithm Solving

This section addresses larger scale examples utilizing the enhanced simulated annealing algorithm (SA), with results illustrated in Figure 8. In Figure 8, the horizontal axis denotes the berth number, while the vertical axis indicates the time spent in port. The yellow rectangles 1–10 represent MASSs, and the blue rectangles 11–40 indicate manned vessels. The length of each rectangle corresponds to the vessel’s size; the starting edge of each rectangle indicates its docking berth number, and the height reflects the duration of berthing at that location.

Furthermore, this section compares and analyzes the solution capabilities of both the improved SA and Gurobi, as detailed in

Table 7. Algorithm comparison results.

Total Number of Vessels/MASS	Total Number of Berths /Class B Berths	Improved SA Algorithm			Gurobi		$\mathit{C}$’s Differential
		$\mathit{C}$ ($)	Average Waiting Time for Berths (Hours)	Runtime (s)	$\mathit{C}$ ($)	Average Waiting Time for Berths (Hours)	Runtime (s)	Gap
3/1	2/1	579,930	3.5	2.5	561,728	3.3	2.5	3.24%
	2/2	463,420	3.2	3.2	439,167	3.1	3.2	5.53%
4/1	3/1	580,300	3.4	3.5	561,728	3.2	4.0	3.31%
	3/2	454,640	3.2	3.7	439,180	3.0	4.1	3.52%
	3/3	414,468	2.8	3.9	391,951	2.7	4.3	5.73%
6/2	4/1	580,000	3.3	6.2	561,728	3.2	8.4	3.25%
	4/2	457,960	3.0	6.4	439,170	2.9	8.6	4.29%
	4/3	411,690	2.7	6.5	391,920	2.7	8.7	5.05%
12/3	6/3	396,210	2.5	8.8	391,920	2.4	10.2	1.09%
20/5	8/3	394,000	2.3	26.7	-	-	-	-
40/10	11/3	382,030	2.2	64.2	-	-	-	-

. For smaller scale examples, there is a deviation in calculation results between these two algorithms that does not exceed 6%. This finding suggests that our proposed algorithm strikes an effective balance between solution quality and computational efficiency. However, for larger scale examples—specifically, when exceeding a total of 20 vessels—Gurobi fails to produce results, as outlined in Table 7. This highlights how our improved algorithm successfully mitigates early termination issues associated with rapid temperature declines during later stages of traditional algorithms through an adaptive cooling strategy. It not only approaches exact solutions effectively for small-scale problems but also exhibits superior computational efficiency and stability when addressing large-scale challenges.

6.2.3. Analysis of the Effectiveness of the Improved Simulated Annealing Algorithm

According to Figure 9, Figure 10 and Figure 11, it is evident that the improved SA demonstrates superior convergence. Upon reaching the 421st iteration, this algorithm outperforms both the genetic algorithm and the particle swarm optimization in terms of computational results. Figure 10 illustrates the iterative outcomes of the GA, while Figure 11 presents those of the PSO. Furthermore, these three algorithms were subjected to additional validation across various scale examples.

Table 8. Comparison results of three algorithms.

Example Size (Number of Vessels/Berths)	Algorithm	$\mathit{C}$ ($)	Average Waiting Time for Berths (Hours)	Running Time (s)	The Number of Convergence Iterations
Small-scale (6/2)	Improved SA	580,000	3.5	6.2	421
	GA	583,000	3.8	8.7	612
	PSO	586,000	4.0	9.1	630
Medium (12/3)	Improved SA	396,210	3.2	8.8	511
	GA	401,300	3.6	11.5	732
	PSO	405,000	3.5	12.3	750
Large-scale (20/5)	Improved SA	394,000	3.0	26.7	920
	GA	401,000	3.3	35.2	1140
	PSO	408,500	3.5	38.7	1200
Supersize (40/10)	Improved SA	382,030	2.8	64.2	1500
	GA	389,000	3.3	80.5	1890
	PSO	395,000	3.2	85.3	1950

indicates that the enhanced SA proposed in this article achieves a more effective solution compared to its counterparts.

Table 8 presents a comparison of the total docking cost of the vessels in the port, the average waiting time for berths for vessels, the execution time, and the number of convergence iterations for the improved SA, GA, and PSO across various scales: small-scale (6/2), medium-scale (12/3), large-scale (20/5), and extra-large scale (40/10). The findings indicate that as the scale of examples increases, both the execution time and the number of convergence iterations for all three algorithms also rise. In most instances, the improved SA demonstrates not only the lowest total cost but also a comparatively shorter average waiting time for berths for vessels.

7. Sensitivity Analysis

7.1. Optimization Effect of Model 1 Under Different Optimization Objectives

This section takes the separated-type berth allocation strategy as an example to compare the effect of single objective optimization and multi-objective optimization. The total docking cost of the vessels in the port is used for single objective optimization, and the total docking cost of the vessels in the port and the waiting time for berths are used for multi-objective optimization, respectively. The solution results are presented in Figure 5 and Figure 12, respectively. It can be found that with multi-objective optimization, the time of vessel docking in the port is shortened by 12% compared with the single-objective optimization strategy (the result is marked in red in Figure 5 and Figure 12, the point where the red dotted line intersects with the red circle represents the total time all vessels spent in the port.). In addition, the berths’ utilization rate, vessels’ average waiting time for berths, and the total docking cost of the vessels in the port of the model are analyzed. As given in

Table 9. Comparison analysis of single-objective and multi-objective optimization.

Optimization Goals	Berth Utilization Rate (%)	Average Waiting Time for Berths (Hours)	$\mathit{C}$ ($)
Single-objective optimization	68.7	4.8	502,580
Multi-objective optimization	73.5	4.1	489,974

, the multi-objective berth allocation optimization model improves the berth utilization rate by 4.8%, shortens the average waiting time for berths of the vessel by 0.7 h, and reduces the total docking cost of the vessels in the port by approximately 3%.

Insight 1: for the separated-type berth allocation operating BAP of MASSs and manned vessels, modeling and optimization from the perspective of port-shipping companies’ collaboration will achieve higher berth allocation efficiency than considering berth allocation separately from the perspective of either the vessel companies or the port.

7.2. Effect of Class B Berths’ Number on the Total Docking Cost of the Vessels of Two Strategies

In this section, the effects of Class B berths’ number are compared and analyzed under two strategies through many experiments.

Figure 13 presents the effects of the number changes of Class B berths on the total docking cost of the vessels under two strategies.

Table 10. The effects of the number of Class B berths updated on the total docking cost of the vessels under two strategies.

Total Number of Port Berths N	Number of Class B Berths	Class B Berths as a Percentage of the Total Number of Berths in the Port	Separated-Type Berth Allocation Strategy: the Total Docking Cost of the Vessels (USD)	Mixed-Type Berth Allocation Strategy: the Total Docking Cost of the Vessels (USD)
2	1	50%	78,681.37	77,678.68
	2	100%	78,681.37	60,748.92
	1	33%	75,522.80	77,678.68
3	2	67%	71,042.04	60,750.58
	3	100%	71,042.04	54,226.00
	1	25%	75,522.80	77,678.68
	2	50%	67,883.47	60,749.38
4	3	75%	67,883.47	54,222.02
	4	100%	67,883.47	54,222.02
	1	20%	75,522.80	77,678.68
	2	40%	67,883.47	60,748.92
5	3	60%	67,883.47	54,222.02
	4	80%	64,934.96	54,222.02
	5	100%	64,934.96	54,222.02
	1	17%	75,522.80	77,678.68
	2	33%	67,883.47	60,748.92
6	3	50%	64,934.96	54,222.02
	4	67%	64,934.96	54,225.65
	5	83%	64,934.96	54,222.02
	6	100%	64,934.96	54,222.02

and Figure 14 present the effects of the number changes of Class B berths on the total docking cost of the vessels in the mixed-type berth allocation strategy (as given in Figure 6).

On the basis of analysis, many results are obtained:

(1)

Insight 2: If the port’s decision is that only one berth will be updated, that means it will own only one Class B berth; then, the separated-type berth allocation strategy (as given in Figure 4) is more economical and more in line with the realistic demand. As given in Figure 13, when the number of Class B berths = 1, the total cost of the separated-type berth allocation strategy for the total docking cost of the vessels in the port is approximately USD 76,200,000, and the total docking cost of the vessels in the port of the mixed-type berth allocation strategy for the total docking cost of the vessels in the port is close to USD 77,700,000. At this time, the total docking cost of the vessels in the port of the mixed-type berth allocation strategy is slightly greater than that of the separated-type berth allocation strategy.

(2)

Insight 3: As given in Figure 13, when the Class B Berths of the port are much greater than 1, it is recommended that the port adopts the mixed-type berth allocation strategy.

As given in Figure 13, when the number of Class B berths > 1, the mixed-type berth allocation strategy will significantly reduce the total docking cost of the vessels in the port by at least 17%; with an increase in the number of Class B berths, the total docking cost of the vessels in the port two berths tends to decrease, but the total docking cost of the vessels in the port of the mixed-type berth allocation strategy is significantly lower than that of the separated-type berth allocation strategy.

(3)

Insight 4: As presented in Table 10 and Figure 14, when the number of Class B berths reaches a certain percentage, the total docking cost of the vessels in the port no longer changes, which indicates that the number of Class B berths at this time is the most appropriate port investment in the number of transformations. When the number of Class B berths increases to three, the total docking cost of the vessels in the port is no longer reduced.

The horizontal coordinate in Figure 14 is the number of Class B berths in the port, the left vertical coordinate is the total number of berths in the port, and the right vertical coordinate is the total docking cost of the vessels.

8. Conclusions

8.1. Summary

This article studies the MASSs’ BAP in mixed operating scenarios involving both MASSs and manned vessels at port terminals. From the perspective of port-shipping companies’ collaboration, this study innovatively computes the total docking cost of the vessels in the port associated with MASSs and further analyzes the time required for manned vessels to loading/unloading at designated berths (Class B berths) allocated for MASS operations. Two distinct berth allocation strategies are proposed from the perspective of port-shipping companies’ collaboration: the separated-type and mixed-type berth allocation strategy. A mixed integer programming model is developed with the objective of minimizing both the total docking cost of the vessels in the port and the waiting time for berths. Additionally, compared with the validation of the berth-allocation strategy and small-scale numerical examples in the single MASS operation scenario in Ref. [17], an improved simulated annealing algorithm is applied innovatively to validate this model using large-scale examples.

Several experiments are conducted to test the effectiveness of the model and to draw insights for commercializing autonomous vessels. The results show the following: Firstly, regarding the BAP associated with the mixed operations of MASSs and manned vessels, a multi-objective modeling and optimization approach that emphasizes the port-shipping companies’ collaboration can achieve much greater berth allocation efficiency compared to strategies that consider either shipping companies or ports in isolation. Secondly, in specific scenarios, e.g., when berth resources are constrained or when there are stringent operating safety requirements, the separated-type berth allocation strategy demonstrates more efficient and safer characteristics. Thirdly, as the number of MASS operations increases, the mixed-type berth allocation strategy can substantially enhance the effectiveness and efficiency of the port under equivalent conditions compared to the separated-type berth allocation strategy. When the number of Class B berths reaches a certain proportion, however, the total docking cost of the vessels in the port will no longer change, which indicates that the number of Class B berths at this time is the most suitable amount of investment for the port, rather than more renovation being better.

8.2. Future Works

BAP study for the mixed operation of MASSs and manned vessels is still at an early stage; there is still much room for improvement in the future. For instance, in terms of model optimization, the MOEA/D algorithm maybe be introduced for Pareto frontier analysis. Additionally, communication failures and cyber-security threats can be considered, and random factors or fault-tolerant systems can be introduced to enhance the robustness and practical adaptability of the model. Moreover, given the differences in port infrastructure and regulatory conditions around the world, the adaptive scalability of the model should also be considered to make it widely applicable to various ports globally. As the operational data of MASSs continues to accumulate, the application of reinforcement learning and hybrid artificial intelligence optimization techniques in berth allocation can also be explored.