Distributionally Robust Programming of Berth-Allocation-with-Crane-Allocation Problem with Uncertain Quay-Crane-Handling Efficiency

Abstract

In order to promote the efficient and intelligent construction of container ports, we focus on the optimization of berth-and-quay-crane (QC) allocation in tidal terminal operations. This paper investigates the quay-crane-profile-(QC-profile)-based assignment problem, and considers the uncertainty in QC profiles regarding QC efficiency for the first time. A mixed-integer programming (MIP) model is established for a discrete berth allocation with a crane-assignment problem (BACAP), considering the tide time window. We aim to minimize the total time loss caused by anchorage and the delay of vessels. Leveraging the theory of uncertainty optimization, the proposed deterministic model is extended into a stochastic programming (SP) model and a distributionally robust optimization (DRO) model, via the consideration of the random QC efficiency. To solve the proposed models, a column generation (CG) algorithm is employed, utilizing the mathematical method and subproblem-solving approach. The numerical experiments with different instances demonstrate that the DRO model yields a smaller variation in the objective function values, and the effectiveness of the CG method. The experimental results verify the robustness of the constructed models, and the efficiency of the proposed algorithm.

1. Introduction

Due to the rising throughput of container terminals worldwide, the significant importance of maritime transport in the global supply chain is well established [1]. Nearly 80% of global trade involves the transportation of goods through maritime ports and terminals [2]. Container terminals, as crucial nodes connecting maritime and land-based logistics, play a pivotal role in the supply chain. However, in 2021, the in-port time for each container vessel increased by 13.7% compared to 2020, leading to severe vessel delays and capacity shortages [3]. In 2021, about 1.95 billion metric tons of cargo were shipped globally, up from some 0.1 billion metric tons in 1980 [4]. Consequently, optimizing container ports is of vital significance to ensuring the stability of the global supply chain. Furthermore, similar to the benefits observed in other transportation sectors, such as aviation and land transportation since the rise of operations research in the 1950s, maritime transportation has gradually emerged as a research hotspot in the field of operations research [5].

In the field of container port optimization, the academic research mainly focuses on berth allocation, QC allocation, empty container repositioning, container pre-marshalling, and crane-scheduling problems [6]. Among them, berth planning and allocation represent the initial stages of terminal operations. Optimal berth decision-making can assist terminal operators in minimizing container transshipment and yard transportation times, leading to reduced traffic congestion at berths and yards, and an enhanced overall terminal service quality. The quay-crane-allocation problem (QCAP), initially introduced by Park and Kim [7], aims to integrate berth allocation and QC scheduling. Recently, it has been proven that the QCAP becomes effective if performed simultaneously with the BAP [8]. Figure 1 illustrates the basic concept of the QCAP, using a satellite image of the Ningbo container terminal as an example. The container-handling area can be divided into three parts: the berth area, the QC area, and the storage area [9]. The berth area encompasses the range from anchorage to the berths, and aims to allocate the berthing positions for vessels, representing the berth-allocation problem (BAP) [10]. The QC area involves the rational allocation of QCs for handling containers of berthing vessels, constituting the QC scheduling problem [11]. The storage area is where inbound containers are stored and can be considered as the port yard, encompassing problems such as empty container repositioning [12] and container pre-marshalling [13]. The BACAP addressed in this study integrates the interdependence between the berth area and the QC area, seeking to determine the vessel’s arrival time, location, and corresponding QC allocation information.

Currently, academic research on the QCAP can be categorized into three types: sequential, integrated, and profile-based approaches [10]. The present study falls within the scope of the profile-based BACAP. The concept of the QC profile was initially introduced by Giallombardo et al. [14], and describes the existence of multiple QC usage plans for each vessel, referred to as QC profiles. For the port authority, it is only necessary to select a QC profile in order to provide the corresponding loading and unloading services for the vessels. The concept of QC profiles aligns well with real-world scenarios, and facilitates the optimization modeling of container terminal operations. In recent years, the academic community has increasingly explored the application of loading/unloading, planning, and problem-solving strategies [15,16].

In container port operations, port authorities can obtain vessel-related information in advance, such as the vessel arrival time and container volumes, and corresponding assumptions are widely used in certain deterministic studies. However, there are still numerous unpredictable factors that render the aforementioned information unreliable. Examples of such uncertainties include the weather conditions, port facility damage, overweight containers, etc. The fluctuations caused by these uncertain factors often lead to unforeseen losses in container terminal operations, and increase the complexity of port management. Moreover, maritime operations often involve significant costs, and even slight fluctuations in these uncertain parameters can result in substantial economic losses. Given this situation, the modeling of, and research into container port optimization under stochastic conditions are gradually becoming the focus of academic research [8,17,18].

Currently, there is limited research on uncertainty-related issues regarding the BACAP based on QC profiles, while uncertainty studies on the BAP are more abundant. The modeling methods primarily focus on robust optimization (RO) [19,20], SP [21,22], fuzzy programming (FP) [23], and fuzzy sets [24,25], among other uncertainty optimization methods. The RO method, developed for the worst-case scenario, features strict constraints that ensure a strong robustness [26,27]. For any uncertain parameter within the uncertainty set, the solutions obtained via RO satisfy the constraints [28]. Considering the high costs involved in maritime operations, the robust optimization theory is particularly suitable for cost optimization and control in the maritime industry. Traditional RO, due to its consideration of the worst-case scenario, is at a disadvantage in terms of the objective function value, compared to the stochastic and deterministic programming models. To overcome this conservatism, DRO has gradually become one of the research directions in the academic community. The objective of the DRO is to optimize the expected value of a probability distribution under the worst-case scenario. Decision-makers can make estimative decisions within a certain range when the distribution information of the stochastic variables is incomplete. However, the DRO theory is currently rarely studied in the fields of the BAP and QCAP.

In terms of solution methods, considering the uncertainty significantly increases the complexity of the BACAP. Currently, most of the research focuses on heuristic algorithms (HAs) for solving the model, while exact algorithms are limited to the application of solvers, such as CPLEX and Gurobi. However, when solving large-scale integer programming problems, solvers often encounter computational bottlenecks, and the computational efficiency is often unsatisfactory. Considering the high costs involved in maritime transportation, even small precision errors can lead to substantial expenses. Therefore, in this study, we employ exact algorithms to solve the BACAP, with discrete berths, and considering quay crane plans. The objective of this problem is to determine the specific arrival time, berth location, and corresponding loading and unloading plans for a series of incoming vessels.

The main contributions and innovations of this study are as follows:

• An MIP model considering the tide time window was constructed to solve the BACAP based on QC profiles, aiming to minimize the total cost of vessels’ anchorage and delay.
• The deterministic model was extended to an SP model and a DRO model, based on the generalized moment information. They are both transformed into chance-constraint programming (CCP) models.
• The CG algorithms employed in this study effectively solve the proposed models, to enhance the efficiency of problem solving, and a rolling horizon heuristic (RHH) algorithm is adopted to generate the initial allocation scheme for the master problem.
• Through extensive numerical experiments, the effectiveness of the proposed model and algorithms under uncertain conditions is validated, with detailed comparative analyses against stochastic and deterministic models. The obtained conclusions serve as reliable reference for decision making by port authorities.

This paper is organized as follows. Section 2 presents a review of the relevant work. Section 3 establishes the deterministic MIP, SP, and DRO models for berth and QC allocation. Section 4 introduces the solution approach. The numerical experiments and analysis are given in Section 5. Section 6 concludes the paper.

2. Review of the Literature

The QCAP problem refers to the situation where, during a specific time period, a designated number of QCs are assigned to each specified vessel, without determining the start and end times of their operations. The QC assignment is typically the next step after berth allocation, with the primary goal of fulfilling the berth plan. In early research, the academic community treated berth allocation and quay-crane assignment as two isolated problems to solve. However, due to the limited number of QCs at the container terminal, when multiple vessels require servicing, factors such as the efficiency of the loading and unloading operations, as well as the back-and-forth movement of QCs between different berths, must be considered. These aforementioned circumstances may cause delay in subsequent operations at the berth, which will affect the arrival, berthing, and handling operations of other vessels. It is precisely because of the close relationship between the BAP and QCAP that integrating them for optimization can help the port authorities achieve a higher operational performance. The study of Park and Kim [7] was the first to consider the combined BACAP, developing a solution procedure to jointly allocate berths and QCs, and making decisions on the berthing position, the start and end time for each vessel, and the number of QCs assigned to each vessel. By integrating the berth-allocation and quay-crane assignments, they aimed to help the terminal achieve a higher performance, and the resulting BACAP is also the focus of this study.

As mentioned earlier, the QCAP can be classified into three categories, based on the type of decision making, including sequential, integrative, and profile-based.

• Sequential QCAP: in the first stage, the number of QCs allocated for each time period is determined but, in the second stage, there is no requirement to propose specific QC allocation schemes. This type of QCAP is referenced in previous works, such as those by Hu et al. [29], Liu et al. [30], and Iris et al. [31].
• Integrated quay crane allocation problem: this concerns the determination of the operational status of each QC during each time interval. The research on such a QCAP is referenced in previous works by Imai et al. [32], Li et al. [33], Agra et al. [34], Hamza et al. [35], and Zheng et al. [36].
• Profile-based QCAP: this line of research introduces the concept of QC profiles, to facilitate the integration of berth allocation and quay crane assignment. This also constitutes the focus of our study. This problem aims to generate a set of feasible QC configurations, based on known container-handling workloads, addressing the joint problem of vessel berthing and terminal QC assignment. The approach involves initially assigning QCs to each vessel in the first stage, and then resolving the corresponding BAP in the second stage. This process is repeated iteratively, until all constraints are satisfied.

Figure 2 illustrates an example of a QC profile utilized for Vessel 5. The primary limitation to addressing planning-based QCAP lies in extending the solutions to accommodate uncertainties. As each QC profile must be determined before vessel berthing and planning formulation, it cannot guarantee the successful completion of all operations under uncertain conditions, making it challenging to extend the application of such quay-crane allocation problems to uncertain scenarios [9]. Currently, there is scarce research employing this QC assignment strategy, and this study contributes to filling this gap.

Currently, scholars have already investigated the BACAP under uncertainty, and some of the findings are summarized in

Table 1. Classification of the literature related to the BACAP under uncertainty.

Reference	Certain	Uncertainty			Uncertainty Description	Decision Variable					QC Constraints		Model	Solution Approach
		AT	OT	Other		B_T	D_T	Pos	S_OT	E_OT	(i)	(ii)		Heuristic	Exact
Giallombardo et al. [14]	√				-	√	√	√				√	MIP	Two-level heuristic	-
Han et al. [39]														-	-
Zhen et al. [40]	√				-	√	√	√				√	MIP	-	CG
Vacca et al. [5]	√				-	√	√	√				√	MIP	-	CG
Lalla-Ruiz et al. [41]	√				-	√	√	√				√	MIP	Biased key genetic	-
Zhen et al. [15]	√				-	√	√	√	√	√		√	MIP	-	CG
Xiang et al. [42]		√	√		Scenario	√		√			√		RO	Grey wolf heuristic	-
Liu et al. [17]		√	√		Scenario	√	√	√			√		SP	Relax-and-fix	-
Wang et al. [16]	√				-	√	√	√	√	√		√	MIP	-	CG
Tan et al. [43]		√	√		Scenario	√	√	√			√		SP	Heuristic	-
Xiang et al. [19]			√		Sets	√	√	√			√		RO	-	Decomposition
Xiang et al. [6]		√	√		Sets	√		√			√		SP	-	C and CG
Park et al. [44]		√			Scenario	√		√			√		RO	Particle swarm
Kolley et al. [45]		√			Scenario	√		√			√		RO	Machine learning
Our study	√			√	Scenario	√	√	√				√	RO/SP	-	CG

Note: AT: arrival time; OT: operation time; B_T: berth time; D_T: departure time; Pos: berth position; S_OT: start of operation time; E_OT: end of operation time; QC constraints (i): integrated; QC constraints (ii): profile. “√” means uncertainty, decisions variables or constraints are considered in the reference.

. To ensure clarity, we have classified the types of QCAP mentioned in each paper into two categories: (i) representing sequential and integrated types, and (ii) representing profile-based types. Regarding uncertainty, from Table 1, it can be observed that the two most common uncertain factors in the BACAP are still the vessel arrival time and the handling time. However, due to the large volume of containers loading and unloading at mainline ports, there is a need for numerous matching high-performance pieces of vertical-handling equipment, to fully accommodate the container workload of vessels. Consequently, the reliability and productivity of QCs at mainline ports are crucial, and significantly impact the overall terminal operational time. In practical operations, it is not always possible to maintain a consistently high level of QC handling efficiency. Various factors, such as issues with the handling equipment itself, terminal operational conditions, interferences from the parallel operations of multiple QCs, the proficiency of operators and truck drivers, and an overload of container handling tasks, can all influence the efficiency of QC loading and unloading operations. However, in the current research, the uncertainties related directly to QCs are mostly limited to operational interruptions caused by technical equipment failures [37,38]. There is scarce consideration of unpredictable events leading to fluctuations and declines in the QC working efficiency, without assuming a complete drop to zero efficiency in uncertain conditions. Therefore, this study sets the QC handling efficiency as a stochastic uncertain variable, allowing for investigations based on a fluctuating efficiency, while still satisfying the container-handling workload as the foundation of the research.

In the selection of uncertain optimization methods, for the BACAP, most of the approaches adopted in different background studies are based on SP, using a set of scenarios and approximate robust models. In this context, uncertainty is often represented by a set of scenarios, each with the same probability of occurrence. This fact makes this fuzzy set particularly suitable for the application of DRO methods, as it considers different combinations of scenario probabilities [9]. Compared to stochastic optimization models, the advantage of using the DRO theory is that it protects decision-makers from the uncertainty inherent to the underlying probability distributions. However, based on the literature survey, it is found that, currently, very few scholars have applied the DRO model to address uncertainty in the field of the BAP and QCAP. Moreover, as indicated by the aforementioned literature review, tidal conditions are crucial considerations, particularly for large vessels with significant draught restrictions, and mainline ports with a high container throughput. From Table 1, it is evident that the existing uncertain research often overlooks the impact of tide time windows on berth allocation or QC assignment plans. Furthermore, the current limitations in similar research include the lack of modeling and solution methods for the QCAP that account for uncertainty in handling plans. This paper innovatively addresses the stochastic BACAP, which considers the random fluctuations in crane consumption quantities due to the crane-handling efficiency in uncertain scenarios.

For the solution of the BACAP, Lee et al. [46] developed a genetic algorithm (GA) to obtain berth schemes. Vacca et al. [5] proposed a model based on exponential variables, and solved the BACAP using a CG method, in one of the first studies to employ an exact branch-and-price algorithm, to find the optimal integer solution. Turkogullar et al. [47] first constructed the integer programming model for the sequence-based QCAP under continuous berth conditions, which only determines the number of allocated QCs in each time interval. Hamza et al. [35] established multiple models for different types of QCAP, including problems considering quayside and discrete berth allocation.

3. Quay-Crane Allocation Models

We first introduce the deterministic berth-allocation problem (DBAP) model, then extend it to the stochastic programming BAP and distributionally robust optimization BAP models. The indices, sets, and parameters used in this paper are mainly listed as follows.

$B$	set of berths, indexed by $b$.
$V$	set of vessels, indexed by $i$ and $j$.
$T$	set of planning times, indexed by $t$.
$G$	set of tide cycles, indexed by $g$.
$P$	set of QC profiles, indexed by $p$.
$Q$	number of QCs owned by the container terminal.
$P_i$	quantity of containers requiring loading and unloading operation for vessel $i$.
$e_i^{arr}$	estimated time of arrival of vessel $i$.
$e_i^{dep}$	estimated time of departure of vessel $i$.
$c_i^1$	cost incurred per unit of time for the anchorage of vessel $i$.
$c_i^2$	cost incurred per unit of time for the delay of vessel $i$.
$\left[{\underset{\_}{t}}_{i,g},{\overline{t}}_{i,g}\right]$	berthing time window for vessel $i$ in the $g^{th}$ tide.
$t_{i,p}=h_{i,p}$	handling time if the vessel $i$ uses the QC profile $p$.
$q_{p,u}$	number of QCs used in the $u^{th}$ time-step of QC profile $p$, which $u\in \left\{1,\dots ,t_{i,p}\right\}$.
$E$	loading and unloading efficiency of the QCs.
$M$	a sufficiently large positive number.

Decision Variables

$x_{i,b}$	binary, equals one if vessel $i$ berths on the $i$ berth, and zero otherwise.
$y_i^{in}$	integer, berthing time of vessel $i$.
$y_i^{out}$	integer, departure time of the vessel $i$.
${\delta }_{i,p}$	binary, equals one if the QC profile $p$ is assigned to the vessel $i$, and zero otherwise.
$w_{i,t}$	binary, equals one if the handling time of the vessel $i$ starts at time $t$.
${\pi }_{i,p,t}$	binary, equals one if vessel $i$ adopts QC profile $p$ at time $t$, and zero otherwise.
${\theta }_{i,j,b}$	binary, equals one if vessel $j$ is assigned to berth $b$ immediately following vessel $i$, and zero otherwise.
${\zeta }_{i,g}^{in}$	binary, equals one if the in-wharf time of vessel $i$ is in the berthing time window of the $g^{th}$ tide cycle.
${\zeta }_{i,g}^{out}$	binary, equals one if thr out-wharf time of vessel $i$ is in the departing time window of the $g^{th}$ tide cycle.

3.1. Problem Description

Before presenting the construction of mathematical models, we first provide a detailed description of the process for the quay-crane-assignment problem (QCAP) based on QC profiles. In this study, we consider a total of $\left|V\right|$ vessels awaiting berthing, $\left|B\right|$ available berths for container vessels to berth, and the port offers $\left|P\right|$ QC profiles for vessel selection. Furthermore, the planning horizon for the entire problem is defined as $\left|T\right|$. Firstly, the vessels will arrive at anchorage at their ETA $e_i^{arr}$. At this point, vessel $i$ is required to await instructions from the port authority for berthing and handling. These arrangements typically specify the berthing time and berth for the vessels, represented by the decision variables $y_i^{in}$ and $x_{i,b}$. Upon entering berth $b$, vessel $i$ is required to select a pre-determined QC profile $p$ for conducting operations. The decision variable for this stage is denoted as ${\delta }_{i,p}$. The QC profiles are predetermined by the port authority, and aim to facilitate the completion of the vessels’ loading and unloading tasks. Additionally, during the vessels’ handling operations, the allocation of QCs to vessel $i$ is determined by the port authority. The total number of QCs required by all concurrently operating QC profiles in the unit of time $t$ must not exceed the total number of $Q$ QCs available at the terminal. Due to the significant variations in the size and workload of container vessels, the QC profiles provided by the port authority may not necessarily be suitable for every berthing vessel. Therefore, this study assumes an hourly handling efficiency $E$ for each QC, and a workload $P_i$ representing the number of containers to be handled for each vessel. Consequently, the selected QC profile for each vessel must satisfy its specific loading and unloading volume requirements. After the vessels complete the loading and unloading operations for the container volume, they are required to wait for the port authority’s clearance for departure from the port. This actual departure time of the vessel $i$ can be denoted as $y_i^{out}$. To ensure the vessel’s successful access to, and departure from, the berths, we consider the impact of tidal windows on this allocation problem. Two decision variables, ${\zeta }_{i,g}^{in}$ and ${\zeta }_{i,g}^{out}$, are defined, to represent whether a vessel begins entering the berth, and starts leaving the berth, within the time window of the tidal cycle, respectively. The variable $\left[{\underset{\_}{t}}_{i,g},{\overline{t}}_{i,g}\right]$ is used to indicate the tidal time window for the arrival and departure of vessel $i$. Additionally, the actual arrival time $y_i^{in}$ and actual departure time $y_i^{out}$ of vessel $i$ should fall within the corresponding tidal time windows for arrival and departure, respectively.

The objective function in this study comprises two components, namely, the vessel anchorage time and delay time. The former refers to the difference between the vessel’s arrival time $e_i^{arr}$ and the actual berthing time $y_i^{in}$, with a coefficient of $c_i^1$. This term primarily accounts for the waiting cost incurred by vessels during their time in anchorage. The latter term represents the delay time, determined by the difference between the vessel’s actual departure time $y_i^{out}$ and the expected departure time $e_i^{dep}$. The expected departure time $e_i^{dep}$ can also be regarded as the ETD for the vessel. Departing later than this time will result in additional costs for the shipowner during subsequent operations, with the coefficient denoted as $c_i^2$. Figure 3 illustrates the detailed activities contained. In Figure 3, the QC profile section in the bottom left corner merits a detailed explanation. For the same vessel $i$, different QC profiles $p$ may result in inconsistent operational times (as shown in Figure 2 with $h_{i,1}=4$; $h_{i,2}=3$; $h_{i,3}=3$). Additionally, this section involves a new index $u$, which will be further elucidated in the subsequent constraint description. Furthermore, a notable characteristic of the problem studied in this study is the uncertainty in the QC handling efficiency $E$ within the vessel’s QC profile. The paper will provide a detailed explanation of the uncertain aspect in the subsequent sections.

3.2. Deterministic BACAP

Currently, extensive research has been conducted in the academic community on the deterministic-berth-allocation-with-crane-assignment problem (DBACAP) models under certain conditions. Numerous models of the DBACAP have been proposed, which are not the focus of this paper. Therefore, this study cites the research findings of Zhen et al. [15]. In that study, Zhen established a QCAP model considering QC profiles, and provided a comprehensive data demonstration of the model. To provide a clearer exposition of the model, this research divides the deterministic model into two parts: the traditional discrete BAP model with tidal conditions, and the QC-related constraints. Based on the above definition, a mathematical model of the DBACAP is formulated, as follows.

3.2.1. Discrete BAP Model

The objective function (1) was to minimize the total cost of anchorage and delay. Objective (1) contained two parts: the first part $c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)$ is the weighted sum of all the vessels’ cost of anchorage for in-wharf and berthing activity, the second part $c_i^2\cdot {\left(y_i^{out}-e_i^{dep}\right)}^{+}$ is the weighted sum of the cost with respect to the vessels’ delay time. The formulation of this objective function is commonly used in the research into the discrete BAP. Constraint (2) ensures that each vessel can berth at the terminal. Constraint (3) ensures that each vessel can only in-wharf and out-wharf once. Constraints (4) and (5) establish the relationship between the arrival and departure time of vessels, and prevent temporal and spatial overlaps, to avoid confliction between two vessels. Constraint (6) indicates that the vessel could only depart after completing its loading and unloading operation. Constraints (7) and (8) represent the constraints on the vessel arrival times. Constraints (9)–(13) are the domains of the decision variables.

$$[DBACAP]\min {\displaystyle \sum _{i\in V}\left[c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)+c_i^2\cdot {\left(y_i^{out}-e_i^{dep}\right)}^{+}\right]}$$

(1)

$${\displaystyle \sum _{b\in B}{x}_{i,b}=1} \forall i\in V$$

(2)

$${\displaystyle \sum _{t\in T}{w}_{i,t}=1} \forall i\in V$$

(3)

$${\displaystyle \sum _{j\in V}{\theta }_{j,i,b}}={\displaystyle \sum _{j\in V}{\theta }_{i,j,b}} \forall i\in V,b\in B$$

(4)

$$y_i^{out}\le y_j^{in}+M\cdot \left(1-{\theta }_{i,j,b}\right) \forall i,j\in V,b\in B,i\ne j$$

(5)

$$y_i^{in}\ge y_i^{in}+{\displaystyle \sum _{p\in P_i}{h}_{i,p}\cdot {\delta }_{i,p}} \forall i\in V$$

(6)

$$y_i^{in}\ge e_i^{arr} \forall i\in V$$

(7)

$${\displaystyle \sum _{t\in T}t\cdot w_{i,t}=}{y}_i^{in} \forall i\in V$$

(8)

$$x_{i,b}\in \{0,1\} \forall i\in V,b\in B$$

(9)

$$w_{i,t}\in \{0,1\} \forall i\in V,t\in T$$

(10)

$${\theta }_{i,j,b}\in \{0,1\} \forall i,j\in V,b\in B$$

(11)

$$y_i^{in},y_i^{out}\in {\mathbb{Z}}^{+} \forall i\in V$$

(12)

$${\delta }_{i,p}\in \{0,1\} \forall i\in V,p\in P$$

(13)

3.2.2. QC-Related Constraints

Constraints (14)–(17) are related to the QCAP. Constraint (14) states that each vessel must select a QC profile for its loading and unloading operations. Constraint (15) defines the relationship between the variables ${\pi }_{i,p,t}$, ${\delta }_{i,p}$, and $w_{i,t}$. Constraint (16) represents the constraint on the number of QCs, ensuring that the total number of used QCs does not exceed the available number of QCs at the port. The working principle of this constraint is extensively explained via an example in Giallombardo et al. [14]. It is worth noting that, due to the zero-based indexing of arrays in computer programming, in the implementation of the code, the lower bound of $u$ in constraint (16) should be modified to $u=\max (t-h_{i,p};0)$, and $q_{p,(t-u+1)}\to q_{p,(t-u)}$. Constraint (17) ensures that the QC profile selected by a vessel should meet its own container volume requirements. Constraint (18) is the domain of the decision variables.

$${\displaystyle \sum _{p\in P}{\delta }_{i,p}=1} \forall i\in V$$

(14)

$${\pi }_{i,p,t}\ge {\delta }_{i,p}+w_{i,t}-1 \forall i\in V,p\in P,t\in T$$

(15)

$${\displaystyle \sum _{i\in V}{\displaystyle \sum _{p\in P_i}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}^t{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\le Q}} \forall t\in T$$

(16)

$${\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\cdot E\ge P_i} \forall i\in V$$

(17)

$${\pi }_{i,p,t}\in \{0,1\} \forall i\in V,p\in P,t\in T$$

(18)

3.2.3. Tide Time Window Constraints

Constraints (19)–(24) are constraints regrading the tide time window. Constraints (23) and (24) ensure that each vessel can only berth or depart once. Constraint (25) is the domain of decision variables.

$$y_i^{in}\ge {\underset{\_}{t}}_{i,g}-M\cdot \left(1-{\zeta }_{i,g}^{in}\right) \forall i\in V,g\in G$$

(19)

$$y_i^{in}\le {\overline{t}}_{i,g}+M\cdot \left(1-{\zeta }_{i,g}^{in}\right) \forall i\in V,g\in G$$

(20)

$$y_i^{out}\ge {\underset{\_}{t}}_{i,g}-M\cdot \left(1-{\zeta }_{i,g}^{out}\right) \forall i\in V,g\in G$$

(21)

$$y_i^{out}\le {\overline{t}}_{i,g}+M\cdot \left(1-{\zeta }_{i,g}^{out}\right) \forall i\in V,g\in G$$

(22)

$${\displaystyle \sum _{g\in G}{\zeta }_{i,g}^{in}=1} \forall i\in V$$

(23)

$${\displaystyle \sum _{g\in G}{\zeta }_{i,g}^{out}=1} \forall i\in V$$

(24)

$${\zeta }_{i,g}^{in},{\zeta }_{i,g}^{out}\in \{0,1\} \forall i\in V,g\in G$$

(25)

There is a non-linear form ${(\cdot )}^{+}$ in objective (1). The un-linearized part ${\left(y_i^{out}-e_i^{dep}\right)}^{+}$ is replaced with the linearization variables ${\eta }_i^{+},{\eta }_i^{-}$. Based on the above auxiliary variables, the linearized $[DBACAP]$ model is as follows:

$$[DBACAP]\min {\displaystyle \sum _{i\in V}\left[c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)+c_i^2\cdot {\eta }_i^{+}\right]},$$

(26)

s.t. constraints (2)–(25).

$$y_i^{out}-e_i^{dep}={\eta }_i^{+}-{\eta }_i^{-}\forall i\in V,$$

(27)

$${\eta }_i^{+},{\eta }_i^{-}\in {\mathbb{Z}}^{+}\forall i\in V.$$

(28)

3.3. BACAP with Uncertain QC Handling Efficiency

3.3.1. Stochastic Programming Model

The $[DBACAP]$ presented in Section 3.2 is a deterministic model, assuming no consideration of technical failures, and the QC operation efficiency is a known parameter. However, in real operational environments, there are numerous uncertainties, with equipment and mechanical failures of QCs being common uncertainties in terminal operations. As a result, the aforementioned assumption is often difficult to hold in practical problems. To address this issue, this study creatively assumes uncertainty in the QC’s handling efficiency $E$ and, subsequently, formulates a SP problem. The uncertainty related to the QC handling efficiency, denoted as $\widehat{E}$, is incorporated in constraint (17). Following the definition by Charnes and Cooper [48], the discrete BAP with uncertain QC operation efficiency proposed in this study falls under the category of chance-constraint programming. Based on the research by Charnes and Cooper [48], we first present the general form of the CCP, as follows:

$$\begin{array}{l}\underset{x\in X}{\min }f(x)\\ \mathrm{s}.\mathrm{t}.\mathbb{P}\left\{g(x,\xi )\le 0\right\}\ge 1-\epsilon ,\end{array}$$

(29)

in which the decision variable $x\in X$, and the functions $f:{ℝ}^n\to ℝ$ and $g:{ℝ}^n\times {ℝ}^k\to {ℝ}^m$ are continuous functions. $X$ represents a compact set in ${ℝ}^n$, $\xi :\Omega \to \Xi$ denotes an uncertainty factor defined on the closed support set $\Xi$, where $\Xi \in {ℝ}^k$ is a $k$ dimensional space comprising all real numbers. Thus, $\xi$ is a random variable on the probability space $(\Omega ,\mathcal{F})$, where $\mathbb{P}$ represents the probability distribution of this random variable $\xi$. The parameter $\epsilon \in (0,1)$ denotes the given confidence level. Based on the constraint (17) and the general form (29), we introduce the confidence level $z_i=1-{\epsilon }_i$, $\forall i\in V$, and rewrite the $[DBACAP]$ as the following stochastic programming $[SPBACAP]$ form:

$$[SPBACAP]\min \theta ,$$

(30)

s.t. constraints (2)–(16), (18)–(25), (27)–(28),

$$\theta \ge {\displaystyle \sum _{i\in V}\left[c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)+c_i^2\cdot {\eta }_i^{+}\right]},$$

(31)

$$\mathbb{P}\left\{{\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\cdot E\ge P_i}\right\}\ge z_i\forall i\in V,$$

(32)

$$z_i\in (0,1)\forall i\in V.$$

(33)

We introduce $\theta$ as a replacement for the previous objective function, and add constraint (31). Constraint (32) represents a chance constraint. Following the standard chance-constraint argumentation [49], the chance constraint can be transformed into a deterministic form. The resulting deterministic equivalent form of constraint (17) is shown in the following formula (34), and the detailed argumentation process can be found in Appendix A.

$${\mathbb{E}}_{\mathbb{P}}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}+{\varphi }^{-1}\left(z_i\right)\cdot \sqrt{\mathbb{D}[\tilde{E}]\cdot {\left({\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\right)}^2}\ge P_i.$$

(34)

In constraint (34), $\varphi (\cdot )$ represents the probability distribution function of the random variable $\xi$, $\mathbb{D}[\tilde{E}]$ is the first central moment of historical data for the QC efficiency, ${\mathbb{E}}_{\mathbb{P}}[\tilde{E}]$ is the first moment information of historical data for the QC efficiency, and $\varphi$ is the probability distribution function of the QC efficiency. Regarding Equation (34), when $\mathbb{D}[\tilde{E}]=0$ and ${\mathbb{E}}_{\mathbb{P}}[\tilde{E}]=E$, it is equivalent to the deterministic constraint (17).

3.3.2. Robust Programming Model

The berth-allocation planning task chain at container terminals operates continuously throughout the day, and the occurrence of uncertainties, such as QC technical failures, can significantly impact the smooth operation of current and future task chains at the port. Therefore, it is essential to consider the worst-case scenario that may arise from the uncertainty in QC efficiency, to enable port management to prepare appropriate contingency measures. In this section, we will further extend the SP model presented in Section 3.3.1 to a distributionally robust chance-constrained (DRCC) model, aiming to obtain berth-allocation schemes that can effectively handle worst-case scenarios. First, based on the research by Ben-Tal et al. [50] and Bertsimas et al. [51], the basic form of the $[DROBACAP]$ model with chance constraints is as follows:

$$\begin{array}{l}\underset{x\in X}{\min }{c}^{\top }x\\ \mathrm{s}.\mathrm{t}.\underset{\mathbb{P}\in \mathcal{F}}{\inf }\mathbb{P}\left\{\tilde{\xi }:a{(x)}^{\top }\tilde{\xi }\le b_i(x),\forall i\in \left|I\right|\right\}\ge 1-\epsilon .\end{array}$$

(35)

Based on the theory presented in Equation (35), we can transform the $[SPBACAP]$ model from Section 3.3.1 into a DRCC model $[DROBACAP]$, as shown below:

$$[DROBACAP]\min \theta ,$$

(36)

s.t. constraints (2)–(16), (18)–(25), (27)–(28), (31), (33),

$$\underset{\mathbb{P}\in \mathcal{F}}{\inf }\mathbb{P}\left\{{\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\cdot \tilde{E}\ge P_i}\right\}\ge z_i\forall i\in V.$$

(37)

We apply the research findings of Calafiore and El-Ghaoui [52] to transform the proposed DRCC constraint (37) into a robust counterpart with convex second-order cone constraints, as shown in Equation (38). For a detailed proof process, please refer to Appendix B.

$$\sqrt{\frac{1-\epsilon }{\epsilon }}{\left|\left|\mathbb{D}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\right|\right|}_2+{\mathbb{E}}_{\mathbb{P}}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\ge P_i\forall i\in V.$$

(38)

4. Solution Approach

4.1. Basic Idea

As the BACAP is an NP-hard problem [53], it is not easy to solve large-scale instances with CPLEX. Moreover, the proposed BACAP model in this paper is a MIP model that involves a large number of binary variables, as there are many approaches to solving robust programming model and chance-constraint programming models. For example, Wang Xin et al. provide two tailored column-and-constraint generation algorithms to solve the robust model [54]. Seyyed et al. apply the interactive fuzzy programming approach to solve the multi-objective model [55]. Seyyed et al. apply the interactive fuzzy programming approach and the BWM method to solve the model [56]. However, in the field of the QCAP, CG is still one of the most applied algorithms. Therefore, this paper employs the CG method to solve the model. The column generation algorithm can effectively solve large-scale integer programming problems without enumerating a vast number of potential decision variables. It decomposes the large-scale integer programming problem into a master problem and subproblems. The basic idea is to identify entering non-basic variables, which can be seen as generating new variables, by solving subproblems. These columns are then added to the master problem. If no entered non-basic variable is found, this implies that the reduced cost (RC) of all the non-basic variables satisfies the optimality conditions. The flowchart of the CG algorithm is illustrated in Figure 4. According to the flowchart shown in Figure 4, the master problem, also known as the restricted master problem (RMP), is defined by a subset of variables. Subsequently, the pricing sub-problem (PSP) is employed, to check if there are variables that have not been considered and result in a negative RC. If such variables exist, their corresponding coefficients are added to the coefficient matrix of the RMP, and the process iterates until all the RC values in the subproblem are greater than, or equal to, zero.

4.2. Restricted Master Problem

For this study, feasible allocation schemes for each vessel i should include the arrival time $y_i^{in}$, departure time $y_i^{out}$, berth information, and crane assignment. The set of feasible allocation schemes for vessel $i$ is defined as $D_i$, indexed by $d$. To construct the RMP in CG method, we introduce a decision variable representing the vessel’s allocation scheme, as well as three parameters representing the time, space, and crane resources occupied by the vessel’s berth assignment.

Indices, Sets and Parameters

$D_i$	set of feasible allocation schemes for vessels, indexed by $d$.
$c_{i,d}$	cost incurred when vessel $i$ adopts the schemes represented by option $d$.
${\beta }_{b,d,t}^i$	binary, equals one if vessel $i$ adopted option $d$, it occupies berth $b$ at time $t$, and zero otherwise.
$q_{d,t}^i$	number of QCs used at time $t$ when vessel $i$ adopts the schemes represented by option $d$.

Decision Variable

${\chi }_{i,b}$

binary, equals one if vessel $i$ adopts option $d$, and zero otherwise.

Furthermore, based on the defined variables and parameters, the $[\mathrm{RMP}]$ is formulated in this paper.

$$[\mathrm{RMP}]\min {\displaystyle \sum _{i\in V}{\displaystyle \sum _{d\in D_i}{c}_{i,d}\cdot {\chi }_{i,d}}}$$

(39)

$${\displaystyle \sum _{d\in D_i}{\chi }_{i,d}}=1\forall i\in V$$

(40)

$${\displaystyle \sum _{i\in V}{\displaystyle \sum _{d\in D_i}{\beta }_{b,d,t}^i\cdot {\chi }_{i,d}}}\le 1\forall b\in B,t\in T$$

(41)

$${\displaystyle \sum _{i\in V}{\displaystyle \sum _{d\in D_i}{q}_{d,t}^i\cdot {\chi }_{i,d}}}\le Q\forall t\in T$$

(42)

$${\chi }_{i,d}\in [0,1]\forall i\in V,d\in D_i$$

(43)

The objective of the $[\mathrm{RMP}]$ is to make the optimal decision within the existing allocation schemes. Constraint (39) minimizes the total cost, where, for each vessel $i$ under each allocation scheme $d$, we have $c_{i,d}=c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)+c_i^2\cdot {\left(y_i^{out}-e_i^{dep}\right)}^{+}$. Constraint (40) ensures that each vessel has one selected allocation plan $d$. Constraint (41) ensures that there are no conflicts in terms of time or berth allocation. Constraint (42) states that the number of cranes used at each time $t$ is less than or equal to the QC capacity. Constraint (43) provides the range of values for ${\chi }_{i,b}$. The $[\mathrm{RMP}]$ in this case consists of $\left|V\right|+\left|B\right|\cdot \left|T\right|+\left|T\right|$ constraints, while this number of feasible allocation schemes for each vessel is quite large. For the problem-solving process, it is not necessary to find all feasible allocation schemes for each vessel. The subsequent pricing subproblem aims to filter out the most suitable allocation schemes. Additionally, for the three proposed models, the constraints of the $[\mathrm{RMP}]$ do not differ significantly. Therefore, the $[\mathrm{RMP}]$ formulated in this chapter applies to all models.

4.3. Pricing Sub-Problem

The objective of the PSP in the column generation algorithm is to find the minimum RC for each vessel $i$ and berth $b$, and add it to the master problem. Based on the constructed MIP model, two solution methods for the PSP are designed in this paper: a mathematical solution method and an enumeration method. Firstly, variables ${\alpha }_i$, ${\epsilon }_{b,t}$, and ${\lambda }_t$ are introduced to represent the dual variables for constraints (40), (41), and (42) in the $[\mathrm{RMP}]$. Then, the RC is formulated as the objective function of the PSP, defined as shown in (44).

$$Z^{\prime }=c_{i,d}-{\alpha }_i-{\displaystyle \sum _{t\in T}{\epsilon }_{b,t}}\cdot {\beta }_{b,d,t}^i-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{p\in P}{\lambda }_t\cdot {\delta }_{i,p}}.}$$

(44)

We set the RC as the objective function of the PSP. The formulation of the $[\mathrm{PSP}]$ is as follows.

$$[PSP]\min \left\{c_i^1\cdot \left(y_i^{in}-e_i^{arr}\right)+c_i^2\cdot {\eta }_i^{+}-{\alpha }_i-{\displaystyle \sum _{t\in T}{\epsilon }_{b,t}}\cdot {\beta }_{b,d,t}^i\right.\left.-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{p\in P}{\lambda }_t\cdot {\delta }_{i,p}}}\right\}$$

(45)

$${\displaystyle \sum _{b\in B}{x}_{i,b}=1}$$

(46)

$${\displaystyle \sum _{t\in T}{w}_{i,t}=1}$$

(47)

$$y_i^{in}\ge y_i^{in}+{\displaystyle \sum _{p\in P_i}{h}_{i,p}\cdot {\delta }_{i,p}}$$

(48)

$$y_i^{in}\ge e_i^{arr}$$

(49)

$$y_i^{in}\le t-M\cdot \left({\beta }_{b,d,t}-1\right)\forall t\in T$$

(50)

$$y_i^{out}\ge t-M\cdot \left(1-{\beta }_{b,d,t}\right)\forall t\in T$$

(51)

$${\displaystyle \sum _{p\in P}{\delta }_{i,p}=1}$$

(52)

$${\pi }_{i,p,t}\ge {\delta }_{i,p}+w_{i,t}-1\forall p\in P,t\in T$$

(53)

$${\displaystyle \sum _{i\in V}{\displaystyle \sum _{p\in P_i}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}^t{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\le Q}}\forall t\in T$$

(54)

$${\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}\cdot E\ge P_i}\forall i\in V$$

(55)

$$y_i^{in}\ge {\underset{\_}{t}}_{i,g}-M\cdot \left(1-{\zeta }_{i,g}^{in}\right)\forall i\in V,g\in G$$

(56)

$$y_i^{in}\le {\overline{t}}_{i,g}+M\cdot \left(1-{\zeta }_{i,g}^{in}\right)\forall i\in V,g\in G$$

(57)

$$y_i^{out}\ge {\underset{\_}{t}}_{i,g}-M\cdot \left(1-{\zeta }_{i,g}^{out}\right)\forall i\in V,g\in G$$

(58)

$$y_i^{out}\ge {\underset{\_}{t}}_{i,g}-M\cdot \left(1-{\zeta }_{i,g}^{out}\right)\forall i\in V,g\in G$$

(59)

$${\displaystyle \sum _{g\in G}{\zeta }_{i,g}^{in}=1}\forall i\in V$$

(60)

$${\displaystyle \sum _{g\in G}{\zeta }_{i,g}^{out}=1}\forall i\in V$$

(61)

$${\displaystyle \sum _{t\in T}t\cdot w_{i,t}=}{y}_i^{in}$$

(62)

$$y_i^{out}-e_i^{dep}={\eta }_i^{+}-{\eta }_i^{-}$$

(63)

$${\eta }_i^{+},{\eta }_i^{-}>0{\eta }_i^{+},{\eta }_i^{-}\in {\mathbb{Z}}^{+}$$

(64)

The objective function of the $[\mathrm{PSP}]$ is given by constraint (45), aiming to minimize the RC. Constraints (50) and (51) establish the relationship between the vessel berthing and the variable ${\beta }_{b,d,t}^i$ in the master problem. The remaining constraints correspond to the constraints in the original deterministic model $[\mathrm{BACAP}]$. Constraint (52) states that each vessel must select a QC profile for its loading and unloading operations. Constraint (53) defines the relationship between the variables ${\pi }_{i,p,t}$, ${\delta }_{i,p}$, and $w_{i,t}$. Constraint (54) represents the constraint on the number of QCs, ensuring that the total number of used QCs does not exceed the available number of QCs at the port. Constraint (55) ensures that the QC profile selected by a vessel should meet its own container volume requirements. Constraints (56)–(62) are tide constraints. Constraints (63) and (64) are linearization operations on the objective function (44). Constraint (56) represents the form of the deterministic model, while constraints (65) and (66) demonstrate the constraints for the SP and DRP, respectively.

$${\mathbb{E}}_{\mathbb{P}}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}+{\varphi }^{-1}\left(z_i\right)\cdot \sqrt{\mathbb{D}[\tilde{E}]\cdot {\left({\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\right)}^2}\ge P_i$$

(65)

$$\sqrt{\frac{1-\epsilon }{\epsilon }}{\left|\left|\mathbb{D}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\right|\right|}_2+{\mathbb{E}}_{\mathbb{P}}[\tilde{E}]\cdot {\displaystyle \sum _{p\in P}{\displaystyle \sum _{u=\max (t-h_{i,p}+1;1)}{q}_{p,(t-u+1)}\cdot {\pi }_{i,p,u}}}\ge P_i$$

(66)

4.4. Solution Procedure

In addition to the RMP and PSP mentioned above, in the process of the CG algorithm, it is necessary to provide a feasible allocation scheme for the master problem. The quality of this feasible solution will also affect the efficiency of the algorithm iterations. Therefore, in this paper, the RHH algorithm is adopted to solve this problem. The basic idea of the RHH algorithm is to divide all vessels requiring berthing operations into subsets of size $nv$ according to their arrival times, considering the lead time (LT) $ut$ and post time (PT) $nt$, to ensure the independence between each subproblem. A set of $\left[\left|V\right|/nv\right]$ subproblems is solved, to obtain an optimal solution. Algorithm 1 process of the RHH algorithm is illustrated as follows.

Algorithm 1 Rolling Horizon Heuristic

$nv\leftarrow$ number of new vessels to consider in each iteration

$nt\leftarrow$ number of time periods to consider for each subproblem after the arrival time of the last vessel

$ut\leftarrow$ number of time periods to consider for each subproblem before the arrival time of the first vessel

$LT=a_{nv}+nt\leftarrow$ limited time horizon of the first subproblem

$U\leftarrow$ number of iterations to cover the planning horizon, $U=\left[\left|V\right|/nv\right]$

Solve the subproblem defined for $nv$ vessels and with time horizon $LT$

For all $k=2,\dots ,U$ do

Fix variables for vessels $1,\dots ,(k-1)\times nv$

$LT\leftarrow a_{k\times nv}+nt$

Solve the restricted mixed integer problem for $k\times nv$ vessels and $LT$ time periods

End for

The RHH algorithmic process of this paper is illustrated in Figure 5. Firstly, the RHH algorithm is used to generate an initial feasible solution for the RMP. Then, the RMP is solved, and the dual variables for each constraint are obtained. Subsequently, the PSP is solved to obtain the minimum RC, and the corresponding column is added to the master problem. When a sufficient number of columns have been found to satisfy the stopping condition of the algorithm, the master problem will be solved again. In this case, the master problem is not the RMP but an MIP problem. The final result represents the optimal solution for the entire problem.

5. Numerical Experiment

5.1. Instance Generation

The instance-related parameters are as follows. The planning horizon $T=168$ with a unit time of 1 hour. The ETA is $e_i^{arr}\sim U(1,168)$, $e_i^{arr}\in {\mathbb{Z}}^{+}$. The $e_i^{dep}$ was generated according to the following rules: $e_i^{dep}=e_i^{arr}+t^{\prime }$, in which $t^{\prime }\in \left[10,20\right]$. Taking into account that vessel delays often result in more severe consequences, the unit time cost of anchorage for vessels, denoted as $c_i^1$, $c_i^1\sim U\left(1000,2000\right)$, and the unit time cost of vessel delays, denoted as $c_i^2$, $c_i^2\sim U\left(2000,3000\right)$. The sufficiently large positive number $M=1000000$. The tide data in this study come from Shanghai Yangshan Port, from 1 February 2023 to 7 February. Detailed data are shown in Appendix C. The tide time window $[{\underset{\_}{t}}_{i,g},{\overline{t}}_{i,g}]$ can be calculated according to the tide and vessel draft. The vessel draft is randomly generated, ranging in $\left[200,300\right]$. We take different vessel type as the basis for generating QC profiles for the container terminal. It is assumed that arriving vessels can be categorized into four types: handysize, mini-Panamax, Panamax, and capsize, and their container capacity and operation time are denoted as $P_i$ and $h_{i,p}$. The occurrence probabilities of four types of vessels are presented in

Table 2. The parameter configuration of the handling requirements corresponding to the vessel type.

Vessel Type	Percent	Workload ${\mathit{P}}_{\mathit{i}}$ (TEU)	QC Volume $\mathit{Q}$	QC Efficiency $\mathit{E}$ (TEU/t)	Handing Time ${\mathit{h}}_{\mathit{i},\mathit{p}}$ (hour)
Handysize	1/2	[1000, 2000]	2–4	50	[2, 10]
Mini-Panamax	1/4	[2000, 3000]	3–5	50	[8, 16]
Panamax	1/8	[3000, 5000]	3–5	50	[12, 20]
Capesize	1/8	[5000, 10,000]	4–6	50	[16, 24]

.

The QC profile and uncertainty-related parameters will now be provided. Based on the parameters in Table 2, the QC profile set $P$ and the corresponding $q_{p,u}$ are generated in the following steps:

For vessel $i$, the minimum workload $U_i$ is calculated based on the vessel-handling capacity $P_i$ and QC efficiency $E$ of each vessel, where $U_i=⌈C_i/E⌉$, and $⌈·⌉$ denotes rounding up to the nearest integer.

The QC occupancy $q_{p,u}$ is set to $r$ for each hour, where $r$ is a random integer from the corresponding vessel’s QC volume in Table 2. The generation of these random numbers is repeated, until the workload requirement $U_i$ is satisfied. For vessel $i$, we will generate up to 10 QC profiles, with at least two QC profiles, and this QC profile information will be added to the port’s QC profile scheme. Depending on the problem size, the number of QC profiles $\left|P\right|$ varies within the range $[2\times |V|,10\times |V|]$.

For the uncertain factor, the QC handling efficiency is assumed to be $E=50$ (TEU/h). For the SP model, the mean of the QC handling efficiency ${\mathbb{E}}_{\mathbb{P}}[\tilde{E}]=50$ (TEU/h), and its variance $\mathbb{D}[\tilde{E}]=100$ (TEU/h).

Regarding algorithm-related parameters, for the RHH algorithm, our study assumes $nt=ut=10$ and $nv=5$.

For the problem scale, we control the problem size using three variables: the number of vessels $\left|V\right|$, the number of berths $\left|B\right|$, and the number of QCs $\left|Q\right|$. We provide data for three different problem sizes, and denote them as G1, G2, and G3. The label G1-1 represents the first instance of the problem size G1. The specific parameters for each instance are presented in

Table 3. The key parameters of the instance scale.

Class	Num. of Vessels $/\mathit{V}/$	Num. of Berths $/\mathit{B}/$	Num. of QCs $/\mathit{Q}/$	Time Planning Horizon $/\mathit{T}/$
G1	10	2	10	168
G2	20	4	15	168
G3	30	6	20	168

.

Finally, the experiments are programmed in C++ (Visual Studio), and solved by CPLEX 12.10. All programs are executed and processed on a PC with the following configuration: 2.70GHz Intel(R) Core™ i7-12700H CPU and 32GB DDR4 3200MHz RAM.

5.2. Algorithm Performance

5.2.1. Comparison with CPLEX

In order to validate the effectiveness of the proposed algorithm, a comparison is made between the algorithm and the direct solving results obtained via CPLEX in terms of the computation time and solution outcomes. A time limit of 1 hour is set for each algorithm in this study. Solutions that exceed this time limit are considered to provide limited assistance in the port berth-allocation task. All computational results are shown in

Table 4. The numerical experimental results of the DBACAP, SPBACAP, and DROBACAP models of CPLEX and CG for solving different problem case sizes.

Instance		DBACAP				SPBACAP				DROBACAP
		t1 (s)	obj1 (USD)	t2 (s)	obj2 (USD)	t3 (s)	obj3 (USD)	t4 (s)	obj4 (USD)	t5 (s)	obj5 (USD)	t6 (s)	Obj6 (USD)
G1	1	29.34	49,527	5.06	49,527	33.23	56,028	6.93	56,028	143.26	84,921	49.19	85,938
	2	56.12	40,582	13.29	40,582	60.18	49,387	13.34	49,387	202.36	72,313	68.17	74,932
	3	16.38	12,083	2.06	12,083	21.69	19,302	5.69	19,302	117.34	30,491	26.39	30,491
	4	129.06	59,032	19.25	59,032	140.36	68,371	41.87	68,371	683.23	83,920	139.69	87,382
	5	352.17	83,910	69.28	83,910	403.69	90,276	102.38	90,276	1238.45	110,293	228.16	115,039
G2	1	412.67	93,028	73.36	93,028	506.87	99,403	103.96	101,293	1839.39	119,384	392.31	125,463
	2	679.19	102,938	121.37	108,374	733.18	125,847	145.68	129,375	3572.19	158,375	743.29	160,432
	3	704.33	73,829	145.36	79,203	802.66	84,038	169.21	84,038	3829.43	100,293	784.16	112,039
	4	587.36	67,362	123.18	67,362	648.21	73,759	133.96	73,759	2637.59	96,743	574.26	105,937
	5	1687.34	129,375	375.92	130,498	1804.34	156,734	452.18	160,192	4683.21	182,637	810.23	192,039
G3	1	3102.75	130,582	638.38	143,029	3320.45	158,463	632.29	162,938	>2 h	-	1453.29	182,930
	2	2930.64	80,283	506.39	83,182	3493.5	95,746	693.12	98,201	>2 h	-	1693.21	140,593
	3	3112.39	99,203	673.17	102,873	3394.12	112,763	634.67	118,372	>2 h	-	1538.34	159,348
	4	3821.96	140,392	738.89	147,384	4023.36	163,746	1029.34	168,473	>2 h	-	2029.66	193,028
	5	3429.34	110,239	673.27	112,635	3648.46	139,583	673.24	144,382	>2 h	-	2312.34	173,593

Note: t1, t3, t5, obj1, obj3, and obj5 represent the time and objective function values used by CPLEX for computation; t2, t4, t6, obj2, obj4, and obj6 represent the time and obtained objective function values for computation via the CG algorithm.

. Based on Table 4, the comparative results of the solving times for the two methods and three models are presented in

Table 5. The computational time comparison of the CPLEX and CG algorithms for solving models.

Group	CPLEX			CG
	Average	Largest	Smallest	Average	Largest	Smallest
CPU time of DBACAP
G1	116.61	352.17	16.38	21.78	69.28	2.06
G2	814.18	1687.34	412.67	167.8	375.92	73.36
G3	3279.41	3821.96	2930.64	646.02	738.89	506.39
CPU time of SPBACAP
G1	131.83	403.69	21.69	34.04	102.38	5.69
G2	899.1	1804.34	506.87	200.1	452.18	103.96
G3	3575.98	3320.45	4023.36	732.53	1029.34	634.67
CPU time of DROBACAP
G1	476.92	1238.45	117.34	102.32	228.16	26.39
G2	3312.36	4683.21	1839.39	660.85	810.23	392.31
G3	>2 h	>2 h	>2 h	1805.36	1453.29	2312.34

. From Table 5, it can be observed that the proposed column generation algorithm significantly outperforms the traditional CPLEX approach, in terms of solving time. When using CPLEX for direct solving, it becomes challenging to obtain optimal solutions within a reasonable time for large-scale datasets (i.e., G3). Moreover, the solving time for the DRCC model experiences a substantial increase, primarily due to the second-order cone constraint (58) derived from the DRCC model. Although similar phenomena exist in the CG algorithm, the rise in the solving time is more gradual compared to direct solving. This is mainly because the CG algorithm does not require the simultaneous solving of multiple second-order cone constraints, resulting in less of an impact on the solving time. According to the specific instances in Table 4, for certain specific data, the CG algorithm often achieves faster solutions than CPLEX, consistent with the findings in Table 5.

Considering different data scales, for small-scale datasets (i.e., G1), the proposed CG algorithm yields results consistent with the precision of the CPLEX algorithm. However, for large-scale datasets (i.e., G3), a certain gap still exists between the results of the CG algorithm and the CPLEX solutions. This discrepancy is more pronounced in the SPBACAP and DROBACAP models, and it may be attributed to two possible reasons. Firstly, it is caused by approximations made during the process of linearizing relevant constraints. Secondly, for large-scale datasets, the subproblem fails to find all feasible columns to add to the master problem, resulting in the final computation being a local optimum solution. Nonetheless, in terms of computational results, the errors between the results obtained via the CG algorithm and CPLEX calculations still remain within satisfactory bounds.

Another notable phenomenon is that the solving time of the CG algorithm is less affected by the problem size compared to CPLEX, as evident from Table 5. The primary reason for this phenomenon is the significant increase in the number of nodes generated by CPLEX’s branch-and-price algorithm with the expansion of the problem scale, leading to a rapid growth in computation. In contrast, for the CG algorithm, as the problem scale increases, the number of additional feasible columns also increases within a polynomial quantity, avoiding exponential growth, and resulting in a more stable increase in CPU time.

5.2.2. CG Algorithm Parameter Analysis

From the previous Table 4 and Table 5, it is evident that there is no significant difference in CPU time between the SP model and the DBACAP model, while the DROBACAP model exhibits a substantial increase in solving time. Therefore, we will compare the efficiency differences of the CG algorithm among different models, and analyze the reasons behind these disparities. Starting from the algorithmic structure, the core of the CG algorithm lies in solving the PSP and the RMP.

The solving speed of the PSP is related to its own constraint complexity, while the solving speed of the RMP is linked to the number of feasible columns, both of which crucially affect the efficiency of the CG algorithm. Apart from the factors inherent to the models themselves, the number of iterations plays a key role in determining the efficiency. Thus, this aspect is initially compared based on the number of iterations. Figure 6 compares the convergence of the deterministic model and the DRO model for three different problem scales. Figure 6a,c,e depict the convergence of the deterministic model, while Figure 6b,d,e show the convergence of the DRO model. Based on the results from Figure 6, there is no significant difference in the number of iterations between the DBACAP model and the DROBACAP model during the column generation (CG) process. For all data sizes, the CG algorithm achieved the optimal solution within 11 iterations. Therefore, it can be inferred that the disparity in algorithm efficiency lies in the solving efficiency of the PSP. Consequently, this study further compares the time taken by the DBACAP and DROBACAP to solve all the subproblems in each iteration. The comparison results are presented in Figure 7.

From Figure 7, it can be observed that there is a significant difference in CPU time between the DROBACAP and DBACAP models in the subproblem-solving process. This disparity is primarily attributed to the substantial increase in the solving time caused by the second-order cone constraints in the DRO model. Based on the conclusions from Figure 6 and Figure 7, it can be deduced that the second-order cone constraints generated during the subproblem modeling have the most significant impact on the efficiency of the CG algorithm. Optimizing this aspect of the approximate algorithm can further enhance the efficiency of the CG method.

5.3. Sensitivity Analysis

In order to validate the robustness of the proposed DRO model, we conduct a fluctuation analysis on it. Considering that the DROBACAP model deals with the worst-case optimal problem, compared to the SPBACAP model and DBACAP model, it can be considered to handle more container volumes $P_i$ under the same problem parameters. Therefore, this paper introduces the fluctuation coefficient $\kappa$ of container volumes, and measures the variation in objective function values between the SP model and the DRO model under three problem sizes when $\kappa \in \{0.8,0.9,1,1.1,1.2\}$. Sensitivity analysis is conducted on the first two sets of data for each data size, and the results are shown in Figure 8. According to the results in Figure 8, it can be observed that the objective function obtained via the SPBACAP model is consistently smaller than the one obtained via the DROBACAP model, which aligns with the general understanding presented in the mathematical derivation earlier. It is worth noting that, with the variation in the fluctuation coefficient $\kappa$, the DRO model exhibits stable changes in the objective function, as shown in Figure 8b–d,f, while the SP model is more sensitive to the changes in $\kappa$. This conclusion demonstrates that the DRO model proposed in this paper offers an excellent resistance to perturbations.

6. Conclusions

This paper investigates the BACAP in container terminals, considering the random fluctuations in QC handling efficiency. Two corresponding models, SPBACAP and DROBACAP, are formulated. Additionally, the CG algorithm is developed to solve the proposed models effectively. The algorithm’s validity and the model’s robustness are demonstrated through numerical experiments, using different scales of real data from the Shanghai Yangshan Port. The main results of this study are as follows:

• A deterministic MIP model can be constructed to solve the BACAP based on QC profiles under tidal conditions, aiming to minimize the vessels’ anchorage and delay costs.
• According to the real operation of the container terminal, this study takes the QC handling efficiency as the uncertain factor to formulate the extension of the model.
• The deterministic model is extended to an SPBACAP model and a DROBACAP model, considering stochastic fluctuations in the QC efficiency. The corresponding robust counterpart theory is applied, to deal with the DRO model.
• The CG algorithm is tailored to solve the models for an efficient solution. By comparing the algorithms, we summarized their characteristics, which could serve as a valuable reference for other scholars conducting research in related fields.
• Verification of the algorithm’s effectiveness through numerical experiments, using real data from the Shanghai Yangshan Port.
• Evaluation of the DRO under different levels of QC handling efficiency fluctuations, confirming its resilience against disturbances. Our model demonstrates a certain level of resilience against unknown risks, as indicated by the results.

However, certain limitations remain in this study. For instance, compared to the number of vessels actually docked in the port, the problem scales achievable by the models are relatively small, limiting their direct application in real-world scenarios. Moreover, the models’ results provide potential risk-avoidance distribution plans, but do not address compensatory measures when risks cannot be fully mitigated. Additionally, the implementation of environmental regulations for vessels in 2023 adds pressure for environmentally friendly solutions, which will become a focus of future research in our academic community. Moreover, in the context of green ports, the research on sustainable solutions for the BACAP under uncertain environments is a highly important and urgent direction. This research encompasses environmental issues, such as fuel consumption during vessel navigation, the energy consumption of relevant equipment while vessels wait in the port, carbon emissions, and other related aspects, which have been studied in deterministic berth-allocation problems. Additionally, attention can be directed toward the research on berth-allocation problems in innovative scenarios, such as shared berth services.