Quay Crane Assignment Problem Research Articles

Optimizing integrated berth allocation and quay crane assignment: A distributionally robust approach

In this research, we have formulated a Two-Stage Distributionally Robust Optimization (TDRO) model within the context of a mean–variance ambiguity set, specifically designed to address the challenges in the Integrated Berth Allocation and Quay Crane Assignment Problem (BACAP). A key consideration in this study is the inherent uncertainty associated with ships’ arrival times. During the initial stage, we derive a baseline schedule governing berth allocation and quay crane assignment. Anticipating potential disruptions arising from uncertain arrival delays, the second stage is meticulously formulated to determine the worst-case expectation of adjustment costs within the mean–variance ambiguity set. Subsequently, we undertake an equivalent transformation, converting the general TDRO model into a Two-Stage Robust Second-Order Cone Programming (TRO-SOCP) model. This transformation facilitates the application of the Column and Constraint Generation (C&CG) algorithm, ensuring the derivation of an exact solution. To address the computational intricacies associated with second-order cone programming, we propose two enhancement strategies for upper and lower bounds, aimed at expediting the solution process. Additionally, to contend with large-scale instances, we introduce a refinement and approximation method, transforming the TDRO model into a Mixed-Integer Programming (MIP) model. Furthermore, extensive numerical experiments are executed on both synthetic and real-life instances to validate the superior performance of our model and algorithms. In terms of the total cost, the TDRO model demonstrates superior performance compared with Two-Stage Stochastic Programming (TSP) and Two-Stage Robust Optimization (TRO) models.

Distributionally robust chance-constrained optimization for the integrated berth allocation and quay crane assignment problem

In this paper, we are the first to propose a distributionally robust chance-constrained (DRCC) optimization model for the integrated berth allocation and quay crane assignment problem (BACAP) in container terminals. In contrast to the classical deterministic BACAP model, we consider the arrival time to be uncertain due to the frequent arrival delays in ports. We then impose a chance constraint that the service time must start after the uncertain arrival time with a probability of at least 1−ϵ, where 1−ϵ represents the target service level in container terminals (ϵ represents the target risk tolerance of the port manager). Under the moment-based ambiguity set, we reformulate the DRCC model into a mixed integer semi-definite programming (MISDP) model. Additionally, we develop an efficient decomposition branch-and-bound algorithm to solve the MISDP model and obtain the exact solution. Fortunately, a special case of the DRCC model arises when the mean and covariance utilized in the ambiguity set are precise, allowing for the transformation of the DRCC model into a mixed integer programming (MIP) model. This conversion significantly reduces the complexity of the problem. Impressively, the solving time of the MISDP model with the decomposition branch-and-bound algorithm is comparable to that of the transformed MIP model. The numerical results show that our model can achieve a schedule with high service at low cost. Meanwhile, we have made an intriguing discovery that the correlation between the target risk tolerance and the actual service level can be depicted as a staircase function regardless of the datasets. This finding offers crucial insights for port management, enabling them to strike a balance between the cost and actual service level by determining an appropriate target service level.

A Proactive-Reactive-Based Approach for Continuous Berth Allocation and Quay Crane Assignment Problems with Hybrid Uncertainty

Port operations have been suffering from hybrid uncertainty, leading to various disruptions in efficiency and tenacity. However, these essential uncertain factors are often considered separately in literature during berth and quay crane assignments, leading to defective, even infeasible schedules. This paper addressed the integrated berth allocation and quay crane assignment problem (BACAP) with stochastic vessel delays under different conditions. A novel approach that combines both proactive and reactive strategies is proposed. First, a mixed-integer programming model is formulated for BACAP with quay crane maintenance activities under the ideal state of no delay. Then, for minor delays, buffer time is added to absorb the uncertainty of the arrival time of vessels. Thus, a robust optimization model for minimizing the total service time of vessels and maximizing the buffer time is developed. Considering that the schedule is infeasible when a vessel is seriously delayed, a reactive model is built to minimize adjustment costs. According to the characteristics of the problem, this article combined local search with the genetic algorithm and proposed an improved genetic algorithm (IGA). Numerical experiments validate the efficiency of the proposed algorithm with CPLEX and Squeaky Wheel Optimization (SWO) in different delay conditions and problem scales. An in-depth analysis presents some management insights on the coefficient setting, uncertainty, and buffer time.

Multi-Port Berth Allocation and Time-Invariant Quay Crane Assignment Problem With Speed Optimization

Multi-Port Berth Allocation and Time-Invariant Quay Crane Assignment Problem With Speed Optimization

The berth allocation and quay crane assignment problem with crane travel and setup times

In this paper, we propose a new approach for including quay crane travel and setup times in the berth allocation and quay crane assignment problem. We first develop a new mixed integer linear programming model (MILP) for the problem without setups (BACASP), in which berthing positions and times are considered as continuous variables. Several groups of valid inequalities are also set forth. Then, for the BACASP with crane travel and setup times, which we denote as BACASP-S, we propose two MILPs: the first is based on the previous BACASP formulation and the second on routing formulations. Due to the complexity of the BACASP-S, we also propose a genetic algorithm and an exact approach which combines various MILPs with the genetic algorithm. All methods and valid inequalities are computationally tested over two different sets of randomly generated instances. According to the results, the models and algorithms can optimally solve, in less than one hour, BACASP-S instances of up to 40 vessels within a quay one kilometer long and a time horizon of one week. Additionally, extensive experiments were conducted on a new large set of instances to assess the effect of various BACASP-S input parameters on the computation effort required to solve the problem. Ceteris paribus, the computational effort required seems to increase with decreasing number of cranes, while vessel processing times and crane setup times seem not to affect it.

Robust optimization for the integrated berth allocation and quay crane assignment problem

AbstractThis paper studies the berth allocation and quay crane assignment problem (denoted by BACAP) under uncertainty. We assume that the ships' arrival and operation time is uncertain in this problem. We merge the proactive and reactive strategies to address the two‐stage robust optimization (denoted by RO) model for the BACAP to obtain a complete schedule with robustness. We obtain the berth allocation and quay crane assignment with a proactive strategy in the first stage. In the second stage, we formulate a rescheduling model with a reactive strategy considering the sensitivity towards the change in the complete schedule. The second stage model is based on the prospect theory, a quantitative way to describe the stakeholders' perception, including the port managers and shipowners, of the deviation from the baseline plan. The two stages are iterated until a favorable schedule with high robustness is found. To illustrate the superiority of the two‐stage robust optimization model with the prospect theory for the complete schedule, we give an intuitive example to compare the performance among the related models. The two‐stage RO model with the prospect theory for the complete schedule can generate a lower cost and higher robustness schedule. As for the solution methods, the column and constraint generation (denoted by C&CG) algorithm is applied to obtain the exact solution for the two‐stage RO model. Moreover, we propose the scenario‐constrained C&CG (denoted by SC) algorithm, which can reduce constraints and variables for the master problem to accelerate the solving process of the two‐stage RO model. In addition, the optimality of the SC algorithm is verified by analyzing the pattern of the occurrence of the worst‐case scenarios. Besides, to tackle the large‐scale instances, we propose the schedule‐fixed (denoted by SF) algorithm, in which the results of the previous iterations are treated as fixed. The SF algorithm can increase computing efficiency with a small gap compared to the optimal solution value. Furthermore, extensive numerical experiments are conducted on both real‐life instances and randomly generated instances to verify the superiority and generality of our model and algorithms.

A new multi-objective model for berth allocation and quay crane assignment problem with speed optimization and air emission considerations (A case study of Rajaee Port in Iran)

A new multi-objective model for berth allocation and quay crane assignment problem with speed optimization and air emission considerations (A case study of Rajaee Port in Iran)

An Enhanced NSGA-II for Solving Berth Allocation and Quay Crane Assignment Problem With Stochastic Arrival Times

The berth allocation and quay crane assignment problem (BACAP) is an important port operation planning problem. To obtain an effective and reliable schedule of berth and quay crane (QC), this study addresses the BACAP with stochastic arrival times of vessels. An efficient method combining scenario generation is presented to simulate the stochastic arrival times. After then, a mixed integer linear programming (MILP) model is established, aiming to minimize the expectation of the vessels’ total stay time in port. A multi-objective constraint-handling (MOCH) strategy is adopted to reformulate the developed model, which converts constraint violations into an objective, thus transforming the single-objective optimization model with complex constraints into a dual-objective optimization model with only easy-handling constraints. Then an enhanced non-dominated sorting genetic algorithm II (ENSGA-II) is proposed to solve the dual-objective model, in which a neighborhood search algorithm and a search bias mechanism are incorporated to strengthen the local exploitation capability. Furthermore, a repair method (RM), penalty function (PF) and the superiority of feasible solutions (SF) strategy for constraint handling are designed respectively and incorporated with genetic algorithm to solve the original single-objective optimization model. Finally, numerical experiments on instances in the literature are conducted to validate the effectiveness of the MOCH and the proposed ENSGA-II. The results show that the average total stay time of vessels is reduced when stochastic arrival times are considered. Comparison results with another two multi-objective methods and three single-objective methods combined with different constraint-handling strategies corroborate the superiority of the proposed ENSGA-II and MOCH.

Effect of Service Priority on the Integrated Continuous Berth Allocation and Quay Crane Assignment Problem after Port Congestion

Container seaport congestion is a challenging problem in improving the service level and optimizing evacuating container vessels after congestion. There is a lack of research on container vessel evacuation strategies for continuous terminals. In this article, the weight of the objective function is regarded as the index for the service priority of vessels. The effects of the service priority on the continuous terminal are analyzed by establishing a mixed integer programming model. The model minimizes the total weighted delay departure time of vessels. Two sets of weight values are adopted, including handling volume of each ship and the squared handling volume, then the optimization results are compared with the unweighted scenario. The model is solved using a genetic algorithm. Lianyungang Port is selected as a case study. The results show that the method using the square of handled container volume is more conducive to ensuring the shipping period of large vessels after congestion. Besides, the quay crane number of large vessels affecting the scheduling strategy is discussed. The method proposed in this article provides a new idea for arranging scheduling strategies in other ports under congestion situations, which can better ensure the planned shipping period of large vessels.

Rolling horizon based robust optimization method of quayside operations in maritime container ports

The quayside operation is a complex system in maritime shipping, which requires an intelligent scheduling approach to cope with complex situations. This paper investigates joint berth allocation and quay crane assignment problem considering uncertain arrival time of vessels and operation efficiency of quay cranes. A nominal model of this problem considering vessel priorities is presented firstly. Meanwhile, a multi-objective robust model is built based on the idea of robust optimization. Besides, control parameters are introduced into the model for levels of uncertainties. Then, an intelligent search algorithm integrated with rolling horizon is designed for solving the problem. Finally, the model and proposed approach are validated by computational results with different instance sizes. Results point out that intelligent algorithm is superior on solution quality and solving time for large-scale examples. Besides, vessel delays show obvious impact on the scheme than uncertain quay crane productivity, especially under the medium uncertainty level.

Discretization-Strategy-Based Solution for Berth Allocation and Quay Crane Assignment Problem

The continuous berth allocation and quay crane assignment problem considers the size of berths and ships, the number of quay cranes, the dynamic ships and non-crossing constraints of quay cranes. In this work, a mixed-integer linear programming model of this problem is established, aiming at minimizing the total stay time and delay penalty of ships. To solve the model, the continuous berth is separated into discrete segments via a proposed discretization strategy. Thereafter, a large neighborhood search algorithm composed of the random removal operator and relaxed sorting-based insertion operator and a backtracking comparison-based constraint repair strategy are proposed. The effectiveness of the model and algorithm presented is verified via real-life instances with different characteristics, and the performances of different combinations of removal operators and insertion operators in the large neighborhood search algorithmic framework are analyzed. Numerical results show that the large neighborhood search algorithm can optimally solve the small-scale instances in a reasonable time. Meanwhile, the results of large-scale instances show that the large neighborhood search algorithm incorporating the discretization strategy is more efficient than other genetic algorithms based on continuous optimization. With the proposed approach, high-quality berth and quay crane allocation results can be obtained efficiently.

Collaboration of vessel speed optimization with berth allocation and quay crane assignment considering vessel service differentiation

Green shipping and efficient information sharing deepen the collaboration between shipping companies and ports, enabling vessel speed optimization (VSO) to be integrated with the berth allocation and quay crane assignment problem (BACAP). However, the literature presented to date usually considers all vessels as a whole and only addresses the vertical conflict between the two levels of the BACAP and the VSO but neglects the horizontal conflict among vessels in each level of the collaborative planning. To describe and address the conflicts within and between levels, we consider the vessel service differentiation (VSD) and propose a BACAP-VSO with VSD (BACAP-VSO-w/-VSD) collaboration mode. To this end, we set up a bi-level multi-objective optimization model. In the upper-level BACAP model, the VSD is implemented by separating vessels into different preferential groups. Multi-objectives are formulated to minimize the service delays of vessels in preferential groups and all vessels. In the lower-level VSO model, the VSD is realized by grouping vessels based on their shipping companies. Multi-objectives are solved to minimize the fuel consumption costs of each shipping company. We develop a nested genetic algorithm incorporated with an inheritance factor and jumping operations for the solution approach. Experimental results show that the proposed collaboration mode can analyze trade-offs among multiple conflicting objectives in any and between both planning levels. Compared to the existing collaboration mode, the proposed collaboration mode further reduces service delays of vessels and fuel consumption costs of shipping companies, improves customer satisfaction, and effectively reduces vessel emissions.

Exact and heuristic methods for the integrated berth allocation and specific time-invariant quay crane assignment problems

The Berth Allocation Problem and the Quay Cranes Assignment Problem are two decisive seaside operations planning problems in maritime terminals. Therefore, integrating these two problems is critical in improving any terminal's performance and enhancing its attractivity. This paper deals with the integrated problem for a continuous quay's topology and dynamic vessels' arrivals. In addition to a berth plan respecting the quay cranes' capacity constraints, the specific time-invariant quay cranes' assignments must also be determined. A novel mixed-integer linear formulation (MILP) and a Variable Neighborhood Search (VNS) approach are presented. The developed approaches are tested on randomly generated instances. In particular, the MILP formulation is compared to existing exact methods. The numerical experiments showed that the MILP formulation provides state-of-the-art results. Furthermore, the VNS approach presents a consistent alternative for the MILP, especially for large-scale instances.

The integrated quay crane assignment and scheduling problems with carbon emissions considerations

The continuous worldwide growth in container terminals’ traffic has pushed further the need for innovative research in ports’ operations. The quay crane assignment and scheduling problems are the prominent focus of such research but lack, to date, carbon emission considerations despite forecasts predicting that emissions from maritime transportation will exponentially increase in the absence of regulatory policies. The objective of this paper aims at filling this gap by solving the integrated quay crane assignment and scheduling problem with consideration of carbon regulatory policies. Two mathematical models of the problem are formulated using mixed integer programming (MIP), to capture the effects of carbon taxation and carbon cap-and-trade, respectively. The sensitivity analysis conducted shows that the primary focus of a port should be minimizing the berthing time of vessels which is highly impacted by the number of QCs assigned to each vessel.

An extended formulation of moldable task scheduling problem and its application to quay crane assignments

In this paper, we study an extended formulation of moldable task scheduling problem (MTSP) motivated by the assignments of quay cranes to vessels. In container terminals, handling time of a vessel depends on the number of quay cranes assigned to that vessel. This characteristic allows us to model quay crane assignment problem (QCAP) as a variant of MTSP. By considering the modeling requirements of various properties of QCAP, we develop an extended formulation of MTSP with specific task to machine assignments. Even though this formulation brings modeling flexibility, it can only be solved for small instances because of its size. For this reason, we provide a generic solution algorithm based on a logic based Benders decomposition by utilizing the extended formulation. There are various characteristics of QCAP observed in different terminals. Accordingly, we implement the proposed decomposition algorithm for contiguous assignments of QCs, uniform QCs as well as the availability of QCs. Computational experiments show that the proposed algorithm is able to solve instances of considerable sizes to optimality and provides a modeling flexibility that allows implementation to different terminal settings.

An almost robust optimization model for integrated berth allocation and quay crane assignment problem

The integrated berth allocation and quay crane assignment problem is an important issue for the operations management in container terminals. This issue primarily considers the assignment of berthing time, position, and the number of quay cranes in each time segment to ships that must be discharged and loaded at terminals. This study examines such a problem by considering uncertainties in the late arrival of ships and inflation of container quantity. Based on historical data, we first divide the uncertainty set into K non-overlapping full-dimensional clusters via K-means clustering, and the weight of each cluster is calculated. Then, we formulate an almost robust model by introducing the weighted max penalty function with the objective of minimizing the total cost, which is caused by the deviations from the expected berthing location and departure time. The concept of robustness index is introduced to investigate the trade-off between the changes in the objective value and the penalty violation. A decomposition method, which contains a deterministic master problem and a stochastic subproblem, is proposed to solve the problem. In each iteration, the subproblem checks the master problem under different realizations, adds scenarios, and cuts into the master problem if needed. Numerical experiments demonstrate that (i) the proposed method can solve the model efficiently, (ii) the robustness index shows that a significant improvement in objective can be achieved at the expense of a small amount of penalty, (iii) the proposed model can handle uncertainties better than the deterministic, fully robust, and worst-case models in terms of total expected cost, total vessel delays, and utilization rates of the berth and quay crane, and (iv) the proposed method becomes more attractive compared with the first come first served approach as the congestion situation or uncertainty degree increase.

A Solution Approach to the Daily Dockworker Planning Problem at a Port Container Terminal

This study focused on vital resources at port container terminals such as quay cranes and dockworkers. We studied the impact of incorporating the dockworker assignment problem (DWAP) into the quay crane assignment problem (QCAP). The aim of this study was to formulate and solve an integrated model for QCAP and DWAP, with the objective of minimizing the total costs of dockworkers, by optimizing workers’ assignment, so that the ships’ costs due to the time spent in the port are not increased. We proposed an integrated solution approach to the studied problem. Our proposed model has been validated on an adequate number of instances based on the real data. Obtained solutions were compared with the solutions obtained by the traditional sequential approach. It was demonstrated that, for all solved instances, our proposed integrated approach resulted in a reduction in the total costs of dockworkers. The major contribution of this study is that this is the first time that these two problems were modeled together. The obtained results show significant savings in the overall costs.

Bi-objective optimization model for integrated planning in container terminal operations

In this paper, we studied an integrated planning optimization model in container operations consisting of tactical and operational level decisions. At the tactical level decision, integration planning includes berth allocation problem, quay crane assignment problem and yard allocation planning. These tactical level decisions are integrated with quay crane scheduling which is an operational level decision at the container terminal. Integration of planning in container terminals is needed because the decision on berth allocation, namely berthing position and berthing time in seaport, is highly dependent on the decision of reserved sub-blocks positions in container yard allocation, the number of quay cranes assigned to each vessel and its scheduling on each vessel bays. This integrated planning decision optimization has two objective functions; namely: (i) minimizing total operational cost, and (ii) maximizing service level. Numerical experiments have been carried out to obtain total cost efficiency and maximum service level resulting from the proposed optimization model.

Mathematical model for quay crane assignment problem with tidal constraints

Objetive: The operations associated with loading and unloading of container ships demand the use of quay cranes that represent one of the most expensive resources of a maritime terminal. Therefore, the assignment of cranes to ships must be optimized. This article proposes a mathematical model to optimize the decision of assignment of cranes to ships, considering the behavior of tides, which is a not commonly considered factor in the scientific literature for similar problems. Metodology: A mixed integer linear mathematical model was designed and tested for the actual case of a container terminal in Buenaventura-Colombia with satisfactory results. Results: With the available capacity the model allows to mobilize up to 2800 containers per half day, while the number of containers per ship in the real case does not exceed 2000 units. Conclusions: The consideration of tides, combined with downtime penalty cost can allow using smaller number of cranes with savings in energy cost.

Integrated proactive and reactive strategies for sustainable berth allocation and quay crane assignment under uncertainty

The berth allocation and quay crane assignment problem (BACAP) is a complex port operation planning problem susceptible to uncertainties, such as vessel arrival time fluctuation to its estimated time of arrival and maritime markets. For promoting reliability and sustainability of container terminals, this paper addresses the optimization of BACAP under the uncertain vessels’ arrival times and fluctuation of loading and unloading volumes. We propose a proactive BACAP strategy considering minimum recovery cost under uncertainty using a reactive strategy. A stochastic programming model is formulated to minimize the basic cost in the baseline schedule, and the recovery cost in real uncertain scenarios. A two-stage meta-heuristic framework based on GA is developed for solving this problem. Numerical experiments and scenario analysis are conducted to validate the effectiveness of the proposed model and the proposed solution approaches.