A survey of berth allocation and quay crane scheduling problems in container terminals

Due to the variety of technical equipments and terminal layouts, research has produced a multitude of optimization models for seaside operations planning in container terminals. To provide a support in modeling problem characteristics and in suggesting applicable algorithms this paper reviews the relevant literature. For this purpose new classification schemes for berth allocation problems and quay crane scheduling problems are developed. Particular focus is put on integrated solution approaches which receive increasing importance for the terminal management.     

A proactive approach for simultaneous berth and quay crane scheduling problem with stochastic arrival and handling time

For a container terminal system, efficient berth and quay crane (QC) schedules have great impact on the improvement of both operation efficiency and customer satisfaction. In this paper we address berth and quay crane scheduling problems in a simultaneous way, with uncertainties of vessel arrival time and container handling time. The berths are of discrete type and vessels arrive dynamically with different service priorities. QCs are allowed to move to other berths before finishing processing on currently assigned vessels, adding more flexibility to the terminal system. A mixed integer programming model is proposed, and a simulation based Genetic Algorithm (GA) search procedure is applied to generate robust berth and QC schedule proactively. Computational experiment shows the satisfied performance of our developed algorithm under uncertainty.     

Optimal berth allocation and time-invariant quay crane assignment in container terminals

Due to the dramatic increase in the world’s container traffic, the efficient management of operations in seaport container terminals has become a crucial issue. In this work, we focus on the integrated planning of the following problems faced at container terminals: berth allocation, quay crane assignment (number), and quay crane assignment (specific). First, we formulate a new binary integer linear program for the integrated solution of the berth allocation and quay crane assignment (number) problems called BACAP. Then we extend it by incorporating the quay crane assignment (specific) problem as well, which is named BACASP. Computational experiments performed on problem instances of various sizes indicate that the model for BACAP is very efficient and even large instances up to 60 vessels can be solved to optimality. Unfortunately, this is not the case for BACASP. Therefore, to be able to solve large instances, we present a necessary and sufficient condition for generating an optimal solution of BACASP from an optimal solution of BACAP using a post-processing algorithm. In case this condition is not satisfied, we make use of a cutting plane algorithm which solves BACAP repeatedly by adding cuts generated from the optimal solutions until the aforementioned condition holds. This method proves to be viable and enables us to solve large BACASP instances as well. To the best of our knowledge, these are the largest instances that can be solved to optimality for this difficult problem, which makes our work applicable to realistic problems.     

A branch-and-price algorithm to solve the integrated berth allocation and yard assignment problem in bulk ports

In this research, two crucial optimization problems of berth allocation and yard assignment in the context of bulk ports are studied. We discuss how these problems are interrelated and can be combined and solved as a single large scale optimization problem. More importantly we highlight the differences in operations between bulk ports and container terminals which highlights the need to devise specific solutions for bulk ports. The objective is to minimize the total service time of vessels berthing at the port. We propose an exact solution algorithm based on a branch and price framework to solve the integrated problem. In the proposed model, the master problem is formulated as a set-partitioning problem, and subproblems to identify columns with negative reduced costs are solved using mixed integer programming. To obtain sub-optimal solutions quickly, a metaheuristic approach based on critical-shaking neighborhood search is presented. The proposed algorithms are tested and validated through numerical experiments based on instances inspired from real bulk port data. The results indicate that the algorithms can be successfully used to solve instances containing up to 40 vessels within reasonable computational time.     

具有预知信息的集装箱码头泊位与岸桥联合调度在线模型

探讨了预知服务需求信息能力下的集装箱码头泊位与岸桥联合调度 over-list 在线模型. 在每个船舶服务请求释放时, 决策者预知后续 k(k \geq 2)个请求的信息,目标为最小化所有请求的最大完工时间. 针对由3个离散泊位组成的混合型泊位与4个岸桥, 以及只有大小两种服务请求的情形, 给出了预知任意 k \geq 2个请求下的竞争比下界; 同时, 对于k=2的特定情形, 给出了具有最优竞争比7/6 的在线策略. 数值实验进一步表明了所设计策略的良好执行性能.     

Modeling and solution of the joint quay crane and truck scheduling problem

This paper addresses the joint quay crane and truck scheduling problem at a container terminal, considering the coordination of the two types of equipment to reduce their idle time between performing two successive tasks. For the unidirectional flow problem with only inbound containers, in which trucks go back to quayside without carrying outbound containers, a mixed-integer linear programming model is formulated to minimize the makespan. Several valid inequalities and a property of the optimal solutions for the problem are derived, and two lower bounds are obtained. An improved Particle Swarm Optimization (PSO) algorithm is then developed to solve this problem, in which a new velocity updating strategy is incorporated to improve the solution quality. For small sized problems, we have compared the solutions of the proposed PSO with the optimal solutions obtained by solving the model using the CPLEX software. The solutions of the proposed PSO for large sized problems are compared to the two lower bounds because CPLEX could not solve the problem optimally in reasonable time. For the more general situation considering both inbound and outbound containers, trucks may go back to quayside with outbound containers. The model is extended to handle this problem with bidirectional flow. Experiment shows that the improved PSO proposed in this paper is efficient to solve the joint quay crane and truck scheduling problem.     

Solution approaches for the soft drink integrated production lot sizing and scheduling problem

In this paper we present a mixed integer programming model that integrates production lot sizing and scheduling decisions of beverage plants with sequence-dependent setup costs and times. The model considers that the industrial process produces soft drink bottles in different flavours and sizes, and it is carried out in two production stages: liquid preparation (stage I) and bottling (stage II). The model also takes into account that the production bottleneck may alternate between stages I and II, and a synchronisation of the production between these stages is required. A relaxation approach and several strategies of the relax-and-fix heuristic are proposed to solve the model. Computational tests with instances generated based on real data from a Brazilian soft drink plant are also presented. The results show that the solution approaches are capable of producing better solutions than those used by the company.     

集装箱码头桥机调度问题基于完工时间下界的算法

关注单船桥机调度问题,指出了单船桥机的闲置会影响码头整体的运作效率。以单个集装箱为任务单位,考虑桥机移动时间、安全距离等约束,建立了最小化桥机完工时间和闲置时间的多目标规划模型。基于完工时间下界的两种不同情况：以重点贝位工作量确定和以平均工作量确定,分别设计了基于邻域搜索的启发式算法和基于贪心策略的“分割贝位”算法,并且证明了在以平均工作量确定下界的情况中该算法不会导致桥机闲置。不同规模、不同下界类型的算例表明：提出的模型与算法得到的桥机调度计划更适合实际生产作业,能够有效地逼近完工时间下界,算法运行速度较现有的研究有显著的提高。     

An effective mathematical formulation for the unidirectional cluster-based quay crane scheduling problem

The quay crane scheduling problem plays an important role in the paradigm of port container terminal management, due to the fact that it closely relates to vessel berthing time. In this paper, we focus on the study of a special strategy for the cluster-based quay crane scheduling problem that forces quay cranes to move unidirectionally during the scheduling. The scheduling problem arising when this strategy is applied is called the unidirectional quay crane scheduling problem in the literature. Different from other researches attempting to construct more sophisticated searching algorithms, in this paper, we seek for a more compact mathematical formulation of the unidirectional cluster-based quay crane scheduling problem that can be easily solved by a standard optimization solver. To assess the performance of the proposed model, commonly accepted benchmark suites are used and the results indicate that the proposed model outperforms the state-of-the-art algorithms designed for the unidirectional cluster-based quay crane scheduling problem.     

Models and algorithms for a staff scheduling problem

We present mathematical models and solution algorithms for a family of staff scheduling problems arising in real life applications. In these problems, the daily assignments to be performed are given and the durations (in days) of the working and rest periods for each employee in the planning horizon are specified in advance, whereas the sequence in which these working and rest periods occur, as well as the daily assignment for each working period, have to be determined. The main objective is the minimization of the number of employees needed to perform all daily assignments in the horizon. We decompose the problem into two steps: the definition of the sequence of working and rest periods (called pattern) for each employee, and the definition of the daily assignment to be performed in each working period by each employee. The first step is formulated as a covering problem for which we present alternative ILP models and exact enumerative algorithms based on these models. Practical experience shows that the best approach is based on the model in which variables are associated with feasible patterns and generated either by dynamic programming or by solving another ILP. The second step is stated as a feasibility problem solved heuristically through a sequence of transportation problems. Although in general this procedure may not find a solution (even if one exists), we present sufficient conditions under which our approach is guaranteed to succeed. We also propose an iterative heuristic algorithm to handle the case in which no feasible solution is found in the second step. We present computational results on real life instances associated with an emergency call center. The proposed approach is able to determine the optimal solution of instances involving up to several hundred employees and a working period of up to 6 months. Mathematics Subject Classification (2000): 90B70, 90C10, 90C27, 90C39, 90C57, 90C59     

A mixed integer programming model for advanced planning and scheduling (APS)

This paper presents a mixed integer programming (MIP) model which succeeds in a system integration of the production planning and shop floor scheduling problems. The proposed advanced planning and scheduling (APS) model explicitly considers capacity constraints, operation sequences, lead times and due dates in a multi-order environment. The objective of the model is to seek the minimum cost of both production idle time and tardiness or earliness penalty of an order. The output of the model is operation schedules with order starting time and finish time. Numerical result shows that the suggested APS model can favorably produce optimal schedules.     

A note on “A mixed integer programming model for advanced planning and scheduling (APS)”

In a recent paper, Chen and Ji [Chen, K., Ji, P., 2007. A mixed integer programming model for advanced planning and scheduling (APS). European Journal of Operational Research 181, 515–522] develop a mixed integer programming model for advanced planning and scheduling problem that considers capacity constraints and precedence relations between the operations. The orders require processing of several operations on eligible machines. The model presented in the above paper works for the case where each operation can be processed on only one machine. However, machine eligibility means that only a subset of machines are capable of processing a job and this subset may include more than one machine. We provide a general model for advanced planning and scheduling problems with machine eligibility. Our model can be used for problems where there are alternative machines that an operation can be assigned to.     

A problem expanding parametric programming method for solving the job shop scheduling problem

A new zero-one integer programming model for the job shop scheduling problem with minimum makespan criterion is presented. The algorithm consists of two parts: (a) a branch and bound parametric linear programming code for solving the job shop problem with fixed completion time; (b) a problem expanding algorithm for finding the optimal completion time. Computational experience for problems having up to thirty-six operations is presented. The largest problem solved was limited by memory space, not computation time. Efforts are under way to improve the efficiency of the algorithm and to reduce its memory requirements.This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047-048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Exact and heuristic solutions of the global supply chain problem with transfer pricing

We examine the example of a multinational corporation that attempts to maximize its global after tax profits by determining the flow of goods, the transfer prices, and the transportation cost allocation between each of its subsidiaries. Vidal and Goetschalckx [Vidal, C.J., Goetschalckx, M., 2001. A global supply chain model with transfer pricing and transportation cost allocation. European Journal of Operational Research 129 (1), 134–158] proposed a bilinear model of this problem and solved it by an Alternate heuristic. We propose a reformulation of this model reducing the number of bilinear terms and accelerating considerably the exact solution. We also present three other solution methods: an implementation of Variable Neighborhood Search (VNS) designed for any bilinear model, an implementation of VNS specifically designed for the problem considered here and an exact method based on a branch and cut algorithm. The solution methods are tested on artificial instances. These results show that our implementation of VNS outperforms the two other heuristics. The exact method found the optimal solution of all small instances and of 26% of medium instances.     

An exact approach for the maximum concurrent k-splittable flow problem

We consider the maximization of a multicommodity flow throughput in presence of constraints on the maximum number of paths to be used. Such an optimization problem is strongly NP-hard, and is known in the literature as the maximum routable demand fraction variant of the k-splittable flow problem. Here we propose an exact approach based on branch and bound rules and on an arc-flow mixed integer programming formulation of the problem. Computational results are provided, and a comparison with a standard commercial solver is proposed.     

A branch and cut algorithm for the hierarchical network design problem

The Hierarchical Network Design Problem consists of locating a minimum cost bi-level network on a graph. The higher level sub-network is a path visiting two or more nodes. The lower level sub-network is a forest connecting the remaining nodes to the path. We optimally solve the problem using an ad hoc branch and cut procedure. Relaxed versions of a base model are solved using an optimization package and, if binary variables have fractional values or if some of the relaxed constraints are violated in the solution, cutting planes are added. Once no more cuts can be added, branch and bound is used. The method for finding valid cutting planes is presented. Finally, we use different available test instances to compare the procedure with the best known published optimal procedure, with good results. In none of the instances we needed to apply branch and bound, but only the cutting planes.     

Hybrid simulated annealing and MIP-based heuristics for stochastic lot-sizing and scheduling problem in capacitated multi-stage production system

This paper addresses lot sizing and scheduling problem of a flow shop system with capacity constraints, sequence-dependent setups, uncertain processing times and uncertain multi-product and multi-period demand. The evolution of the uncertain parameters is modeled by means of probability distributions and chance-constrained programming (CCP) theory. A new mixed-integer programming (MIP) model with big bucket time approach is proposed to formulate the problem. Due to the complexity of problem, two MIP-based heuristics with rolling horizon framework named non-permutation heuristic (NPH) and permutation heuristic (PH) have been performed to solve this model. Also, a hybrid meta-heuristic based on a combination of simulated annealing, firefly algorithm and proposed heuristic for scheduling is developed to solve the problem. Additionally, Taguchi method is conducted to calibrate the parameters of the meta-heuristic and select the optimal levels of the algorithm’s performance influential factors. Computational results on a set of randomly generated instances show the efficiency of the hybrid meta-heuristic against exact solution algorithm and heuristics.     

具有机器适用限制的分布式置换流水车间问题的模型与算法

考虑具有机器适用限制的多个不同置换流水车间的调度问题. 机器适用限制指的是每个工件只能分配到其可加工工厂集合. 所有置换流水车间拥有的机器数相同但是具有不同的加工能力. 首先, 针对该问题建立了基于位置的混合整数线性规划模型; 进而, 对一般情况和三种特殊情况给出了具有较小近似比的多项式时间算法. 其次, 基于NEH方法提出了启发式算法NEHg, 并给出了以NEHg为上界的分支定界算法. 最后, 通过例子说明了NEHg启发式算法和分支定界算法的计算过程, 并进行大量的实验将NEHg与NEH算法结果进行比较, 从而验证了NEHg算法的有效性.     

呼叫中心坐席人员排班问题优化模型与算法研究

国内呼叫企业在保证每个坐席人员周内当值相同班次情况下,通过灵活安排周内当班日期与班次来制定排班方案。针对该实际排班场景,构建问题整数规划模型。通过对问题数据特征及优化性质分析,以及对班次人力有效满足区段电话服务需求的量化指标表征,分别提出两个构造性启发式算法。使用企业实例数据对模型算法进行计算实验。实验结果显示,整数规划模型适合于求解小规模排班问题最优解,而启发式算法能够以小计算成本获得大规模排班问题优化解。最后讨论保证员工上班规律性的同班次用工制度对企业人力成本控制的影响。     

Heuristic approaches for solving transit vehicle scheduling problem with route and fueling time constraints

Electric bus scheduling problem can be defined as vehicle scheduling problem with route and fueling time constraints (VSPRFTC). Every vehicle’s travel miles (route time) after charging is limited, thus the vehicle must be recharged after taking several trips and the minimal charging time (fueling time) must be satisfied. A multiple ant colony algorithm (ACA) was presented to solve VSPRFTC based on ACA used to solve traveling salesman problem (TSP), a new metaheuristic approach inspired by the foraging behavior of real colonies of ants. The VSPRFTC considered in this paper minimizes a multiple, hierarchical objective function: the first objective is to minimize the number of tours (or vehicles) and the second is to minimize the total deadhead time. New improvement of ACA as well as detailed operating steps was provided on the basis of former algorithm. Then in order to settle contradiction between accelerating convergence and avoiding prematurity or stagnation, improvement on route construction rule and Pheromone updating rule was adopted. A group feasible trip sets (blocks) had been produced after the process of applying ACA. In dealing with the fueling time constraint a bipartite graphic model and its optimization algorithm are developed for trip set connecting in a hub and spoke network system to minimize the number of vehicle required. The maximum matching of the bipartite graph is obtained by calculating the maximum inflow with the Ford–Fulkerson algorithm. At last, an example was analyzed to demonstrate the correctness of the application of this algorithm. It proved to be more efficient and robust in solving this problem.