A local branching-based algorithm for the quay crane scheduling problem under unidirectional schedules

The quay crane scheduling problem (QCSP) is at the basis of a major logistic process in maritime container terminals: the process of discharging/loading containers from/on berthed vessels. Several groups of containers, laying in one or more stowage portions of a containership, have to be assigned to multiple cranes and discharge/loading operations have to be optimally sequenced, under some complicating constraints imposed by the practical working rules of quay cranes. The QCSP has been the object of a great deal of research work since the last decade and it is focused in this paper, with the aim of consolidating a promising solution approach based upon the combination of specialized branch & bound (B&B) and heuristic algorithms. A cost-effective solution technique that incorporates the local branching method within a refined B&B algorithm is proposed and its effectiveness is assessed by numerical comparisons against the latest algorithm available in literature.     

Yard crane scheduling in port container terminals

Yard cranes are the most popular container handling equipment for loading containers onto or unloading containers from trucks in container yards of land scarce port container terminals. However, such equipment is bulky, and very often generates bottlenecks in the container flow in a terminal because of their slow operations. Hence, it is essential to develop good yard crane work schedules to ensure a high terminal throughput. This paper studies the problem of scheduling a yard crane to perform a given set of loading/unloading jobs with different ready times. The objective is to minimize the sum of job waiting times. A branch and bound algorithm is proposed to solve the scheduling problem optimally. Efficient and effective algorithms are proposed to find lower bounds and upper bounds. The performance of the proposed branch and bound algorithm is evaluated by a set of test problems generated based on real life data. The results show that the algorithm can find the optimal sequence for most problems of realistic sizes.     

Minimizing total tardiness on a two-machine re-entrant flowshop

We present a branch and bound algorithm for a two-machine re-entrant flowshop scheduling problem with the objective of minimizing total tardiness. In the re-entrant flowshop considered here, all jobs must be processed twice on each machine, that is, each job should be processed on machine 1, machine 2 and then machine 1 and machine 2. By regarding a job as a pair of sub-jobs, each of which represents a pass through the two machines, we develop dominance properties, a lower bound and heuristic algorithms for the problem, and use these to develop a branch and bound algorithm. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems and results are reported. Results of the experiments show that the suggested branch and bound algorithm can solve problems with up to 20 sub-jobs in a reasonable amount of CPU time, and the average percentage gap of the heuristic solutions is about 13%.     

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。     

A heuristic for scheduling in a two-stage hybrid flowshop with renewable resources shared among the stages

In this paper we propose a heuristic for solving the problem of resource constrained preemptive scheduling in the two-stage flowshop with one machine at the first stage and parallel unrelated machines at the second stage, where renewable resources are shared among the stages, so some quantities of the same resource can be used at different stages at the same time. Availability of every resource at any moment is limited and resource requirements of jobs are arbitrary. The objective is minimization of makespan. The problem is NP-hard. The heuristic first sequences jobs on the machine at stage 1 and then solves the preemptive scheduling problem at stage 2. Priority rules which depend on processing times and resource requirements of jobs are proposed for sequencing jobs at stage 1. A column generation algorithm which involves linear programming, a tabu search algorithm and a greedy procedure is proposed to minimize the makespan at stage 2. A lower bound on the optimal makespan in which sharing of the resources between the stages is taken into account is also derived. The performance of the heuristic evaluated experimentally by comparing the solutions to the lower bound is satisfactory.     

A hybrid two-stage transportation and batch scheduling problem

We study the coordinated scheduling problem of hybrid batch production on a single batching machine and two-stage transportation connecting the production, where there is a crane available in the first-stage transportation that transports jobs from the warehouse to the machine and there is a vehicle available in the second-stage transportation to deliver jobs from the machine to the customer. As the job to be carried out is big and heavy in the steel industry, it is reasonable assumed that both the crane and the vehicle have unit capacity. The batching machine processes a batch of jobs simultaneously. Each batch occur a setup cost. The objective is to minimize the sum of the makespan and the total setup cost. We prove that this problem is strongly NP-hard. A polynomial time algorithm is proposed for a case where the job transportation times are identical on the crane or the vehicle. An efficient heuristic algorithm for the general problem is constructed and its tight worst-case bound is analyzed. In order to further verify the performance of the proposed heuristics, we develop a lower bound on the optimal objective function. Computational experiments show that the heuristic algorithm performs well on randomly generated problem instances.     

考虑任务优先约束的同类岸桥作业调度优化

分析了以箱组为任务对象QCSP与以整贝为任务对象QCSP的异同,指出前者更能均衡各岸桥作业负荷,并减少船舶装卸作业时间。考虑到岸桥具有作业效率差异的特点,将其视为同类平行机调度问题,同时结合任务优先约束、岸桥作业不可相互穿越与安全距离等特有约束,建立了更加符合实际的以箱组为任务对象的岸桥作业调度混合整数规划模型,其优化目标是最小化装卸作业的makespan。针对模型求解的复杂度,设计了一种遗传算法,对算法搜索空间进行了讨论,并推导了问题的低界。实验算例表明所建立的模型能够反映岸桥作业调度过程中作业效率差异及任务优先约束现象,其算法能够在允许的运算时间内获得稳定的满意解,并且优化结果要全面优于以整贝为任务对象QCSP的调度方案。     

A branch and bound algorithm to minimize total tardiness of jobs in a two identical-parallel-machine scheduling problem with a machine availability constraint

This research focuses on the problem of scheduling jobs on two identical parallel machines that are not continuously available with the objective of minimizing total tardiness. After processing a given number of jobs, each machine requires a preventive maintenance task, during which the machine cannot process jobs. We present dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as an upper bound obtained from a heuristic algorithm. Performance of the algorithm is evaluated through a series of computational experiments on randomly generated instances and results are reported.     

Crane scheduling in container yards with inter-crane interference

This paper examines the problem of scheduling multiple yard cranes to perform a given set of jobs with different ready times in a yard zone with only one bi-directional travelling lane. Due to sharing of the travelling lane among two or more yard cranes, inter-crane interference, a planned move of a yard crane blocked by the other yard cranes, may happen. The scheduling problem is formulated as an integer program. It is noted that the scheduling problem is NP-complete. This research develops a dynamic programming-based heuristic to solve the scheduling problem and an algorithm to find lower bounds for benchmarking the schedules found by the heuristic. Computational experiments are carried out to evaluate the performance of the heuristic and the results show that the heuristic can indeed find effective solutions for the scheduling problem, with the heuristic solutions on average 7.3% above their lower bounds.     

A note on makespan minimization in two-stage flexible flow shops with uniform machines

We consider the two-stage flexible flow shop makespan minimization problem with uniform parallel machines. Soewandi and Elmaghraby [Soewandi, H., Elmaghraby, S., 2003. Sequencing on two-stage hybrid flowshops with uniform machines to minimize makespan. IIE Transaction 35, 467–477] developed a heuristic (S–E) and derived a machine speed-dependent worst-case ratio bound for it. We point out that this bound works well when the uniform machines have approximately equal speeds but is not indicative of the performance of the S–E heuristic when the machine speeds are in a wide range. Motivated by this observation, we propose an alternative tight machine-speed dependent worst-case bound for the S–E heuristic that works well when the machine speeds vary significantly. We then combine the two speed-dependent ratio bounds into a speed-independent bound. Our findings facilitate the narrowing of the gap between experimental performance and worst-case bound for the S–E heuristic.     

需要安装时间的平行多功能机排序问题的启发式算法

考虑需要安装时间的平行多功能机排序问题。在该模型中,每个工件对应机器集合的一个子集,其只能在这个子集中的任一台机器上加工,称这个子集为该工件的加工集合;工件分组,同组工件具有相同的加工时间和加工集合,不同组中的工件在同一台机器上连续加工需要安装时间,目标函数为极小化最大完工时间。对该问题NP-难的一般情况设计启发式算法:首先按照特定的规则将所有工件组都整组地安排到各台机器上,然后通过在各机器间转移工件不断改进当前最大完工时间。通过与下界的比较检验算法的性能,大量的计算实验表明,算法是实用而有效的。     

Performance of scheduling algorithms for no-wait flowshops with parallel machines

We study the performance of scheduling algorithms for a manufacturing system, called the ‘no-wait flowshop’, which consists of a certain number of machine centers. Each center has one or more identical parallel machines. Each job is processed by at most one machine in each center. The problem of finding the minimum finish time schedule is considered here in a flowshop consisting of two machine centers. Heuristic algorithms are presented and are analyzed in the worst case performance context. For the case of two centers, one with a single machine and the other with m, two heuristics are presented with tight performance guarantees of 3 − (1/m) and 2. When both centers have m machines, a heuristic is presented with an upper bound performance guarantee of . It is also shown that this bound can be reduced to 2(1 + ε). For the flowshop with any number of machines in each center, we provide a heuristic algorithm with an upper bound performance guarantee that depends on the relative number of machines in the centers.     

Two-Stage,Hybrid Flowshop Scheduling Problem

This paper describes the two-stage flowshop problem when there are identical multiple machines at each stage, and shows that the problem is NP-complete. An efficient heuristic algorithm is developed for finding an approximate solution of a special case when there is only one machine at stage 2. The effectiveness of the proposed heuristic algorithm in finding a minimum makespan schedule is empirically evaluated and found to increase with the increase in the number of jobs.     

An algorithm for single machine sequencing with deadlines to minimize total weighted completion time

Each of n jobs is to be processed without interruption on a single machine. Each job becomes available for processing at time zero, has a deadline by which it must be completed and has a positive weight. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times. In this paper a branch and bound algorithm for the problem is presented which incorporates lower bounds that are obtained using a new technique called the multiplier adjustment method. Firstly several dominance conditions are derived. Then a heuristic is described and sufficient conditions for its optimality are given. The lower bound is obtained by performing a Lagrangean relaxation of the deadline constraints; the Lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimal solution of the relaxed problem, thus yielding a lower bound. The algorithm is tested on problems with up to fifty jobs.     

An algorithm for single machine sequencing with release dates to minimize total weighted completion time

Each of n jobs is to be processed without interruption on a single machine which can handle only one job at a time. Each job becomes available for processing at its release date, requires a processing time and has a positive weight. Given a processing order of the jobs, the earliest completion time for each job can be computed. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times. In this paper a branch and bound algorithm for the problem is derived. Firstly a heuristic is presented which is used in calculating the lower bound. Then the lower bound is obtained by performing a Lagrangean relaxation of the release date constraints; the Lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimum solution of the relaxed problem thus yielding a lower bound. A method to increase the lower bound by deriving improved constraints to replace the original release date constraints is given. The algorithm, which includes several dominance rules, is tested on problems with up to fifty jobs. The computational results indicate that the version of the lower bound using improved constraints is superior to the original version.     

考虑成本的最大延迟时间同类机调度问题

假定生产时机器成本是固定的,研究了一类考虑成本的同类机调度问题,调度的目标是在给定加工完所有作业的总预算的成本限制下最小化最大作业延迟时间。为该类问题构建了混合整数规划模型。通过设计相关规则在机器成本预算内来选择加工机器,以及对传统的LPT(最长加工时间优先)、ECT(最早完工时间优先)、EDD(最早工期优先)等算法进行改进,提出了一个启发式算法H,并理论证明了该算法在同型机和同类机下的最坏误差界。通过算例说明了算法的执行情况,同时也考虑了给定总预算不同的多种情形,采用大量随机数据实验验证了算法的有效性。     

Scheduling a hybrid assembly-differentiation flowshop to minimize total flow time

This study considers a hybrid assembly-differentiation flowshop scheduling problem (HADFSP), in which there are three production stages, including components manufacturing, assembly, and differentiation. All the components of a job are processed on different machines at the first stage. Subsequently, they are assembled together on a common single machine at the second stage. At the third stage, each job of a particular type is processed on a dedicated machine. The objective is to find a job schedule to minimize total flow time (TFT). At first, a mixed integer programming (MIP) model is formulated and then some properties of the optimal solution are presented. Since the NP-hardness of the problem, two fast heuristics (SPT-based heuristic and NEH-based heuristic) and three hybrid meta-heuristics (HGA-VNS, HDDE-VNS and HEDA-VNS) are developed for solving medium- and large-size problems. In order to evaluate the performances of the proposed algorithms, a lower bound for the HADFSP with TFT criteria (HADFSP-TFT) is established. The MIP model and the proposed algorithms are compared on randomly generated problems. Computational results show the effectiveness of the MIP model and the proposed algorithms. The computational analysis indicates that, in average, the HDDE-VNS performs better and more robustly than the other two meta-heuristics, whereas the NEH heuristic consume little time and could reach reasonable solutions.     

Scheduling crossover cranes at container terminals during seaside peak times

This paper deals with crane scheduling in a block of a container terminal equipped with two crossover cranes. We concentrate on the situation when all requests in this block refer to the seaside. This seaside peak time occurs when a container ship is berthed and has to be served as quickly as possible because berthing time is a major cost driver for both, the shipping company and the port operator. In this case, the two cranes highly interfere. We formulate this problem as a MIP, prove its \(\mathcal {NP}\)-hardness and introduce a lower bound. Further, we provide several heuristic solution procedures based on priority rules. A comparison of all procedures concludes this paper and gives insights on how to solve this problem in practice.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

A branch and bound algorithm for the two-stage assembly scheduling problem

This paper develops a branch and bound algorithm for the two-stage assembly scheduling problem. In this problem, there are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule the jobs on the machines so that the maximum completion time, or makespan, is minimized. A lower bound based on solving an artificial two-machine flow shop problem is derived. Also, several dominance theorems are established and incorporated into the branch and bound algorithm. Computational experience with the algorithm is reported for problems with up to 8000 jobs and 10 first-stage machines.