A branch‐and‐bound algorithm for scheduling of new product development projects

In this paper, we consider scheduling problem in a new product development project. Each research and development project consists of a set of activities in which each activity may be executed in several ways. Each way of execution of an activity is a different technology, called an alternative technology, which may fail due to technical risks. In this work, we focus on alternative technologies for scheduling activities to maximize the expected net present value. We assume that the activity payoff is obtained as soon as any single technology is completed successfully. Therefore, we analyze the problem and develop a two‐phase solution procedure, consisting of a branch‐and‐bound algorithm and a recursive search procedure. The efficiency of the proposed algorithm has been evaluated on a set of randomly generated test instances.     

An improved branch‐and‐bound clustering approach for data partitioning

In this paper, we are concerned with clustering algorithms for vertical partitioning. In particular, we examine the use of a branch‐and‐bound scheme. An existing algorithm using such a scheme may produce infeasible solutions to some problems. We adopt the same branch‐and‐bound scheme and develop a new branching strategy to avoid infeasibility. Illustrative examples are used to demonstrate the effectiveness of our new approach. In addition, we also show how to formulate the horizontal partitioning problem such that the same algorithm can be applied.     

Lower bound algorithms for multiprocessor task scheduling with ready times

In this paper, we deal with multiprocessor task scheduling with ready times and prespecified processor allocation. We consider an on‐line scenario where tasks arrive over time, and, at any point in time, the scheduler only has knowledge of the released tasks. An application of this problem arises in wavelength division multiplexing broadcasting where the main future will be in the so‐called one‐to‐many transmission. We propose algorithms to find lower bounds of the minimum makespan, and present experiments on various scenarios.     

Graphics processing unit‐accelerated bounding for branch‐and‐bound applied to a permutation problem using data access optimization

Branch‐and‐bound (B&B) algorithms are attractive methods for solving to optimality combinatorial optimization problems using an implicit enumeration of a dynamically built tree‐based search space. Nevertheless, they are time‐consuming when dealing with large problem instances. Therefore, pruning tree nodes (subproblems) is traditionally used as a powerful mechanism to reduce the size of the explored search space. Pruning requires to perform the bounding operation, which consists of applying a lower bound function to the subproblems generated during the exploration process. Preliminary experiments performed on the Flow‐Shop scheduling problem (FSP) have shown that the bounding operation consumes over 98% of the execution time of the B&B algorithm. In this paper, we investigate the use of graphics processing unit (GPU) computing as a major complementary way to speed up the search. We revisit the design and implementation of the parallel bounding model on GPU accelerators. The proposed approach enables data access optimization. Extensive experiments have been carried out on well‐known FSP benchmarks using an Nvidia Tesla C2050 GPU card. Compared to a CPU‐based single core execution using an Intel Core i7‐970 processor without GPU, speedups higher than 100 times faster are achieved for large problem instances. At an equivalent peak performance, GPU‐accelerated B&B is twice faster than its multi‐core counterpart. Copyright © 2013 John Wiley & Sons, Ltd.     

Work stealing with private integer–vector–matrix data structure for multi‐core branch‐and‐bound algorithms

In this paper, the focus is put on multi‐core branch‐and‐bound algorithms for solving large‐scale permutation‐based optimization problems. We investigate five work stealing (WS) strategies with a new data structure called integer–vector–matrix (IVM). In these strategies, each thread has a private IVM allowing the local management of a set of subproblems enumerated using a factorial system. The WS strategies differ in the way the victim thread is selected and the granularity of stolen work units (intervals of factoradics). To assess the efficiency of the private IVM‐based WS approach, the five WS strategies have been extensively experimented on the flowshop scheduling permutation problem and compared with their conventional linked‐list‐based counterparts. The obtained results demonstrate that the IVM‐based WS outperforms the linked‐list‐based one in terms of CPU time, memory usage and number of performed WS operations. Copyright © 2016 John Wiley & Sons, Ltd.     

A branch‐and‐cut algorithm for the Team Orienteering Problem

The Team Orienteering Problem aims at maximizing the total amount of profit collected by a fleet of vehicles while not exceeding a predefined travel time limit on each vehicle. In the last years, several exact methods based on different mathematical formulations were proposed. In this paper, we present a new two‐index formulation with a polynomial number of variables and constraints. This compact formulation, reinforced by connectivity constraints, was solved by means of a branch‐and‐cut algorithm. The total number of instances solved to optimality is 327 of 387 benchmark instances, 26 more than any previous method. Moreover, 24 not previously solved instances were closed to optimality.     

航天测控调度分枝定界算法

通过分析航天测控调度问题的测控需求,建立了航天测控调度整数规划模型,引入了拉格朗日松弛思想并与分枝定界算法结合,设计了基于拉格朗日松弛的分枝定界算法求解航天测控调度问题。通过对两个场景的仿真实验,得到了两个场景的航天测控调度问题最优值,验证了基于拉格朗日松弛的分枝定界算法的有效性。     

分段堆场预测调度研究

针对船舶分段移动计划主要依靠人为预测及人工调度的现状,提出BP神经网络来预测一个周期内进出场分段的数量,并研究建立以分段移动度最小为目标的优化模型,模型综合考虑了分段在堆场中的停放位置及进、出场路径。通过分支定界法选择分段在堆场中停放位置的最优方案,并构建启发式规则来确定分段在堆场中的最优进、出场路径,从而实现对模型的求解。以某船厂实际数据为例,对模型在堆场调度问题中的应用进行了实例验证,结果表明,所研究方法可求解得出较优的堆场作业计划,并实现堆场资源的高效利用。     

A solution approach for log truck scheduling based on composite pricing and branch and bound

The logging truck scheduling problem is one of the most complex routing problems where both pick‐up and delivery operations are included. It consists in finding one feasible route for each vehicle in order to satisfy the demands of the customers and in such a way that the total transport cost is minimized. We use a mathematical formulation of the log truck scheduling problem where each column represents a feasible route. We generate a large pool of columns based on solving a transportation problem. Then we apply a composite pricing algorithm, which mainly consists of pricing the pool of columns and maintain an active set of these, for solving the LP relaxed model. A branch and price approach is used to obtain integer solutions in which we apply composite pricing to generate new columns. Numerical results from case studies at Swedish forestry companies are presented.     

A tabu method for a two-agent single-machine scheduling with deterioration jobs

In recent 10 years, the multi-agent idea applied in scheduling issues has received continuing attention. However, the study of the multi-agent scheduling with deteriorating jobs is relatively limited. In light of this, this paper deliberates upon a two-agent single-machine scheduling problem with deteriorating jobs. Taking the proposed model, the actual processing time of a job from both the first agent and the second agent is modeled as a linearly increasing function of its starting time. The goal of this paper is to minimize the total weighted number of tardy jobs of the first agent subject to the condition that the maximum lateness of the second agent is allowed to have an upper bound. The complexity of the model concerned in the paper is claimed as an NP-hard one. Following that, several dominance rules and a lower bound are proposed to be applied in a branch-and-bound algorithm for the optimal solution, and a tabu algorithm is applied to find near-optimal solutions for the problem. The simulation results obtained from all the proposed algorithms are also reported.     

最小费用充电站选址问题的分支定界算法

电动汽车的充电站选址问题是当前社会的热点问题,其实质是组合优化中经典的NP-hard问题。基于最小开设费用对充电站选址问题进行研究,首先对该问题进行了数学建模,进而研究了该问题的数学性质并给予相应的证明,利用这些性质减小问题的规模,从而降低问题的求解难度;然后设计了上下界子算法以及降阶子算法,基于这些子算法提出了一种可以快速缩小问题规模同时得到最优解的分支定界算法,降低了时间复杂度,同时可以对解空间进行大量剪枝加快求解速度;最后通过分析和求解一个示例来进一步阐述所提算法的原理和执行过程。     

求解阻塞混流生产机器人制造单元调度问题的分支定界算法

针对阻塞混流生产机器人制造单元调度问题,为了同时优化机器人运行顺序和工件加工顺序,提出了分支定界算法。首先,定义机器人活动,将双排序转化为单排序;其次,构建顺序插入规则生成可行解;最后,依据顺序插入规则,设计了分支过程。通过计算随机生成算例,计算结果表明：工作站个数为3时,分支定界算法得到的目标函数值与CPLEX相同,但平均运行时间比CPLEX降低38.58%,证实了分支定界算法的有效性;工作站个数大于3时,与CPLEX相比,在同等时间内,有85.19%的算例搜索到更好解,因此,对于大规模情形,分支定界算法更有价值。     

非凸二次规划全局极小问题的新型分枝定界算法

针对求解多面集上二次函数的全局近似最优解问题,利用逐步缩小对偶间隙的处理办法,提出了一个新型分枝定界算法。新算法的主要改进之处是利用了Lagrange 对偶性获取下界。最后,用构造和随机产生的问题实例,对提出的新算法和传统的分枝定界算法做了初步的数值比较实验。计算实验表明算法对求解中大规模非凸二次规划问题的有效性。     

A branch‐and‐price approach to k‐clustering minimum biclique completion problem

We consider the problem of finding k‐bipartite subgraphs, called “clusters,” in a bipartite graph , such that each vertex i of S appears in exactly one of the subgraphs, every vertex j of T appears in each cluster in which at least one of its neighbors appears, and the total number of edges needed to complete each cluster (i.e. to become a biclique) is minimized. This problem has been shown to be strongly NP‐hard and is known as the k‐clustering minimum biclique completion problem. It has been applied to bundling channels for multicast transmissions. The application consists of finding k‐multicast sessions that partition the set of demands, given a set of demands of services from clients. Each service should belong to a single multicast session, while each client can appear in more than one session. We extend previous work by developing a branch‐and‐price algorithm that embeds a new metaheuristic based on variable neighborhood infeasible search and a branching rule that exploits the problem structure. The metaheuristic can also efficiently solve the pricing subproblem. In addition to the random instances used in the literature, we present structured instances generated using the MovieLens dataset collected by the GroupLens Research Project. Extensive computational results show that our branch‐and‐price algorithm outperforms the approaches proposed in the literature.     

A heuristic-based hybrid genetic-variable neighborhood search algorithm for task scheduling in heterogeneous multiprocessor system

Effective task scheduling, which is essential for achieving high performance in a heterogeneous multiprocessor system, remains a challenging problem despite extensive studies. In this article, a heuristic-based hybrid genetic-variable neighborhood search algorithm is proposed for the minimization of makespan in the heterogeneous multiprocessor scheduling problem. The proposed algorithm distinguishes itself from many existing genetic algorithm (GA) approaches in three aspects. First, it incorporates GA with the variable neighborhood search (VNS) algorithm, a local search metaheuristic, to exploit the intrinsic structure of the solutions for guiding the exploration process of GA. Second, two novel neighborhood structures are proposed, in which problem-specific knowledge concerned with load balancing and communication reduction is utilized respectively, to improve both the search quality and efficiency of VNS. Third, the proposed algorithm restricts the use of GA to evolve the task-processor mapping solutions, while taking advantage of an upward-ranking heuristic mostly used by traditional list scheduling approaches to determine the task sequence assignment in each processor. Empirical results on benchmark task graphs of several well-known parallel applications, which have been validated by the use of non-parametric statistical tests, show that the proposed algorithm significantly outperforms several related algorithms in terms of the schedule quality. Further experiments are carried out to reveal that the proposed algorithm is able to maintain high performance within a wide range of parameter settings.     

Generating subproblems in branch and bound algorithms for parallel machines scheduling problem

A branch and bound algorithm (B&B) has been widely used in various discrete and combinatorial optimization fields. To obtain optimal solutions as soon as possible for scheduling problems, three tools, which are branching, bounding and dominance rules, have been developed in the B&B algorithm. One of these tools, a branching is a method for generating subproblems and directly determines size of solution to be searched in the B&B algorithm. Therefore, it is very important to devise effective branching scheme for the problem.In this note, a survey of branching schemes is performed for parallel machines scheduling (PMS) problems with n independent jobs and m machines and new branching schemes that can be used for identical and unrelated PMS problems, respectively, are suggested. The suggested branching methods show that numbers of generated subproblems are much smaller than that of other methods developed earlier and therefore, it is expected that they help to reduce a lot of CPU time required to obtain optimal solutions in the B&B algorithm.     

Data‐aware task scheduling on heterogeneous hybrid memory multiprocessor systems

In this paper, we propose a method about task scheduling and data assignment on heterogeneous hybrid memory multiprocessor systems for real‐time applications. In a heterogeneous hybrid memory multiprocessor system, an important problem is how to schedule real‐time application tasks to processors and assign data to hybrid memories. The hybrid memory consists of dynamic random access memory and solid state drives when considering the performance of solid state drives into the scheduling policy. To solve this problem, we propose two heuristic algorithms called improvement greedy algorithm and the data assignment according to the task scheduling algorithm, which generate a near‐optimal solution for real‐time applications in polynomial time. We evaluate the performance of our algorithms by comparing them with a greedy algorithm, which is commonly used to solve heterogeneous task scheduling problem. Based on our extensive simulation study, we observe that our algorithms exhibit excellent performance and demonstrate that considering data allocation in task scheduling is significant for saving energy. We conduct experiments on two heterogeneous multiprocessor systems. Copyright © 2016 John Wiley & Sons, Ltd.     

分支定界算法的分布并行化研究

介绍了一种专用于计算分支定界算法的机群计算平台,其中所使用的分布并行策略减少了分支定界算法计算时间复杂度,减小了问题的规模;可以把计算平台机群中的任何一台计算机上计算出的当前全局最佳本分值,实时地广播给所有其他并行的计算机,并作为它们新的最佳本分值,实现分支节点的快速并行淘汰;应用启发式算法修改了分支定界算法,提高了分支节点的淘汰效率。选用旅行商问题实例作为测试基准。计算表明,在保证求得最优解的前提下,该平台能很好地提高分支定界算法的效率。     

An Exact Method to Minimize the Number of Tardy Jobs in Single Machine Scheduling

This paper considers the problem of scheduling n jobs on a single machine to minimize the number of tardy (or late) jobs. Each job has a release date, a processing time and a due date. The general case with non-equal release dates and different due dates is considered. Using new and efficient lower bounds and several dominance rules, a branch and bound scheme is proposed based on the definition of a master sequence, i.e. a sequence containing at least one optimal sequence. With this procedure, 95% of 140-job instances are optimally solved in a maximum of one-hour CPU time.