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.     

Single machine interfering jobs problem with flowtime objective

Interfering jobs problems (or multi agents scheduling problems) are an emergent topic in the scheduling literature. In these decision problems, two or more sets of jobs have to be scheduled, each one with its own criteria. More specifically, we focus on a problem in which jobs belonging to two sets have to be scheduled in a single machine in order to minimize the total flowtime of the jobs in one set, while the total flowtime of the jobs in the other set should not exceed a given constant \(\epsilon \). This problem is known to be weakly NP-hard, and, in the literature, a dynamic programming (DP) algorithm has been proposed to find optimal solutions. In this paper, we first analyse the distribution of solutions of the problem in order to establish its empirical hardness. Next, a novel encoding scheme and a set of properties associated to the neighbourhood of this scheme are presented. These properties are used to develop both exact and approximate methods, i.e. a branch and bound (B&B) method, several constructive heuristics, and different versions of a genetic algorithm (GA). The computational experience carried out shows that the proposed B&B is more efficient than the existing DP algorithm. The results also show the advantages of the proposed encoding scheme, as the approximate methods yield close-to-optimum solutions for big-sized instances where exact methods are not feasible.     

Minimizing the number of tardy jobs under piecewise-linear deterioration

In this paper, we consider a single machine scheduling problem with piecewise-linear deterioration where its objective is to minimize the number of tardy jobs, in which the processing time of each job depends on its starting time where all the jobs have a specific deterioration rate. The problem is known to be NP-hard; therefore a Branch and Bound algorithm and a heuristic algorithm with O(n²) are proposed. The proposed heuristic algorithm has been utilized for solving large scale problems and upper bound of the B&B algorithm. Computational experiments on 1840 problems demonstrate that the Branch and Bound procedure can solve problems with 28 jobs and 85.4% of all the sample problems optimally showing the high capability of the proposed procedure. Also it is shown that the average value of the ratio of optimal answer to the heuristic algorithm result with the objective ∑(1-_Ui)$\sum (1-U_i)$ is at last 1.08 which is more efficient in contrast to other proposed algorithms in related studies in the literature. According to high efficacy of the heuristic algorithm, large scale samples are also being solved and the results are presented. A specific form of this problem is also being considered and it is proven that the B&B procedure can handle problems with more jobs even up to 44 jobs.     

Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

A scheduling problem with unrelated parallel machines, sequence and machine-dependent setup times, due dates and weighted jobs is considered in this work. A branch-and-bound algorithm (B&B) is developed and a solution provided by the metaheuristic GRASP is used as an upper bound. We also propose a set of instances for this type of problem. The results are compared to the solutions provided by two mixed integer programming models (MIP) with the solver CPLEX 9.0. We carry out computational experiments and the algorithm performs extremely well on instances with up to 30 jobs.     

A branch and bound algorithm for the response time variability problem

The response time variability problem (RTVP) is an NP-hard scheduling problem that has been studied intensively recently and has a wide range of real-world applications in mixed-model assembly lines, multithreaded computer systems, network environments and others. The RTVP arises whenever products, clients or jobs need to be sequenced in order to minimise the variability in the time between two successive points at which they receive the necessary resources. To date, the best exact method for solving this problem is a mixed integer linear programming (MILP) model, which solves to optimality most of instances with up to 40 units to be scheduled in a reasonable amount of time. The goal of this paper is to increase the size of the instances that can be solved to optimality. We have designed an algorithm based on the branch and bound (B&B) technique to take advantage of the particular features of the problem. Our computational experiments show that the B&B algorithm is able to solve larger instances with up to 55 units to optimality in a reasonable time.     

Heuristics for the two‐machine scheduling problem with a single server

In this paper, the NP‐hard two‐machine scheduling problem with a single server is addressed. The problem consists of a given set of jobs to be scheduled on two identical parallel machines, where each job must be processed on one of the machines, and prior to processing, the job is set up on its machine using one server; the latter is shared between the two machines. An ant colony optimization (ACO) algorithm is introduced for the problem and its performance was assessed by comparing with an exact solution (branch and bound [B&B]), a genetic algorithm (GA), and simulated annealing (SA). The computational results reflected the superiority of “ACO” in large problems, with a performance similar to SA and GA in smaller problems, while solving the tested problems within a reasonable computational time.     

Hadoop下资源匹配最大集作业调度算法

针对目前层级队列作业调度算法中资源占比高的作业执行效率低的问题,提出一种资源匹配最大集算法。该算法分析作业特征,引入完成度、等待时间、优先级、重调度次数为紧迫值因子,优先考虑资源占比高或等待时间长的作业,以改善作业公平性;采用双队列结构在可用资源总量内优先选择高紧迫值作业,在不同资源占比作业集比较中选择作业数最大集,以实现调度平衡。在与最大最小公平(Max-min fairness)算法的实例对比中发现,该算法可降低作业集平均等待时间、提高资源利用率。实验对比结果表明,该算法可将不同资源占比的单一类型作业集执行时间缩短18.73%,其中资源占比高的作业执行时间缩短27.26%;在混合型作业集中对应的执行时间可分别缩短22.36%与30.28%。所提算法能有效减少资源占比高作业的等待,提高作业整体执行效率。     

A Truncated Sum of Processing‐Times–Based Learning Model for a Two‐Machine Flowshop Scheduling Problem

Scheduling with learning effects has gained increasing attention in recent years. A well‐known learning model is called “sum‐of‐processing‐times‐based learning” in which the actual processing time of a job is a nonincreasing function of the jobs already processed. However, the actual processing time of a given job drops to zero precipitously when the normal job processing times are large. Moreover, the concept of learning process is relatively unexplored in a flowshop environment. Motivated by these observations, this article addresses a two‐machine flowshop problem with a truncated learning effect. The objective is to find an optimal schedule to minimize the total completion time. First, a branch‐and‐bound algorithm incorporating with a dominance property and four lower bounds is developed to derive the optimal solution. Then three simulated annealing algorithms are also proposed for near‐optimal solution. The experimental results indicated that the branch‐and‐bound algorithm can solve instances up to 18 jobs, and the proposed simulated annealing algorithm performs well in item of CPU time and error percentage. © 2011 Wiley Periodicals, Inc.     

Heuristic algorithms for two-machine flowshop with availability constraints

The majority of the scheduling studies carry a common assumption that machines are available all the time. However, machines may not always be available in the scheduling period due to breakdown or preventive maintenance. Taking preventive maintenance activity into consideration, we dealt with the two-machine flowshop scheduling problem with makespan objective. The preventive maintenance policy in this paper was dependent on the number of finished jobs. The integer programming model was proposed. We combined two recent constructive heuristics, HI algorithm and H algorithm, with Johnson’s algorithm, and named the combined heuristic H&J algorithm. We also developed a constructive heuristic, HD, with time complexities O(n²). Based on the difference in job processing times on two machines, both H&J and HD showed good performance, and the latter was slightly better. The HD algorithm was able to obtain the optimality in 98.88% of cases. We also employed the branch and bound (B&B) algorithm to obtain the optimum. With a good upper bound and a modified lower bound, the proposed B&B algorithm performed significantly effectively.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.     

Integrated production and outbound distribution scheduling problems with job release dates and deadlines

In this paper, we study an integrated production and outbound distribution scheduling model with one manufacturer and one customer. The manufacturer has to process a set of jobs on a single machine and deliver them in batches to the customer. Each job has a release date and a delivery deadline. The objective of the problem is to issue a feasible integrated production and distribution schedule minimizing the transportation cost subject to the production release dates and delivery deadline constraints. We consider three problems with different ways how a job can be produced and delivered: non-splittable production and delivery (NSP–NSD) problem, splittable production and non-splittable delivery problem and splittable production and delivery problem. We provide polynomial-time algorithms that solve special cases of the problem. One of these algorithms allows us to compute a lower bound for the NP-hard problem NSP–NSD, which we use in a branch-and-bound (B&B) algorithm to solve problem NSP–NSD. The computational results show that the B&B algorithm outperforms a MILP formulation of the problem implemented on a commercial solver.     

Weak-restriction bi-objective optimization algorithm for scheduling with rejection on non-identical batch processing machines

We investigate the problem of scheduling a set of jobs with arbitrary sizes and unequal weights on a set of parallel batch machines with non-identical capacities. The objective is to minimize the makespan of the accepted jobs and the total rejection penalty of the rejected jobs, simultaneously. To address the studied problem, a Pareto-based ant colony optimization algorithm with the first job selection probability (FPACO) is proposed. A weak-restriction selection strategy is proposed to obtain the desirability of candidate jobs. Two objective-oriented heuristic information and pheromone matrices are designed, respectively, to record the experience in different search dimensions. Moreover, a local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with four existing algorithms through extensive simulation experiments. The experimental results indicate that the proposed algorithm outperforms all of the compared algorithms within a reasonable time.     

A genetic algorithm-based approach for solving the resource-sharing and scheduling problem

We introduce a heuristic that is based on a unique genetic algorithm (GA) to solve the resource-sharing and scheduling problem (RSSP). This problem was previously formulated as a continuous-time mixed integer linear programming model and was solved optimally using a branch-and-bound (B&B) algorithm. The RSSP considers the use of a set of resources for the production of several products. Producing each product requires a set of operations with precedence relationships among them. Each operation can be performed using alternative modes which define the subset of the resources needed, and an operation may share different resources simultaneously. The problem is to select a single mode for each operation and accordingly to schedule the resources, while minimizing the makespan time. The GA we propose is based on a new encoding schema that adopts the structure of a DNA in nature. In our experiments we compared the effectiveness and runtime of our GA versus a B&B algorithm and two truncated B&B algorithms that we developed on a set of 118 problem instances. The results demonstrate that the GA solved all the problems (10 runs each), and reaches optimality in 75% of the runs, had an average deviation of less than 1% from the optimal makespan, and a runtime that was much less sensitive to the size of the problem instance.     

An MO‐GVNS algorithm for solving a multiobjective hybrid flow shop scheduling problem

This paper addresses the multiobjective hybrid flow shop (MOHFS) scheduling problem. In the MOHFS problem considered here, we have a set of jobs that must be performed in a set of stages. At each stage, we have a set of unrelated parallel machines. Some jobs may skip stages. The evaluation criteria are the minimizations of makespan, the weighted sum of the tardiness, and the weighted sum of the earliness. For solving it, an algorithm based on the multiobjective general variable neighborhood search (MO‐GVNS) metaheuristic, named adapted MO‐GVNS, is proposed. This work also presents and compares the results obtained by the adapted MO‐GVNS with those of four other algorithms: multiobjective reduced variable neighborhood search, nondominated sorting genetic algorithm II (NSGA‐II), and NSGA‐III, and another MO‐GVNS from the literature. The results were evaluated based on the Hypervolume, Epsilon, and Spacing metrics, and statistically validated by the Levene test and confidence interval charts. The results showed the efficiency of the proposed algorithm for solving the MOHFS problem.     

A hybrid memetic algorithm for maximizing the weighted number of just-in-time jobs on unrelated parallel machines

This paper presents a hybrid memetic algorithm for the problem of scheduling n jobs on m unrelated parallel machines with the objective of maximizing the weighted number of jobs that are completed exactly at their due dates. For each job, due date, weight, and the processing times on different machines are given. It has been shown that when the numbers of machines are a part of input, this problem is NP-hard in the strong sense. At first, the problem is formulated as an integer linear programming model. This model is practical to solve small-size problems. Afterward, a hybrid memetic algorithm is implemented which uses two heuristic algorithms as constructive algorithms, making initial population set. A data oriented mutation operator is implemented so as to facilitate memetic algorithm search process. Performance of all algorithms including heuristics (H1 and H2), hybrid genetic algorithm and hybrid memetic algorithm are evaluated through computational experiments which showed the capabilities of the proposed hybrid algorithm.     

A heuristic method for two-stage hybrid flow shop with dedicated machines

This paper considers a two-stage hybrid flow shop scheduling problem with dedicated machines, in which the first stage contains a single common critical machine, and the second stage contains several dedicated machines. Each job must be first processed on the critical machine in stage one and depending on the job type, the job will be further processed on the dedicated machine of its type in stage two. The objective is to minimize the makespan. To solve the problem, a heuristic method based on branch and bound (B&B) algorithm is proposed. Several lower bounds are derived and four constructive heuristics are used to obtain initial upper bounds. Then, three dominance properties are employed to enhance the performance of the proposed heuristic method. Extensive computational experiments on two different problem categories each with various problem configurations are conducted. The results show that the proposed heuristic method can produce very close-to-optimal schedules for problems up to 100 jobs and five dedicated machines within 60 s. The comparisons with solutions of two other meta-heuristic methods also prove the better performance of the proposed heuristic method.     

Ant colony optimization algorithm for scheduling jobs with fuzzy processing time on parallel batch machines with different capacities

We study the problem of scheduling on parallel batch processing machines with different capacities under a fuzzy environment to minimize the makespan. The jobs have non-identical sizes and fuzzy processing times. After constructing a mathematical model of the problem, we propose a fuzzy ant colony optimization (FACO) algorithm. Based on the machine capacity constraint, two candidate job lists are adopted to select the jobs for building the batches. Moreover, based on the unoccupied space of the solution, heuristic information is designed for each candidate list to guide the ants. In addition, a fuzzy local optimization algorithm is incorporated to improve the solution quality. Finally, the proposed algorithm is compared with several state-of-the-art algorithms through extensive simulated experiments and statistical tests. The comparative results indicate that the proposed algorithm can find better solutions within reasonable time than all the other compared algorithms.     

Solving a bi-objective flowshop scheduling problem by a Multi-objective Immune System and comparing with SPEA2+ and SPGA

This paper presents a bi-objective flowshop scheduling problem with sequence-dependent setup times. The objective functions are to minimize the total completion time and the total earliness/tardiness for all jobs. An integer programming model is developed for the given problem that belongs to an NP-hard class. Thus, an algorithm based on a Multi-objective Immune System (MOIS) is proposed to find a locally Pareto-optimal frontier of the problem. To prove the efficiency of the proposed MOIS, different test problems are solved. Based on some comparison metrics, the computational results of the proposed MOIS is compared with the results obtained using two well-established multi-objective genetic algorithms, namely SPEA2+ and SPGA. The related results show that the proposed MOIS outperforms genetic algorithms, especially for the large-sized problems.     

A heuristic approach for single-machine scheduling with due dates and class setups

The single-machine sequence-independent class setup scheduling problem is examined in this paper. It is assumed that jobs are classified into classes and a setup is required between jobs of different classes, but not of the same class. Furthermore, this setup time is fixed and depends only on the current job. Since the problem is NP-hard, a heuristic algorithm is proposed to find an approximate schedule that minimizes the maximum lateness on a set of jobs. The algorithm can easily be modified to solve the maximum tardiness problems as well. The accuracy of the heuristic algorithm in generating near optimal solutions is empirically evaluated.Scope and purposeFor batch manufacturing, it maybe desirable to produce many items of the same type, or class, at the same run in order to save the setup cost. However, committing facilities to long production runs for one product may inevitably make others tardy. Small batch size may conform urgent jobs to their delivery date, but one of the consequences would be the loss of productive efficiency due to numerous setups. Therefore, scheduling is basically a trade-off between the inherently conflicting efficiency measure and due-date compliance. This paper considers a single-machine scheduling problem in which jobs are classified into classes and a setup is required between jobs of different classes. The setup time is fixed and depends only on the current job. This problem is called a sequence-independent class setup problem and is NP-complete.