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%.     

Two-machine flowshop scheduling problems involving a batching machine with transportation or deterioration consideration

This paper considers two scheduling problems for a two-machine flowshop where a single machine is followed by a batching machine. The first problem is that there is a transporter to carry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the single machine. For the first problem with minimizing the makespan, we formulate it as a mixed integer programming model and then prove that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst case performance is analyzed. The computational experiments are carried out and the numerical results show that the heuristic algorithm is effective. For the second problem, we derive the optimal algorithms with polynomial time for minimizing the makespan, the total completion time and the maximum lateness, respectively.     

Scheduling unit-time tasks in renewable resource constrained flowshops

In this paper we address the problem of scheduling unit time jobs in a flowshop to minimize makespan under a discrete renewable resource constraint. We propose algorithms having complexityO(n) for special cases of two machine and three machine problems. For the three machineN P-hard problem we develop a branch and bound procedure and several upper and lower bounds. We test the efficiency of the procedure and report our computational experience.Suna Kondakci was a visiting assistant professor at the Industrial Engineering Department, Purdue University, when the paper was revised.     

Minimising maximum lateness in a two-machine flowshop

The objective of this paper is to develop a branch and bound algorithm for the problem of minimising maximum lateness in a two-machine flowshop, subject to release dates and time lag constraints. The importance of this NP-hard problem is twofold, it arises as a strong relaxation of the classical permutation flowshop problem, and it generalises several well studied two-machine flowshop problems. Computational experiments performed on a large set of randomly generated problems show that our algorithm can solve to optimality large size instances.     

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.     

Algorithms for two-machine flow-shop sequencing with precedence constraints

This paper considers the problem of minimizing the maximum completion time in a two-machine flow-shop for which precedence constraints on the jobs are specified. If one job has precedence over another, then the second of these jobs cannot be processed on a machine until the first of them is completed on that machine. A powerful new lower bounding rule which uses Lagrangean relaxation is developed. Then several dominance theorems are presented which are used to eliminate some jobs from the problem. The lower bound is used in three branch and bound algorithms, two of which employ well-known branching rules while the third uses a new branching rule. The algorithms are compared and tested on problems with up to 80 jobs.     

Parallel machine scheduling with a common due window

In this paper, we consider a machine scheduling problem where jobs should be completed at times as close as possible to their respective due dates, and hence both earliness and tardiness should be penalized. Specifically, we consider the problem with a set of independent jobs to be processed on several identical parallel machines. All the jobs have a given common due window. If a job is completed within the due window, then there is no penalty. Otherwise, there is either a job-dependent earliness penalty or a job-dependent tardiness penalty depending on whether the job is completed before or after the due window. The objective is to find an optimal schedule with minimum total earliness–tardiness penalty. The problem is known to be NP-hard. We propose a branch and bound algorithm for finding an optimal schedule of the problem. The algorithm is based on the column generation approach in which the problem is first formulated as a set partitioning type formulation and then in each branch and bound iteration the linear relaxation of this formulation is solved by the standard column generation procedure. Our computational experiments show that this algorithm is capable of solving problems with up to 40 jobs and any number of machines within a reasonable computational time.     

Heuristics for a two-stage hybrid flowshop scheduling problem with ready times and a product-mix ratio constraint

This paper focuses on the scheduling problem of minimizing makespan for a given set of jobs in a two-stage hybrid flowshop subject to a product-mix ratio constraint. There are identical parallel machines at the first stage of the hybrid flowshop, while there is a single batch-processing machine at the second stage. Ready times of the jobs (at the first stage) may be different, and a given product-mix ratio of job types should be kept in each batch at the second stage. We present three types of heuristic algorithms: forward scheduling algorithms, backward scheduling algorithms, and iterative algorithms. To evaluate performance of the suggested algorithms, a series of computational experiments are performed on randomly generated test problems and results are reported.     

Minimizing total tardiness on a single machine with unequal release dates

This study addresses the problem of minimizing total tardiness on a single machine with unequal release dates. Dominance properties established in previous literatures and herein are adopted to develop branch and bound and heuristic procedures. Computational experiments were conducted to evaluate the approaches. The results revealed that the branch and bound algorithm is efficient in solving hard problems and easy problems that involve up to 50 and 500 jobs, respectively. The computational effectiveness of the heuristic is also reported.     

带有运输且加工具有灵活性的无等待流水作业排序问题

研究一类带有运输且加工具有灵活性的两阶段无等待流水作业排序问题, 其中每阶段只有一台机器, 每个工件有两道工序需要依次在两台机器上加工, 工件在两台机器上的加工及两道工序之间不允许等待. 给出两种近似算法, 并分别分析其最坏情况界. 第一种算法是排列排序, 证明了最坏情况界不超过5/2; 第二种算法将工件按照两道工序加工时间之和的递增顺序排序, 证明其最坏情况界不超过2. 最后, 通过数值模拟比较算法的性能. 对问题中各参数取不同值的情况, 分别生成若干个实例, 用算法得到的解与最优解的下界作比值, 通过分析这些比值的最大值、最小值和平均值来比较上述两个算法的性能.     

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.     

Minimizing maximum lateness of jobs in inventory constrained scheduling

This paper focuses on a machine scheduling problem having applications in truck scheduling at transshipment terminals. Jobs increase and decrease, respectively, the level of a central inventory. Naturally, jobs decreasing the inventory level can be processed only if the level of the inventory is high enough not to drop below zero. We consider the problem to find a schedule for jobs such that the maximum lateness among all jobs is minimized. We develop properties of optimal solutions, lower bounds, and heuristic methods in order to find upper bounds. These are incorporated in four branch and bound algorithms that are based on fixing sequences of jobs in forward or backward direction in two different types of representations. By means of a computational study, we compare these approaches with each other in order to show their efficiency.     

Minimizing due date related performance measures on two batch processing machines

This paper considers a scheduling problem in a two-machine flowshop of two batch processing machines. On each batch processing machine, jobs are processed in a batch, and each batch is allowed to contain jobs up to the maximum capacity of the associated machine. The scheduling problem is analyzed with respect to three due date related objectives including maximum tardiness, number of tardy jobs and total tardiness. In the analysis, several solution properties are characterized and based upon these properties, three efficient polynomial time algorithms are developed for minimizing the due date related measures.     

Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs

Motivated by just-in-time manufacturing, we consider a single machine scheduling problem with dual criteria, i.e., the minimization of the total weighted earliness subject to minimum number of tardy jobs. We discuss several dominance properties of optimal solutions. We then develop a heuristic algorithm with time complexity O(n³) and a branch and bound algorithm to solve the problem. The computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium-size problems.     

Scheduling the maintenance on a single machine

This paper considers a single machine scheduling problem with preventive maintenance. In many cases, a machine must be maintained after it continuously works for a period of time. But most papers in the literature ignore non-availability of the machine. For this reason, this paper studies the problem of scheduling processing of jobs and maintenance of machine simultaneously. The objective is to minimise total completion time of jobs. The problem is proved to be NP-hard in the strong sense. Three heuristic algorithms and a branch and bound algorithm are proposed. Computational experiments are done to evaluate the effectiveness of the algorithms.     

Resolution of a scheduling problem in a flowshop robotic cell

We develop in this paper a generic and precise identification of a scheduling problem in a flexible manufacturing system. We consider a flowshop robotic cell that processes several jobs. We assume that there is no intermediate buffer between machines. So, jobs may be blocked when downstream machines are busy. We present an integer programming model to determine the sequence of jobs that minimizes the makespan criterion. In order to solve large size problems, we propose a genetic algorithm (GA). Finally, computational experiments are proposed in order to compare the makespan returned by the GA to a lower bound.     

A two-machine flowshop scheduling problem with a truncated sum of processing-times-based learning function

Scheduling with learning effects has received continuing attention in the recent days. However, it can be found that the actual processing time of a given job drops to zero precipitously as the job has a big processing time or the number of jobs increases. Moreover, most researchers paid more attention to single-machine settings, and the flowshop settings then are relatively unexplored. Motivated by these observations, we consider a two-machine total completion time flowshop problem in which the actual job processing time is a function depending on the jobs that have already been processed and a control parameter. In this paper, we develop a branch-and-bound and a genetic heuristic-based algorithm for the problem. In addition, the experimental results of all proposed algorithms are also provided.     

Makespan minimization in machine dominated flowshop

In a flowshop scheduling problem, a set of jobs is processed by a set of machines. The jobs follow the same sequence in all machines. We study the flowshop scheduling problem under a new case of machine dominance that is often found in the manufacturing of computers and electronic devices. We provide a formula for makespan value for a given sequence, show that the makespan value depends only on certain jobs in the sequence, and present an algorithm that finds a sequence with minimum makespan. Numerical examples of the solution approaches are provided.     

Dynamic Programming State-Space Relaxation for Single-Machine Scheduling

The problem of sequencing jobs on a single machine to minimize total cost is considered. Machine capacity constraints require that, at any time, at most one job is processed. Also, no machine idle-time between processing jobs is allowed. In contrast to most research, it is not assumed that the cost is a non-decreasing function of completion time. A dynamic programming formulation of the problem is presented. Since the number of states required by this formulation is prohibitively large, the possibilities for branch and bound algorithms are explored. It is shown that the dynamic programming formulation can be relaxed by mapping the state-space onto a smaller state-space and performing the recursion on this smaller state-space, thereby giving a lower bound. Techniques for improving this lower bound through the use of penalties and through the use of state-space modifiers are discussed. Computational results are presented for the problem in which each job has a due date, and the objective is to minimize the sum of holding costs for jobs completed before their due date and tardiness costs for jobs completed after their due date.     

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.