Minimizing the makespan in open‐shop scheduling problems with a convex resource consumption function

We consider open‐shop scheduling problems where operation‐processing times are a convex decreasing function of a common limited nonrenewable resource. The scheduler's objective is to determine the optimal job sequence on each machine and the optimal resource allocation for each operation in order to minimize the makespan. We prove that this problem is NP‐hard, but for the special case of the two‐machine problem we provide an efficient optimization algorithm. We also provide a fully polynomial approximation scheme for solving the preemptive case. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Scheduling for parallel dedicated machines with a single server

This paper examines scheduling problems in which the setup phase of each operation needs to be attended by a single server, common for all jobs and different from the processing machines. The objective in each situation is to minimize the makespan. For the processing system consisting of two parallel dedicated machines we prove that the problem of finding an optimal schedule is N P‐hard in the strong sense even if all setup times are equal or if all processing times are equal. For the case of m parallel dedicated machines, a simple greedy algorithm is shown to create a schedule with the makespan that is at most twice the optimum value. For the two machine case, an improved heuristic guarantees a tight worst‐case ratio of 3/2. We also describe several polynomially solvable cases of the later problem. The two‐machine flow shop and the open shop problems with a single server are also shown to be N P‐hard in the strong sense. However, we reduce the two‐machine flow shop no‐wait problem with a single server to the Gilmore—Gomory traveling salesman problem and solve it in polynomial time. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 304–328, 2000     

Stochastic scheduling subject to machine breakdowns: The preemptive‐repeat model with discounted reward and other criteria

We consider the problem of scheduling a set of jobs on a single machine subject to random breakdowns. We focus on the preemptive‐repeat model, which addresses the situation where, if a machine breaks down during the processing of a job, the work done on the job prior to the breakdown is lost and the job will have to be started from the beginning again when the machine resumes its work. We allow that (i) the uptimes and downtimes of the machine follow general probability distributions, (ii) the breakdown process of the machine depends upon the job being processed, (iii) the processing times of the jobs are random variables following arbitrary distributions, and (iv) after a breakdown, the processing time of a job may either remain a same but unknown amount, or be resampled according to its probability distribution. We first derive the optimal policy for a class of problems under the criterion to maximize the expected discounted reward earned from completing all jobs. The result is then applied to further obtain the optimal policies for other due date‐related criteria. We also discuss a method to compute the moments and probability distributions of job completion times by using their Laplace transforms, which can convert a general stochastic scheduling problem to its deterministic equivalent. The weighted squared flowtime problem and the maintenance checkup and repair problem are analyzed as applications. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Proportionate flow shop: New complexity results and models with due date assignment

It is known that the proportionate flow shop minimum makespan $Fm/prpt/C_{\max }$ problem is solved optimally by any permutation job sequence. We show that the $Fm/prpt/C_{\max }$ problem is at least ordinary NP‐hard when missing operations are allowed and present some solvable cases. We then consider the standard proportionate flow shop problem (with no missing operations) and show that the solution algorithms for a class of single‐machine due date assignment problems can be extended/generalized to the corresponding proportionate flow shop problems. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 98–106, 2015     

Asymptotically optimal linear time algorithms for two‐stage and three‐stage flexible flow shops

We consider two‐stage and three‐stage flexible flow shops with parallel machines in each stage and the minimum makespan objective. We develop linear time algorithms for these problems with absolute worst‐case bounds either sharper or no worse than the currently available ones and we accomplish this with lower complexity algorithms. We also demonstrate the asymptotic optimality of a class of algorithms for multistage flexible flow shop problems (which includes the proposed algorithms) under certain probabilistic assumptions on the job processing times. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 259–268, 2000     

Production scheduling with history‐dependent setup times

We consider a parallel‐machine scheduling problem with jobs that require setups. The duration of a setup does not depend only on the job just completed but on a number of preceding jobs. These setup times are referred to as history‐dependent. Such a scheduling problem is often encountered in the food processing industry as well as in other process industries. In our model, we consider two types of setup times—a regular setup time and a major setup time that becomes necessary after several “hard‐to‐clean” jobs have been processed on the same machine. We consider multiple objectives, including facility utilization, flexibility, number of major setups, and tardiness. We solve several special cases assuming predetermined job sequences and propose strongly polynomial time algorithms to determine the optimal timing of the major setups for given job sequences. We also extend our analysis to develop pseudopolynomial time algorithms for cases with additional objectives, including the total weighted completion time, the total weighted tardiness, and the weighted number of tardy jobs. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

链约束下加工时间恶化的Flow Shop排序问题

讨论作业具有线性加工时间,作业间具有链约束的两台处理机流水作业排序问题,目标函数为极小化完工时间。在作业加工时间简单线性恶化下,提出作业的非负开始和停止延迟恶化率,构造了满足约束条件的复合作业。在此基础上,给出作业间具有平行链约束的两台处理机流水作业排序问题的最优多项式算法。     

A bicriterion approach to common flow allowances due window assignment and scheduling with controllable processing times

We investigate a single‐machine scheduling problem for which both the job processing times and due windows are decision variables to be determined by the decision maker. The job processing times are controllable as a linear or convex function of the amount of a common continuously divisible resource allocated to the jobs, where the resource allocated to the jobs can be used in discrete or continuous quantities. We use the common flow allowances due window assignment method to assign due windows to the jobs. We consider two performance criteria: (i) the total weighted number of early and tardy jobs plus the weighted due window assignment cost, and (ii) the resource consumption cost. For each resource consumption function, the objective is to minimize the first criterion, while keeping the value of the second criterion no greater than a given limit. We analyze the computational complexity, devise pseudo‐polynomial dynamic programming solution algorithms, and provide fully polynomial‐time approximation schemes and an enhanced volume algorithm to find high‐quality solutions quickly for the considered problems. We conduct extensive numerical studies to assess the performance of the algorithms. The computational results show that the proposed algorithms are very efficient in finding optimal or near‐optimal solutions. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 41–63, 2017     

Flow‐shop batch scheduling with identical processing‐time jobs

In many practical situations of production scheduling, it is either necessary or recommended to group a large number of jobs into a relatively small number of batches. A decision needs to be made regarding both the batching (i.e., determining the number and the size of the batches) and the sequencing (of batches and of jobs within batches). A setup cost is incurred whenever a batch begins processing on a given machine. This paper focuses on batch scheduling of identical processing‐time jobs, and machine‐ and sequence‐independent setup times on an m‐machine flow‐shop. The objective is to find an allocation to batches and their schedule in order to minimize flow‐time. We introduce a surprising and nonintuitive solution for the problem. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Complexity and workload considerations in product mix decisions under the theory of constraints

The literature on the product mix decision (or master production scheduling) under the Theory of Constraints (TOC), which was developed in the past two decades, has addressed this problem as a static operational decision. Consequently, the developed solution techniques do not consider the system's dynamism and the associated challenges arising from the complexity of operations during the implementation of master production schedules. This paper aims to address this gap by developing a new heuristic approach for master production scheduling under the TOC philosophy that considers the main operational factors that influence actual throughput after implementation of the detailed schedule. We examine the validity of the proposed heuristic by comparison to Integer Linear Programming and two heuristics in a wide range of scenarios using simulation modelling. Statistical analyses indicate that the new algorithm leads to significantly enhanced performance during implementation for problems with setup times. The findings show that the bottleneck identification approach in current methods in the TOC literature is not effective and accurate for complex operations in real‐world job shop systems. This study contributes to the literature on master production scheduling and product mix decisions by enhancing the likelihood of achieving anticipated throughput during the implementation of the detailed schedule. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 357–369, 2015     

Scheduling jobs,with exponentially distributed processing times,on two machines of a flow shop

This paper treats the problem of sequencing n jobs on two machines in a “flow shop.” (That is, each job in the shop is required to flow through the same sequence of the machines.) The processing time of a given job on a given machine is assumed to be distributed exponentially, with a known mean. The objective is to minimize the expected job completion time. This paper proves an optimal ordering rule, previously conjectured by Talwar [10]. A formula is also derived through Markov Chain analysis, which evaluates the expected job completion time for any given sequence of the jobs. In addition, the performance of a heuristic rule is discussed in the light of the optimal solution.     

Flow shop scheduling with earliness,tardiness, and intermediate inventory holding costs

We consider the problem of scheduling customer orders in a flow shop with the objective of minimizing the sum of tardiness, earliness (finished goods inventory holding), and intermediate (work‐in‐process) inventory holding costs. We formulate this problem as an integer program, and based on approximate solutions to two different, but closely related, Dantzig‐Wolfe reformulations, we develop heuristics to minimize the total cost. We exploit the duality between Dantzig‐Wolfe reformulation and Lagrangian relaxation to enhance our heuristics. This combined approach enables us to develop two different lower bounds on the optimal integer solution, together with intuitive approaches for obtaining near‐optimal feasible integer solutions. To the best of our knowledge, this is the first paper that applies column generation to a scheduling problem with different types of strongly ????‐hard pricing problems which are solved heuristically. The computational study demonstrates that our algorithms have a significant speed advantage over alternate methods, yield good lower bounds, and generate near‐optimal feasible integer solutions for problem instances with many machines and a realistically large number of jobs. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Minimizing mean absolute deviation of completion times about a common due date

We consider a single machine scheduling problem in which the objective is to minimize the mean absolute deviation of job completion times about a common due date. We present an algorithm for determining multiple optimal schedules under restrictive assumptions about the due date, and an implicit enumeration procedure when the assumptions do not hold. We also establish the similarity of this problem to the two parallel machines mean flow time problem.     

Scheduling jobs with piecewise linear decreasing processing times

We study the problems of scheduling a set of nonpreemptive jobs on a single or multiple machines without idle times where the processing time of a job is a piecewise linear nonincreasing function of its start time. The objectives are the minimization of makespan and minimization of total job completion time. The single machine problems are proved to be NP‐hard, and some properties of their optimal solutions are established. A pseudopolynomial time algorithm is constructed for makespan minimization. Several heuristics are derived for both total completion time and makespan minimization. Computational experiments are conducted to evaluate their efficiency. NP‐hardness proofs and polynomial time algorithms are presented for some special cases of the parallel machine problems. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 531–554, 2003     

Counterexamples to optimal permutation schedules for certain flow shop problems

It is well known that a minimal makespan permutation sequence exists for the n × 3 flow shop problem and for the n × m flow shop problem with no inprocess waiting when processing times for both types of problems are positive. It is shown in this paper that when the assumption of positive processing times is relaxed to include nonnegative processing times, optimality of permutation schedules cannot be guaranteed.     

Bicriteria scheduling on a single batching machine with job transportation and deterioration considerations

We study a single batching machine scheduling problem with transportation and deterioration considerations arising from steel production. A set of jobs are transported, one at a time, by a vehicle from a holding area to the single batching machine. The machine can process several jobs simultaneously as a batch. The processing time of a job will increase if the duration from the time leaving the holding area to the start of its processing exceeds a given threshold. The time needed to process a batch is the longest of the job processing times in the batch. The problem is to determine the job sequence for transportation and the job batching for processing so as to minimize the makespan and the number of batches. We study four variations (P1, P2, P3, P4) of the problem with different treatments of the two criteria. We prove that all the four variations are strongly NP‐hard and further develop polynomial time algorithms for their special cases. For each of the first three variations, we propose a heuristic algorithm and analyze its worst‐case performance. For P4, which is to find the Pareto frontier, we provide a heuristic algorithm and an exact algorithm based on branch and bound. Computational experiments show that all the heuristic algorithms perform well on randomly generated problem instances, and the exact algorithm for P4 can obtain Pareto optimal schedules for small‐scale instances. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 269–285, 2014     

Makespan minimization in the two‐machine flowshop batch scheduling problem

In this paper we consider a practical scheduling problem commonly arising from batch production in a flexible manufacturing environment. Different part‐types are to be produced in a flexible manufacturing cell organized into a two‐stage production line. The jobs are processed in batches on the first machine, and the completion time of a job is defined as the completion time of the batch containing it. When processing of all jobs in a batch is completed on the first machine, the whole batch of jobs is transferred intact to the second machine. A constant setup time is incurred whenever a batch is formed on any machine. The tradeoff between the setup times and batch processing times gives rise to the batch composition decision. The problem is to find the optimal batch composition and the optimal schedule of the batches so that the makespan is minimized. The problem is shown to be strongly NP‐hard. We identify some special cases by introducing their corresponding solution methods. Heuristic algorithms are also proposed to derive approximate solutions. We conduct computational experiments to study the effectiveness of the proposed heuristics. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 128–144, 2000     

A robust policy for serial agile production systems

One of the major problems in modeling production systems is how to treat the job arrival process. Restrictive assumptions such as Markovian arrivals do not represent real world systems, especially if the arrival process is generated by job departures from upstream workstations. Under these circumstances, cost‐effective policies that are robust with respect to the nature of the arrival process become of interest. In this paper, we focus on minimizing the expected total holding and setup costs in a two‐stage produce‐to‐order production system operated by a cross‐trained worker. We will show that if setup times are insignificant in comparison with processing times, then near‐optimal policies can be generated with very robust performances with respect to the arrival process. We also present conditions under which these near‐optimal policies can be obtained by using only the arrival and service rates. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Scheduling with centralized and decentralized batching policies in concurrent open shops

This article considers batch scheduling with centralized and decentralized decisions. The context of our study is concurrent open shop scheduling where the jobs are to be processed on a set of independent dedicated machines, which process designated operations of the jobs in batches. The batching policy across the machines can be centralized or decentralized. We study such scheduling problems with the objectives of minimizing the maximum lateness, weighted number of tardy jobs, and total weighted completion time, when the job sequence is determined in advance. We present polynomial time dynamic programming algorithms for some cases of these problems and pseudo‐polynomial time algorithms for some problems that are NP‐hard in the ordinary sense. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 58: 17–27, 2011     

Review of the ordered and proportionate flow shop scheduling research

Ordered flow shop models have appeared in the literature since the mid‐1970s and proportionate flow shop models have appeared since the early 1980s. We provide a detailed review of these models along with some analysis and potential topics for future research. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013