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     

Batch scheduling with step‐deteriorating processing times to minimize flowtime

Both topics of batch scheduling and of scheduling deteriorating jobs have been very popular among researchers in the last two decades. In this article, we study a model combining these two topics. We consider a classical batch scheduling model with unit‐jobs and batch‐independent setup times, and a model of step‐deterioration of processing times. The objective function is minimum flowtime. The optimal solution of the relaxed version (allowing non‐integer batch sizes) is shown to have a unique structure consisting of two consecutive decreasing arithmetic sequences of batch sizes. We also introduce a simple and efficient rounding procedure that guarantees integer batch sizes. The entire solution procedure requires an effort of O(n) (where nis the number of jobs.) © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

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     

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     

On‐line algorithms for minimizing makespan on batch processing machines

We consider problem of scheduling jobs on‐line on batch processing machines with dynamic job arrivals to minimize makespan. A batch machine can handle up to B jobs simultaneously. The jobs that are processed together from a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. In the first part of this paper, we address the single batch processing machine scheduling problem. First we deal with two variants: the unbounded model where B is sufficiently large and the bounded model where jobs have two distinct arrival times. For both variants, we provide on‐line algorithms with worst‐case ratio (the inverse of the Golden ratio) and prove that these results are the best possible. Furthermore, we generalize our algorithms to the general case and show a worst‐case ratio of 2. We then consider the unbounded case for parallel batch processing machine scheduling. Lower bound are given, and two on‐line algorithms are presented. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 241–258, 2001     

An FPTAS for scheduling a two‐machine flowshop with one unavailability interval

We study a deterministic two‐machine flowshop scheduling problem with an assumption that one of the two machines is not available in a specified time period. This period can be due to a breakdown, preventive maintenance, or processing unfinished jobs from a previous planning horizon. The problem is known to be NP‐hard. Pseudopolynomial dynamic programming algorithms and heuristics with worst case error bounds are given in the literature to solve the problem. They are different for the cases when the unavailability interval is for the first or second machine. The existence of a fully polynomial time approximation scheme (FPTAS) was formulated as an open conjecture in the literature. In this paper, we show that the two cases of the problem under study are equivalent to similar partition type problems. Then we derive a generic FPTAS for the latter problems with O(n⁵/ε⁴) time complexity. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Scheduling parallel dedicated machines with the speeding‐up resource

We consider a problem of scheduling jobs on m parallel machines. The machines are dedicated, i.e., for each job the processing machine is known in advance. We mainly concentrate on the model in which at any time there is one unit of an additional resource. Any job may be assigned the resource and this reduces its processing time. A job that is given the resource uses it at each time of its processing. No two jobs are allowed to use the resource simultaneously. The objective is to minimize the makespan. We prove that the two‐machine problem is NP‐hard in the ordinary sense, describe a pseudopolynomial dynamic programming algorithm and convert it into an FPTAS. For the problem with an arbitrary number of machines we present an algorithm with a worst‐case ratio close to 3/2, and close to 3, if a job can be given several units of the resource. For the problem with a fixed number of machines we give a PTAS. Virtually all algorithms rely on a certain variant of the linear knapsack problem (maximization, minimization, multiple‐choice, bicriteria). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

A branch‐and‐price algorithm for parallel machine scheduling with time windows and job priorities

This paper presents a branch‐and‐price algorithm for scheduling n jobs on m nonhomogeneous parallel machines with multiple time windows. An additional feature of the problem is that each job falls into one of ρ priority classes and may require two operations. The objective is to maximize the weighted number of jobs scheduled, where a job in a higher priority class has “infinitely” more weight or value than a job in a lower priority class. The methodology makes use of a greedy randomized adaptive search procedure (GRASP) to find feasible solutions during implicit enumeration and a two‐cycle elimination heuristic when solving the pricing subproblems. Extensive computational results are presented based on data from an application involving the use of communications relay satellites. Many 100‐job instances that were believed to be beyond the capability of exact methods, were solved within minutes. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

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     

Optimal job splitting on a multi‐slot machine with applications in the printing industry

In this article, we define a scheduling/packing problem called the Job Splitting Problem, motivated by the practices in the printing industry. There are n types of items to be produced on an m‐slot machine. A particular assignment of the types to the slots is called a “run” configuration and requires a setup cost. Once a run begins, the production continues according to that configuration and the “length” of the run represents the quantity produced in each slot during that run. For each unit of production in excess of demand, there is a waste cost. Our goal is to construct a production plan, i.e., a set of runs, such that the total setup and waste cost is minimized. We show that the problem is strongly NP‐hard and propose two integer programming formulations, several preprocessing steps, and two heuristics. We also provide a worst‐case bound for one of the heuristics. Extensive tests on real‐world and randomly generated instances show that the heuristics are both fast and effective, finding near‐optimal solutions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

General multiprocessor task scheduling

Most papers in the scheduling field assume that a job can be processed by only one machine at a time. Namely, they use a one‐job‐on‐one‐machine model. In many industry settings, this may not be an adequate model. Motivated by human resource planning, diagnosable microprocessor systems, berth allocation, and manufacturing systems that may require several resources simultaneously to process a job, we study the problem with a one‐job‐on‐multiple‐machine model. In our model, there are several alternatives that can be used to process a job. In each alternative, several machines need to process simultaneously the job assigned. Our purpose is to select an alternative for each job and then to schedule jobs to minimize the completion time of all jobs. In this paper, we provide a pseudopolynomial algorithm to solve optimally the two‐machine problem, and a combination of a fully polynomial scheme and a heuristic to solve the three‐machine problem. We then extend the results to a general m‐machine problem. Our algorithms also provide an effective lower bounding scheme which lays the foundation for solving optimally the general m‐machine problem. Furthermore, our algorithms can also be applied to solve a special case of the three‐machine problem in pseudopolynomial time. Both pseudopolynomial algorithms (for two‐machine and three‐machine problems) are much more efficient than those in the literature. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 57–74, 1999     

Two‐agent scheduling on a single sequential and compatible batching machine

We study two‐agent scheduling on a single sequential and compatible batching machine in which jobs in each batch are processed sequentially and compatibility means that jobs of distinct agents can be processed in a common batch. A fixed setup time is required before each batch is started. Each agent seeks to optimize some scheduling criterion that depends on the completion times of its own jobs only. We consider several scheduling problems arising from different combinations of some regular scheduling criteria, including the maximum cost (embracing lateness and makespan as its special cases), the total completion time, and the (weighted) number of tardy jobs. Our goal is to find an optimal schedule that minimizes the objective value of one agent, subject to an upper bound on the objective value of the other agent. For each problem under consideration, we provide either a polynomial‐time or a pseudo‐polynomial‐time algorithm to solve it. We also devise a fully polynomial‐time approximation scheme when both agents’ scheduling criteria are the weighted number of tardy jobs.     

Server scheduling on parallel dedicated machines with fixed job sequences

We consider server scheduling on parallel dedicated machines to minimize the makespan. Each job has a loading operation and a processing operation. The loading operation requires a server that serves all the jobs. Each machine has a given set of jobs to process, and the processing sequence is known and fixed. We design a polynomial‐time algorithm to solve the two‐machine case of the problem. When the number of machines is arbitrary, the problem becomes strongly NP‐hard even if all the jobs have the same processing length or all the loading operations require a unit time. We design two heuristic algorithms to treat the case where all the loading times are unit and analyze their performance.     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Signature based analysis of m‐Consecutive‐k‐out‐of‐n: F systems with exchangeable components

In this article, we study reliability properties of m‐consecutive‐k‐out‐of‐n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature‐based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m‐consecutive‐k‐out‐of‐n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Minimizing makespan on a single batch processing machine with nonidentical job sizes

We deal with the problem of minimizing makespan on a single batch processing machine. In this problem, each job has both processing time and size (capacity requirement). The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is just the processing time of the longest job in the batch. An approximation algorithm with worst‐case ratio 3/2 is given for the version where the processing times of large jobs (with sizes greater than 1/2) are not less than those of small jobs (with sizes not greater than 1/2). This result is the best possible unless P = NP. For the general case, we propose an approximation algorithm with worst‐case ratio 7/4. A number of heuristics by Uzosy are also analyzed and compared. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 226–240, 2001     

Approximation schemes for single‐machine scheduling with a fixed maintenance activity to minimize the total amount of late work

We consider the problem of scheduling n independent and simultaneously available jobs without preemption on a single machine, where the machine has a fixed maintenance activity. The objective is to find the optimal job sequence to minimize the total amount of late work, where the late work of a job is the amount of processing of the job that is performed after its due date. We first discuss the approximability of the problem. We then develop two pseudo‐polynomial dynamic programming algorithms and a fully polynomial‐time approximation scheme for the problem. Finally, we conduct extensive numerical studies to evaluate the performance of the proposed algorithms. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 172–183, 2016     

A m‐parallel crane scheduling problem with a non‐crossing constraint

In this paper, we study a m‐parallel machine scheduling problem with a non‐crossing constraint motivated by crane scheduling in ports. We decompose the problem to allow time allocations to be determined once crane assignments are known and construct a backtracking search scheme that manipulates domain reduction and pruning strategies. Simple approximation heuristics are developed, one of which guarantees solutions to be at most two times the optimum. For large‐scale problems, a simulated annealing heuristic that uses random neighborhood generation is provided. Computational experiments are conducted to test the algorithms. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.     

Single machine just‐in‐time scheduling problems with two competing agents

In scheduling problems with two competing agents, each one of the agents has his own set of jobs to be processed and his own objective function, and both share a common processor. In the single‐machine problem studied in this article, the goal is to find a joint schedule that minimizes the total deviation of the job completion times of the first agent from a common due‐date, subject to an upper bound on the maximum deviation of job completion times of the second agent. The problem is shown to be NP‐hard even for a nonrestrictive due‐date, and a pseudopolynomial dynamic program is introduced and tested numerically. For the case of a restrictive due‐date (a sufficiently small due‐date that may restrict the number of early jobs), a faster pseudopolynomial dynamic program is presented. We also study the multiagent case, which is proved to be strongly NP‐hard. A simple heuristic for this case is introduced, which is tested numerically against a lower bound, obtained by extending the dynamic programming algorithm. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 1–16, 2014     

Reliability of a 2‐dimensional k‐within‐consecutive‐r × s‐out‐of‐m × n:F system

A 2‐dimensional rectangular (cylindrical) k‐within‐consecutive‐r × s‐out‐of‐m × n:F system is the rectangular (cylindrical) m × n‐system if the system fails whenever k components in a r × s‐submatrix fail. This paper proposes a recursive algorithm for the reliability of the 2‐dimensional k‐within‐consecutive‐r × s‐out‐m × n:F system, in the rectangular case and the cylindrical case. This algorithm requires min ( O (mk^r(n?s)), O (nk^s(m?r))), and O (mk^rn) computing time in the rectangular case and the cylindrical case, respectively. The proposed algorithm will be demonstrated and some numerical examples will be shown. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 625–637, 2001.