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 equivalence between the proportionate flow shop and single‐machine scheduling problems

We derive sufficient conditions which, when satisfied, guarantee that an optimal solution for a single‐machine scheduling problem is also optimal for the corresponding proportionate flow shop scheduling problem. We then utilize these sufficient conditions to show the solvability in polynomial time of numerous proportionate flow shop scheduling problems with fixed job processing times, position‐dependent job processing times, controllable job processing times, and also problems with job rejection. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 595–603, 2015     

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     

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     

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     

A single-machine scheduling problem with random processing times

We consider the problem of scheduling n jobs with random processing times on a single machine in order to minimize the expected variance of the completion times. We prove a number of results, including one to the effect that the optimal schedule must be V shaped when the jobs have identical means or variances or have exponential processing times.     

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.     

Single‐machine scheduling with deadlines to minimize the total weighted late work

We consider scheduling a set of jobs with deadlines to minimize the total weighted late work on a single machine, where the late work of a job is the amount of processing of the job that is scheduled after its due date and before its deadline. This is the first study on scheduling with the late work criterion under the deadline restriction. In this paper, we show that (i) the problem is unary NP‐hard even if all the jobs have a unit weight, (ii) the problem is binary NP‐hard and admits a pseudo‐polynomial‐time algorithm and a fully polynomial‐time approximation scheme if all the jobs have a common due date, and (iii) some special cases of the problem are polynomially solvable.     

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     

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 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     

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     

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.     

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     

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     

An exact branch‐and‐price algorithm for multitasking scheduling on unrelated parallel machines

We consider the multitasking scheduling problem on unrelated parallel machines to minimize the total weighted completion time. In this problem, each machine processes a set of jobs, while the processing of a selected job on a machine may be interrupted by other available jobs scheduled on the same machine but unfinished. To solve this problem, we propose an exact branch‐and‐price algorithm, where the master problem at each search node is solved by a novel column generation scheme, called in‐out column generation, to maintain the stability of the dual variables. We use a greedy heuristic to obtain a set of initial columns to start the in‐out column generation, and a hybrid strategy combining a genetic algorithm and an exact dynamic programming algorithm to solve the pricing subproblems approximately and exactly, respectively. Using randomly generated data, we conduct numerical studies to evaluate the performance of the proposed solution approach. We also examine the effects of multitasking on the scheduling outcomes, with which the decision maker can justify making investments to adopt or avoid multitasking.     

Single machine scheduling with a restricted rate‐modifying activity

In this paper we consider the problem of scheduling a set of jobs on a single machine on which a rate‐modifying activity may be performed. The rate‐modifying activity is an activity that changes the production rate of the machine. So the processing time of a job is a variable, which depends on whether it is scheduled before or after the rate‐modifying activity. We assume that the rate‐modifying activity can take place only at certain predetermined time points, which is a constrained case of a similar problem discussed in the literature. The decisions under consideration are whether and when to schedule the rate‐modifying activity, and how to sequence the jobs in order to minimize some objectives. We study the problems of minimizing makespan and total completion time. We first analyze the computational complexity of both problems for most of the possible versions. The analysis shows that the problems are NP‐hard even for some special cases. Furthermore, for the NP‐hard cases of the makespan problem, we present a pseudo‐polynomial time optimal algorithm and a fully polynomial time approximation scheme. For the total completion time problem, we provide a pseudo‐polynomial time optimal algorithm for the case with agreeable modifying rates. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A branch‐and‐price approach for the stochastic generalized assignment problem

In this article, we address a stochastic generalized assignment machine scheduling problem in which the processing times of jobs are assumed to be random variables. We develop a branch‐and‐price (B&P) approach for solving this problem wherein the pricing problem is separable with respect to each machine, and has the structure of a multidimensional knapsack problem. In addition, we explore two other extensions of this method—one that utilizes a dual‐stabilization technique and another that incorporates an advanced‐start procedure to obtain an initial feasible solution. We compare the performance of these methods with that of the branch‐and‐cut (B&C) method within CPLEX. Our results show that all B&P‐based approaches perform better than the B&C method, with the best performance obtained for the B&P procedure that includes both the extensions aforementioned. We also utilize a Monte Carlo method within the B&P scheme, which affords the use of a small subset of scenarios at a time to estimate the “true” optimal objective function value. Our experimental investigation reveals that this approach readily yields solutions lying within 5% of optimality, while providing more than a 10‐fold savings in CPU times in comparison with the best of the other proposed B&P procedures. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 131–143, 2014     

Machine scheduling with pickup and delivery

The coordination of production, supply, and distribution is an important issue in logistics and operations management. This paper develops and analyzes a single‐machine scheduling model that incorporates the scheduling of jobs and the pickup and delivery arrangements of the materials and finished jobs. In this model, there is a capacitated pickup and delivery vehicle that travels between the machine and the storage area, and the objective is to minimize the makespan of the schedule. The problem is strongly NP‐hard in general but is solvable in polynomial time when the job processing sequence is predetermined. An efficient heuristic is developed for the general problem. The effectiveness of the heuristic is studied both analytically and computationally. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

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.