Note: “Simultaneous optimization of efficiency and performance balance measures in single-machine scheduling problems”

n independent jobs are to be scheduled nonpreemptively on a single machine so as to minimize some performance measure. Federgruen and Mosheiov [2] show that a large class of such scheduling problems can be optimized by solving either a single instance or a finite sequence of instances of the so-called SQC problem, in which all the jobs have a fixed or controllable common due date and the sum of general quasiconvex functions of the job completion times is to be minimized. In this note we point out that this is not always true. In particular, we show that the algorithm proposed in [2] does not always find a global optimal schedule to the problem of minimizing the weighted sum of the mean and variance of job completion times. © 1996 John Wiley & Sons, Inc.     

The single-machine absolute-deviation early-tardy problem with random completion times

We consider sequencing n jobs on a single machine subject to job completion times arising from either machine breakdowns or other causes. The objective is to minimize an expected weighted combination of due dates, completion times, earliness, and tardiness penalties. The determination of optimal distinct due dates or optimal common due dates for a given schedule is investigated. The scheduling problem for a fixed common due date is considered when random completion times arise from machine breakdowns. The optimality of a V-shaped about (a point) T sequence is established when the number of machine breakdowns follows either a Poisson or a geometric distribution and the duration of a breakdown has an exponential distribution. © 1996 John Wiley & Sons, Inc.     

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.     

Single machine parallel batch scheduling subject to precedence constraints

We consider the single machine parallel batch scheduling problems to minimize makespan and total completion time, respectively, under precedence relations. The complexities of these two problems are reported as open in the literature. In this paper, we settle these open questions by showing that both problems are strongly NP‐hard, even when the precedence relations are chains. When the processing times of jobs are directly agreeable or inversely agreeable with the precedence relations, there is an O(n²) time algorithm to minimize the makespan. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

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.     

Scheduling maintenance and semiresumable jobs on a single machine

The majority of scheduling literature assumes that the machines are available at all times. In this paper, we study single machine scheduling problems where the machine maintenance must be performed within certain intervals and hence the machine is not available during the maintenance periods. We also assume that if a job is not processed to completion before the machine is stopped for maintenance, an additional setup is necessary when the processing is resumed. Our purpose is to schedule the maintenance and jobs to minimize some performance measures. The objective functions that we consider are minimizing the total weighted job completion times and minimizing the maximum lateness. In both cases, maintenance must be performed within a fixed period T, and the time for the maintenance is a decision variable. In this paper, we study two scenarios concerning the planning horizon. First, we show that, when the planning horizon is long in relation to T, the problem with either objective function is NP-complete, and we present pseudopolynomial time dynamic programming algorithms for both objective functions. In the second scenario, the planning horizon is short in relation to T. However, part of the period T may have elapsed before we schedule any jobs in this planning horizon, and the remaining time before the maintenance is shorter than the current planning horizon. Hence we must schedule one maintenance in this planning horizon. We show that the problem of minimizing the total weighted completion times in this scenario is NP-complete, while the shortest processing time (SPT) rule and the earliest due date (EDD) rule are optimal for the total completion time problem and the maximum lateness problem respectively. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 845–863, 1999     

Scheduling about a large common due date with tolerance to minimize mean absolute deviation of completion times

In this article the problem of minimizing the mean absolute deviation (MAD) of job completion times about an unrestricted given common due date with tolerance in the n-job, single-machine scheduling environment is considered. We describe some optimality conditions and show that the unrestricted version of the MAD problem with an arbitrary due date tolerance is polynomial by proposing a polynomial-time algorithm for identifying an optimal schedule. © 1994 John Wiley & Sons, Inc.     

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.     

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     

A note on combined job selection and sequencing problems

We investigate the solvability of two single‐machine scheduling problems when the objective is to identify among all job subsets with cardinality k,1≤k≤n, the one that has the minimum objective function value. For the single‐machine minimum maximum lateness problem, we conclude that the problem is solvable in O(n²) time using the proposed REMOVE algorithm. This algorithm can also be used as an alternative to Moore's algorithm to solve the minimum number of tardy jobs problem by actually solving the hierarchical problem in which the objective is to minimize the maximum lateness subject to the minimum number of tardy jobs. We then show that the REMOVE algorithm cannot be used to solve the general case of the single‐machine total‐weighted completion time problem; we derive sufficient conditions among the job parameters so that the total weighted completion time problem becomes solvable in O(n²) time. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 449–453, 2013     

Lower bounds for single-machine scheduling problems

This article addresses deterministic, nonpreemptive scheduling of n jobs with unequal release times on a single machine to minimize the sum of job completion times. This problem is known to be NP-hard. The article compares six available lower bounds in the literature and shows that the lower bound based on the optimal solution to the preemptive version of the problem is the dominant lower bound.     

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     

Single-machine scheduling with early and tardy completion costs

We address a single-machine scheduling problem in which penalties are assigned for early and tardy completion of jobs. These penalties are common in industrial settings where early job completion can cause the cash commitment to resources in a time frame earlier than needed, giving rise to early completion penalties. Tardiness penalties arise from a variety of sources, such as loss of customer goodwill, opportunity costs of lost sales, and direct cash penalties. Accounting for earliness cost makes the performance measure nonregular, and this nonregularity has apparently discouraged researchers from seeking solutions to this problem. We found that it is not much more difficult to design an enumerative search for this problem than it would be if the performance measure were regular. We present and demonstrate an efficient timetabling procedure which can be embedded in an enumerative algorithm allowing the search to be conducted over the domain of job permutations.© 1993 John Wiley & Sons, Inc.     

Minimizing the weighted sum of quadratic completion times on a single machine

This article discusses the scheduling problem of minimizing the weighted sum of quadratic completion times on a single machine. It establishes links between orderings of adjacent and nonadjacent jobs that lead to a powerful branch and bound method. Computational results show that this method clearly outperforms the state of the art algorithm. © 1995 John Wiley & Sons. Inc.     

Flowshop/no-idle or no-wait scheduling to minimize the sum of completion times

This paper deals with flowshop/sum of completion times scheduling problems, working under a “no-idle” or a “no-wait” constraint, the former prescribes for the machines to work continuously without idle intervals and the latter for the jobs to be processed continuously without waiting times between consecutive machines. Under either of the constraints the problem is unary NP-Complete for two machines. We prove some properties of the optimal schedule for n/2/F, no-idle/σC_i. For n/m/P, no-idle/σC_i, and n/m/P, no-wait/σC_i, with an increasing or decreasing series of dominating machines, we prove theorems that are the basis for polynomial bounded algorithms. All theorems are demonstrated numerically.     

Scheduling with resource-dependent release dates—a comparison of two different resource consumption functions

We consider the problem of scheduling a set of jobs on a single machine where the release time of a job is related to the amount of resource consumed. The objective is to minimize the total resource consumption with a control on the completion times of the jobs. Four different variants of the problem are studied: (i) minimization of the total resource consumption subject to a common deadline for all jobs, (ii) minimization of the total resource consumption subject to a constraint on the total completion time of the jobs, (iii) minimization of the weighted total resource consumption and maximum job completion time, and (iv) minimization of the weighted total resource consumption and the total job completion time. We compare the common resource consumption function with the function where the resource consumed is proportional to the processing time of the job. We show that these two different resource consumption functions can give rise to very different solution methods and different computational complexities for the problem. © 1994 John Wiley & Sons, Inc.     

Single-machine scheduling with dynamic arrivals: Decomposition results and an improved algorithm

This article considers the single-machine dynamic scheduling problem where the jobs have different arrival times and the objective is to minimize the sum of completion times. This problem is known to be strongly NP-hard. We develop decomposition results for this problem such that a large problem can be solved by combining optimal solutions for several smaller problems. The decomposition results can be used with any implicit enumeration method to develop an optimal algorithm. Our computational experiment indicates that the computational efficiency of the currently best available branch-and-bound algorithm can be improved with the use of our decomposition results. © 1996 John Wiley & Sons, Inc.     

Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date

We consider a single-machine scheduling problem in which all jobs have the same due date and penalties are assessed for both early and late completion of jobs. However, earliness and tardiness are penalized at different rates. The scheduling objective is to minimize either the weighted sum of absolute deviations (WSAD) or the weighted sum of squared deviations (WSSD). For each objective we consider two versions of the problem. In the unconstrained version an increase in the due date does not yield any further decrease in the objective function. We present a constructive algorithm for the unconstrained WSAD problem and show that this problem is equivalent to the two-parallel, nonidentical machine, mean flow-time problem. For the unconstrained WSSD and the constrained WSAD and WSSD problems we propose implicit enumeration procedures based on several dominance conditions. We also report on our computational experience with the enumeration procedures.     

An evaluation of scheduling heuristics for dynamic single-processor scheduling with early/tardy costs

In an endeavor to broaden the application of scheduling models to decisions involving the use of a manager's time we use simulation to investigate the performance of a number of simple algorithms (including eight priority rules and a construction heuristic) in a dynamic setting with tasks arriving (randomly) and scheduling decisions being made, over time. We compare these simple methods relative to a bound that uses an adjacent pairwise interchange algorithm. We model uncertainty in task durations, and costs being incurred for early and tardy task completion (representative of JIT settings). In addition to evaluating the efficacy of the scheduling rules and various preemption strategies (using ANOVA), we highlight the managerial implications of the effects of eight environmental parameters. © 1996 John Wiley & Sons, Inc.     

Optimization of schedule stability and efficiency under processing time variability and random machine breakdowns in a job shop environment

The ability to cope with uncertainty in dynamic scheduling environments is becoming an increasingly important issue. In such environments, any disruption in the production schedule will translate into a disturbance of the plans for several external activities as well. Hence, from a practical point of view, deviations between the planned and realized schedules are to be avoided as much as possible. The term stability refers to this concern. We propose a proactive approach to generate efficient and stable schedules for a job shop subject to processing time variability and random machine breakdowns. In our approach, efficiency is measured by the makespan, and the stability measure is the sum of the variances of the realized completion times. Because the calculation of the original measure is mathematically intractable, we develop a surrogate stability measure. The version of the problem with the surrogate stability measure is proven to be NP‐hard, even without machine breakdowns; a branch‐and‐bound algorithm is developed for this problem variant. A tabu search algorithm is proposed to handle larger instances of the problem with machine breakdowns. The results of extensive computational experiments indicate that the proposed algorithms are quite promising in performance. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011