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.     

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

We consider the problem of scheduling N jobs on M parallel machines so as to minimize the maximum earliness or tardiness cost incurred for each of the jobs. Earliness and tardiness costs are given by general (but job-independent) functions of the amount of time a job is completed prior to or after a common due date. We show that in problems with a nonrestrictive due date, the problem decomposes into two parts. Each of the M longest jobs is assigned to a different machine, and all other jobs are assigned to the machines so as to minimize their makespan. With these assignments, the individual scheduling problems for each of the machines are simple to solve. We demonstrate that several simple heuristics of low complexity, based on this characterization, are asymptotically optimal under mild probabilistic conditions. We develop attractive worst-case bounds for them. We also develop a simple closed-form lower bound for the minimum cost value. The bound is asymptotically accurate under the same probabilistic conditions. In the case where the due date is restrictive, the problem is more complex only in the sense that the set of initial jobs on the machines is not easily characterized. However, we extend our heuristics and lower bounds to this general case as well. Numerical studies exhibit that these heuristics perform excellently even for small- or moderate-size problems both in the restrictive and nonrestrictive due-date case. © 1997 John Wiley & Sons, Inc.     

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.     

Minimizing the sum of absolute lateness in single-machine and multimachine scheduling

Kanet addressed the problem of scheduling n jobs on one machine so as to minimize the sum of absolute lateness under a restrictive assumption on their common due date. This article extends the results to the problem of scheduling n jobs on m parallel identical processors in order to minimize the sum of absolute lateness. Also, a heuristic algorithm for a more general version with no restriction on the common due date, for the problem of n-job single-machine scheduling is presented and its performance is reported.     

Single- and multiple-processor models for minimizing completion time variance

This article concerns the scheduling of n jobs around a common due date, so as to minimize the average total earliness plus total lateness of the jobs. Optimality conditions for the problem are developed, based on its equivalence to an easy scheduling problem. It seems that this problem inherently has a huge number of optimal solutions and an algorithm is developed to find many of them. The model is extended to allow for the availability of multiple parallel processors and an efficient algorithm is developed for that problem. In this more general case also, the algorithm permits great flexibility in finding an optimal schedule.     

Scheduling stochastic jobs with asymmetric earliness and tardiness penalties

We consider a stochastic counterpart of the well-known earliness-tardiness scheduling problem with a common due date, in which n stochastic jobs are to be processed on a single machine. The processing times of the jobs are independent and normally distributed random variables with known means and known variances that are proportional to the means. The due dates of the jobs are random variables following a common probability distribution. The objective is to minimize the expectation of a weighted combination of the earliness penalty, the tardiness penalty, and the flow-time penalty. One of our main results is that an optimal sequence for the problem must be V-shaped with respect to the mean processing times. Other characterizations of the optimal solution are also established. Two algorithms are proposed, which can generate optimal or near-optimal solutions in pseudopolynomial time. The proposed algorithms are also extended to problems where processing times do not satisfy the assumption in the model above, and are evaluated when processing times follow different probability distributions, including general normal (without the proportional relation between variances and means), uniform, Laplace, and exponential. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44, 531–557, 1997.     

A proof for the longest-job-first policy in one-machine scheduling

We consider a one-machine scheduling problem with earliness and tardiness penalties. All jobs are assigned a common due date and the objective is to minimize the total penalty due to job earliness and tardiness. We are interested in finding the optimal combination of the common due-date value and the job sequence. Despite the fact that this problem in general is very hard to solve, we prove that there exists at least a common property for all optimal solutions: The first job in an optimal sequence is one of the longest jobs. We also prove that this property holds for a general class of unimodal penalty functions.     

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.     

Due-date assignment and early/tardy scheduling on identical parallel machines

This article examines the problem of simultaneously assigning a common due date to a set of independent jobs and scheduling them on identical parallel machines in such a way that the costs associated with the due date and with the earliness or tardiness of the jobs are minimized. We establish that, for certain values of the due-date cost, an optimal schedule for this problem is also optimal for an early/tardy scheduling problem studied by Emmons. We discuss the solution properties for the two problems, and show that both problems are NP-hard even for two machines. We further show that these problems become strongly NP-hard if the number of machines is allowed to be arbitrary. We provide a dynamic programming solution for the problems, the complexity of which indicates that the problems can be solved in pseudopolynomial time as long as the number of machines remains fixed. Finally, we present the results of a limited computational study. © 1994 John Wiley & Sons, Inc.     

Integrated production scheduling and preventive maintenance planning for a single machine under a cumulative damage failure process

This paper finds the optimal integrated production schedule and preventive maintenance plan for a single machine exposed under a cumulative damage process, and investigates how the optimal preventive maintenance plan interacts with the optimal production schedule. The goal is to minimize the total tardiness. The optimal policy possesses the following properties: Under arbitrary maintenance plan when jobs have common processing time, and different due dates, the optimal production schedule is to order the jobs by earliest due date first rule; and when jobs have common due date and different processing times, the optimal production schedule is shortest processing time first. The optimal maintenance plan is of control limit type under any arbitrary production schedule when machine is exposed under a cumulative damage failure process. Numerical studies on the optimal maintenance control limit of the maintenance plan indicate that as the number of jobs to be scheduled increases, the effect of jobs due dates on the optimal maintenance control limit diminishes. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Minimizing weighted earliness and tardiness penalties in single‐machine scheduling with idle time permitted

In this paper, a single‐machine scheduling problem with weighted earliness and tardiness penalties is considered. Idle time between two adjacent jobs is permitted and due dates of jobs could be unequal. The dominance rules are utilized to develop a relationship matrix, which allows a branch‐and‐bound algorithm to eliminate a high percentage of infeasible solutions. After combining this matrix with a branching strategy, a procedure to solve the problem is proposed. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 760–780, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10039     

Scheduling multiple products on parallel machines with setup costs

We consider a class of production scheduling models with m identical machines in parallel and k different product types. It takes a time p_i to produce one unit of product type i on any one of the machines. There is a demand stream for product type i consisting of n_i units with each unit having a given due date. Before a machine starts with the production of a batch of products of type i a setup cost c is incurred. We consider several different objective functions. Each one of the objective functions has three components, namely a total setup cost, a total earliness cost, and a total tardiness cost. In our class of problems we find a relatively large number of problems that can be solved either in polynomial time or in pseudo‐polynomial time. The polynomiality or pseudo‐polynomiality is achieved under certain special conditions that may be of practical interest; for example, a regularity pattern in the string of due dates combined with earliness and tardiness costs that are similar for different types of products. The class of models we consider includes as special cases discrete counterparts of a number of inventory models that have been considered in the literature before, e.g., Wagner and Whitin (Manage Sci 5 (1958), 89–96) and Zangwill (Oper Res 14 (1966), 486–507; Manage Sci 15 (1969), 506–527). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Vehicle routing problems with regular objective functions on a path

We investigate the problem of scheduling a fleet of vehicles to visit the customers located on a path to minimize some regular function of the visiting times of the customers. For the single‐vehicle problem, we prove that it is pseudopolynomially solvable for any minsum objective and polynomially solvable for any minmax objective. Also, we establish the NP‐hardness of minimizing the weighted number of tardy customers and the total weighted tardiness, and present polynomial algorithms for their special cases with a common due date. For the multivehicle problem involving n customers, we show that an optimal solution can be found by solving or O(n) single‐vehicle problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 34–43, 2014     

A hybrid algorithm for the one machine sequencing problem to minimize total tardiness

In a recent paper, Hamilton Emmons has established theorems relating to the order in which pairs of jobs are to be processed in an optimal schedule to minimize the total tardiness of performing n jobs on one machine. Using these theorems, the algorithm of this paper determines the precedence relationships among pairs of jobs (whenever possible) and eliminates the first and the last few jobs in an optimal sequence. The remaining jobs are then ordered by incorporating the precedence relationships in a dynamic programming framework. Propositions are proved which considerably reduce the total computation involved in the dynamic programming phase. Computational results indicate that the solution time goes up less than linearly with the size (n) of the problem. The median solution time for solving 50 job problems was 0.36 second on UNIVAC 1108 computer.     

CON/SLK due date assignment and scheduling on a single machine with two agents

We consider scheduling problems involving two agents (agents A and B), each having a set of jobs that compete for the use of a common machine to process their respective jobs. The due dates of the A‐jobs are decision variables, which are determined by using the common (CON) or slack (SLK) due date assignment methods. Each agent wants to minimize a certain performance criterion depending on the completion times of its jobs only. Under each due date assignment method, the criterion of agent A is always the same, namely an integrated criterion consisting of the due date assignment cost and the weighted number of tardy jobs. Several different criteria are considered for agent B, including the maxima of regular functions (associated with each job), the total (weighted) completion time, and the weighted number of tardy jobs. The overall objective is to minimize the performance criterion of agent A, while keeping the objective value of agent B no greater than a given limit. We analyze the computational complexity, and devise polynomial or pseudo‐polynomial dynamic programming algorithms for the considered problems. We also convert, if viable, any of the devised pseudopolynomial dynamic programming algorithms into a fully polynomial‐time approximation scheme. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 416–429, 2016     

On scheduling a single machine to minimize a piecewise linear objective function: A compact MIP formulation

We study the scheduling situation in which a set of jobs subjected to release dates and deadlines are to be performed on a single machine. The objective is to minimize a piecewise linear objective function ∑_jF_j where F_j(C_j) corresponds to the cost of the completion of job j at time C_j. This class of function is very large and thus interesting both from a theoretical and practical point of view: It can be used to model total (weighted) completion time, total (weighted) tardiness, earliness and tardiness, etc. We introduce a new Mixed Integer Program (MIP) based on time interval decomposition. Our MIP is closely related to the well‐known time‐indexed MIP formulation but uses much less variables and constraints. Experiments on academic benchmarks as well as on real‐life industrial problems show that our generic MIP formulation is efficient. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early-tardy penalties

We examine the problem of scheduling n jobs with a common due date on a single machine. The processing time of each job is a random variable, which follows an arbitrary distribution with a known mean and a known variance. The machine is not reliable; it is subject to stochastic breakdowns. The objective is to minimize the expected sum of squared deviations of job completion times from the due date. Two versions of the problem are addressed. In the first one the due date is a given constant, whereas in the second one the due date is a decision variable. In each case, a general form of the deterministic equivalent of the stochastic scheduling problem is obtained when the counting process related to the machine uptime distribution is a generalized Poisson process. A sufficient condition is derived under which optimal sequences are V-shaped with respect to mean processing times. Other characterizations of optimal solutions are also established. Based on the optimality properties, algorithms with pseudopolynomial time complexity are proposed to solve both versions of the problem. © 1996 John Wiley & Sons, Inc.     

Minimizing mean tardiness and earliness in single-machine scheduling problems with unequal due dates

We consider a single-machine scheduling problem with the objective of minimizing the mean (or equivalently, total) tardiness and earliness when due dates may differ among jobs. Some properties of the optimal solution are discussed, and these properties are used to develop both optimal and heuristic algorithms. Results of computational tests indicate that optimal solutions can be found for problems with up to 20 jobs, and that two of the heuristic procedures provide optimal or very near optimal solutions in many instances. © 1994 John Wiley & Sons, Inc.     

Single batch machine scheduling with deliveries

We consider the problem of scheduling a set of n jobs on a single batch machine, where several jobs can be processed simultaneously. Each job j has a processing time p_j and a size s_j. All jobs are available for processing at time 0. The batch machine has a capacity D. Several jobs can be batched together and processed simultaneously, provided that the total size of the jobs in the batch does not exceed D. The processing time of a batch is the largest processing time among all jobs in the batch. There is a single vehicle available for delivery of the finished products to the customer, and the vehicle has capacity K. We assume that K = rD, where and r is an integer. The travel time of the vehicle is T; that is, T is the time from the manufacturer to the customer. Our goal is to find a schedule of the jobs and a delivery plan so that the service span is minimized, where the service span is the time that the last job is delivered to the customer. We show that if the jobs have identical sizes, then we can find a schedule and delivery plan in time such that the service span is minimum. If the jobs have identical processing times, then we can find a schedule and delivery plan in time such that the service span is asymptotically at most 11/9 times the optimal service span. When the jobs have arbitrary processing times and arbitrary sizes, then we can find a schedule and delivery plan in time such that the service span is asymptotically at most twice the optimal service span. We also derive upper bounds of the absolute worst‐case ratios in both cases. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 470–482, 2015     

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