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.     

Minimization of expected variance of completion times on single machine for stochastic jobs

This article deals with the problem of scheduling jobs with random processing times on single machine in order to minimize the expected variance of job completion times. Sufficient conditions for the existence of V-shaped optimal sequences are derived separately for general and ordered job processing times. It is shown that when coefficient of variation of random processing times are bounded by a certain value, an optimal sequence is V-shaped. © 1997 John Wiley & Sons, Inc.     

On n/1/F̄ dynamic deterministic problems

We consider sequencing of n jobs which will arrive intermittently and are to be processed on a single machine; the arrival and the processing times of each jobs are assumed known. A schedule is to be developed that minimizes the mean flow time. Two models are considered: (i) when no pre-emption or inserted idle time is allowed in the schedule, and (ii) when pre-emption is allowed but the jobs follow a pre-empt-repeat discipline We illustrate that Cobham's and Phipp's SPT dispatching rule does not guarantee the optimum F? even for the non-preemptive model We propose a branch and bound algorithm for both models and discuss our computational experience We also examine the relative performances of the optimum nonpre-emptive sequence, and the optimum pre-empt-repeat sequence over that resulting from SPT dispatching rule on a large number of sets of jobs of varying sizes and tightness.     

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     

Acceptance sampling,imperfect production,and the optimality of zero defects

Acceptance sampling is often used to monitor the quality of raw materials and components when product testing is destructive, time-consuming, or expensive. In this paper we consider the effect of a buyer-imposed acceptance sampling policy on the optimal batch size and optimal quality level delivered by an expected cost minimizing supplier. We define quality as the supplier's process capability, i.e., the probability that a unit conforms to all product specifications, and we assume that unit cost is an increasing function of the quality level. We also assume that the supplier faces a known and constant “pass-through” cost, i.e., a fixed cost per defective unit passed on to the buyer. We show that the acceptance sampling plan has a significant impact on the supplier's optimal quality level, and we derive the conditions under which zero defects (100% conformance) is the policy that minimizes the supplier's expected annual cost. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 515–530, 1997     

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     

Johnson's approximate method for the 3 × n job shop problem

The effectiveness of Johnson's Approximate Method (JAM) for the 3 × n job shop scheduling problems was examined on 1,500 test cases with n ranging from 6 to 50 and with the processing times A_i, B_i, C_i (for item i on machines A, B, C) being uniformly and normally distributed. JAM proved to be quite effective for the case B_i ? max (A_i, C_i) and optimal for B_i, ? min (A_i, C_i).     

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     

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     

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.     

Mathematical aspects of the 3 × n job-shop sequencing problem

The paper discusses mathematical properties of the well-known Bellman-Johnson 3 × n sequencing problem. Optimal rules for some special cases are developed. For the case min B_i ≥ maxA_j we find an optimal sequence of the 2 × n problem for machines B and C and move one item to the front of the sequence to minimize (7); when min B_i ≥ max C_j we solve a 2 × n problem for machines A and B and move one item to the end of the optimal sequence so as to minimize (9). There is also given a sufficient optimality condition for a solution obtained by Johnson's approximate method. This explains why this method so often produces an optimal solution.     

To wait or not to wait: Optimal ordering under lead time uncertainty and forecast updating

There has been a dramatic increase over the past decade in the number of firms that source finished product from overseas. Although this has reduced procurement costs, it has increased supply risk; procurement lead times are longer and are often unreliable. In deciding when and how much to order, firms must consider the lead time risk and the demand risk, i.e., the accuracy of their demand forecast. To improve the accuracy of its demand forecast, a firm may update its forecast as the selling season approaches. In this article we consider both forecast updating and lead time uncertainty. We characterize the firm's optimal procurement policy, and we prove that, with multiplicative forecast revisions, the firm's optimal procurement time is independent of the demand forecast evolution but that the optimal procurement quantity is not. This leads to a number of important managerial insights into the firm's planning process. We show that the firm becomes less sensitive to lead time variability as the forecast updating process becomes more efficient. Interestingly, a forecast‐updating firm might procure earlier than a firm with no forecast updating. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Optimal betting strategies for favorable games

We examine the problem of a gambler interested in maximizing the expected value of a convex utility function of his fortune after n plays of a game. We allow any probability distribution to rule the outcome of each play, and this distribution may change from play to play according to a Markov process. We present results regarding the existence of an optimal policy and its structural dependence on the gambler's fortune. The well-known results of Bellman and Kalaba for exponential and logarithmic utility functions and coin-tossing games are generalized. We also examine the situation of general stale spaces and show that the same structural results hold.     

A dynamic model for component testing

We consider the component testing problem of a system where the main feature is that the component failure rates are not constant parameters, but they change in a dynamic fashion with respect to time. More precisely, each component has a piecewise-constant failure-rate function such that the lifetime distribution is exponential with a constant rate over local intervals of time within the overall mission time. There are several such intervals, and the rates change dynamically from one interval to another. We note that these lifetime distributions can also be used in a more general setting to approximate arbitrary lifetime distributions. The optimal component testing problem is formulated as a semi-infinite linear program. We present an algorithmic procedure to compute optimal test times based on the column-generation technique and illustrate it with a numerical example. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 187–197, 1997     

Optimal pricing for a short life‐cycle product when customer price‐sensitivity varies over time

Technology products often experience a life‐cycle demand pattern that resembles a diffusion process, with weak demand in the beginning and the end of the life cycle and high demand intensity in between. The customer price‐sensitivity also changes over the life cycle of the product. We study the prespecified pricing decision for a product that exhibits such demand characteristics. In particular, we determine the optimal set of discrete prices and the times to switch from one price to another, when a limited number of price changes are allowed. Our study shows that the optimal prices and switching times show interesting patterns that depend on the product's demand pattern and the change in the customers' price sensitivity over the life cycle of the product. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Markov decision processes with restricted observations: Finite horizon case

In this article we consider a Markov decision process subject to the constraints that result from some observability restrictions. We assume that the state of the Markov process under consideration is unobservable. The states are grouped so that the group that a state belongs to is observable. So, we want to find an optimal decision rule depending on the observable groups instead of the states. This means that the same decision applies to all the states in the same group. We prove that a deterministic optimal policy exists for the finite horizon. An algorithm is developed to compute policies minimizing the total expected discounted cost over a finite horizon. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44 : 439–456, 1997     

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     

Information disclosure with endogenous channel structure

This article investigates the firms' optimal quality information disclosure strategies in a supply chain, wherein the supplier may encroach into the retail channel to sell products directly to end consumers. We consider two disclosure formats, namely, retailer disclosure (R-C) and supplier disclosure (S-C), and examine the optimal disclosure format from each firm's perspective. We show that either firm prefers to delegate the disclosure option to its partner when the supplier cannot encroach. However, the threat of supplier encroachment dramatically alters the firm's preference of disclosure. The supplier may prefer the S-C format to the R-C format when the entry cost is low and the disclosure cost is high to achieve a higher quality information transparency. Meanwhile, the retailer may prefer the R-C format to the S-C format when the entry cost is intermediate to deter the possible encroachment of the supplier. In this sense, the firms' preferences of disclosure format can be aligned due to the threat of supplier encroachment. The consumer surplus is always higher under the S-C format while either disclosure format can lead to a higher social welfare. We also consider an alternative scenario under which the supplier encroaches after the product quality information is disclosed. An interesting observation appears that the supplier may encroach when the product quality is low but foregoes encroachment when the product quality gets higher.     

Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem

Scheduling a set of n jobs on a single machine so as to minimize the completion time variance is a well‐known NP‐hard problem. In this paper, we propose a sequence, which can be constructed in O(n log n) time, as a solution for the problem. Our primary concern is to establish the asymptotical optimality of the sequence within the framework of probabilistic analysis. Our main result is that, when the processing times are randomly and independently drawn from the same uniform distribution, the sequence is asymptotically optimal in the sense that its relative error converges to zero in probability as n increases. Other theoretical results are also derived, including: (i) When the processing times follow a symmetric structure, the problem has 2^{⌊(n−1)/2⌋} optimal sequences, which include our proposed sequence and other heuristic sequences suggested in the literature; and (ii) when these 2^{⌊(n−1)/2⌋} sequences are used as approximate solutions for a general problem, our proposed sequence yields the best approximation (in an average sense) while another sequence, which is commonly believed to be a good approximation in the literature, is interestingly the worst. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 373–398, 1999     

Scheduling jobs that are subject to failure propagation

A single machine is available to process a collection of stochastic tasks. Processing is interrupted when the machine breaks down. We introduce a new model of breakdowns that more realistically incorporates the effects that job processing may have on the machine. This failure propagation model is equivalent to a Bayesian formulation in which learning about breakdown rates occurs as jobs evolve. Optimal scheduling policies are described in terms of Gittins indices and these indices are characterised in two special cases. For example, we obtain conditions which ensure that an optimal policy will only preempt a job's processing either at its completion or at a machine breakdown. We also bound the value lost by simplistic modelling which ignores the learning phenomenon. © 1996 John Wiley & Sons, Inc.