Lawrence M. Wein

Scheduling semiconductor wafer fabrication

The impact that scheduling can have on the performance of semi-conductor wafer fabrication facilities is assessed. The performance measure considered is the mean throughput time (sometimes called cycle time, turnaround time or manufacturing interval) for a lot of wafers. A variety of input control and sequencing rules are evaluated using a simulation model of a representative, but fictitious, semiconductor wafer fabrication. Certain of these scheduling rules are derived by restricting attention to the sub-set of stations that are heavily utilized, and by using a Brownian network model, which approximates a multi-class queuing network model with dynamic control capability. Three versions of the wafer fabrication model, which differ only by the number of servers present at particular stations, are studied. The three versions have one, two, and four stations, respectively, that are heavily utilized (near 90% utilization). The simulation results indicate that scheduling has a significant impact on average throughput time, with larger improvements coming from discretionary imput control than from lot sequencing. The effects that specific sequencing rules have are highly dependent on both the type of input control used and the number of bottleneck stations in the fabrication. >

Emergency response to a smallpox attack: The case for mass vaccination

In the event of a smallpox bioterrorist attack in a large U.S. city, the interim response policy is to isolate symptomatic cases, trace and vaccinate their contacts, quarantine febrile contacts, but vaccinate more broadly if the outbreak cannot be contained by these measures. We embed this traced vaccination policy in a smallpox disease transmission model to estimate the number of cases and deaths that would result from an attack in a large urban area. Comparing the results to mass vaccination from the moment an attack is recognized, we find that mass vaccination results in both far fewer deaths and much faster epidemic eradication over a wide range of disease and intervention policy parameters, including those believed most likely, and that mass vaccination similarly outperforms the existing policy of starting with traced vaccination and switching to mass vaccination only if required.

An Inverse-Optimization-Based Auction Mechanism to Support a Multiattribute RFQ Process

We consider a manufacturer who uses a reverse, or procurement, auction to determine which supplier will be awarded a contract. Each bid consists of a price and a set of non-price attributes (e.g., quality, lead time). The manufacturer is assumed to know the parametric form of the suppliers’ cost functions (in terms of the non-price attributes), but has no prior information on the parameter values. We construct a multi-round open-ascending auction mechanism, where the manufacturer announces a slightly different scoring rule (i.e., a function that ranks the bids in terms of the price and non-price attributes) in each round. Via inverse optimization, the manufacturer uses the bids from the first several rounds to learn the suppliers’ cost functions, and then in the final round chooses a scoring rule that attempts to maximize his own utility. Under the assumption that suppliers submit their myopic best-response bids in the last round, and do not distort their bids in the earlier rounds (i.e., they choose their minimum-cost bid to achieve any given score), our mechanism indeed maximizes the manufacturer’s utility within the open-ascending format. We also discuss several enhancements that improve the robustness of our mechanism with respect to the model’s informational and behavioral assumptions.

Emergency response to an anthrax attack

We developed a mathematical model to compare various emergency responses in the event of an airborne anthrax attack. The system consists of an atmospheric dispersion model, an age-dependent dose–response model, a disease progression model, and a set of spatially distributed two-stage queueing systems consisting of antibiotic distribution and hospital care. Our results underscore the need for the extremely aggressive and timely use of oral antibiotics by all asymptomatics in the exposure region, distributed either preattack or by nonprofessionals postattack, and the creation of surge capacity for supportive hospital care via expanded training of nonemergency care workers at the local level and the use of federal and military resources and nationwide medical volunteers. The use of prioritization (based on disease stage and/or age) at both queues, and the development and deployment of modestly rapid and sensitive biosensors, while helpful, produce only second-order improvements.

Dynamic scheduling of a multiclass make-to-stock queue

Motivated by make-to-stock production systems, we consider a scheduling problem for a single server queue that can process a variety of different job classes. After jobs are processed, they enter a finished goods inventory that services customer demand. The scheduling problem is to dynamically decide which job class, if any, to serve next in order to minimize the long-run expected average cost incurred per unit of time, which includes linear costs which may differ by class for backordering and holding finished goods inventory. Under the heavy traffic condition that the server must be busy the great majority of the time in order to satisfy customer demand, the scheduling problem is approximated by a dynamic control problem involving Brownian motion. The Brownian control problem is solved, and its solution is interpreted in terms of the queueing system to obtain a scheduling policy. A simulation experiment is performed that demonstrates the policys effectiveness.

Analysis of a Forecasting-Production-Inventory System with Stationary Demand

We consider a production stage that produces a single item in a make-to-stock manner. Demand for finished goods is stationary. In each time period, an updated vector of demand forecasts over the forecast horizon becomes available for use in production decisions. We model the sequence of forecast update vectors using the Martingale model of forecast evolution developed by Graves et al. (1986, 1998) and Heath and Jackson (1994). The production stage is modeled as a single-server, discrete-time, continuous-state queue. We focus on a modified base-stock policy incorporating forecast information and use an approximate analysis rooted in heavy traffic theory and random walk theory to obtain a closed-form expression for the (forecast-corrected) base-stock level that minimizes the expected steady-state inventory holding and backorder costs. This expression, which is shown to be accurate under certain conditions in a simulation study, sheds some light on the interrelationships among safety stock, stochastic correlated demand, inaccurate forecasts, and random and capacitated production in forecasting-production-inventory systems.

Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network

We consider a multiclass closed queueing network with two single-server stations. Each class requires service at a particular station, and customers change class after service according to specified probabilities. There is a general service time distribution for each class. The problem is to schedule the two servers to maximize the long-run average throughput of the network. By assuming a large customer population and nearly balanced loading of the two stations, the scheduling problem can be approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved exactly and the solution is interpreted in terms of the queueing network to obtain a scheduling rule. We conjecture, quite naturally, that the resulting scheduling rule is asymptotically optimal under heavy traffic conditions, but no attempt is made to prove that. The scheduling rule is a static priority policy that computes an index for each class and awards higher priority at station 1 respectively, station 2 to classes with the smaller respectively, larger values of this index. An analytical comparison of this rule to any other static policy is also obtained. An example is given that illustrates the procedure and demonstrates its effectiveness.

Quantifying the Routes of Transmission for Pandemic Influenza

Motivated by the desire to assess nonpharmaceutical interventions for pandemic influenza, we seek in this study to quantify the routes of transmission for this disease. We construct a mathematical model of aerosol (i.e., droplet-nuclei) and contact transmission of influenza within a household containing one infected. An analysis of this model in conjunction with influenza and rhinovirus data suggests that aerosol transmission is far more dominant than contact transmission for influenza. We also consider a separate model of a close expiratory event, and find that a close cough is unlikely (≈1% probability) to generate traditional droplet transmission (i.e., direct deposition on the mucous membranes), although a close, unprotected and horizontally-directed sneeze is potent enough to cause droplet transmission. There are insufficient data on the frequency of close expiratory events to assess the relative importance of aerosol transmission and droplet transmission, and it is prudent to leave open the possibility that droplet transmission is important until proven otherwise. However, the rarity of close, unprotected and horizontally-directed sneezes—coupled with the evidence of significant aerosol and contact transmission for rhinovirus and our comparison of hazard rates for rhinovirus and influenza—leads us to suspect that aerosol transmission is the dominant mode of transmission for influenza.

Due-date setting and priority sequencing in a multiclass M/G.1 queue

The problem of simultaneous due-date setting and priority sequencing is analyzed in the setting of a multiclass M/G/1 queueing system. The objective is to minimize the weighted average due-date lead time (due-date minus arrival date) of jobs subject to a constraint on either the fraction of tardy jobs or the average job tardiness. Several parametric and nonparametric due-date setting policies are proposed that depend on the class of arriving job, the state of the queueing system at the time of the jobs arrival, and the sequencing policy (the weighted shortest expected processing time rule) that is used. In a simulation experiment performed on a two-class M/M/1 system, these policies outperformed traditional due-date setting policies, and due-date setting had a larger impact on performance than priority sequencing.

Optimal scheduling of radiotherapy and angiogenic inhibitors

We incorporate a previously validated mathematical model of a vascularized tumor into an optimal control problem to determine the temporal scheduling of radiotherapy and angiogenic inhibitors that maximizes the control of a primary tumor. Our results reveal that optimal antiangiogenic monotherapy gives a large initial injection to attain a 20: 1 ratio of tumor cell volume to supporting vasculature volume. It thereafter maintains this 20: 1 ratio via a continuous dose rate that is intensified over time. The optimal radiation monotherapy schedule is characterized by amodest dose intensification over time. The best performance is achieved by our optimal combination regimen, where the antiangiogenic treatment again maintains a constant tumor-to-vasculature ratio, but is administered in a dose-intensified manner only during the latter portion of the radiation fractionation schedule.