We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization, one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to singlestage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than single-stage optimization.For network flow and design under demand uncertainty, we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is -hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable.Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed "budget for demand uncertainty." Using a budget of uncertainty, we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector.We generalize the approach to multicommodity network flow and design, and give applications to lot-sizing and locationtransportation problems. By projecting out second-stage flow variables, we define an upper bounding problem for the two-stage min-max-min optimization problem. Finally, we present computational results comparing the proposed twostage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization.
This software tool locates and computes the intensity of radiation skin dose resulting from fluoroscopically guided interventional procedures. It is comprised of multiple modules. Using standardized body specific geometric values, a software module defines a set of male and female patients arbitarily positioned on a fluoroscopy table. Simulated X-ray angiographic (XA) equipment includes XRII and digital detectors with or without bi-plane configurations and left and right facing tables. Skin dose estimates are localized by computing the exposure to each 0.01 × 0.01 m(2) on the surface of a patient irradiated by the X-ray beam. Digital Imaging and Communications in Medicine (DICOM) Structured Report Dose data sent to a modular dosimetry database automatically extracts the 11 XA tags necessary for peak skin dose computation. Skin dose calculation software uses these tags (gantry angles, air kerma at the patient entrance reference point, etc.) and applies appropriate corrections of exposure and beam location based on each irradiation event (fluoroscopy and acquistions). A physicist screen records the initial validation of the accuracy, patient and equipment geometry, DICOM compliance, exposure output calibration, backscatter factor, and table and pad attenuation once per system. A technologist screen specifies patient positioning, patient height and weight, and physician user. Peak skin dose is computed and localized; additionally, fluoroscopy duration and kerma area product values are electronically recorded and sent to the XA database. This approach fully addresses current limitations in meeting accreditation criteria, eliminates the need for paper logs at a XA console, and provides a method where automated ALARA montoring is possible including email and pager alerts.
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