Single Binary Variable Research Articles

Mixed-integer linear optimization for optimal lift-gas allocation with well-separator routing

The lift-gas allocation problem with well-separator routing constraints is a mixed-integer nonlinear program of considerable complexity. To this end, a mixed-integer linear formulation (compact) is obtained by piecewise-linearizing the nonlinear curves, using binary variables to express the linearization and routing decisions. A new formulation (integrated) combining the decisions on linearization and routing is developed by using a single binary variable. The structures of both formulations are explored to generate lifted cover cuts. Numerical tests show that the solution of the integrated formulation using cutting-plane generation is faster in spite of having more variables than the compact formulation.

Dose-Finding Based on Multiple Toxicities in a Soft Tissue Sarcoma Trial

The scientific goal of a phase I oncology trial of a new chemotherapeutic agent is to find a dose with an acceptable level of toxicity. For ethical reasons, dose-finding is done adaptively, with doses chosen for successive cohorts of patients based on the data obtained from previous cohorts. Typically, patients are at risk for several qualitatively different toxicities, each occurring at several possible severity levels. In this article, we describe how we addressed the dose-finding problem in a phase I trial of gemcitabine for treatment of soft tissue sarcoma. The oncologists planning the trial wanted to account for differences in importance among the toxicities that they had identified. They also requested that the dose-finding method utilize the fact that a low-grade toxicity observed at a given dose, although not dose-limiting, provides a warning that a higher grade of that toxicity is likely to occur at a higher dose. Conventional phase I methods reduce each type of toxicity to an indicator of its occurrence at or above a severity level considered dose-limiting, define “toxicity” as the maximum of these indicators, and base dose-finding on that single binary variable. Because conventional methods do not address the aforementioned concerns, we developed a Bayesian method for dose-finding in the sarcoma trial based on a vector of correlated, ordinal-valued toxicities with severity levels varying with dose. We also developed a method for jointly eliciting the prior, a vector of weights quantifying the clinical importance of each level of each type of toxicity, and a target total toxicity burden acceptable to the physicians. Our method assigns each cohort the dose with a current posterior mean total toxicity burden closest to the target. The elicitation process is iterative, with the oncologists repeatedly shown the algorithm's behavior and asked to adjust their weights to ensure that the statistical decisions reflect appropriate clinical behavior. We describe how this methodology has worked in the sarcoma trial, present simulations and sensitivity analyses of the trial under several clinical scenarios, and provide guidelines for general application.