Knowledge that Transforms

Mathematical programming models with noisy, erroneous, or incomplete data are common in operations research applications. Difficulties with such data are typically dealt with reactively—through sensitivity analysis—or proactively—through stochastic programming formulations. In this paper, we characterize the desirable properties of a solution to models, when the problem data are described by a set of scenarios for their value, instead of using point estimates. A solution to an optimization model is defined as: solution robust if it remains “close” to optimal for all scenarios of the input data, and model robust if it remains “almost” feasible for all data scenarios. We then develop a general model formulation, called robust optimization (RO), that explicitly incorporates the conflicting objectives of solution and model robustness. Robust optimization is compared with the traditional approaches of sensitivity analysis and stochastic linear programming. The classical diet problem illustrates the issues. Robust optimization models are then developed for several real-world applications: power capacity expansion; matrix balancing and image reconstruction; air-force airline scheduling; scenario immunization for financial planning; and minimum weight structural design. We also comment on the suitability of parallel and distributed computer architectures for the solution of robust optimization models.

This is the second half of a two-part survey on arc routing problems. The first part appeared in the March–April 1995 issue of this journal. Here, the rural postman problem (RPP) is reviewed. The paper is organized as follows: applications, the undirected RPP, the directed RPP, the stacker crane problem, and the capacitated arc routing problem.

Two medical applications of linear programming are described in this paper. Specifically, linear programming-based machine learning techniques are used to increase the accuracy and objectivity of breast cancer diagnosis and prognosis. The first application to breast cancer diagnosis utilizes characteristics of individual cells, obtained from a minimally invasive fine needle aspirate, to discriminate benign from malignant breast lumps. This allows an accurate diagnosis without the need for a surgical biopsy. The diagnostic system in current operation at University of Wisconsin Hospitals was trained on samples from 569 patients and has had 100% chronological correctness in diagnosing 131 subsequent patients. The second application, recently put into clinical practice, is a method that constructs a surface that predicts when breast cancer is likely to recur in patients that have had their cancers excised. This gives the physician and the patient better information with which to plan treatment, and may eliminate the need for a prognostic surgical procedure. The novel feature of the predictive approach is the ability to handle cases for which cancer has not recurred (censored data) as well as cases for which cancer has recurred at a specific time. The prognostic system has an expected error of 13.9 to 18.3 months, which is better than prognosis correctness by other available techniques.

This paper introduces a new two-stage assembly scheduling problem. There are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule jobs on the machines so that the makespan is minimized. We show that the search for an optimal solution may be restricted to permutation schedules. The problem is proved to be NP-hard in the strong sense even when m = 2. A schedule associated with an arbitrary permutation of jobs is shown to provide a worst-case ratio bound of two, and a heuristic with a worst-case ratio bound of 2 − 1/m is presented. The compact vector summation technique is applied for finding approximation solutions with worst-case absolute performance guarantees.

This paper reviews the literature on quantitatively-oriented approaches for determining lot sizes when production or procurement yields are random. We discuss issues related to the modeling of costs, yield uncertainty, and performance in the context of systems with random yields. We provide a review of the existing literature, concentrating on descriptions of the types of problems that have been solved and important structural results. We identify a variety of shortcomings of the literature in addressing problems encountered in practice, and suggest directions for future research.