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IntegerProgramming - A Tutorial Guide to Mixed-Integer...

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A Tutorial Guide to Mixed-Integer Programming Models and Solution Techniques J. Cole Smith and Z. Caner Ta¸ skın Department of Industrial and Systems Engineering University of Florida Gainesville, FL 32611 [email protected], [email protected] December 30, 2006 Abstract Mixed-integer programming theory provides a mechanism for optimizing decisions that take place in complex systems, including those encountered in biology and medi- cine. This chapter is intended for researchers and practitioners wanting an introduction to the field of mixed-integer programming. We begin by discussing basic mixed-integer programming formulation principles and tricks, especially with regards to the use of binary variables to form logical statements. We then discuss two core techniques, branch-and-bound and cutting plane algorithms, used to solve mixed-integer programs. We illustrate the use of mixed-integer programs in the context of several medical ap- plications, and close with a featured study on Intensity Modulated Radiation Therapy planning. 1 Introduction This chapter describes the use of mixed-integer programming in optimizing complex systems, such as those arising in biology, medicine, transportation, telecommunications, sports, and national security. Consider, for instance, an emergency that results in 100 injuries. A triage center is established to administer first aid and assign victims to one of three nearby hospitals, each of which is capable of handling a limited number of patients. Each hospital may have varying equipment and staff levels, and each will be located at a different distance from the emergency. The optimization problem that arises is to assign patients to hospitals in a way that maximizes the effectiveness of care that can be given to the victims, while obeying physical capacity restrictions imposed by the hospitals. Experts often attempt to solve these problems based on intuition and experience, but the resulting solution is almost invariably suboptimal due to the inherent complexities of such problems. In applications of critical importance, there is sufficient motivation to turn to mathematical techniques that can provably obtain a “best” solution. 1
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Mixed-integer programming is a subset of the broader field of mathematical programming. Mathematical programming formulations include a set of variables, which represent actions that can be taken in the system being modelled. One then attempts to optimize (either in the minimization or maximization sense) a function of these variables, which maps each possible set of decisions into a single score that assesses the quality of the solution. These scores are often in units of currency representing total cost incurred or revenue gained etc. The limitations of the system are included as a set of constraints, which are usually stated by restricting functions of the decision variables to be equal to, not more than, or not less than, a certain numerical value. Another type of constraint can simply restrict the set of values to which a variable might be assigned.
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