A Tutorial Guide to MixedInteger 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
Mixedinteger 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 mixedinteger programming. We begin by discussing basic mixedinteger
programming formulation principles and tricks, especially with regards to the use of
binary variables to form logical statements.
We then discuss two core techniques,
branchandbound and cutting plane algorithms, used to solve mixedinteger programs.
We illustrate the use of mixedinteger 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 mixedinteger 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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Mixedinteger 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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 Linear Programming, Optimization, objective function, objective function value

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