Integer optimization

Abstract

In linear optimization, we assume that all of the decision variables are continuous. However, in reality, some are not. For example, the number of crude oil tankers used to transport crude oil from the Middle East to the US is an integer, and rounding down 6.5 ships to 6 ships can lead to considerable errors. Therefore, a natural extension to linear optimization models is integer linear optimization models, which are the same as linear optimization models except that the decision variables can only take integer values. We also have mixed-integer linear optimization models, in which some decision variables can only take integer values, and the others are continuous. We often use "integer optimization models" to refer to integer linear optimization models or both integer linear optimization models and mixed-integer linear optimization models. However, one should keep in mind that integer linear optimization models are not linear; in other words, integer linear optimization models belong to the category of nonlinear optimization models.We use ℤ+ to represent the set of non-negative integers. Hence, x ∈ ℤ + means that x is non-negative and can only take integer values.