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Model Building in Mathematical Programming, 5th Edition by H. Paul Williams

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Chapter 9

Building integer programming models I

9.1 The uses of discrete variables

When integer variables are used in a mathematical programming model, they may serve a number of purposes. These are described below.

9.1.1 Indivisible (discrete) quantities

This is the obvious use mentioned at the beginning of Chapter 8 where we wish to use a variable to represent a quantity that can only come in whole numbers such as aeroplanes, cars, houses or people.

9.1.2 Decision variables

Variables are frequently used in integer programming (IP) to indicate which of a number of possible decisions should be made. Usually, these variables can only take the two values, zero or one. Such variables are known as zero–one (0–1) (or binary) variables. For example, δ = 1 indicates that a depot should be built and δ = 0 indicates that a depot should not be built. We usually adopt the convention of using the Greek letter ‘δ’ for 0 − 1 variables and reserve Latin letters for continuous (real or rational) variables.

It is easy to ensure that a variable, which is also specified to be integer, can only take the two values 0 or 1 by giving the variable a simple upper bound (SUB) of 1. (All variables are assumed to have a simple lower bound of 0 unless it is stated to the contrary).

Although decision variables are usually 0 − 1, they need not always be. For example we might have γ = 0 indicates that no depot should be built; γ = 1 indicates that a depot of type A should be built; γ = 2 indicates that a depot ...

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