QABE Lecture 11
Linear Programming I: Solving problems in
a world of constraints
School of Economics, UNSW
2011
Contents
1
Introduction
1
2
The Business Headache
2
3
Introduction to Linear programming
3
3.1
Equations of two variables
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.2
Linear inequalities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.3
Systems of linear inequalities
. . . . . . . . . . . . . . . . . . . . . . . . .
4
3.4
Graphing the system
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
Linear Programming
4
4.1
Terminology
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4.2
Application
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1
Introduction
We move now away from the world of matrix algebra and into the world of
linear pro
gramming
. It is natural to think that this would require us to do some kind of
computer
programming(!), but this is not the case.
In fact the word ‘programming’ is an arte
fact of the historical background of the techniques we’ll be studying rather than saying
something about our method.
You see, we will be dealing with the very common problem that anyone faces when
they must attempt to maximize (or minimize) some quantity, subject to a number of
constraints
. These constraints might include the minimum production level that must
be attained, or the maximum number of days that can be worked, or the limit of what
an individual can carry at one time.
1
The reason why this process is called ‘linear
programming’ is that for a great many problems, the constraint and ‘success’ equations
are linear in nature, and the ‘programming’ bit comes from the names given to various
wartime schedules of activity that were the outcome of solving just these kinds of
problems during World War II. They called them ‘programmes’.
1
An account by Chris Bonnington, expedition leader of the first ascent of the Western Face of Mt
Everest, describes how he had to solve exactly our kind of linear programming problem; he had to get
an amount of equipment, provisions and oxygen up the mountain to support his climbers with all kinds
of constraints, one of which was the amount that any one climber could carry in a single load.
1
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ECON 1202/ECON 2291: QABE
c School of Economics, UNSW
Before jumping straight into the solution method of these problems, we need to spend
some time considering equations where the left and righthand sides don’t necessarily
equal
each other, but instead must be greaterthan, lessthan, or some combination of
these with ‘equalto’.
These are known as
inequalities
.
We look at these because
they are generally how our constraints will be given. Knowing how to deal with such
constraints, we’ll be wellplaced to solve bigger linear programming problems.
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 One '11
 LorettiIsabellaDobrescu
 Economics, Operations Research, Linear Programming, Optimization, ECON 1202/ECON, QABE Lecture

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