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QABE_Notes11 - QABE Lecture 11 Linear Programming I Solving...

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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 war-time 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 right-hand sides don’t necessarily equal each other, but instead must be greater-than, less-than, or some combination of these with ‘equal-to’. 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 well-placed to solve bigger linear programming problems.
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