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Intro - Chapter 1 Introduction Optimization problems are...

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Chapter 1 Introduction Optimization problems are abundant in every day life, and all of us face such problems fre- quently, although we may not always be aware of the fact! Obvious examples are, for instance, the use of your GPS to find a shortest route from your home to your work place in the morning, or the scheduling of trains on the rail connections between Waterloo and Toronto. There are however many more, less obvious examples. How, for example, does the region of Waterloo determine the structure of its electricity network? How are schedules for buses determined? And how does a company set the price for a newly developed product? All of these questions are optimization problems, and in this chapter we show that many of these questions admit a mathematical formulation. 1.1 Linear Programming In this section we introduce linear programming, an efficient and versatile method to optimize (minimize or maximize) a linear function subject to a system of linear inequality and/or equality constraints. As unimpressive as this may sound, linear programming is a very powerful tool that is used in practice to solve instances of optimization problems arising in applications in most branches of industry. In fact, a recent survey (see also [3]) of Fortune 500 firms shows that 85% of all respondents use linear programming in their operations. The roots of linear programming 7
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8 CHAPTER 1. INTRODUCTION can be traced back at least a couple of hundred years to the work of Fourier on solutions of sys- tems of linear inequalities. The name linear programming , however, was coined only recently, in the late 1930’s and early 1940’s when the Russian mathematician Leonid Kantorovich and the American George Dantzig formally defined the technique. George Dantzig also developed the Simplex algorithm which to this date remains one of the most popular methods to solve lin- ear programs. Dantzig who worked as a mathematical advisor for the U.S. Air Force initially applied linear programming to solve logistical problems arising in the military. It did, however, not take long for industry to realize the technique’s potential, and its use is wide-spread today. In this section we will see two typical examples of optimization problems that can be solved via linear programming. 1.1.1 A production example Production problems are probably among the most frequent applications of linear programming. In a typical such application, we are given a company that produces a number of different prod- ucts from a set of resources. Producing a unit of a product requires the use of a certain amount of each resource, and can be sold at a certain price on the market. The company has limited amounts of each of the resources available, and is interested in maximizing its revenue from sales. How much of each product should be produced?
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