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Chapter 8 Linear programming (I) 8.1 Introduction Decision making is a process that has to be carried out in many areas of life. Usually there is a particular aim in making one decision rather than another. Two particular aims that are often considered in commerce are maximising profit and minimising loss. During, and after, the Second World War a group of American mathematicians developed some mathematical methods to help with decision making. They sought to produce mathematical models of situations in which all the requirements, constraints and objectives were expressed as algebraic equations. They then developed methods for obtaining the optimal solution – the maximum or minimum value of a required function. In this chapter we will study problems for which all the algebraic expressions are linear, that is, of the form (a number) x + (a number) y ; For example, Profit = 4 x + 3 y is a linear equation in x and y , and 16 x + 18 y ≤ 25 is a linear inequation or inequality in x and y . This area of mathematics is called linear programming . Linear programming methods are some of the most widely used method employed to solve management and economic problems. They have been applied in a variety of contexts, some of which will be discussed in this chapter, with enormous savings in money and resources. 71
CHAPTER 8. LINEAR PROGRAMMING (I) 72 8.2Formulating linear programming problems .