LP Problem Guide

May 14th, 2015
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Many applications in business and economics involve a process called optimization, in which we are required to find the minimum cost, the maximum profit, or the minimum use of resources. In this section we discuss one type of optimization problem called linear programming.

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LP Problem Guide486CHAPTER 9LINEAR PROGRAMMING9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLESMany applications in business and economics involve a process called optimization, in which we are required to find the minimum cost, the maximum profit, or the minimum use of resources. In this section we discuss one type of optimization problem called linear programming. A two-dimensional linear programming problem consists of a linear objective function and a system of linear inequalities called constraints. The objective function gives the quantity that is to be maximized (or minimized), and the constraints determine the set of feasible solutions.For example, consider a linear programming problem in which we are asked to maximize the value ofFigure 9.10yFeasiblesolutionsz ax byxThe objective function has itsoptimal value at one of the verticesof the region determined by theconstraints.Theorem 9.1Optimal Solution of aLinear Programming ProblemEXAMPLE 1Objective functionsubject to a set of constraints that determine the region indicated in Figure 9.10. Because every point in the region satisfies each constraint, it is not clear how we should go about finding the point that yields a maximum value of z. Fortunately, it can be shown that if there is an optimal solution, it must occur at one of the vertices of the region. In other words, we can find the maximum value by testing z at each of the vertices, as illustrated in Example 1.If a linea

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