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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.2

# Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.2

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9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES Many applications in business and economics involve a process called optimization, in which it is required to find the minimum cost, the maximum profit, or the minimum use of resources. In this section, one type of optimization problem called linear programming is discussed. 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 you are asked to maximize the value of Objective function subject 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 to go about finding the point that yields a maximum value of z . Fortunately, it can be shown that if there is an op- timal solution, it must occur at one of the vertices of the region. In other words, you can find the maximum value by testing z at each of the vertices , as illustrated in Example 1. EXAMPLE 1 Solving a Linear Programming Problem Find the maximum value of Objective function subject to the following constraints. Solution The constraints form the region shown in Figure 9.11. At the four vertices of this region, the objective function has the following values. x 2 y 1 x 1 2 y 4 y 0 x 0 z 5 3 x 1 2 y z 5 ax 1 by 522 CHAPTER 9 LINEAR PROGRAMMING Figure 9.10 x y The objective function has its optimal value at one of the vertices of the region determined by the constraints. Feasible solutions Theorem 9.1 Optimal Solution of a Linear Programming Problem If a linear programming problem has a solution, it must occur at a vertex of the set of feasible solutions. If the problem has more than one solution, then at least one of them must occur at a vertex of the set of feasible solutions. In either case, the value of the objective function is unique. Constraints

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At At At (Maximum value of z ) At So, the maximum value of z is 8, and this occurs when and REMARK : In Example 1, try testing some of the interior points in the region. You will see that the corresponding values of z are less than 8. To see why the maximum value of the objective function in Example 1 must occur at a vertex, consider writing the objective function in the form This equation represents a family of lines, each of slope Of these infinitely many lines, you want the one that has the largest z -value, while still intersecting the region deter- mined by the constraints. In other words, of all the lines whose slope is you want the one that has the largest y -intercept and intersects the given region, as shown in Figure 9.12. It should be clear that such a line will pass through one (or more) of the vertices of the region.
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.2

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