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Unformatted text preview: 72 Chapter 3 Analyzing Solutions of Linear Programs In the jargon of operations research, a plot of profit versus capacity is known as a para metric analysis. We hav e seen that, in a linear program, a parametric analysis on one or mo re righthandside values exhibits a diseconomy of scale. 3.17. LINEAR PROGRAMS AN D PLANE GEOME TRY Our attention now turns to a new topic. In this section and the next, geometry is used to provide insight into linear programs. These sections help you to: • Visualize the set of feasible solutions to a linear program. • Strengthen your understanding of the Perturbation Theorem. • See what determ ines the “allowable increase” and “allowable decrease” in right handside values and in objective coefficients. • Learn how degeneracy can qualify the Perturbation Theorem. • Discover the role that “extreme points” play in linear programming. We will begin with twodimensional (planar) geometry. To do so, we turn our att ention to a linear program that has only two decision variables. This linear program is: Program 3.4. Maximize {2 A + 3 B} subject to the constraints A s 6 , A + B 7 , 2B 9, —A + 3 B 9, A B 0. As we shall see, the optimal solution to this linear program sets A = 3 and B = 4. Cartesian Coordinates Let us consider how Cartesian coordinates identify each pair (A, B) of real number s with a point in the plane. Im agine that this page is flat. Imagine also the plane that includes this page but whose extent is infinite. And imagine that you place a dot somewhere on this page; call that dot the origin. Then, locate each pair (A, B) of numbers in this plane by this recipe: • Starting at the origin, walk A units to the right. • Then walk B units toward the top of the page. • Place the pair (A, B) at the point where you ended. Cartesian coordinates locate the pair (0, 0) at the origin. Cartesian coordinates locate the pair (3, 1) three units to the right of the origin and 1 unit above it, and so forth. The Feasible Region The set of all feasible solutions to a linear program is called its feasible region. Figure 3.5 uses Cartesian coordinates to represent this feasible region as the shaded portion of the plane. For instance, the pair (5, 1) lies within the shade d region because it is feasible t o set A 5 and B = 1. 3.17. Linear Programs and Plane Geometry 73 7 6 2B 9 5 4. 3 2 A0 1 B 1 Figure 3.5 The feasible region for Program 3.4. To indicate how Figure 3.5 was constructed, we begin with a single constraint, namely, with the constraint A + B 7. Let us observe that: • The points (A, B) that satisfy the constraint A + B 7 form a region whose bound ary is the line on which A + B = 7. • Two points determine a line. The line A + B = 7 includes the points (7, 0) and (0,7)....
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 Fall '09
 MarinaEpelman
 Optimization, feasible region, Linear Programs, objective vector

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