LINEAR PROGRAMMING J

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1 9. Sstems of Linear Inequalities 9. Linear Programming Involving Two Variables 9. The Simple Method: Maimization 9.4 The Simple Method: Minimization 9.5 The Simple Method: Mied Constraints John von Neumann LINEAR PROGRAMMING J ohn von Neumann was born in Budapest, Hungar, where his father was a successful banker. John s genius was recognized at an earl age. B the age of ten, his mathematical knowledge was so great that instead of attending regular classes he studied privatel under the direction of leading Hungarian mathematicians. At the age of twent-one, he acquired two degrees, one in chemical engineering at Zurich, and the other a Ph.D. in mathematics from the Universit of Budapest. He spent some time teaching at the Universit of Berlin, and then, in 90, accepted a visiting professorship at Princeton Universit. In 9, John von Neumann and Albert Einstein were among the first full professors to be appointed to the newl organized Institute for Advanced Stud at Princeton. During World War II, von Neumann was a consultant at Los Alamos, and his research helped in the development of the atomic bomb. In 954, President Eisenhower appointed him to the Atomic Energ Commission. von Neumann is considered to be the father of modern game theor a branch of mathematics that deals with strategies and decision making. Much of his results concerning game theor were published in a length paper in 944 titled Theor of Games and Economic Behavior, written with Oskar Morgenstern. In 955, John von Neumann was diagnosed with cancer he died in 957 at the age of 5. Man stories are told of his mental abilities. Even during the final months of his life, as his brother read to him in German from Goethe s Faust, each time a page was turned John would recite from memor the continuation of the passage on the following page. 9. SYSTEMS OF LINEAR INEQUALITIES The following statements are inequalities in two variables. 6 and 6 An ordered pair (a, b) is a solution of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel. For instance, (, ) is a solution of the inequalit 6 because 6. The graph of an inequalit is the collection of all solutions of the inequalit. To sketch the graph of an inequalit such as 6 we begin b sketching the graph of the corresponding equation

2 SECTION 9. SYSTEMS OF LINEAR INEQUALITIES 479 The graph of the equation will normall separate the plane into two or more regions. In each such region, one of the following must be true. () All points in the region are solutions of the inequalit. () No points in the region are solutions of the inequalit. Thus, we can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region. Sketching the Graph of an Inequalit in Two Variables. Replace the inequalit sign b an equal sign, and sketch the graph of the resulting equation. (We use a dashed line for < or > and a solid line for or.). Test one point in each of the regions formed b the graph in Step. If the point satisfies the inequalit, then shade the entire region to denote that ever point in the region satisfies the inequalit. In this section, we will work with linear inequalities of the form a b c a b c a b c a b c. The graph of each of these linear inequalities is a half-plane ling on one side of the line a b c. The simplest linear inequalities are those corresponding to horizontal or vertical lines, as shown in Eample. EXAMPLE Sketching the Graph of a Linear Inequalit Sketch the graphs of (a) and (b). Solution (a) The graph of the corresponding equation is a vertical line. The points that satisf the inequalit are those ling to the right of this line, as shown in Figure 9.. (b) The graph of the corresponding equation is a horizontal line. The points that satisf the inequalit are those ling below (or on) this line, as shown in Figure 9.. Figure 9. Figure 9. = =

3 480 CHAPTER 9 LINEAR PROGRAMMING EXAMPLE Sketching the Graph of a Linear Inequalit Sketch the graph of. Solution The graph of the corresponding equation is a line, as shown in Figure 9.. Since the origin (0, 0) satisfies the inequalit, the graph consists of the half-plane ling above the line. (Tr checking a point below the line. Regardless of which point ou choose, ou will see that it does not satisf the inequalit.) Figure 9. = (0, 0) For a linear inequalit in two variables, we can sometimes simplif the graphing procedure b writing the inequalit in slope-intercept form. For instance, b writing in the form we can see that the solution points lie above the line, as shown in Figure 9.. Similarl, b writing the inequalit 5 in the form 5 we see that the solutions lie below the line 5. Sstems of Inequalities Man practical problems in business, science, and engineering involve sstems of linear inequalities. Here is an eample of such a sstem

4 SECTION 9. SYSTEMS OF LINEAR INEQUALITIES 48 A solution of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem. For instance,, 4 is a solution of this sstem because and 4 satisf each of the four inequalities in the sstem. The graph of a sstem of inequalities in two variables is the collection of all points that are solutions of the sstem. For instance, the graph of the sstem above is the region shown in Figure 9.4. Note that the point, 4 lies in the region because it is a solution of the sstem of inequalities. To sketch the graph of a sstem of inequalities in two variables, we first sketch the graph of each individual inequalit (on the same coordinate sstem) and then find the region that is common to ever graph in the sstem. For sstems of linear inequalities, it is helpful to find the vertices of the solution region, as shown in the following eample. Figure (, 4) (, 4) is a solution because it satisfies the sstem of inequalities. EXAMPLE Solving a Sstem of Inequalities Sketch the graph (and label the vertices) of the solution set of the following sstem. Solution We have alread sketched the graph of each inequalit in Eamples and. The triangular region common to all three graphs can be found b superimposing the graphs on the same coordinate plane, as shown in Figure 9.5. To find the vertices of the region, we find the points of intersection of the boundaries of the region. Verte A:, 4 Verte B : 5, Verte C :, Obtained b finding Obtained b finding Obtained b finding the point of the point of the point of intersection of intersection of intersection of...

5 48 CHAPTER 9 LINEAR PROGRAMMING Figure 9.5 < > C = (, ) B = (5, ) A = (, 4) For the triangular region shown in Figure 9.5, each point of intersection of a pair of boundar lines corresponds to a verte. With more complicated regions, two border lines can sometimes intersect at a point that is not a verte of the region, as shown in Figure 9.6. In order to keep track of which points of intersection are actuall vertices of the region, we suggest that ou make a careful sketch of the region and refer to our sketch as ou find each point of intersection. When solving a sstem of inequalities, ou should be aware that the sstem might have no solution. For instance, the sstem has no solution points because the quantit cannot be both less than and greater than, as shown in Figure 9.7. Figure 9.6 (Not a verte) Figure > Border lines can intersect at a point that is not a verte. + < No Solution

6 SECTION 9. SYSTEMS OF LINEAR INEQUALITIES 48 Another possibilit is that the solution set of a sstem of inequalities can be unbounded. For instance, the solution set of forms an infinite wedge, as shown in Figure 9.8. Figure > + < Unbounded Region Applications Our last eample in this section shows how a sstem of linear inequalities can arise in an applied problem. EXAMPLE 4 Solution An Application of a Sstem of Inequalities The liquid portion of a diet is to provide at least 00 calories, 6 units of vitamin A, and 90 units of vitamin C dail. A cup of dietar drink X provides 60 calories, units of vitamin A, and 0 units of vitamin C. A cup of dietar drink Y provides 60 calories, 6 units of vitamin A, and 0 units of vitamin C. Set up a sstem of linear inequalities that describes the minimum dail requirements for calories and vitamins. We let number of cups of dietar drink X number of cups of dietar drink Y. Then, to meet the minimum dail requirements, the following inequalities must be satisfied. For calories: For vitamin A: 66 6 For vitamin C:

7 484 CHAPTER 9 LINEAR PROGRAMMING The last two inequalities are included because and cannot be negative. The graph of this sstem of linear inequalities is shown in Figure 9.9. Figure (0, 6) (, 4) (, ) (9, 0) An point inside the region shown in Figure 9.9 (or on its boundar) meets the minimum dail requirements for calories and vitamins. For instance, cups of dietar drink X and cups of dietar drink Y suppl 00 calories, 48 units of vitamin A, and 90 units of vitamin C. SECTION 9. EXERCISES In Eercises 6, match the linear inequalit with its graph. [The graphs are labeled (a) (f).] (a) (b) (c) 4 (d) (e) (f) In Eercises 7, sketch the graph of the given linear inequalit In Eercises, sketch the graph of the solution of the given sstem of linear inequalities

8 SECTION 9. EXERCISES A person plans to invest no more than $0,000 in two different 0 interest-bearing accounts. Each account is to contain at 0 0 least $5000. Moreover, one account should have at least twice the amount that is in the other account. Find a sstem of inequalities to describe the various amounts that can be de posited in each account, and sketch the graph of the sstem Two tpes of tickets are to be sold for a concert. One tpe.. 0 costs $5 per ticket and the other tpe costs $5 per ticket. The promoter of the concert must sell at least 5,000 tickets 0 including 8000 of the $5 tickets and 4000 of the $5 tickets. 9 6 Moreover, the gross receipts must total at least $75,000 in order for the concert to be held. Find a sstem of inequalities describing the different numbers of tickets that can be sold, In Eercises 6, derive a set of inequalities to describe the given and sketch the graph of the sstem. region. 4. A dietitian is asked to design a special diet supplement using. Rectangular region with vertices at,, 5,, 5, 7, and two different foods. Each ounce of food X contains 0 units, 7. of calcium, 5 units of iron, and 0 units of vitamin B. Each ounce of food Y contains 0 units of calcium, 0 units of 4. Parallelogram with vertices at 0, 0, 4, 0,, 4, and 5, 4. iron, and 0 units of vitamin B. The minimum dail requirements in the diet are 00 units of calcium, 50 units of iron, 5. Triangular region with vertices at 0, 0, 5, 0, and,. 6. Triangular region with vertices at, 0,, 0, and 0,. 7. A furniture compan can sell all the tables and chairs it produces. Each table requires hour in the assembl center and and 00 units of vitamin B. Find a sstem of inequalities describing the different amounts of food X and food Y that can be used in the diet, and sketch the graph of the sstem. hours in the finishing center. Each chair requires hours 4. Rework Eercise 4 using minimum dail requirements of in the assembl center and hours in the finishing center. 80 units of calcium, 60 units of iron, and 80 units of The compan s assembl center is available hours per da, vitamin B. and its finishing center is available 5 hours per da. If is the number of tables produced per da and is the number of chairs, find a sstem of inequalities describing all possible production levels. Sketch the graph of the sstem. 8. A store sells two models of a certain brand of computer. Because of the demand, it is necessar to stock at least twice as man units of model A as units of model B. The cost to the store for the two models is $800 and $00, respectivel. The management does not want more than $0,000 in computer inventor at an one time, and it wants at least four model A computers and two model B computers in inventor at all times. Devise a sstem of inequalities describing all possible inventor levels, and sketch the graph of the sstem.