Here are some guidelines for solving a linear programming problem in two variables in which an objective function is to be maximized or minimized.

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1 Appendi F. Linear Programming F F. Linear Programming Linear Programming: A Graphical Approach Man applications in business and economics involve a process called optimization, in which ou are asked to find the minimum or maimum value of a quantit. In this appendi, ou will stud an optimization strateg called linear programming. A two-dimensional linear programming problem consists of a linear objective function and a sstem of linear inequalities called constraints. The objective function gives the quantit that is to be maimized (or minimized), and the constraints determine the set of feasible solutions. For eample, suppose ou are asked to maimize the value of z a b subject to a set of constraints that determines the region in Figure F.. Because ever point in the shaded region satisfies each constraint, it is not clear how ou should find the point that ields a maimum value of z. Fortunatel, it can be shown that when there is an optimal solution, it must occur at one of the vertices. So, ou can find the maimum value of z b testing z at each of the vertices. Optimal of a Linear Programming Problem If a linear programming problem has a solution, then it must occur at a verte of the set of feasible solutions. If there is more than one solution, then at least one of them must occur at such a verte. In either case, the value of the objective function is unique. What ou should learn Solve linear programming problems. Use linear programming to model and solve real-life problems. Wh ou should learn it Linear programming is a powerful tool used in business and industr to manage resources effectivel in order to maimize profits or minimize costs. For instance, Eercise on page F9 shows how to use linear programming to analze the profitabilit of two models of snowboards. Here are some guidelines for solving a linear programming problem in two variables in which an objective function is to be maimized or minimized. Solving a Linear Programming Problem. Sketch the region corresponding to the sstem of constraints. (The points inside or on the boundar of the region are feasible solutions.). Find the vertices of the region.. Test the objective function at each of the vertices and select the values of the variables that optimize the objective function. For a bounded region, both a minimum and a maimum value will eist. (For an unbounded region, if an optimal solution eists, it will occur at a verte.) Feasibl e solutions Figure F.

2 F Appendi F Sstems of Inequalities Eample Solving a Linear Programming Problem Find the maimum value of z subject to the following constraints. = (, ) + = (, ) = (, ) (, ) = The constraints form the region shown in Figure F.. At the four vertices of this region, the objective function has the following values. Figure F. At 共, 兲: z 共兲 共兲 At 共, 兲: z 共兲 共兲 At 共, 兲: z 共兲 共兲 8 Maimum value of z At 共, 兲: z 共兲 共兲 So, the maimum value of z is 8, and this value occurs when and. Stud Tip Now tr Eercise 7. In Eample, tr testing some of the interior points in the region. You will see that the corresponding values of z are less than 8. Here are some eamples. At 共, 兲: z 共兲 共兲 At 共, 兲: z 共兲 共 兲 9 At 共, 兲: z 共 兲 共 兲 Remember that a verte of a region can be found using a sstem of linear equations. The sstem will consist of the equations of the lines passing through the verte. To see wh the maimum value of the objective function in Eample must occur at a verte, consider writing the objective function in the form z Famil of lines where z兾 is the -intercept of the objective function. This equation represents a famil of lines, each of slope. Of these infinitel man lines, ou want the one that has the largest z-value while still intersecting the region determined b the constraints. In other words, of all the lines with a slope of, ou want the one that has the largest -intercept and intersects the given region, as shown in Figure F.. It should be clear that such a line will pass through one (or more) of the vertices of the region. Figure F.

3 Appendi F. F Linear Programming The net eample shows that the same basic procedure can be used to solve a problem in which the objective function is to be minimized. Eample Solving a Linear Programming Problem Find the minimum value of z 7 where and, subject to the following constraints. 7 The region bounded b the constraints is shown in Figure F.. B testing the objective function at each verte, ou obtain the following. At 共, 兲: z 共兲 7共兲 Minimum value of z At 共, 兲: z 共兲 7共兲 8 (, ) (, ) (, ) At 共, 兲: z 共兲 7共兲 At 共, 兲: z 共兲 7共兲 (, ) At 共, 兲: z 共兲 7共兲 (, ) At 共, 兲: z 共兲 7共兲 So, the minimum value of z is, and this value occurs when and. Now tr Eercise 9. Eample Solving a Linear Programming Problem Find the maimum value of z 7 where and, subject to the following constraints. 7 This linear programming problem is identical to that given in Eample above, ecept that the objective function is maimized instead of minimized. Using the values of z at the vertices shown in Eample, ou can conclude that the maimum value of z is, and that this value occurs when and. Now tr Eercise. Figure F. (, )

4 F Appendi F Sstems of Inequalities It is possible for the maimum (or minimum) value in a linear programming problem to occur at two different vertices. For instance, at the vertices of the region shown in Figure F.7, the objective function z has the following values. At 共, 兲: z 共兲 共兲 (, ) At 共, 兲: z 共兲 共兲 8 (, ) z = for an point along this line segment. At 共, 兲: z 共兲 共兲 Maimum value of z At 共, 兲: z 共兲 共兲 Maimum value of z At 共, 兲: z 共兲 共兲 (, ) In this case, ou can conclude that the objective function has a maimum value (of ) not onl at the vertices 共, 兲 and 共, 兲, but also at an point on the line segment connecting these two vertices, as shown in Figure F.7. Note that b rewriting the objective function as (, ) (, ) Figure F.7 z ou can see that its graph has the same slope as the line through the vertices 共, 兲 and 共, 兲. Some linear programming problems have no optimal solutions. This can occur when the region determined b the constraints is unbounded. Eample An Unbounded Region Find the maimum value of z where and, subject to the following constraints. ⱖ ⱖ 7 ⱕ 7 (, ) The region determined b the constraints is shown in Figure F.8. For this unbounded region, there is no maimum value of z. To see this, note that the point 共, 兲 lies in the region for all values of. B choosing large values of, ou can obtain values of z 共兲 共兲 that are as large as ou want. So, there is no maimum value of z. For the vertices of the region, the objective function has the following values. So, there is a minimum value of z, z, which occurs at the verte 共, 兲. At 共, 兲: z 共兲 共兲 At 共, 兲: z 共兲 共兲 At 共, 兲: z 共兲 共兲 Now tr Eercise. Minimum value of z (, ) (, ) Figure F.8

5 Appendi F. F Linear Programming Applications Eample shows how linear programming can be used to find the maimum profit in a business application. Eample Optimizing Profit A manufacturer wants to maimize the profit from selling two tpes of boed chocolates. A bo of chocolate covered creams ields a profit of $., and a bo of chocolate covered cherries ields a profit of $.. Market tests and available resources have indicated the following constraints.. The combined production level should not eceed boes per month.. The demand for a bo of chocolate covered cherries is no more than half the demand for a bo of chocolate covered creams.. The production level of a bo of chocolate covered creams is less than or equal to boes plus three times the production level of a bo of chocolate covered cherries. What is the maimum monthl profit? How man boes of each tpe should be produced per month to ield the maimum monthl profit? Let be the number of boes of chocolate covered creams and be the number of boes of chocolate covered cherries. The objective function (for the combined profit) is given b P.. The three constraints translate into the following linear inequalities... Because neither nor can be negative, ou also have the two additional constraints of and. Figure F.9 shows the region determined b the constraints. To find the maimum profit, test the value of P at each verte of the region. At 共, 兲: P.共兲 共兲 At 共8, 兲: P.共8兲 共兲 Maimum profit Boes of chocolate covered cherries. (, ) (, ) At 共, 兲: P.共兲 共兲 87 At 共, 兲: Now tr Eercise 9. (, ) P.共兲 共兲 9 So, the maimum monthl profit is $, and it occurs when the monthl production consists of 8 boes of chocolate covered creams and boes of chocolate covered cherries. (8, ) 8 Boes of chocolate covered creams Figure F.9

6 F Appendi F Sstems of Inequalities In Eample, suppose the manufacturer improves the production of chocolate covered creams so that a profit of $. per bo is obtained. The maimum profit can now be found using the objective function P.. B testing the values of P at the vertices of the region, ou find that the maimum profit is now $9, which occurs when and. Eample Optimizing Cost As in Eample 9 on page F7, let be the number of cups of dietar drink X and let be the number of cups of dietar drink Y. For Calories: For Vitamin A: For Vitamin C: 9 (, ) (, ) (, ) (9, ) 8 Cups of drink X Figure F. The graph of the region determined b the constraints is shown in Figure F.. To determine the minimum cost, test C at each verte of the region. At 共, 兲: C.共兲.共兲.9 At 共, 兲: C.共兲.共兲.7 At 共, 兲: C.共兲.共兲. 8 The cost C is given b C... Liquid Portion of a Diet Cups of drink Y The minimum dail requirements from the liquid portion of a diet are calories, units of vitamin A, and 9 units of vitamin C. A cup of dietar drink X costs $. and provides calories, units of vitamin A, and units of vitamin C. A cup of dietar drink Y costs $. and provides calories, units of vitamin A, and units of vitamin C. How man cups of each drink should be consumed each da to minimize the cost and still meet the dail requirements? Minimum value of C At 共9, 兲: C.共9兲.共兲.8 So, the minimum cost is $. per da, and this cost occurs when three cups of drink X and two cups of drink Y are consumed each da. Now tr Eercise. Technolog Tip You can check the points of the vertices of the constraints b using a graphing utilit to graph the equations that represent the boundaries of the inequalities. Then use the intersect feature to confirm the vertices.

7 Appendi F. Linear Programming F7 F. Eercises For instructions on how to use a graphing utilit, see Appendi A. Vocabular and Concept Check In Eercises, fill in the blank(s).. In the process called, ou are asked to find the minimum or maimum value of a quantit.. The of a linear programming problem gives the quantit that is to be maimized or minimized.. The of a linear programming problem determine the set of.. To solve a linear programming problem, ou test the objective function at which points in the region representing the sstem of constraints? Procedures and Problem Solving Solving a Linear Programming Problem In Eercises, find the minimum and maimum values of the objective function and where the occur, subject to the indicated constraints. (For each eercise, the graph of the region determined b the constraints is provided.). :. : z z 8 7. : 8. : z 7 z 7 See Eercise. See Eercise. 9. :. : z z 9 8 Figure for 9 Figure for. :. : z. z See Eercise 9. See Eercise.. :. : z 7 z :. : z z See Eercise. See Eercise.

8 F8 Appendi F Sstems of Inequalities Solving a Linear Programming Problem In Eercises 7, sketch the region determined b the constraints. Then find the minimum and maimum values of the objective function and where the occur, subject to the indicated constraints. 7. : 8. : z z :. : z z 8. :. : z z See Eercise 9. See Eercise.. :. : z z See Eercise 9. See Eercise.. :. : z z : 8. : z z See Eercise. See Eercise. 9. :. : z z See Eercise. See Eercise. Eploration In Eercises, perform the following. (a) Graph the region bounded b the following constraints. (b) Graph the objective function for the given maimum value of z on the same set of coordinate aes as the graph of the constraints. (c) Use the graph to determine the feasible point or points that ield the maimum. Eplain how ou arrived at our answer. Objective Function Maimum. z z. z z. z z. z z A Problem with an Unusual Characteristic In Eercises 8, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. The objective function is to be maimized in each case.. :. : z z. 7. : 8. : z z 7 9. Accounting An accounting firm has 8 hours of staff time and 9 hours of reviewing time available each week. The firm charges $ for an audit and $ for a ta return. Each audit requires hours of staff time and 8 hours of review time. Each ta return requires. hours of staff time and hours of review time. (a) What numbers of audits and ta returns will ield the maimum revenue? (b) What is the maimum revenue?

9 Appendi F. Linear Programming F9. (p. F) A manufacturer produces two models of snowboards. The amounts of time (in hours) required for assembling, painting, and packaging the two models are as follows. Model A Model B Assembling. Painting Packaging.7. The total amounts of time available for assembling, painting, and packaging are hours, hours, and hours, respectivel. The profits per unit are $ for model A and $ for model B. (a) How man of each model should be produced to maimize profit? (b) What is the maimum profit?. Agriculture A farming cooperative mies two brands of cattle feed. Brand X costs $ per bag and contains two units of nutritional element A, two units of nutritional element B, and two units of nutritional element C. Brand Y costs $ per bag and contains one unit of nutritional element A, nine units of nutritional element B, and three units of nutritional element C. The minimum requirements for nutritional elements A, B, and C are units, units, and units, respectivel. (a) Find the number of bags of each brand that should be mied to produce a miture having a minimum cost per bag. (b) What is the minimum cost?. Business A pet suppl compan mies two brands of dr dog food. Brand X costs $ per bag and contains eight units of nutritional element A, one unit of nutritional element B, and two units of nutritional element C. Brand Y costs $ per bag and contains two units of nutritional element A, one unit of nutritional element B, and seven units of nutritional element C. Each bag of mied dog food must contain at least units, units, and units of nutritional elements A, B, and C, respectivel. (a) Find the numbers of bags of brands X and Y that should be mied to produce a miture meeting the minimum nutritional requirements and having a minimum cost per bag. (b) What is the minimum cost? Conclusions True or False? In Eercises and, determine whether the statement is true or false. Justif our answer.. When an objective function has a maimum value at the adjacent vertices, 7 and 8,, ou can conclude that it also has a maimum value at the points.,. and 7.8,... When solving a linear programming problem, if the objective function has a maimum value at two adjacent vertices, then ou can assume that there are an infinite number of points that will produce the maimum value. Think About It In Eercises 8, find an objective function that has a maimum or minimum value at the indicated verte of the constraint region shown below. (There are man correct answers.). The maimum occurs at verte A.. The maimum occurs at verte B. 7. The maimum occurs at verte C. 8. The minimum occurs at verte C. Think About It In Eercises 9 and, determine values of t such that the objective function has a maimum value at each indicated verte. 9. :. : z t z t (a), (b), A(, ) C(, ) B(, ) (a), (b),