Linear Programming

Transcription

1 97887_9.qp // :6 AM Page Linear Programming Sstems of Linear Inequalities Linear Programming Involving Two The Simple Method: Maimization The Simple Method: Minimization The Simple Method: Mied Constraints Product Advertising (p. 75) Minimizing Steel Waste (p. 69) Optimization Software (p. 5) Restocking ATMs (p. 5) Target Heart Rate (p. 8) Clockwise from top left, Dmitrijs Dmitrijevs/Shutterstock.com; Gerald Bernard/Shutterstock.com; Marquis/Shutterstock.com; wavebreakmedia ltd/shutterstock.com; Grigoreva Liubov Dmitrievna/Shutterstock.com

2 Chapter 9 Linear Programming 9. Sstems of Linear Inequalities Sketch the graph of a linear inequalit. Sketch the graph of a sstem of linear inequalities. LINEAR INEQUALITIES AND THEIR GRAPHS 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 begin b sketching the graph of the corresponding equation 6. The graph of the equation separates the plane into two regions. In each region, one of the following two statements must be true.. All points in the region are solutions of the inequalit.. No point in the region is a solution of the inequalit. So, ou can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region. REMARK When possible, use test points that are convenient to substitute into the inequalit, such as,. Sketching the Graph of an Inequalit in Two. Replace the inequalit sign with an equal sign, and sketch the graph of the resulting equation. 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, ou will work with linear inequalities of the following forms. 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. When the line is dashed, the points on the line are not solutions of the inequalit; when the line is solid, the points on the line are solutions of the inequalit. The simplest linear inequalities are those corresponding to horizontal or vertical lines, as shown in Eample on the net page.

3 9. Sstems of Linear Inequalities 5 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. 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.. Because the origin, satisfies the inequalit, the graph consists of the half-plane ling above the line. (Tr checking a point below the line to see that it does not satisf the inequalit.) (, ) = Figure 9. For a linear inequalit in two variables, ou can sometimes simplif the graphing procedure b writing the inequalit in slope-intercept form. For instance, b writing < in the form > ou can see that the solution points lie above the line, as shown in Figure 9..

4 6 Chapter 9 Linear Programming 9 6 Figure 9. (, ) (, ) is a solution because it satisfies the sstem of inequalities. SYSTEMS OF INEQUALITIES Man practical problems in business, science, and engineering involve sstems of linear inequalities. An eample of such a sstem is shown below. 5 A solution of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem. For instance,, is a solution of this sstem because and 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.. Note that the point, lies in the shaded region because it is a solution of the sstem of inequalities. To sketch the graph of a sstem of inequalities in two variables, 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. This region represents the solution set of the sstem. For sstems of linear inequalities, it is helpful to find the vertices of the solution region, as shown in Eample. Solving a Sstem of Inequalities Sketch the graph (and label the vertices) of the solution set of the sstem shown below. < > SOLUTION You have alread sketched the graph of each of these inequalities 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, find the points of intersection of the boundaries of the region. Verte A:, 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... < 6 > 6 C(, ) B(5, ) A(, ) 6 Figure 9.5

5 9. Sstems of Linear Inequalities 7 (Not a verte) Border lines can intersect at a point that is not a verte. Figure 9.6 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. To determine which points of intersection are actuall vertices of the region, sketch the region and refer to our sketch as ou find each point of intersection. When solving a sstem of inequalities, 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 > + < Figure 9.7 No Solution Another possibilit is that the solution set of a sstem of inequalities can be unbounded. For instance, consider the following sstem. < > The graph of the inequalit < is the half-plane that lies below the line. The graph of the inequalit > is the half-plane that lies above the line. The intersection of these two half-planes is an infinite wedge that has a verte at,, as shown in Figure 9.8. This unbounded region represents the solution set. + = + = Figure 9.8 (, ) Unbounded Region

6 8 Chapter 9 Linear Programming The last eample in this section shows how a sstem of linear inequalities can arise in an applied problem. An Application of a Sstem of Inequalities REMARK An point inside the shaded region (or on its boundar) shown in Figure 9.9 meets the minimum dail requirements for calories and vitamins. For instance, cups of dietar drink X and cups of dietar drink Y suppl calories, 8 units of vitamin A, and 9 units of vitamin C. The liquid portion of a diet is to provide at least calories, 6 units of vitamin A, and 9 units of vitamin C dail. A cup of dietar drink X provides 6 calories, units of vitamin A, and units of vitamin C. A cup of dietar drink Y provides 6 calories, 6 units of vitamin A, and units of vitamin C. Set up a sstem of linear inequalities that describes the minimum dail requirements for calories and vitamins. SOLUTION Let number of cups of dietar drink X and number of cups of dietar drink Y. To meet the minimum dail requirements, the inequalities listed below must be satisfied. For calories: 6 6 For vitamin A: 6 6 For vitamin C: 9 The last two inequalities are included because and cannot be negative. The graph of this sstem of linear inequalities is shown in Figure (, 6) (, ) (, ) (9, ) 6 8 Figure 9.9 LINEAR ALGEBRA APPLIED A heart rate monitor watch is designed to ensure that a person eercises at a health pace. It displas transmissions from a chest monitor that measures the electrical activit of the heart. Researchers have identified target heart rate ranges, or zones, for achieving specific results from eercise. Fitness facilities and cardiolog treadmill rooms often displa wall charts like the one shown here, to help determine whether a person s eercise heart rate is in a health range. In Eercise 58, ou will write a sstem of inequalities for a person s target heart rates during eercise. Heart rate (bpm) Eercise Target Zone Chart Anaerobic Threshold Target Heart Rate Fat Burning Recover Age (ears) Aleksandr Markin/Shutterstock.com

7 9. Eercises 9 9. Eercises Identifing the Graph of a Linear Inequalit In Eercises 6, match the linear inequalit with its graph. [The graphs are labeled (a) (f).]. > > (a) (c) (e) (f) Sketching the Graph of a Linear Inequalit In Eercises 7 6, sketch the graph of the linear inequalit <. > > <.. > > < (b) (d) Solving a Sstem of Inequalities In Eercises 7, determine whether each ordered pair is a solution of the sstem of linear inequalities > < 8 < 7 (a), (a), (b), (b) 6, (c), (c) 8, (d), (d), 9.. (a), (a), (b), (b), (c), (c), (d), (d), Solving a Sstem of Inequalities In Eercises, sketch the graph of the solution of the sstem of linear inequalities < 6 8. > < >. 6 < > < 5. < 6. < 6 > 5 > 9 < < 6 > > > 6 > > 6 < > >

8 Chapter 9 Linear Programming Writing a Sstem of Inequalities In Eercises 5 8, derive a set of inequalities that describes the region Writing a Sstem of Inequalities In Eercises 9 5, derive a set of inequalities that describes the region. 9. Rectangular region with vertices at,, 5,, 5, 7, and, Parallelogram with vertices at,,,,,, and 5,. 5. Triangular region with vertices at,, 5,, and,. 5. Triangular region with vertices at,,,, and,. 5. Reasoning Consider the inequalit 8 < 5. Without graphing, determine whether the solution points lie in the half-plane above or below the boundar line. Eplain our reasoning. 55. Investment A person plans to invest no more than $, in two different interest-bearing accounts. Each account is to contain at least $5. Moreover, one account should have at least twice the amount that is in the other account. Find a sstem of inequalities describing the various amounts that can be deposited in each account, and sketch the graph of the sstem Consider the following sstem of inequalities a b c d > e where a > and b >. Find values of a, b, c, d, and e such that (a) the origin is a solution of the sstem and (b) the origin is not a solution of the sstem. 56. Laptop Inventor A store sells two models of laptop computers. Because of the demand, it is necessar to stock at least twice as man units of the Pro Series as units of the Delue Series. The costs to the store of the two models are $8 and $, respectivel. The management does not want more than $, in laptop inventor at an one time, and it wants at least four Pro Series models and two Delue Series models in inventor at all times. Devise a sstem of inequalities describing all possible inventor levels, and sketch the graph of the sstem. 57. Furniture Production A furniture compan produces tables and chairs. Each table requires hour in the assembl center and hours in the finishing center. Each chair requires hours in the assembl center and hours in the finishing center. The assembl center is available hours per da, and the finishing center is available 5 hours per da. Let and be the numbers of tables and chairs produced per da, respectivel. Find a sstem of inequalities describing all possible production levels, and sketch the graph of the sstem. 58. Target Heart Rate One formula for a person s maimum heart rate is, where is the person s age in ears for 7. When a person eercises, it is recommended that the person strive for a heart rate that is at least 5% of the maimum and at most 75% of the maimum. (Source: American Heart Association) (a) Write a sstem of inequalities that describes the region corresponding to these heart rate recommendations. (b) Sketch a graph of the region in part (a). (c) Find two solutions of the sstem and interpret their meanings in the contet of the problem. 59. Diet Supplement A dietitian designs a special diet supplement using two different foods. Each ounce of food X contains units of calcium, 5 units of iron, and units of vitamin B. Each ounce of food Y contains units of calcium, units of iron, and units of vitamin B. The minimum dail requirements for the diet are units of calcium, 5 units of iron, and units of vitamin B. Find a sstem of inequalities describing the different amounts of food X and food Y that the dietitian can prescribe. Sketch the graph of the sstem. 6. Rework Eercise 59 using minimum dail requirements of 8 units of calcium, 6 units of iron, and 8 units of vitamin B.

9 9. Linear Programming Involving Two 9. Linear Programming Involving Two Find a maimum or minimum of an objective function subject to a sstem of constraints. Find an optimal solution to a real-world linear programming problem. Figure 9. Feasible solutions SOLVING A LINEAR PROGRAMMING PROBLEM Man applications in business and economics involve a process called optimization, which is used to find the minimum cost, the maimum profit, or the minimum use of resources. In this section, 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 Objective function subject to a set of constraints that determines the region in Figure 9.. Because ever point in the region satisfies each constraint, it is not clear how to go about finding 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 of the region. This means that ou can find the maimum value b testing z at each of the vertices. THEOREM 9. Optimal Solution of a Linear Programming Problem If a linear programming problem has an optimal solution, then it must occur at a verte of the set of feasible solutions. If the problem has more than one optimal solution, then at least one of them must occur at a verte of the set of feasible solutions. In either case, the value of the objective function is unique. A linear programming problem can include hundreds, and sometimes even thousands, of variables. However, in this section, ou will solve linear programming problems that involve onl two variables. The graphical method for solving a linear programming problem in two variables is outlined below. Graphical Method for Solving a Linear Programming Problem To solve a linear programming problem involving two variables b the graphical method, use the following steps.. 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 maimum value will eist. (For an unbounded region, if an optimal solution eists, then it will occur at a verte.)

10 Chapter 9 Linear Programming Solving a Linear Programming Problem Find the maimum value of z Objective function subject to the constraints listed below. Constraints + = (, ) = (, ) = (, ) (, ) = Figure 9. SOLUTION The constraints form the region shown in Figure 9.. At the four vertices of this region, the objective function has the values listed below. At, : z At, : z At, : z 8 (Maimum value of z) At, : z So, the maimum value of z is 8, and this occurs when and. 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, : z 5 At At, : z, : z 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. 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 whose slope is, ou want the one that has the largest -intercept and intersects the specified region, as shown in Figure 9.. Such a line will pass through one (or more) of the vertices of the region. z = + z = 8 z = 6 z = z = Figure 9.

11 9. Linear Programming Involving Two The graphical method for solving a linear programming problem will work whether the objective function is to be maimized or minimized. The steps used are precisel the same in either case. Once ou have evaluated the objective function at the vertices of the set of feasible solutions, simpl choose the largest value as the maimum and the smallest value as the minimum. For instance, the same test used in Eample to find the maimum value of z can be used to conclude that the minimum value of z is, and that this occurs at the verte,. Solving a Linear Programming Problem + = = 9 (5, 6) (5, ) (, ) + = 7 5 (, ) (7, ) Figure 9. (, 5) 5 (, ) (6, ) (, ) (, ) (5, ) 6 Figure 9. Find the maimum value of the objective function z 6 Objective function where and, subject to the constraints Constraints SOLUTION The region bounded b the constraints is shown in Figure 9.. B testing the objective function at each verte, ou obtain At, : At, : z 6 z 6 66 At 5, 6: At 5, : z z 5 6 (Maimum value of z) At 7, : z So, the maimum value of z is, and this occurs when 5 and. Minimizing an Objective Function Find the minimum value of the objective function z 5 7 Objective function where and, subject to the constraints 6 5 Constraints 5 7. SOLUTION The region bounded b the constraints is shown in Figure 9.. B testing the objective function at each verte, ou obtain At, : z 5 7 (Minimum value of z) At, : z At, 5: z 5 75 At 6, : z At 5, : z At, : z So, the minimum value of z is, and this occurs when and.

12 Chapter 9 Linear Programming (, ) (, ) (, ) Figure 9.5 z = for an point along this line. (5, ) (5, ) When solving a linear programming problem, it is possible that the maimum (or minimum) value occurs at two different vertices. For instance, at the vertices of the region shown in Figure 9.5, the objective function z Objective function has the following values. At, : z At, : z 8 At, : z (Maimum value of z) At 5, : z 5 (Maimum value of z) At 5, : z 5 In this case, the objective function has a maimum value (of ) not onl at the vertices, and 5,, but also at an point on the line segment connecting these two vertices. Some linear programming problems have no optimal solution. This can occur when the region determined b the constraints is unbounded. Eample illustrates such a problem. An Unbounded Region Find the maimum value of z Objective function where and, subject to the constraints 7 Constraints 7. SOLUTION The region determined b the constraints is shown in Figure (, ) (, ) (, ) 5 Figure 9.6 REMARK For this problem, the objective function does have a minimum value, z, which occurs at the verte,. For this unbounded region, there is no maimum value of z. To see this, note that the point, lies in the region for all values of. B choosing large values of, ou can obtain values of z that are as large as ou want. So, there is no maimum value of z.

13 97887_9.qp // 7: AM Page 5 9. Linear Programming Involving Two 5 APPLICATION An Application: Optimal Cost Eample in Section 9. set up a sstem of linear equations for the following problem. The liquid portion of a diet is to provide at least calories, 6 units of vitamin A, and 9 units of vitamin C dail. A cup of dietar drink X provides 6 calories, units of vitamin A, and units of vitamin C. A cup of dietar drink Y provides 6 calories, 6 units of vitamin A, and units of vitamin C. Now, suppose that dietar drink X costs $. per cup and drink Y costs $.5 per cup. How man cups of each drink should be consumed each da to minimize the cost and still meet the dail requirements? SOLUTION Begin b letting be the number of cups of dietar drink X and be the number of cups of dietar drink Y. Moreover, to meet the minimum dail requirements, the inequalities listed below must be satisfied. For calories: 6 6 For vitamin A: 6 For vitamin C: C..5. (, 6) At 共, 6兲: C.共兲.5共6兲.9 At 共, 兲: C.共兲.5共兲.7 At 共, 兲: C.共兲.5共兲.66 (, ) (, ) Objective function The graph of the region corresponding to the constraints is shown in Figure 9.7. To determine the minimum cost, test C at each verte of the region, as follows. 8 Constraints The cost C is 6 ⱖ ⱖ 6 ⱖ 9 ⱖ ⱖ (9, ) Figure (Minimum value of C) At 共9, 兲: C.共9兲.5共兲.8 So, the minimum cost is $.66 per da, and this occurs when three cups of drink X and two cups of drink Y are consumed each da. LINEAR ALGEBRA APPLIED Financial institutions that replenish automatic teller machines (ATMs) need to take into account a vast arra of variables and constraints to keep the machines stocked appropriatel. Demand for cash machines fluctuates with such factors as weather, economic conditions, customer withdrawing habits, da of the week, and even road construction. Further complicating the matter, government regulations require penalt fees for depositing and withdrawing mone from the U.S. Federal Reserve in the same week. To address this comple problem, a compan can use high-end optimization software to set up and solve a linear programming problem with several variables and constraints. The compan determines an equation for the objective function to minimize total cash in ATMs, while establishing constraints on travel routes, penalt fees, cash limits for trucks, and so on. The optimal solution generated b the software allows the compan to build detailed ATM restocking schedules. wavebreakmedia ltd/shutterstock.com

14 6 Chapter 9 Linear Programming 9. Eercises Solving a Linear Programming Problem In Eercises and, find the minimum and maimum values of each objective function and where the occur, subject to the given constraints.. Objective function:. Objective function: (a) z 8 (a) z (b) z 5.5 (b) z 6 5 Solving a Linear Programming Problem In Eercises 8, sketch the region determined b the indicated constraints. Then find the minimum and maimum values of each objective function and where the occur, subject to the constraints.. Objective function:. Objective function: (a) z 7 (a) z 5 5 (b) z 5 (b) z Objective function: 6. Objective function: (a) z 5 (a) z 5 (b) z 7 (b) z (, 5) (, ) (, ) (, ) (, ) (, ) (5, ) (, ) 5 7. Objective function: 8. Objective function: (a) z (a) z (b) z (b) z Maimizing an Objective Function In Eercises 9, maimize the objective function subject to the constraints 5 and, where,. 9. z. z 5. z. z Recognizing Unusual Characteristics 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. (In each problem, the objective function is to be maimized.). Objective function:. Objective function: z.5 z 5 5. Objective function: 6. Objective function: z z 7. Objective function: 8. Objective function: z z

15 9. Eercises 7 9. Optimal Profit The costs to a store for two models of Global Positioning Sstem (GPS) receivers are $8 and $. The $8 model ields a profit of $5 and the $ model ields a profit of $. Market tests and available resources indicate the following constraints. (a) The merchant estimates that the total monthl demand will not eceed units. (b) The merchant does not want to invest more than $8, in GPS receiver inventor. What is the optimal inventor level for each model? What is the optimal profit?. A compan tring to determine an optimal profit finds that the objective function has a maimum value at the vertices shown in the graph. 8 6 (, ) (5, 7) 6 8 (a) Can ou conclude that it also has a maimum value at the point, 9? Eplain. (b) Can ou conclude that it also has a maimum value at the point 6, 6? Eplain. (c) Find two additional points that maimize the objective function.. Optimal Cost A farming cooperative mies two brands of cattle feed. Brand X costs $ per bag, and brand Y costs $5 per bag. Research and available resources have indicated the following constraints. (a) Brand X contains two units of nutritional element A, two units of element B, and two units of element C. (b) Brand Y contains one unit of nutritional element A, nine units of element B, and three units of element C. (c) The minimum requirements for nutrients A, B, and C are units, 6 units, and units, respectivel. What is the optimal number of bags of each brand that should be mied? What is the optimal cost?. Optimal Cost Rework Eercise given that brand Y now costs $ per bag, and it now contains one unit of nutritional element A, twelve units of element B, and four units of element C.. Optimal Revenue An accounting firm charges $5 for an audit and $5 for a ta return. The times (in hours) required for staffing and reviewing each are shown in the table. The firm has 9 hours of staff time and 55 hours of review time available each week. What numbers of audits and ta returns will bring in an optimal revenue?. Optimal Revenue The accounting firm in Eercise lowers its charge for an audit to $. What numbers of audits and ta returns will bring in an optimal revenue? 5. Optimal Profit A fruit grower has 5 acres of land for raising crops A and B. The profit is $85 per acre for crop A and $5 per acre for crop B. Research and available resources indicate the following constraints. (a) It takes da to trim an acre of crop A and das to trim an acre of crop B, and there are das per ear available for trimming. (b) It takes. da to pick an acre of crop A and. da to pick an acre of crop B, and there are das per ear available for picking. What is the optimal acreage for each fruit? What is the optimal profit? 6. Determine, for each given verte, t-values such that the objective function has a maimum value at the verte. Objective function: z t (a), 6 (b), 5 (c) 6, (d), Finding an Objective Function In Eercises 7, 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.) 7. Maimum at verte A 8. Maimum at verte B 9. Maimum at verte C. Minimum at verte C Component Audit Ta Return Staffing 75.5 Reviewing A(, ) B(, ) 5 6 C(5, )

16 8 Chapter 9 Linear Programming 9. The Simple Method: Maimization Write the simple tableau for a linear programming problem. Use pivoting to find an improved solution. Use the simple method to solve a linear programming problem that maimizes an objective function. Use the simple method to optimize an application. REMARK Note that for a linear programming problem in standard form, the objective function is to be maimized, not minimized. (Minimization problems will be discussed in Sections 9. and 9.5.) THE SIMPLEX TABLEAU For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. For problems involving more than two variables or large numbers of constraints, it is better to use methods that are adaptable to technolog. One such method is called the simple method, developed b George Dantzig in 96. It provides a sstematic wa of eamining the vertices of the feasible region to determine the optimal value of the objective function. Suppose ou want to find the maimum value of z 6, where and, subject to the following constraints Because the left-hand side of each inequalit is less than or equal to the right-hand side, there must eist nonnegative numbers s, s, and s that can be added to the left side of each equation to produce the following sstem of linear equations. s s 7 5 s 9 The numbers s, s, and s are called slack variables because the represent the slack in each inequalit. Standard Form of a Linear Programming Problem A linear programming problem is in standard form when it seeks to maimize the objective function z c c... c n n subject to the constraints a a... a n n b a a... a n n b a m a m.... a mn n b m where i and b i. After adding slack variables, the corresponding sstem of constraint equations is a a... a n n s b a a... a n n s b. a m a m... a mn n where s i. s m b m

17 9. The Simple Method: Maimization 9 A basic solution of a linear programming problem in standard form is a solution,,..., n, s, s,..., s m of the constraint equations in which at most m variables are nonzero, and the variables that are nonzero are called basic variables. A basic solution for which all variables are nonnegative is called a basic feasible solution. The simple method is carried out b performing elementar row operations on a matri called the simple tableau. This tableau consists of the augmented matri corresponding to the constraint equations together with the coefficients of the objective function written in the form c c... c n n s s... s m z. In the tableau, it is customar to omit the coefficient of z. For instance, the simple tableau for the linear programming problem z 6 s s 7 5 s 9 is as follows. s s s b Objective function Constraints s s s Current -value For this initial simple tableau, the basic variables are s, s, and s, and the nonbasic variables are and. Note that the basic variables are labeled to the right of the simple tableau net to the appropriate rows. This technique is important as ou proceed through the simple method. It helps keep track of the changing basic variables, as shown in Eample. Because and are the nonbasic variables in this initial tableau, the have an initial value of zero, ielding a current z-value of zero. From the columns that are farthest to the right, ou can see that the basic variables have initial values of s, s 7, and s 9. So the current solution is,, s, s 7, and s 9. This solution is a basic feasible solution and is often written as,, s, s, s,,, 7, 9. The entr in the lower right corner of the simple tableau is the current value of z. Note that the bottom-row entries under and are the negatives of the coefficients of and in the objective function z 6. z To perform an optimalit check for a solution represented b a simple tableau, look at the entries in the bottom row of the tableau. If an of these entries are negative (as above), then the current solution is not optimal.

18 5 Chapter 9 Linear Programming PIVOTING Once ou have set up the initial simple tableau for a linear programming problem, the simple method consists of checking for optimalit and then, if the current solution is not optimal, improving the current solution. (An improved solution is one that has a larger z-value than the current solution.) To improve the current solution, bring a new basic variable into the solution, the entering variable. This implies that one of the current basic variables (the departing variable) must leave, otherwise ou would have too man variables for a basic solution. Choose the entering and departing variables as follows.. The entering variable corresponds to the smallest (the most negative) entr in the bottom row of the tableau, ecluding the b-column.. The departing variable corresponds to the smallest nonnegative ratio b i a ij in the column determined b the entering variable, when a ij >.. The entr in the simple tableau in the entering variable s column and the departing variable s row is called the pivot. Finall, to form the improved solution, appl Gauss-Jordan elimination to the column that contains the pivot, as illustrated in Eample. (This process is called pivoting.) Pivoting to Find an Improved Solution Use the simple method to find an improved solution for the linear programming problem represented b the tableau shown below. s s s b s 7 s 5 9 s 6 The objective function for this problem is z 6. SOLUTION Note that the current solution,, s, s 7, s 9 corresponds to a z-value of. To improve this solution, ou determine that entering variable, because 6 is the smallest entr in the bottom row. is the s s s b s s s To see wh ou choose as the entering variable, remember that z 6. So, it appears that a unit change in produces a change of 6 in z, whereas a unit change in produces a change of onl in z.

19 9. The Simple Method: Maimization 5 REMARK In the event of a tie when choosing entering and/or departing variables, an of the tied variables ma be chosen. To find the departing variable, locate the b i s that have corresponding positive elements in the entering variables column and form the ratios, 7 7, and Here the smallest nonnegative ratio is, so ou can choose variable s as the departing s s s b s Departing 7 s 5 9 s 6 Note that the pivot is the entr in the first row and second column. Now, use Gauss-Jordan elimination to obtain the improved solution shown below. 5 6 Before Pivoting After Pivoting R R 5R R 6R R The new tableau now appears as follows. s s s b 6 s s 6 66 Note that has replaced in the basic variables column and the improved solution,, s, s, s,,, 6, 5 has a z-value of z In Eample, the improved solution is not et optimal because the bottom row still has a negative entr. So, appl another iteration of the simple method to improve the solution further, as follows. Choose as the entering variable. Moreover, the smaller of the ratios 6 8 and 57 5 is 5, so s is the departing variable. Gauss-Jordan elimination produces the following matrices s R R R R R R R

20 5 Chapter 9 Linear Programming So, the new simple tableau is as follows In this tableau, there is still a negative entr in the bottom row. So, choose s as the entering variable and s as the departing variable, as shown in the net tableau. s s s Departing s s s 7 7 s 7 7 b b One more iteration of the simple method produces the tableau shown below. (Tr checking this.) s 5 (5, 6) 5 (5, ) (, ) 5 (, ) (7, ) Figure Maimum z-value In this tableau, there are no negative elements in the bottom row. So, the optimal solution is determined to be with,, s, s, s 5,,,, z Because the linear programming problem in Eample involved onl two decision variables, ou could have used a graphical solution technique, as in Eample, Section 9.. Notice in Figure 9.8 that each iteration in the simple method corresponds to moving from a given verte to an adjacent verte with an improved z-value., z s, z 66 s 8 s b 5, 6 z 6 s 5, z

21 THE SIMPLEX METHOD 9. The Simple Method: Maimization 5 The steps involved in the simple method can be summarized as follows. The Simple Method (Standard Form) To solve a linear programming problem in standard form, use the following steps.. Convert each inequalit in the set of constraints to an equation b adding slack variables.. Create the initial simple tableau.. Locate the most negative entr in the bottom row, ecluding the b-column. This entr is called the entering variable, and its column is called the entering column. (If ties occur, an of the tied entries can be used to determine the entering column.). Form the ratios of the entries in the b-column with their corresponding positive entries in the entering column. (If all entries in the entering column are or negative, then there is no maimum solution.) The departing row corresponds to the smallest nonnegative ratio b i /a ij. (For ties, choose either row.) The entr in the departing row and the entering column is called the pivot. 5. Use elementar row operations to change the pivot to and all other entries in the entering column to. This process is called pivoting. 6. When all entries in the bottom row are zero or positive, this is the final tableau. Otherwise, go back to Step. 7. If ou obtain a final tableau, then the linear programming problem has a maimum solution. The maimum value of the objective function is given b the entr in the lower right corner of the tableau. Note that the basic feasible solution of an initial simple tableau is,,..., n, s, s,..., s m,,...,, b, b,..., b m. This solution is basic because at most m variables are nonzero (namel, the slack variables). It is feasible because each variable is nonnegative. In the net two eamples, the use of the simple method to solve a problem involving three decision variables is illustrated. LINEAR ALGEBRA APPLIED There are man commerciall available optimization software packages to aid operations researchers in allocating resources to maimize profits and minimize costs. Man of these programs use the simple method as their foundation. As ou will see in the net section, the simple method can also be applied to minimization problems. In addition to optimizing the objective function, the software packages often contain built-in sensitivit reporting to analze how changes or errors in the data will affect the outcome. Marcin Balcerzak/Shutterstock.com

22 5 Chapter 9 Linear Programming The Simple Method with Three Decision Use the simple method to find the maimum value of z subject to the constraints where,, and. SOLUTION Using the basic feasible solution Objective function the initial simple tableau for this problem is as follows. (Tr checking these computations, and note the tie that occurs when choosing the first entering variable.) s s s b s s 5 s Departing s s s b s Departing 5 5 5,,, s, s, s,,,,, This implies that the optimal solution is,,, s, s, s 5,, 5,,, s s s and the maimum value of z is 5. Note that s. The optimal solution ields a maimum value of z 5 provided that and 5 5,,. Notice that these values satisf the constraints giving equalit in the first and third constraints, et the second constraint has a slack of. 5 b s s

23 Occasionall, the constraints in a linear programming problem will include an equation. In such cases, add a slack variable called an artificial variable to form the initial simple tableau. Technicall, this new variable is not a slack variable (because there is no slack to be taken). Once ou have determined an optimal solution in such a problem, check to see that an equations in the original constraints are satisfied. Eample illustrates such a case. The Simple Method with Three Decision Use the simple method to find the maimum value of z subject to the constraints 6 Objective function where,, and. SOLUTION Using the basic feasible solution,,, s, s, s,,,, 6,, the initial simple tableau for this problem is as follows. (Note that s is an artificial variable, rather than a slack variable.) s s s b s Departing 6 s s 5 5 s Departing 7 65 s 5 9. The Simple Method: Maimization 55 s s s b 5 5 s s s b s This implies that the optimal solution is,,, s, s, s, 8,,,, and the maimum value of z is 5. (This solution satisfies the equation provided in the constraints, because 8.)

24 56 Chapter 9 Linear Programming APPLICATIONS Eample shows how the simple method can be used to maimize profits in a business application. A Business Application: Maimum Profit A manufacturer produces three tpes of plastic fitures. The times required for molding, trimming, and packaging are shown in the table. (Times are given in hours per dozen fitures, and profits are given in dollars per dozen fitures.) The maimum amounts of production time that the manufacturer can allocate to each component are as follows. Molding:, hours Trimming: 6 hours Packaging: hours How man dozen of each tpe of fiture should be produced to obtain a maimum profit? SOLUTION Let,, and represent the numbers of dozens of tpes A, B, and C fitures, respectivel. The objective function (to be maimized) is Profit P 6 5 where,, and. Moreover, using the information in the table, ou can construct the constraints listed below., 6 So, the initial simple tableau with the basic feasible solution,,, s, s, s,,,,, 6, is shown on the net page. Process Tpe A Tpe B Tpe C Molding Trimming Packaging Profit $ $6 $5

25 , s Departing 6 s s 6 5 Now, to this initial tableau, appl the simple method as shown. b 6 6 s Departing 8 96, s Departing 9 99,6 9. The Simple Method: Maimization 57 s s 6 s s s s s s s b b s s s s b , From this final simple tableau, the maimum profit is $,, and this is obtained b using the production levels listed below. Tpe A: 6 dozen units Tpe B: 5 dozen units Tpe C: 8 dozen units

26 58 Chapter 9 Linear Programming Eample 5 shows how the simple method can be used to maimize the audience in an advertising campaign. A Business Application: Media Selection The advertising alternatives for a compan include television, radio, and newspaper advertisements. The costs and estimates of audience coverage are provided in the table. The local newspaper limits the number of weekl advertisements from a single compan to ten. Moreover, in order to balance the advertising among the three tpes of media, no more than half of the total number of advertisements should occur on the radio, and at least % should occur on television. The weekl advertising budget is $8,. How man advertisements should be run in each of the three tpes of media to maimize the total audience? SOLUTION To begin, let,, and represent the numbers of advertisements on television, in the newspaper, and on the radio, respectivel. The objective function (to be maimized) is z,, 8, Objective function where,, and. The constraints for this problem are as follows. 6 8,.5. A more manageable form of this sstem of constraints is as follows So, the initial simple tableau is as follows. s s s s b 6 8 s Departing s s 9 s,, 8, Television Newspaper Radio Cost per advertisement $ $6 $ Audience per advertisement,, 8, Constraints

27 Now, to this initial tableau, appl the simple method, as follows. s Departing 9 7 s s, 5 9, 7 6 s Departing s 5,,, 9. The Simple Method: Maimization 59 s s s s s s s s b 9 b 6 s s s s 8 8, 9 8 7, 7 6,,5, b s From this final simple tableau, ou can see that the maimum weekl audience for an advertising budget of $8, is z,5, Maimum weekl audience and this occurs when,, and. The results are summarized as follows. Media Number of Advertisements Cost Audience Television $8, Newspaper $6, Radio $ 5, Total 8 $8,,5,

28 6 Chapter 9 Linear Programming 9. Eercises Writing a Simple Tableau In Eercises, write the simple tableau for the linear programming problem. You do not need to solve the problem. (In each case, the objective function is to be maimized.). Objective function:. Objective function: z z 8 5,,. Objective function:. Objective function: z z 6 9 8,,, Standard Form In Eercises 5 8, eplain wh the linear programming problem is not in standard form. 5. (Minimize) 6. (Maimize) Objective function: Objective function: z z 6,, 7. (Maimize) 8. (Maimize) Objective function: Objective function: z z 5,, 6 6, Pivoting In Eercises 9 and, use one iteration of pivoting to find an improved solution for the linear programming problem represented b the given tableau. 9. s s s b s 6 s 8 s 5. s s s b s 5 6 s 5 s Using the Simple Method In Eercises, use the simple method to solve the linear programming problem. (In each case, the objective function is to be maimized.). Objective function:. Objective function: z z 8,,. Objective function:. Objective function: z z ,, 5. Objective function: 6. Objective function: z z ,,, 7. Objective function: 8. Objective function: z 5 z 7,, 9. Objective function:. Objective function: z 5 8 z ,, 5 6,

29 9. Eercises 6. Objective function: z 5,,. Objective function: z ,,,. Objective function: z 7,,,. Objective function: z ,,, Using Artificial In Eercises 5 and 6, use an artificial variable to solve the linear programming problem, and check our solution. 5. Objective function: z ,, 6. Objective function: z 9 6 8,,, 7. Optimal Profit A grower raises crops X, Y, and Z. The profit is $6 per acre for crop X, $ per acre for crop Y, and $ per acre for crop Z. Research and available resources indicate the following constraints. (a) The grower has 5 acres of land available. (b) The costs per acre of producing crops X, Y, and Z are $, $8, and $, respectivel. (c) The grower s total cost cannot eceed $,. Use the simple method to find the optimal number of acres of land for each crop. What is the maimum profit? 8. Consider the following initial simple tableau. s s s b 5 s 5 s 9 8 s (a) List the objective function and constraints corresponding to the tableau. (b) What solution does the tableau represent? Is it optimal? Eplain. (c) To find an improved solution using an iteration of the simple method, which entering and departing variables would ou select? Eplain. 9. Optimal Profit A fruit juice compan makes two special drinks b blending apple and pineapple juices. The profit per liter is $.6 for the first drink and $.5 for the second drink. The portions of apple and pineapple juice in each drink are shown in the table. Ingredient First drink Second drink Apple juice % 6% Pineapple juice 7% % There are liters of apple juice and 5 liters of pineapple juice available. Use the simple method to find the number of liters of each drink that should be produced in order to maimize the profit.. Optimal Profit Rework Eercise 9 given that the second drink was changed to contain 8% apple juice and % pineapple juice.

30 6 Chapter 9 Linear Programming. Optimal Profit A manufacturer produces three models of biccles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table below. Model A Model B Model C Assembling.5 Painting.5 Packaging.75.5 The total times available for assembling, painting, and packaging are 6 hours, 95 hours, and 5 hours, respectivel. The profits per unit are $5 for model A, $5 for model B, and $55 for model C. What is the optimal production level for each model? What is the optimal profit?. Optimal Profit Suppose that in Eercise the total times available for assembling, painting, and packaging are hours, 5 hours, and 5 hours, respectivel, and that the profits per unit are $8 for model A, $5 for model B, and $5 for model C. What is the optimal production level for each model? What is the optimal profit?. Investment An investor has up to $5, to invest in three tpes of investments. Tpe A investments pa 8% annuall and have a risk factor of. Tpe B investments pa % annuall and have a risk factor of.6. Tpe C investments pa % annuall and have a risk factor of.. To have a well-balanced portfolio, the investor imposes the following conditions. The average risk factor should be no greater than.5. Moreover, at least one-fourth of the total portfolio is to be allocated to tpe A investments and at least one-fourth of the portfolio is to be allocated to tpe B investments. How much should be allocated to each tpe of investment to obtain a maimum return?. Investment An investor has up to $5, to invest in three tpes of investments. Tpe A investments pa 6% annuall and have a risk factor of. Tpe B investments pa % annuall and have a risk factor of.6. Tpe C investments pa % annuall and have a risk factor of.8. To have a well-balanced portfolio, the investor imposes the following conditions. The average risk factor should be no greater than.5. Moreover, at least one-half of the total portfolio is to be allocated to tpe A investments and at least one-fourth of the portfolio is to be allocated to tpe B investments. How much should be allocated to each tpe of investment to obtain a maimum return? Unbounded Solutions In the simple method, it ma happen that in selecting the departing variable all the calculated ratios are negative. This indicates an unbounded solution. Demonstrate this in Eercises 5 and (Maimize) 6. (Maimize) Objective function: Objective function: z z, Other Optimal Solutions If the simple method terminates and one or more variables not in the final basis have bottom-row entries of zero, bringing these variables into the basis will determine other optimal solutions. Demonstrate this in Eercises 7 and (Maimize) 8. (Maimize) Objective function: Objective function: z.5 z 5 5 5,, 5 Using Technolog In Eercises 9 and, use a graphing utilit or software program to maimize the objective function subject to the constraints where,,,. 9. z 7 6. z. 5, True or False? In Eercises and, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the tet. If a statement is false, provide an eample that shows the statement is not true in all cases or cite an appropriate statement from the tet.. After creating the initial simple tableau, the entering column is chosen b locating the most negative entr in the bottom row.. If all entries in the entering column are or negative, then there is no maimum solution.

31 9. The Simple Method: Minimization 9. The Simple Method: Minimization 6 Determine the dual of a linear programming problem that minimizes an objective function. Use the simple method to solve a linear programming problem that minimizes an objective function. THE DUAL OF A MINIMIZATION PROBLEM In Section 9., the simple method was applied onl to linear programming problems in standard form where the objective function was to be maimized. In this section, this procedure will be etended to linear programming problems in which the objective function is to be minimized. A minimization problem is in standard form when the objective function w c c... c n n is to be minimized, subject to the constraints a a... a n n b a a... a n n b a m a m.... a mn n b m where i and b i. The basic procedure used to solve such a problem is to convert it to a maimization problem in standard form, and then appl the simple method as discussed in Section 9.. Consider Eample 5 in Section 9., where geometric methods were used to solve the minimization problem shown below. Minimization Problem: Find the minimum value of w..5 subject to the constraints Objective function Constraints where and. To solve this problem using the simple method, it must be converted to a maimization problem. The first step is to form the augmented matri for this sstem of inequalities. To this augmented matri, add a last row that represents the coefficients of the objective function, as follows Net, form the transpose of this matri b interchanging its rows and columns Note that the rows of this matri are the columns of the first matri, and vice versa. Now, interpret this new matri as a maimization problem, as shown on the net page.