The Berlin Airlift ( ) was an operation by the United States and Great. Linear Programming. Preview Exercises

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1 790 Chapter 7 Sstems of Equations an Inequalities 108. The reason that sstems of linear inequalities are appropriate for moeling health weight is ecause guielines give health weight ranges, rather than specific weights, for various heights I graphe the solution set of Ú + an Ú 1 without using test points. In Eercises , write a sstem of inequalities for each graph Preview Eercises Eercises will help ou prepare for the material covere in the net section a. Graph the solution set of the sstem: + Ú 6 8 Ú List the points that form the corners of the graphe region in part (a). c. Evaluate 3 + at each of the points otaine in part () a. Graph the solution set of the sstem: = Write a sstem of inequalities whose solution set inclues ever point in the rectangular coorinate sstem Sketch the graph of the solution set for the following sstem of inequalities: Ú n + 1n 6 0, 7 0 m + 1m 7 0, 7 0. Ú 0 Ú List the points that form the corners of the graphe region in part (a). c. Evaluate + 5 at each of the points otaine in part () Bottle water an meical supplies are to e shippe to survivors of an earthquake plane.the ottle water weighs 0 pouns per container an meical kits weigh 10 pouns per kit. Each plane can carr no more than 80,000 pouns. If represents the numer of ottles of water to e shippe per plane an represents the numer of meical kits per plane, write an inequalit that moels each plane s 80,000-poun weight restriction. Ojectives Section 7.6 Write an ojective function escriing a quantit that must e maimize or minimize. Use inequalities to escrie limitations in a situation. Use linear programming to solve prolems. Linear Programming West Berlin chilren at Tempelhof airport watch fleets of U.S. airplanes ringing in supplies to circumvent the Soviet lockae. The airlift egan June 8, 198 an continue for 15 months. The Berlin Airlift ( ) was an operation the Unite States an Great Britain in response to militar action the former Soviet Union: Soviet troops close all roas an rail lines etween West German an Berlin, cutting off suppl routes to the cit. The Allies use a mathematical technique evelope uring Worl

2 Section 7.5 Sstems of Inequalities 791 War II to maimize the amount of supplies transporte. During the 15-month airlift, 78,8 flights provie asic necessities to lockae Berlin, saving one of the worl s great cities. In this section, we will look at an important application of sstems of linear inequalities. Such sstems arise in linear programming, a metho for solving prolems in which a particular quantit that must e maimize or minimize is limite other factors. Linear programming is one of the most wiel use tools in management science. It helps usinesses allocate resources to manufacture proucts in a wa that will maimize profit. Linear programming accounts for more than 50% an perhaps as much as 90% of all computing time use for management ecisions in usiness. The Allies use linear programming to save Berlin. Write an ojective function escriing a quantit that must e maimize or minimize. Ojective Functions in Linear Programming Man prolems involve quantities that must e maimize or minimize. Businesses are intereste in maimizing profit. An operation in which ottle water an meical kits are shippe to earthquake survivors nees to maimize the numer of survivors helpe this shipment. An ojective function is an algeraic epression in two or more variales escriing a quantit that must e maimize or minimize. EXAMPLE 1 Writing an Ojective Function Bottle water an meical supplies are to e shippe to survivors of an earthquake plane. Each container of ottle water will serve 10 people an each meical kit will ai 6 people. If represents the numer of ottles of water to e shippe an represents the numer of meical kits, write the ojective function that moels the numer of people that can e helpe. Solution Because each ottle of water serves 10 people an each meical kit ais 6 people, we have The numer of 10 times the numer people helpe is of ottles of water plus 6 times the numer of meical kits. = Using z to represent the numer of people helpe, the ojective function is z = Unlike the functions that we have seen so far, the ojective function is an equation in three variales. For a value of an a value of, there is one an onl one value of z. Thus, z is a function of an. Check Point 1 A compan manufactures ookshelves an esks for computers. Let represent the numer of ookshelves manufacture ail an the numer of esks manufacture ail. The compan s profits are $5 per ookshelf an $55 per esk. Write the ojective function that moels the compan s total ail profit, z, from ookshelves an esks. (Check Points through are relate to this situation, so keep track of our answers.) Use inequalities to escrie limitations in a situation. in Linear Programming Ieall, the numer of earthquake survivors helpe in Eample 1 shoul increase without restriction so that ever survivor receives water an meical kits. However, the planes that ship these supplies are suject to weight an volume restrictions. In linear programming prolems, such restrictions are calle constraints. Each

3 79 Chapter 7 Sstems of Equations an Inequalities constraint is epresse as a linear inequalit. The list of constraints forms a sstem of linear inequalities. EXAMPLE Writing a Constraint Each plane can carr no more than 80,000 pouns. The ottle water weighs 0 pouns per container an each meical kit weighs 10 pouns. Let represent the numer of ottles of water to e shippe an the numer of meical kits.write an inequalit that moels this constraint. Solution Because each plane can carr no more than 80,000 pouns, we have The total weight of the water ottles plus the total weight of the meical kits must e less than or equal to 80,000 pouns ,000. Each ottle weighs 0 pouns. Each kit weighs 10 pouns. The plane s weight constraint is moele the inequalit ,000. Check Point To maintain high qualit, the compan in Check Point 1 shoul not manufacture more than a total of 80 ookshelves an esks per a. Write an inequalit that moels this constraint. In aition to a weight constraint on its cargo, each plane has a limite amount of space in which to carr supplies. Eample 3 emonstrates how to epress this constraint. EXAMPLE 3 Writing a Constraint Each plane can carr a total volume of supplies that oes not ecee 6000 cuic feet. Each water ottle is 1 cuic foot an each meical kit also has a volume of 1 cuic foot. With still representing the numer of water ottles an the numer of meical kits, write an inequalit that moels this secon constraint. Solution Because each plane can carr a volume of supplies that oes not ecee 6000 cuic feet, we have The total volume of the water plus the total volume of the meical kits must e less than or equal to 6000 cuic feet Each ottle is 1 cuic foot. Each kit is 1 cuic foot. The plane s volume constraint is moele the inequalit In summar, here s what we have escrie so far in this ai-to-earthquakevictims situation: z = This is the ojective function moeling the numer of people helpe with ottles of water an meical kits ,000 These are the constraints ase on each plane s weight an volume limitations. Check Point 3 To meet customer eman, the compan in Check Point 1 must manufacture etween 30 an 80 ookshelves per a, inclusive. Furthermore, the compan must manufacture at least 10 an no more than 30 esks per a. Write an inequalit that moels each of these sentences. Then summarize what ou have escrie aout this compan writing the ojective function for its profits an the three constraints.

4 Section 7.6 Linear Programming 793 Use linear programming to solve prolems. Solving Prolems with Linear Programming The prolem in the earthquake situation escrie previousl is to maimize the numer of survivors who can e helpe, suject to each plane s weight an volume constraints. The process of solving this prolem is calle linear programming, ase on a theorem that was proven uring Worl War II. Solving a Linear Programming Prolem Let z = a + e an ojective function that epens on an. Furthermore, z is suject to a numer of constraints on an. If a maimum or minimum value of z eists, it can e etermine as follows: 1. Graph the sstem of inequalities representing the constraints.. Fin the value of the ojective function at each corner, or verte, of the graphe region. The maimum an minimum of the ojective function occur at one or more of the corner points. EXAMPLE Solving a Linear Programming Prolem Determine how man ottles of water an how man meical kits shoul e sent on each plane to maimize the numer of earthquake victims who can e helpe. Solution We must maimize z = suject to the following constraints: , = 80, (0, 6000) (000, 000) + = (0, 0) (000, 0) Figure 7.9 The region in quarant I representing the constraints ,000 an Step 1 Graph the sstem of inequalities representing the constraints. Because (the numer of ottles of water per plane) an (the numer of meical kits per plane) must e nonnegative, we nee to graph the sstem of inequalities in quarant I an its ounar onl. To graph the inequalit ,000, we graph the equation = 80,000 as a soli lue line (Figure 7.9). Setting = 0, the -intercept is 000 an setting = 0, the -intercept is Using (0, 0) as a test point, the inequalit is satisfie, so we shae elow the lue line, as shown in ellow in Figure 7.9. Now we graph first graphing + = 6000 as a soli re line. Setting = 0, the -intercept is Setting = 0, the -intercept is Using (0, 0) as a test point, the inequalit is satisfie, so we shae elow the re line, as shown using green vertical shaing in Figure 7.9. We use the aition metho to fin where the lines = 80,000 an + = 6000 intersect (0, 6000) (000, 000) (0, 0) (000, 0) Figure = 80,000 + = 6000 No change Multipl -10. " " A: = 80, = -60, = 0,000 = 000 Back-sustituting 000 for in + = 6000, we fin = 000, so the intersection point is (000, 000). The sstem of inequalities representing the constraints is shown the region in which the ellow shaing an the green vertical shaing overlap in Figure 7.9. The graph of the sstem of inequalities is shown again in Figure The re an lue line segments are inclue in the graph.

5 79 Chapter 7 Sstems of Equations an Inequalities (0, 6000) (000, 000) (0, 0) (000, 0) Figure 7.30 (repeate) Step Fin the value of the ojective function at each corner of the graphe region. The maimum an minimum of the ojective function occur at one or more of the corner points. We must evaluate the ojective function, z = , at the four corners, or vertices, of the region in Figure Corner (, ) Ojective Function z 10 6 (0, 0) z = = 0 (000, 0) z = = 0,000 (000, 000) z = =,000 ; maimum (0, 6000) z = = 36,000 Thus, the maimum value of z is,000 an this occurs when = 000 an = 000. In practical terms, this means that the maimum numer of earthquake survivors who can e helpe with each plane shipment is,000. This can e accomplishe sening 000 water ottles an 000 meical kits per plane. Check Point For the compan in Check Points 1 3, how man ookshelves an how man esks shoul e manufacture per a to otain maimum profit? What is the maimum ail profit? EXAMPLE 5 Solving a Linear Programming Prolem (0,.5) (0, 0) 1 + = 5 (3, 1) (, 0) = 3 Figure 7.31 The graph of + 5 an - in quarant I Fin the maimum value of the ojective function suject to the following constraints: Solution We egin graphing the region in quarant I 1 Ú 0, Ú 0 forme the constraints. The graph is shown in Figure Now we evaluate the ojective function at the four vertices of this region. Ojective function: z Thus, the maimum value of z is 7, an this occurs when = 3 an = 1. We can see wh the ojective function in Eample 5 has a maimum value that occurs at a verte solving the equation for. z = + = +z Slope = At 10, 0: z = # = 0 At 1, 0: z = # + 0 = At (3, 1): z = 3+1=7 At 10,.5: z = # =.5 -intercept = z z = + Ú 0, Ú 0 c Maimum value of z This is the ojective function of Eample 5. Solve for. Recall that the slope-intercept form of a line is = m +.

6 Section 7.6 Linear Programming In this form, z represents the -intercept of the ojective function. The equation escries infinitel man parallel lines, each with slope -. The process in linear programming involves fining the maimum z-value for all lines that intersect the region etermine the constraints. Of all the lines whose slope is -, we re looking for the one with the greatest -intercept that intersects the given region. As we see in Figure 7.3, such a line will pass through one (or possil more) of the vertices of the region. Check Point 5 Fin the maimum value of the ojective function suject to the constraints Ú 0, Ú 0, + Ú 1, + 6. z = Figure 7.3 The line with slope - with the greatest -intercept that intersects the shae region passes through one of its vertices. Eercise Set 7.6 Practice Eercises In Eercises 1, fin the value of the ojective function at each corner of the graphe region. What is the maimum value of the ojective function? What is the minimum value of the ojective function? 1. Ojective Function z = (1, ) (, 10) (7, 5) (8, 3) 3. Ojective Function z = (0, 8) (, 9) (0, 0) (8, 0). Ojective Function z = 3 + (, 10) (3, ) (5, 1) (8, 6) (7, ). Ojective Function z = (0, 9) (0, 0) (3, 0) (, ) In Eercises 5 1, an ojective function an a sstem of linear inequalities representing constraints are given. a. Graph the sstem of inequalities representing the constraints.. Fin the value of the ojective function at each corner of the graphe region. c. Use the values in part () to etermine the maimum value of the ojective function an the values of an for which the maimum occurs. 5. Ojective Function z = 3 + Ú 0, Ú 0 c Ú 6. Ojective Function 7. Ojective Function 8. Ojective Function 9. Ojective Function 10. Ojective Function 11. Ojective Function z = + 3 Ú 0, Ú 0 c z = + Ú 0, Ú 0 c Ú 3 z = + 6 Ú 0, Ú 0 c Ú -10 z = c Ú - Ú -3 z = c Ú z = + Ú 0, Ú Ú

7 796 Chapter 7 Sstems of Equations an Inequalities 1. Ojective Function 13. Ojective Function 1. Ojective Function Application Eercises Ú 0, Ú 0 + Ú 10 + Ú A television manufacturer makes rear-projection an plasma televisions. The profit per unit is $15 for the rear-projection televisions an $00 for the plasma televisions. a. Let = the numer of rear-projection televisions manufacture in a month an let = the numer of plasma televisions manufacture in a month. Write the ojective function that moels the total monthl profit.. The manufacturer is oun the following constraints: Equipment in the factor allows for making at most 50 rear-projection televisions in one month. Equipment in the factor allows for making at most 00 plasma televisions in one month. The cost to the manufacturer per unit is $600 for the rear-projection televisions an $900 for the plasma televisions. Total monthl costs cannot ecee $360,000. Write a sstem of three inequalities that moels these constraints. c. Graph the sstem of inequalities in part (). Use onl the first quarant an its ounar, ecause an must oth e nonnegative.. Evaluate the ojective function for total monthl profit at each of the five vertices of the graphe region. [The vertices shoul occur at (0, 0), (0, 00), (300, 00), (50, 100), an (50, 0).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit manufacturing rear-projection televisions each month an plasma televisions each month. The maimum monthl profit is $. 16. a. A stuent earns $10 per hour for tutoring an $7 per hour as a teacher s ai. Let = the numer of hours each week spent tutoring an let = the numer of hours each week spent as a teacher s ai. Write the ojective function that moels total weekl earnings.. The stuent is oun the following constraints: To have enough time for stuies, the stuent can work no more than 0 hours per week. The tutoring center requires that each tutor spen at least three hours per week tutoring. z = + The tutoring center requires that each tutor spen no Ú 0, Ú 0 more than eight hours per week tutoring. + 3 Ú 6 Write a sstem of three inequalities that moels these + Ú 3 constraints. + 9 z = c. Graph the sstem of inequalities in part (). Use onl the first quarant an its ounar, ecause an are nonnegative. Ú 0, Ú 0. Evaluate the ojective function for total weekl earnings + 7 at each of the four vertices of the graphe region. [The + 10 vertices shoul occur at (3, 0), (8, 0), (3, 17), an (8, 1).] z = e. Complete the missing portions of this statement: The stuent can earn the maimum amount per week tutoring for hours per week an working as a teacher s ai for hours per week. The maimum amount that the stuent can earn each week is $. Use the two steps for solving a linear programming prolem, given in the o on page 793, to solve the prolems in Eercises A manufacturer prouces two moels of mountain iccles. The times (in hours) require for assemling an painting each moel are given in the following tale: Moel A Assemling 5 Painting 3 Moel B The maimum total weekl hours availale in the asseml epartment an the paint epartment are 00 hours an 108 hours, respectivel. The profits per unit are $5 for moel A an $15 for moel B. How man of each tpe shoul e prouce to maimize profit? 18. A large institution is preparing lunch menus containing foos A an B. The specifications for the two foos are given in the following tale: Units of Carohrates per Ounce Units of Protein per Ounce Units of Fat Foo per Ounce A 1 1 B Each lunch must provie at least 6 units of fat per serving, no more than 7 units of protein, an at least 10 units of carohrates. The institution can purchase foo A for $0.1 per ounce an foo B for $0.08 per ounce. How man ounces of each foo shoul a serving contain to meet the ietar requirements at the least cost? 19. Foo an clothing are shippe to survivors of a natural isaster. Each carton of foo will fee 1 people, while each carton of clothing will help 5 people. Each 0-cuic-foot o of foo weighs 50 pouns an each 10-cuic-foot o of clothing weighs 0 pouns. The commercial carriers transporting foo an clothing are oun the following constraints: The total weight per carrier cannot ecee 19,000 pouns. The total volume must e less than 8000 cuic feet. How man cartons of foo an clothing shoul e sent with each plane shipment to maimize the numer of people who can e helpe?

8 Section 7.6 Linear Programming On June, 198, the former Soviet Union locke all lan an water routes through East German to Berlin. A gigantic airlift was organize using American an British planes to ring foo, clothing, an other supplies to the more than million people in West Berlin. The cargo capacit was 30,000 cuic feet for an American plane an 0,000 cuic feet for a British plane. To reak the Soviet lockae, the Western Allies ha to maimize cargo capacit, ut were suject to the following restrictions: No more than planes coul e use. The larger American planes require 16 personnel per flight, oule that of the requirement for the British planes. The total numer of personnel availale coul not ecee 51. The cost of an American flight was $9000 an the cost of a British flight was $5000. Total weekl costs coul not ecee $300,000. Fin the numer of American an British planes that were use to maimize cargo capacit. 1. A theater is presenting a program for stuents an their parents on rinking an riving. The procees will e onate to a local alcohol information center. Amission is $.00 for parents an $1.00 for stuents. However, the situation has two constraints: The theater can hol no more than 150 people an ever two parents must ring at least one stuent. How man parents an stuents shoul atten to raise the maimum amount of mone?. You are aout to take a test that contains computation prolems worth 6 points each an wor prolems worth 10 points each. You can o a computation prolem in minutes an a wor prolem in minutes. You have 0 minutes to take the test an ma answer no more than 1 prolems. Assuming ou answer all the prolems attempte correctl, how man of each tpe of prolem must ou answer to maimize our score? What is the maimum score? 3. In 1978, a ruling the Civil Aeronautics Boar allowe Feeral Epress to purchase larger aircraft. Feeral Epress s options inclue 0 Boeing 77s that Unite Airlines was retiring an/or the French-uilt Dassault Fanjet Falcon 0. To ai in their ecision, eecutives at Feeral Epress analze the following ata: Boeing 77 Falcon 0 Direct Operating Cost $100 per hour $500 per hour Paloa,000 pouns 6000 pouns Feeral Epress was face with the following constraints: Hourl operating cost was limite to $35,000. Total paloa ha to e at least 67,000 pouns. Onl twent 77s were availale. Given the constraints, how man of each kin of aircraft shoul Feeral Epress have purchase to maimize the numer of aircraft? Writing in Mathematics. What kins of prolems are solve using the linear programming metho? 5. What is an ojective function in a linear programming prolem? 6. What is a constraint in a linear programming prolem? How is a constraint represente? 7. In our own wors, escrie how to solve a linear programming prolem. 8. Descrie a situation in our life in which ou woul reall like to maimize something, ut ou are limite at least two constraints. Can linear programming e use in this situation? Eplain our answer. Critical Thinking Eercises Make Sense? In Eercises 9 3, etermine whether each statement makes sense or oes not make sense, an eplain our reasoning. 9. In orer to solve a linear programming prolem, I use the graph representing the constraints an the graph of the ojective function. 30. I use the coorinates of each verte from m graph representing the constraints to fin the values that maimize or minimize an ojective function. 31. I nee to e ale to graph sstems of linear inequalities in orer to solve linear programming prolems. 3. An important application of linear programming for usinesses involves maimizing profit. 33. Suppose that ou inherit $10,000. The will states how ou must invest the mone. Some (or all) of the mone must e investe in stocks an ons. The requirements are that at least $3000 e investe in ons, with epecte returns of $0.08 per ollar, an at least $000 e investe in stocks, with epecte returns of $0.1 per ollar. Because the stocks are meium risk, the final stipulation requires that the investment in ons shoul never e less than the investment in stocks. How shoul the mone e investe so as to maimize our epecte returns? 3. Consier the ojective function z = A + B 1A 7 0 an B 7 0 suject to the following constraints: + 3 9, -, Ú 0, an Ú 0. Prove that the ojective function will have the same maimum value at the vertices (3, 1) an (0, 3) if A = 3 B. Group Eercises 35. Group memers shoul choose a particular fiel of interest. Research how linear programming is use to solve prolems in that fiel. If possile, investigate the solution of a specific practical prolem. Present a report on our finings, incluing the contriutions of George Dantzig, Narenra Karmarkar, an L. G. Khachion to linear programming. 36. Memers of the group shoul interview a usiness eecutive who is in charge of eciing the prouct mi for a usiness. How are prouction polic ecisions mae? Are other methos use in conjunction with linear programming? What are these methos? What sort of acaemic ackgroun, particularl in mathematics, oes this eecutive have? Present a group report aressing these questions, emphasizing the role of linear programming for the usiness.

9 798 Chapter 7 Sstems of Equations an Inequalities Preview Eercises Eercises will help ou prepare for the material covere in the first section of the net chapter. 37. Solve the sstem: + + z = 19 c + z = 13 z = 5. What makes it fairl eas to fin the solution? 38. Solve the sstem: w z = z = z = 1 z = 3. Epress the solution set in the form 51w,,, z6. What makes it fairl eas to fin the solution? 39. Consier the following arra of numers: B R. Rewrite the arra as follows: Multipl each numer in the top row - an a this prouct to the corresponing numer in the ottom row. Do not change the numers in the top row. Chapter Summar 7 Summar, Review, an Test DEFINITIONS AND CONCEPTS EXAMPLES 7.1 Sstems of Linear Equations in Two Variales a. Two equations in the form A + B = C are calle a sstem of linear equations. A solution of the sstem E. 1, p. 78 is an orere pair that satisfies oth equations in the sstem.. Sstems of linear equations in two variales can e solve eliminating a variale, using the sustitution metho (see the o on page 730) or the aition metho (see the o on page 73). c. Some linear sstems have no solution an are calle inconsistent sstems; others have infinitel man solutions. The equations in a linear sstem with infinitel man solutions are calle epenent. For etails, see the o on page 73.. Functions of Business Revenue Function Cost Function Profit Function R1 = 1price per unit sol C1 = fie cost + 1cost per unit prouce P1 = R1 - C1 The point of intersection of the graphs of R an C is the reak-even point. The -coorinate of the point reveals the numer of units that a compan must prouce an sell so that the mone coming in, the revenue, is equal to the mone going out, the cost.the -coorinate gives the amount of mone coming in an going out. E., p. 730; E. 3, p. 73; E., p. 733 E. 5, p. 73; E. 6, p. 735 E. 9, p. 70; Figure 7.8, p Sstems of Linear Equations in Three Variales a. Three equations in the form A + B + Cz = D are calle a sstem of linear equations in three variales. E. 1, p. 78 A solution of the sstem is an orere triple that satisfies all three equations in the sstem.. A sstem of linear equations in three variales can e solve eliminating variales. Use the aition metho to eliminate an variale, reucing the sstem to two equations in two variales. Use sustitution or the aition metho to solve the resulting sstem in two variales. Details are foun in the o on page 79. c. Three points that o not lie on a line etermine the graph of a quaratic function Use the three given points to create a sstem of three equations. Solve the sstem to fin a,, an c. E., p. 79; E. 3, p. 751 = a + + c. E., p. 75