Q:1 Productco produces three products. Each product requires labor, lumber, and paint. The resource requirements, unit price, and variable cost (exclusive of raw materials) for each product are given in Table 70. Currently, 900 labor hours, 1,550 gallons of paint, and 1,600 board feet of lumber are available. Additional labor can be purchased at \$6 per hour, additional paint at \$2 per gallon, and additional lumber at \$3 per board foot. For the following two sets of priorities, use preemptive goal programming to determine an optimal production schedule.

For set 1: Priority 1 Obtain profit of at least \$10,500.

Priority 2 Purchase no additional labor.

Priority 3 Purchase no additional paint.

Priority 4 Purchase no additional lumber.

Q:2 Consider a maximization problem with the optimal tableau in Table 73. The optimal solution to this LP is z 10, x3 3, x4 5, x1 x2 0. Determine the second-best bfs to this LP.(Hint: Show that the second-best solution must be a bfs that is one pivot away from the optimal solution.)

Q:3 A hospital outpatient clinic performs four types of operations. The profit per operation, as well as the minutes of X-ray time and laboratory time used are given in Table 72. The clinic has 500 private rooms and 500 intensive care rooms. Type 1 and Type 2 operations require a patient to stay in an intensive care room for one day while Type 3 and Type 4 operations require a patient to stay in a private room for one day. Each day the hospital is required to perform at least 100 operations of each type. The hospital has set the following goals:

Goal 1 Earn a daily profit of at least \$100,000.

Goal 2 Use at most 50 hours daily of X-ray time.

Goal 3 Use at most 40 hours daily of laboratory time.

Q:4 Suppose we have obtained the tableau in Table 75 for a maximization problem. State conditions on a1, a2, a3, b, c1, and c2 that are required to make the following statements true:

a The current solution is optimal, and there are alternative optimal solutions.

b The current basic solution is not a basic feasible solution

Q:5 Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are given in Figure 31. Town 1 is at (10, 20); town 2 is at (60, 20); town 3 is at (40, 30); town 4 is at (80, 60). Town 1 averages 20 fires per year; town 2, 30 fires; town 3, 40 fires; and town 4, 25 fires. The county wants to build the fire station in a location that minimizes the average distance that a fire engine must travel to respond to a fire. Since most roads run in either an east–west or a north–south direction, we assume that the fire engine can only do the same. Thus, if the fire station were located at (30, 40) and a fire occurred at town 4, the fire engine would have to travel (80  30)  (60  40) 70 miles to the fire. Use linear programming to determine where the fire station should be located. (Hint: If the fire station is to be located at the point (x, y) and there is a town at the point (a, b), define variables e, w, n, s (east, west, north, south) that satisfy the equations x  a w  e and y  b n  s. It should now be easy to obtain the correct LP formulation.)

Q:6 During the 1972 football season, the games shown in Table 76 were played by the Miami Dolphins, the Buffalo Bills, and the New York Jets. Suppose that on the basis of these games, we want to rate these three teams. Let M Miami rating, J Jets rating, and B Bills rating. Given values of M, J, and B, you would predict that when, for example, the Bills play Miami, Miami is expected to win by M  B points. Thus, for the first Miami–Bills game, your prediction would have been in error by |M  B 1| points. Show how linear programming can be used to determine ratings for each team that minimize the sum of the prediction errors for all games.

Q:7 A bus company believes that it will need the following number of bus drivers during each of the next five years: year 1—60 drivers; year 2—70 drivers; year 3—50 drivers; year 4—65 drivers; year 5—75 drivers. At the beginning of each year, the bus company must decide how many drivers should be hired or fired. It costs \$4,000 to hire a driver and \$2,000 to fire a driver. A driver’s salary is \$10,000 per year. At the beginning of year 1, the company has 50 drivers. A driver hired at the beginning of a year may be used to meet the current year’s requirements and is paid full salary for the current year. Formulate an LP to minimize the bus company’s salary, hiring, and firing costs over the next five years.

Q:8 During the next four quarters, Dorian Auto must meet (on time) the following demands for cars: quarter 1—4,000; quarter 2—2,000; quarter 3—5,000; quarter 4—1,000. At the beginning of quarter 1, there are 300 autos in stock, and the company has the capacity to produce at most 3,000 cars per quarter. At the beginning of each quarter, the company can change production capacity by one car. It costs \$100 to increase quarterly production capacity. It costs \$50 per quarter to maintain one car of production capacity (even if it is unused during the current quarter). The variable cost of producing a car is \$2,000. A holding cost of \$150 per car is assessed against each quarter’s ending inventory. It is required that at the end of quarter 4, plant capacity must be at least 4,000 cars. Formulate an LP to minimize the total cost incurred during the next four quarters.

Q:9 Ghostbusters, Inc., exorcises (gets rid of) ghosts. During each of the next three months, the company will receive the following number of calls from people who want their ghosts exorcised: January, 100 calls; February, 300 calls; March, 200 calls. Ghostbusters is paid \$800 for each ghost exorcised during the month in which the customer calls. Calls need not be responded to during the month they are made, but if a call is responded to one month after it is made, then Ghostbusters loses \$100 in future goodwill, and if a call is responded to two months after it is made, Ghostbusters loses \$200 in goodwill. Each employee of Ghostbusters can exorcise 10 ghosts during a month. Each employee is paid a salary of \$4,000 per month. At the beginning of January, the company has 8 workers. Workers can be hired and trained (in 0 time) at a cost of \$5,000 per worker. Workers can be fired at a cost of \$4,000 per worker. Formulate an LP to maximize Ghostbusters’ profit (revenue less costs) over the next three months. Assume that all calls must be handled by the end of March

Q:10 Carco uses robots to manufacture cars. The following demands for cars must be met (not necessarily on time, but all demands must be met by end of quarter 4): quarter 1— 600; quarter 2—800; quarter 3—500; quarter 4—400. At the beginning of the quarter, Carco has two robots. Robots can be purchased at the beginning of each quarter, but a maximum of two per quarter can be purchased. Each robot can build as many as 200 cars per quarter. It costs \$5,000 to purchase a robot. Each quarter, a robot incurs \$500 in maintenance costs (even if it is not used to build any cars). Robots can also be sold at the beginning of each quarter for \$3,000. At the end of each quarter, a holding cost of \$200 per car is incurred. If any demand is backlogged, then a cost of \$300 per car is incurred for each quarter the demand is backlogged. At the end of quarter 4, Carco must have at least two robots. Formulate an LP to minimize the total cost incurred in meeting the next four quarters’ demands for cars.

Q:11 Suppose the bfs for an optimal tableau is degenerate, and a nonbasic variable in row 0 has a zero coefficient. Show by example that either of the following cases may hold:

Case 1 The LP has more than one optimal solution.

Case 2 The LP has a unique optimal solution.

Q:12 Show that if the contribution to profit for trains is between \$1.50 and \$3, the current basis remains optimal. If the contribution to profit for trains is \$2.50, then what would be the new optimal solution?  Show that if available carpentry hours remain between 60 and 100, the current basis remains optimal. If between 60 and 100 carpentry hours are available, would Giapetto still produce 20 soldiers and 60 trains?  Show that if the weekly demand for soldiers is at least 20, then the current basis remains optimal, and Giapetto should still produce 20 soldiers and 60 trains.

Q:13 For the Dorian Auto problem (Example 2 in Chapter 3), a Find the range of values on the cost of a comedy ad for which the current basis remains optimal.

b Find the range of values on the cost of a football ad for which the current basis remains optimal.

c Find the range of values for required HIW exposures for which the current basis remains optimal. Determine the new optimal solution if 28 million HIW exposures are required.

d Find the range of values for required HIM exposures for which the current basis remains optimal. Determine

Q:14 Consider the diet problem discussed in Section 3.4. Use the LINDO output in Figure 8 to answer the following questions. a If a Brownie costs 30¢, then what would be the new optimal solution to the problem?

b If a bottle of cola cost 35¢, then what would be the new optimal solution to the problem?

c If at least 8 oz of chocolate were required, then what would be the cost of the optimal diet?

d If at least 600 calories were required, then what would be the cost of the optimal diet?

e If at least 9 oz of sugar were required, then what would be the cost of the optimal diet?

Q:15 Mondo produces motorcycles at three plants. At each plant, the labor, raw material, and production costs (excluding labor cost) required to build a motorcycle are as shown in Table 7. Each plant has sufficient machine capacity to produce up to 750 motorcycles per week. Each of Mondo’s workers can work up to 40 hours per week and is paid \$12.50 per hour worked. Mondo has a total of 525 workers and now owns 9,400 units of raw material. Each week, at least 1,400 Mondos must be produced. Let x1 motorcycles produced at plant 1; x2 motorcycles produced at plant 2; and x3 motorcycles produced at plant 3. The LINDO output in Figure 10 enables Mondo to minimize the variable cost (labor production) of meeting demand. Use the output to answer the following questions: a What would be the new optimal solution to the problem if the production cost at plant 1 were only \$40?

Q:16 Steelco uses coal, iron, and labor to produce three types of steel. The inputs (and sales price) for one ton of each type of steel are shown in Table 8. Up to 200 tons of coal can be purchased at a price of \$10 per ton. Up to 60 tons of iron can be purchased at \$8 per ton, and up to 100 labor hours can be purchased at \$5 per hour. Let x1 tons of steel 1 produced; x2 tons of steel 2 produced; and x3 tons of steel 3 produced.

Q:17 In Problem 4 of Section 5.2, how much should Gepbab be willing to pay for another unit of capacity at plant 1? 5 In Problem 5 of Section 5.2, suppose that Mondo could purchase an additional unit of raw material at a cost of \$6. Should the company do it? Explain. 6 In Problem 6 of Section 5.2, what is the most that Steelco should be willing to pay for an extra ton of coal? In Problem  of , what is the most that Carco should be willing to pay for an extra ton of steel?  In Problem of  what is the most that Carco should be willing to pay to rent an additional Type 1 machine for one day? In Problem what is the most that one should be willing to pay for an additional ounce of chocolate?

Q:18 In Problem 6 of Section , what is the most that Steelco should be willing to pay for an extra ton of iron?  In Problem 6 of Section 5.2, what is the most that Steelco should be willing to pay for an extra hour of labor? 9 In Problem  of Section 5.2, suppose that a new customer wishes to buy a pair of shoes during month 1 for \$70. Should Shoeco oblige him?

Q:19 In Problem 7 of Section 5.2, what is the most the company would be willing to pay for having one more worker at the beginning of month 1?  In solving part (c) of Example 8, a manager reasons as follows: The average cost of producing a car is \$11,600 up to 1,000 cars. Therefore, if a customer is willing to pay me \$25,000 for a car, I should certainly fill his order. What is wrong with this reasoning?

Q:20 Use the LINDO PARA command to graph the optimal z-value for  as a function of b4. 2 Use the PARA command to graph the optimal z-value for Example 2 as a function of b1. Then answer the same questions for b2, b3, and b4, respectively. 3 For the Giapetto example of Section 3.1, graph the optimal z-value as a function of x2’s objective function coefficient. Also graph the optimal z-value as a function of b1, b2, and b3.

Q:21 For Example 1, suppose that we increase the sales price of a product. Show that in the new optimal solution, the amount produced of that product cannot decrease. 6 For , suppose that we increase the cost of producing a type of car. Show that in the new optimal solution to the LP, the number of cars produced of that type cannot increase.

Q:22 Consider the Sailco problem (Example 12 in Chapter 3). Suppose we want to consider how profit will be affected if we change the number of sailboats that can be produced each month with regular-time labor. How can we use the PARA command to answer this question? (Hint: Let c change in number of sailboats that can be produced each month with regular-time labor. Change the right-hand side of some constraints to 40 c and add another constraint to the problem.)

Q:23 HAL produces two types of computers: PCs and VAXes. The computers are produced in two locations: New York and Los Angeles. New York can produce up to 800 computers and Los Angeles up to 1,000 computers. HAL can sell up to 900 PCs and 900 VAXes. The profit associated with each production site and computer sale is as follows: New York—PC, \$600; VAX, \$800; Los Angeles—PC, \$1,000; VAX, \$1,300. The skilled labor required to build each computer at each location is as follows: New York— PC, 2 hours; VAX, 2 hours; Los Angeles—PC, 3 hours; VAX, 4 hours. A total of 4,000 hours of labor are available. Labor is purchased at a cost of \$20 per hour. Let XNP PCs produced in New York XLP PCs produced in Los Angeles XNV VAXes produced in New York XLV VAXes produced in Los Angeles Use the LINDO printout in Figure 17 to answer the following questions:

a If 3,000 hours of skilled labor were available, what would be HAL’s profit?

b Suppose an outside contractor offers to increase the capacity of New York to 850 computers at a cost of \$5,000. Should HAL hire the contractor?

c By how much would the profit for a VAX produced in Los Angeles have to increase before HAL would want to produce VAXes in Los Angeles? d What is the most HAL should pay for an extra hour of labor?

Q:24 Vivian’s Gem Company produces two types of gems: Types 1 and 2. Each Type 1 gem contains 2 rubies and 4 diamonds. A Type 1 gem sells for \$10 and costs \$5 to produce. Each Type 2 gem contains 1 ruby and 1 diamond. A Type 2 gem sells for \$6 and costs \$4 to produce. A total of 30 rubies and 50 diamonds are available. All gems that are produced can be sold, but marketing considerations

Q:25 Wivco produces product 1 and product 2 by processing raw material. Up to 90 lb of raw material may be purchased at a cost of \$10/lb. One pound of raw material can be used to produce either 1 lb of product 1 or 0.33 lb of product 2. Using a pound of raw material to produce a pound of product 1 requires 2 hours of labor or 3 hours to produce 0.33 lb of product 2. A total of 200 hours of labor are available, and at most 40 pounds of product 2 can be sold. Product 1 sells for \$13/lb and product 2, \$40/lb. Let RM pounds of raw material processed

P1 pounds of raw material used to produce product 1

P2 pounds of raw material used to produce product 2

Q:26 Giapetto, Inc., sells wooden soldiers and wooden trains. The resources used to produce a soldier and train are shown in Table 11. A total of 145,000 board feet of lumber and 90,000 hours of labor are available. As many as 50,000 soldiers and 50,000 trains can be sold, with trains selling for \$55 and soldiers for \$32. In addition to producing trains and soldiers itself, Giapetto can buy (from an outside supplier) extra soldiers at \$27 each and extra trains at \$50 each. Let SM thousands of soldiers manufactured SB thousands of soldiers bought at \$27 TM thousands of trains manufactured TB thousands of trains bought at \$50 Then Giapetto can maximize profit by solving the LP in the LINDO printout in Figure 22. Use this printout to answer the following questions. (Hint: Think about the units of the constraints and objective function.)

Q:27 Wivco produces two products: 1 and 2. The relevant data are shown in Table 12. Each week, up to 400 units of raw material can be purchased at a cost of \$1.50 per unit. The company employs four workers, who work 40 hours per week. (Their salaries are considered a fixed cost.) Workers are paid \$6 per hour to work overtime. Each week, 320 hours of machine time are available.

Q:28 a If overtime cost only \$4 per hour, would Wivco use it?  If each unit of product 1 sold for \$15.50, would the current basis remain optimal? What would be the new optimal solution? What is the most that Wivco should be willing to pay for another unit of raw material?  How much would Wivco be willing to pay for another hour of machine time? If each worker were required (as part of the regular workweek) to work 45 hours per week, what would the company’s profits be? f Explain why the shadow price of row (1) is 0.10. (Hint: If the right-hand side of (1) were increased from 50 to 51, then in the absence of advertising for product 1, 51 units of product 1 could now be sold each week.)

Q:29 Machinco produces four products, requiring time on two machines and two types (skilled and unskilled) of labor. The amount of machine time and labor (in hours) used by each product and the sales prices are given in Table 13. Each month, 700 hours are available on machine 1 and 500 hours on machine 2. Each month, Machinco can purchase up to 600 hours of skilled labor at \$8 per hour and up to 650 hours of unskilled labor at \$6 per hour. Formulate an LP that will enable Machinco to maximize its monthly profit. Solve this LP and use the output to answer the following questions:

a By how much does the price of product 3 have to increase before it becomes optimal to produce it?

b If product 1 sold for \$290, then what would be the new optimal solution to the problem?

c What is the most Machinco would be willing to pay for an extra hour of time on each machine?

d What is the most Machinco would be willing to pay for an extra hour of each type of labor?

e If up to 700 hours of skilled labor could be purchased each month, then what would be Machinco’s monthly profits?

Q:30 Autoco has three assembly plants located in various parts of the country. The first plant (built in 1937 and located in Norwood, Ohio) requires 2 hours of labor and 1 hour of machine time to assemble one automobile. The second plant (built in 1958 and located in Bakersfield, California) requires 1.5 hours of labor and 1.5 hours of machine time to assemble one automobile. The third plant (built in 1981 and located in Kingsport, Tennessee) requires 1.1 hours of labor and 2.5 hours of machine time to assemble one automobile. The firm pays \$30 per hour of labor and \$10 per hour of machine time at each of its plants. The first plant has a capacity of 1,000 hours of machine time per day; the second, 900 hours; and the third, 2,000 hours. The manufacturer’s production target is 1,800 automobiles per day.

Q:31 Machinco produces four products, requiring time on two machines and two types (skilled and unskilled) of labor. The amount of machine time and labor (in hours) used by each product and the sales prices are given in Table 13. Each month, 700 hours are available on machine 1 and 500 hours on machine  Each month, Machinco can purchase up to 600 hours of skilled labor at \$8 per hour and up to 650 hours of unskilled labor at \$6 per hour. Formulate an LP that will enable Machinco to maximize its monthly profit. Solve this LP and use the output to answer the following questions:  By how much does the price of product  have to increase before it becomes optimal to produce it?  If product 1 sold for \$290, then what would be the new optimal solution to the problem?  What is the most Machinco would be willing to pay for an extra hour of time on each machine? What is the most Machinco would be willing to pay for an extra hour of each type of labor? If up to 700 hours of skilled labor could be purchased each month, then what would be Machinco’s monthly profits?

Q:32 Solve Review Problem 24 of Chapter 3 on LINDO and answer the following questions:  For which type of DRGs should the hospital seek to increase demand?  What resources are in excess supply? Which resources should the hospital expand?  What is the most the hospital should be willing to pay additional nurses?

Q:33 Old Macdonald’s 200-acre farm sells wheat, alfalfa, and beef. Wheat sells for \$30 per bushel, alfalfa sells for \$200 per bushel, and beef sells for \$300 per ton. Up to 1,000 bushels of wheat and up to 1,000 bushels of alfalfa can be sold, but demand for beef is unlimited. If an acre of land is devoted to raising wheat, alfalfa, or beef, the yield and the required labor are given in Table 15. As many as 2,000 hours of labor can be purchased at \$15 per hour. Each acre devoted to beef requires 5 bushels of alfalfa. The LINDO output in Figure 24 shows how to maximize profit, use it to answer the following questions

Q:34 Show that if the contribution to profit for trains is between \$1.50 and \$3, the current basis remains optimal. If the contribution to profit for trains is \$2.50, what would be the new optimal solution?  Show that if available carpentry hours remain between 60 and 100, the current basis remains optimal. If between 60 and 100 carpentry hours are available, then would Giapetto still produce 20 soldiers and 60 trains?

Q:35 Show that if the weekly demand for soldiers is at least 20, the current basis remains optimal, and Giapetto should still produce 20 soldiers and 60 trains.  For the Dorian Auto problem (Example 2 in Chapter 3),  Find the range of values of the cost of a comedy ad for which the current basis remains optimal.  Find the range of values of the cost of a football ad for which the current basis remains optimal.

Q:36 Find the range of values of required HIW exposures for which the current basis remains optimal. Determine the new optimal solution if 28 million HIW exposures are required.  Find the range of values of required HIM exposures for which the current basis remains optimal. Determine the new optimal solution if 24 million HIM exposures are required.  Find the shadow price of each constraint. f If 26 million HIW exposures are required, determine the new optimal z-value.

Q:37 For the following LP, x2 and s1 are basic variables in the optimal tableau. Use the formulas of this section to determine the optimal tableau

max z

x1 x2 s.t. 2×1 +x2

s.t. x1+ x2

s.t. x1, x2

Q:38 In the Dakota problem, show that the current basis remains optimal if c3, the price of chairs, satisfies 15  c3  22.5. If c3 21, find the new optimal solution. Also, if c3 25, find the new optimal solution.  If c1 55 in the Dakota problem, show that the new optimal solution does not produce any desks.

Q:39 In the Dakota problem, show that if the amount of lumber (board ft) available (b1) satisfies b1 24, the current basis remains optimal. If b1 30, find the new optimal solution.  Show that if tables sell for \$50 and use 1 board ft of lumber, 3 finishing hours, and 1.5 carpentry hours, the current basis for the Dakota problem will no longer be optimal. Find the new optimal solution.

Q:40 Dakota Furniture is considering manufacturing home computer tables. A home computer table sells for \$36 and uses 6 board ft of lumber, 2 finishing hours, and 2 carpentry hours. Should the company manufacture any home computer tables? 6 Sugarco can manufacture three types of candy bar. Each candy bar consists totally of sugar and chocolate. The compositions of each type of candy bar and the profit earned from each candy bar are shown in Table 10. Fifty oz of sugar and 100 oz of chocolate are available. After defining xi to be the number of Type i candy bars manufactured, Sugarco should solve the following LP

Q:41 For what values of Type 1 candy bar profit does the current basis remain optimal? If the profit for a Type 1 candy bar were 7¢, what would be the new optimal solution to Sugarco’s problem? For what values of Type 2 candy bar profit would the current basis remain optimal? If the profit for a Type 2 candy bar were 13¢, then what would be the new optimal solution to Sugarco’s problem? c For what amount of available sugar would the current basis remain optimal?

Q:42 Show that as long as soldiers (x1) contribute between \$2 and \$4 to profit, the current basis remains optimal. If soldiers contribute \$3.50 to profit, find the new optimal solution to the Giapetto problem.  Show that as long as trains (x2) contribute between \$1.50 and \$3.00 to profit, the current basis remains optimal. c Show that if between 80 and 120 finishing hours are available, the current basis remains optimal. Find the new optimal solution to the Giapetto problem if 90 finishing hours are available.

Q:43 Show that as long as the demand for soldiers is at least 20, the current basis remains optimal. e Giapetto is considering manufacturing toy boats. A toy boat uses 2 carpentry hours and 1 finishing hour. Demand for toy boats is unlimited. If a toy boat contributes \$3.50 to profit, should Giapetto manufacture any toy boats?

Q:44 Find the range of values of the cost of a comedy ad (currently \$50,000) for which the current basis remains optimal.  Find the range of values of the number of required HIW exposures (currently 28 million) for which the current basis remains optimal. If 40 million HIW exposures were required, what would be the new optimal solution?  Suppose an ad on a news program costs \$110,000 and reaches 12 million HIW and 7 million HIM. Should Dorian advertise on the news program? 9 Show that if the right-hand side of the ith constraint is increased by , then the right-hand side of the optimal tableau is given by (original right-hand side of the optimal tableau) (column i of B1 ).

Q:45 If the cost of a brownie is 70¢ and a piece of cheesecake costs 60¢, does the current basis remain optimal?  If the cost of a brownie is 20¢ and a piece of cheesecake is \$1, does the current basis remain optimal?  If the fat requirement is reduced to 3 oz and the calorie requirement is increased to 800 calories, does the current basis remain optimal?

Q:46 600 calories, does the current basis remain optimal?  If the price of a bottle of soda is 15¢ and a piece of cheesecake is 60¢, show that the current basis remains optimal. What will be the new optimal solution to the diet problem? 6 If 8 oz of chocolate and 60 calories are required, show that the current basis remains optimal. The following questions refer to the Dakota problem.

Q:47 Suppose that the price of a desk is \$65, a table is \$25, and a chair is \$18. Show that the current basis remains optimal. What is the new optimal z-value?  Suppose that 60 board ft of lumber and 23 finishing hours are available. Show that the current basis remains optimal.  Suppose 40 board ft of lumber, 21 finishing hours, and 8.5 carpentry hours are available. Show that the current basis remains optimal.

Q:48 Prove the Case 1 result for right-hand sides. Use the fact that if a constraint is nonbinding in the optimal solution, then its slack or excess variable is in the optimal basis, and the corresponding column of B1 will have a single 1 and all other elements equal to 0. 13 In this problem, we sketch a proof of the 100% Rule for right-hand sides. Consider an LP with two constraints and right-hand sides b1 and b2. Suppose that if only one right-hand side is changed, the current basis remains optimal for L1  b1  U1 and L2  b2  U2. Suppose we change the right-hand sides to b1  b1 b1 and b2  b2 b2.

Q:49 Find the dual of  in Chapter 3 (an auto company) and give an economic interpretation of the dual problem.  Find the dual of Example 2 in Chapter 3 (Dorian Auto) and give an economic interpretation of the dual problem.

Q:50 The following questions refer to the Giapetto problem (see Problem 7 of Section 6.3).

a Find the dual of the Giapetto problem.

b Use the optimal tableau of the Giapetto problem to determine the optimal dual solution.

c Verify that the Dual Theorem holds in this instance.

Q:51 In this problem, we use weak duality to prove Lemma 3. a Show that Lemma 3 is equivalent to the following: If the dual is feasible, then the primal is bounded. (Hint: Do you remember, from plane geometry, what the contrapositive is?)  Use weak duality to show the validity of the form of Lemma 3 given in part (a). (Hint: If the dual is feasible, then there must be a dual feasible point having a w-value of, say, wo. Now use weak duality to show that the primal is bounded.)

Q:52 Find and interpret the shadow prices  If 18 finishing hours were available, what would be Dakota’s revenue? (It can be shown by the methods of Section 6.3 that if 16  finishing hours  24, the current basis remains optimal.) 3 If 9 carpentry hours were available, what would be Dakota’s revenue? (For  carpentry hours  10, the current basis remains optimal.) 314 CHAPTER 6 Sensitivity Analysis and Duality  1 Shadow Prices for Normal Max Problem 6.8 Shadow Prices 315 4 If 30 board feet of lumber were available, what would be Dakota’s revenue? (For 24 the current basis remains optimal.) 5 If 30 carpentry hours were available, why couldn’t the shadow price for the carpentry constraint be used to determine the new z-value?

Q:53 Suppose we are working with a min problem and increase the right-hand side of a  constraint. What can happen to the optimal z-value? Suppose we are working with a min problem and increase the right-hand side of a  constraint. What can happen to the optimal z-value?

Q:54 A company manufactures two products (1 and 2). Each unit of product 1 can be sold for \$15, and each unit of product 2 for \$25. Each product requires raw material and two types of labor (skilled and unskilled) (see Table 29). Currently, the company has available 100 hours of skilled labor, 70 hours of unskilled labor, and 30 units of raw material. Because of marketing considerations, at least 3 units of product 2 must be produced. a Explain why the company’s goal is to maximize revenue.

Q:55 7 For the Dorian problem (see Problem 8 of Section 6.3), answer the following questions:

a What would Dorian’s cost be if 40 million HIW exposures were required?

b What would Dorian’s cost be if only 20 million HIM exposures were required?

Q:56 Assuming the current basis remains optimal (it does), what would the company’s revenue be if 35 units of raw material were available? With the current basis optimal, what would the company’s revenue be if 80 hours of skilled labor were available? With the current basis optimal, what would the company’s new revenue be if at least 5 units of product 2 were required? How about if at least 2 units of product 2 were required?

Q:57 For the Dorian problem (see Problem 8 of Section 6.3), answer the following questions:

a What would Dorian’s cost be if 40 million HIW exposures were required?

b What would Dorian’s cost be if only 20 million HIM exposures were required?

Q:58 For the Dakota problem, suppose that 22 finishing hours and 9 carpentry hours are available. What would be the new optimal z-value? [Hint: Use the 100% Rule to show that the current basis remains optimal, and mimic (34)–(36).]  For the diet problem, suppose at least 8 oz of chocolate and at least 9 oz of sugar are required (with other requirements remaining the same). What is the new optimal z-value?

Q:59 For the Dakota problem, suppose computer tables sell for \$35 and use 6 board feet of lumber, 2 hours of finishing time, and 1 hour of carpentry time. Is the current basis still optimal? Interpret this result in terms of shadow prices. The following questions refer to the Sugarco problem (Problem 6 of Section 6.3): a For what values of profit on a Type 1 candy bar does the current basis remain optimal? b If a Type 1 candy bar used 0.5 oz of sugar and 0.75 oz of chocolate, would the current basis remain optimal? c A Type 4 candy bar is under consideration. A Type 4 candy bar yields a 10¢ profit and uses 2 oz of sugar and 1 oz of chocolate. Does the current basis remain optimal?

Q:60 Suppose, in the Dakota problem, a desk still sells for \$60 but now uses 8 board ft of lumber, 4 finishing hours, and 15 carpentry hours. Determine whether the current basis remains optimal. What is wrong with the following reasoning? The change in the column for desks leaves the second and third dual constraints unchanged and changes the first to 8y1 4y2 15y3 60 Because y1 0, y2 10, y3 10 satisfies the new dual constraint, the current basis remains optimal.

Q:61 Let x [x1 x2 x3 s1 s2 s3] be a primal feasible point for the Dakota problem and y [ y1 y2 y3 e1 e2 e3] be a dual feasible point. a Multiply the ith constraint (in standard form) of the primal by yi and sum the resulting constraints. b Multiply the jth dual constraint (in standard form) by xj and sum them. 6.11 The Dual Simplex Method 329 c Compute: part (a) answer minus part (b) answer. d Use the part (c) answer and the Dual Theorem to show that if x is primal optimal and y is dual optimal, then (38) and (39) hold. e Use the part (c) answer to show that if (38) and (39) both hold, then x is primal optimal and y is dual optimal. (Hint: Look at Lemma 2.)

Q:62 fficiency of the town’s four elementary schools. The three outputs of the schools are defined to be Output 1 average reading score Output 2 average mathematics score Output 3 average self-esteem score The three inputs to the schools are defined to be Input 1 average educational level of mothers (defined by highest grade completed—12 high school graduate; 16 college graduate, and so on). Input 2 number of parent visits to school (per child) Input 3 teacher to student ratio The relevant information for the four schools is given

Q:63 5 Explain why the amount of each output produced by the composite hospital obtained by averaging hospitals 1 and 3 (with the absolute value of the dual prices as weights) is at least as large as the amount of the corresponding output produced by hospital  (Hint: Price out variables t1, t2, and t3, and use the fact that the coefficient of these variables in row 0 of the optimal tableau must equal 0.) 6 Explain why the dual price for the 8w1 15w2 1 constraint must equal the optimal z-value for the hospital 2 LP.

Q:64 Use the LINDO output in Figure 13 to answer the following questions

: a What is the most that Leary should pay for an additional hour of labor?

b What is the most that Leary should pay for an additional acre of land?

c If only 40 acres of land were available, what would be Leary’s profit?

d If the price of wheat dropped to \$26, what would be the new optimal solution?

e Farmer Leary is considering growing barley. Demand for barley is unlimited. An acre yields 4 bushels of barley and requires 3 hours of labor. If barley sells for \$30 per bushel, should Leary produce any barley?

Q:65Farmer Leary grows wheat and corn on his 45-acre farm. He can sell at most 140 bushels of wheat and 120 bushels of corn. Each planted acre yields either 5 bushels of wheat or 4 bushels of corn. Wheat sells for \$30 per bushel, and corn sells for \$50 per bushel. Six hours of labor are needed to harvest an acre of wheat, and 10 hours are needed to harvest an acre of corn. As many as 350 hours of labor can be purchased at \$10 per hour. Let A1 acres planted with wheat A2 acres planted with corn L hours of labor that are purchased

Q:66 Zales Jewelers uses rubies and sapphires to produce two types of rings. A Type 1 ring requires 2 rubies, 3 sapphires, and 1 hour of jeweler’s labor. A Type 2 ring requires 3 rubies, 2 sapphires, and 2 hours of jeweler’s labor. Each Type 1 ring sells for \$400, and each Type 2 ring sells for \$500. All rings produced by Zales can be sold. Zales now has 100 rubies, 120 sapphires, and 70 hours of jeweler’s.

Q:67Radioco manufactures two types of radios. The only scarce resource that is needed to produce radios is labor. The company now has two laborers. Laborer 1 is willing to work up to 40 hours per week and is paid \$5 per hour. Laborer  is willing to work up to 50 hours per week and is paid \$6 per hour. The price as well as the resources required to build each type of radio are given in Table 61. a Letting xi be the number of type i radios produced each week.

Q:68For what values of the price of a Type 1 radio would the current basis remain optimal?

c For what values of the price of a Type 2 radio would the current basis remain optimal?

d If laborer 1 were willing to work only 30 hours per week, would the current basis remain optimal?

e If laborer 2 were willing to work as many as 60 hours per week, would the current basis remain optimal?

f If laborer 1 were willing to work an additional hour, what is the most that Radioco should pay?

Q:69 Write down the dual to Beerco’s LP and find its optimal solution.  Find the range of values of the price of ale for which the current basis remains optimal.  Find the range of values of the price of beer for which the current basis remains optimal. Find the range of values of the amount of available corn for which the current basis remains optimal.  Find the range of values of the amount of available hops for which the current basis remains optimal.  Find the range of values of the amount of available malt for which the current basis remains optimal.

Q:70Suppose Beerco is considering manufacturing malt liquor. A barrel of malt liquor requires 0.5 lb of corn, 3 lb of hops, and 3 lb of malt and sells for \$50. Should Beerco manufacture any malt liquor? h Suppose we express the Beerco constraints in ounces. Write down the new LP and its dual. i What is the optimal solution to the dual of the new LP? (Hint: Think about what happens to cBVB. Use the idea of shadow prices to explain why the dual to the original LP (pounds) and the dual to the new LP (ounces) should have different optimal solutions.)

Q:71Wivco produces two products: 1 and 2. The relevant data are shown in Table 64. Each week, as many as 400 units of raw material can be purchased at a cost of \$1.50 per unit. The company employs four workers, who work 40 hours per week (their salaries are considered a fixed cost). Workers can be asked to work overtime and are paid \$6 per hour for overtime work. Each week, 320 hours of machine time are available. In the absence of advertising, 50 units of product 1 and 60 units of product 2 will be demanded each week. Advertising can be used to stimulate demand for each product. Each dollar spent on advertising product 1 increases its demand by 10 units; each dollar spent for product 2 increases its demand by 15 units. At most \$100 can be spent on advertising. Define P1 number of units of product 1 produced each week P2 number of units of product 2 produced each week OT number of hours of overtime labor used each week RM number of units of raw material purchased each week A1 dollars spent each week on advertising product 1 A2 dollars spent each week on advertising product 2.

Q:72If overtime were only \$4 per hour, would Wivco use it? b If each unit of product 1 sold for \$15.50, would the current basis remain optimal? What would be the new optimal solution? c What is the most that Wivco should be willing to pay for another unit of raw material? d How much would Wivco be willing to pay for another hour of machine time? e If each worker were required (as part of the regular workweek) to work 45 hours per week, what would the company’s profits be?

Q:73Explain why the shadow price of row (1) is 0.10. (Hint: If the right-hand side of (1) were increased from 50 to 51, then in the absence of advertising for product 1, 51 units could now be sold each week.) g Wivco is considering producing a new product (product 3). Each unit sells for \$17 and requires 2 hours of labor, 1 unit of raw material, and 2 hours of machine time. Should Wivco produce any of product 3?

Q:74The following question concerns the Rylon example discussed in Section 3.9. After defining RB ounces of Regular Brute produced annually LB ounces of Luxury Brute produced annually RC ounces of Regular Chanelle produced annually LC ounces of Luxury Chanelle produced annually RM pounds of raw material purchased annually the LINDO output in Figure 15 was obtained for this problem. Use this output to answer the following questions: a Interpret the shadow price of each constraint. b If the price of RB were to increase by 50¢.

Q:75 If 8,000 laboratory hours were available each year, but only 2,000 lb of raw material were available each year, would Rylon’s profits increase or decrease? [Hint: Use the 100% Rule to show that the current basis remains optimal. Then use reasoning analogous to (34)–(37) to determine the new objective function value.] d Rylon is considering expanding its laboratory capacity.

Q:76 1 For a cost of \$10,000 (incurred now), annual laboratory capacity can be increased by 1,000 hours. Option 2 For a cost of \$200,000 (incurred now), annual laboratory capacity can be increased by 10,000 hours. Suppose that all other aspects of the problem remain unchanged and that future profits are discounted, with the interest rate being 11 1 9 % per year. Which option, if any, should Rylon choose?

Q:77The following questions pertain to the Finco investment example of Section 3.11. The LINDO output for this problem is shown in Figure 17.  If Finco has \$2,000 more on hand at time 0, by how much would their time 3 cash increase?  Observe that if Finco were given a dollar at time 1, the cash available for investment at time 1 would now be 0.5A 1.2C 1.08S0 1. Use this fact and the shadow price of Constraint 2 to determine by how much Finco’s time 3 cash position would increase if an extra dollar were available at time 1.

Q:78 By how much would Finco’s time 3 cash on hand change if Finco were given an extra dollar at time 2? d If investment D yielded \$1.80 at time 3, would the current basis remain optimal?  Suppose that a super money market fund yielded 25% for the period between time 0 and time 1. Should Finco invest in this fund at time 0?

Q:79Show that if the investment limitations of \$75,000 on investments A, B, C, and D were all eliminated, the current basis would remain optimal. (Knowledge of the 100% Rule is required for this part.) What would be the new optimal z-value? A new investment (investment F) is under consideration. One dollar invested in investment F generates the following cash flows: time 0, \$1.00; time 1, \$1.10; time 2, \$0.20; time 3, \$0.10. Should Finco invest in investment F?

Q:80The following questions pertain to the Star Oil capital budgeting example of Section 3.6. The LINDO output for this problem is shown in Figure 16. a Find and interpret the shadow price for each constraint. b If the NPV of investment 1 were \$5 million, would the optimal solution to the problem change? c If the NPV of investment 2 and investment 4 were each decreased by 25%, would the optimal solution to the problem change? (This part requires knowledge of the 100% Rule.)

Q:81Suppose that Star Oil’s investment budget were changed to \$50 million at time 0 and \$15 million at time 1. Would Star be better off? (This part requires knowledge of the 100% Rule.) e Suppose a new investment (investment 6) is available. Investment 6 yields an NPV of \$10 million and requires a cash outflow of \$5 million at time 0 and \$10 .

Q:82 The following questions pertain to the Finco investment example of Section 3.11. The LINDO output for this problem is shown in Figure 17. a If Finco has \$2,000 more on hand at time 0, by how much would their time 3 cash increase?  Observe that if Finco were given a dollar at time 1, the cash available for investment at time 1 would now be 0.5A 1.2C 1.08S0 1. Use this fact and the shadow price of Constraint 2 to determine by how much Finco’s time 3 cash position would increase if an extra dollar were available at time 1.

Q:83By how much would Finco’s time 3 cash on hand change if Finco were given an extra dollar at time 2? d If investment D yielded \$1.80 at time 3, would the current basis remain optimal?  Suppose that a super money market fund yielded 25% for the period between time 0 and time 1. Should Finco invest in this fund at time 0?

Q:84Show that if the investment limitations of \$75,000 on investments A, B, C, and D were all eliminated, the current basis would remain optimal. (Knowledge of the 100% Rule is required for this part.) What would be the new optimal z-value? g A new investment (investment F) is under consideration. One dollar invested in investment F generates the following cash flows: time 0, \$1.00; time 1, \$1.10; time 2, \$0.20; time 3, \$0.10. Should Finco invest in investment F?

Q:86The following questions pertain to the Star Oil capital budgeting example of Section 3.6. The LINDO output for this problem is shown in Figure 16.  Find and interpret the shadow price for each constraint.  If the NPV of investment 1 were \$5 million, would the optimal solution to the problem change? If the NPV of investment 2 and investment 4 were each decreased by 25%, would the optimal solution to the problem change? (This part requires knowledge of the 100% Rule.)

Q:87 6 Consider an LP with three  constraints. The righthand sides are 10, 15, and 20, respectively. In the optimal tableau, s2 is a basic variable in the second constraint, which has a right-hand side of 12. Determine the range of values of b2 for which the current basis remains optimal. (Hint: If rhs of Constraint 2 is 15 , this should help in finding the rhs of the optimal tableau.

Q:88Use LINDO to solve the Sailco problem of Section 3.10. Then use the output to answer the following questions:  If month 1 demand decreased to 35 sailboats, what would be the total cost of satisfying the demands during the next four months?  If the cost of producing a sailboat with regular-time labor during month 1 were \$420, what would be the new optimal solution?  Suppose a new customer is willing to pay \$425 for a sailboat. If his demand must be met during month 1, should Sailco fill the order? How about if his demand must be met during month 4?

Q:89A company supplies goods to three customers, who each require 30 units. The company has two warehouses. Warehouse 1 has 40 units available, and warehouse 2 has 30 units available. The costs of shipping 1 unit from warehouse to customer are shown in Table 7. There is a penalty for each unmet customer unit of demand: With customer 1, a penalty cost of \$90 is incurred; with customer 2, \$80; and with customer 3, \$110. Formulate a balanced transportation problem to minimize the sum of shortage and shipping costs. 2 Referring to Problem 1, suppose that extra units could be purchased and shipped to either warehouse for a total cost of \$100 per unit and that all customer demand must be met. Formulate a balanced transportation problem to minimize the sum of purchasing and shipping costs.

Q:90A shoe company forecasts the following demands during the next six months: month 1—200; month 2—260; month 3—240; month 4—340; month 5—190; month 6—150. It costs \$7 to produce a pair of shoes with regular-time labor (RT) and \$11 with overtime labor (OT). During each month, regular production is limited to 200 pairs of shoes, and overtime production is limited to 100 pairs. It costs \$1 per month to hold a pair of shoes in inventory. Formulate a balanced transportation problem to minimize the total cost of meeting the next six months of demand on time.

Q:91Steelco manufactures three types of steel at different plants. The time required to manufacture 1 ton of steel (regardless of type) and the costs at each plant are shown in Table 8. Each week, 100 tons of each type of steel (1, 2, and 3) must be produced. Each plant is open 40 hours per week. a Formulate a balanced transportation problem to minimize the cost of meeting Steelco’s weekly requirements. b Suppose the time required to produce 1 ton of steel depends on the type of steel as well as on the plant at which it is produced (see Table 9, page 372). Could a transportation problem still be formulated?

Q:92A bank has two sites at which checks are processed. Site 1 can process 10,000 checks per day, and site 2 can process 6,000 checks per day. The bank processes three types of checks: vendor, salary, and personal. The processing cost per check depends on the site (see Table 11). Each day, 5,000 checks of each type must be processed. Formulate a balanced transportation problem to minimize the daily cost of processing checks.

Q:93The U.S. government is auctioning off oil leases at two sites: 1 and 2. At each site, 100,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil. Government rules state that no bidder can receive more than 40% of the land being auctioned. Cliff has bid \$1,000/acre for site 1 land and \$2,000/acre for site 2 land. Blake has bid \$900/acre for site 1 land and \$2,200/acre for site 2 land. Alexis has bid \$1,100/acre for site 1 land and \$1,900/acre for site 2 land. Formulate a balanced transportation model to maximize the government’s revenue.

Q:94The Ayatola Oil Company controls two oil fields. Field 1 can produce up to 40 million barrels of oil per day, and field 2 can produce up to 50 million barrels of oil per day. At field 1, it costs \$3 to extract and refine a barrel of oil; at field 2, the cost is \$2. Ayatola sells oil to two countries: England and Japan. The shipping cost per barrel is shown in Table 12. Each day, England is willing to buy up to 40 million barrels (at \$6 per barrel), and Japan is willing to buy up to 30 million barrels (at \$6.50 per barrel). Formulate a balanced transportation problem to maximize Ayatola’s profits.

Q:95For the examples and problems of this section, discuss whether it is reasonable to assume that the proportionality assumption holds for the objective function.  Touche Young has three auditors. Each can work as many as 160 hours during the next month, during which time three projects must be completed. Project 1 will take 130 hours; project 2, 140 hours; and project 3, 160 hours. The amount per hour that can be billed for assigning each auditor to each project is given in Table 13. Formulate a balanced transportation problem to maximize total billings during the next month.

Q:96Explain how each of the following would modify the formulation of the Sailco problem as a balanced transportation problem:

a Suppose demand could be backlogged at a cost of \$30/sailboat/month. (Hint: Now it is permissible to ship from, say, month 2 production to month 1 demand.)

b If demand for a sailboat is not met on time, the sale is lost and an opportunity cost of \$450 is incurred.

c Sailboats can be held in inventory for a maximum of two months.

d At a cost of \$440/sailboat, Sailco can purchase up to 10 sailboats/month from a subcontractor.

Q:97Use the northwest corner method to find a bfs for Problems 1, 2, and 3 of Section 7.1. 2 Use the minimum-cost method to find a bfs for Problems 4, 7, and 8 of Section 7.1. (Hint: For a maximization problem, call the minimum-cost method the maximumprofit method or the maximum-revenue method.)

Q:98The following problems refer to the Powerco example.

1 Determine the range of values of c14 for which the current basis remains optimal.

2 Determine the range of values of c34 for which the current basis remains optimal.

3 If s2 and d3 are both increased by 3, what is the new optimal solution?

4 If s3 and d3 are both decreased by 2, what is the new optimal solution?

Q:99 1 Five employees are available to perform four jobs. The time it takes each person to perform each job is given in Table 50. Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs. 2† Doc Councillman is putting together a relay team for the 400-meter relay. Each swimmer must swim 100 meters of breaststroke, backstroke, butterfly, or freestyle.

Q:100 Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert island with Jennifer Aniston, Courteney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility measures” in Table 52 indicate how much happiness each couple would experience if they spent all their time together. The happiness earned by a couple is proportional to the fraction of time they spend together. For example, if Freddie and Gwyneth spend half their time together, they earn happiness of  1 2 (9) 4.5.

a Let xij be the fraction of time that the ith man spends with the jth woman. The goal of the eight people is to maximize the total happiness of the people on the island. Formulate an LP whose optimal solution will yield the optimal values of the xij’s.

Q :101 Explain why the optimal solution in part (a) will have four xij 1 and twelve xij 0. The optimal solution requires that each person spend all his or her time with one person of the opposite sex, so this result is often referred to as the Marriage Theorem.  Determine the marriage partner for each person.  Do you think the Proportionality Assumption of linear programming is valid in this situation?

Q:102Greydog Bus Company operates buses between Boston and Washington, D.C. A bus trip between these two cities takes 6 hours. Federal law requires that a driver rest for four or more hours between trips. A driver’s workday consists of two trips: one from Boston to Washington and one from Washington to Boston. Table 54 gives the departure times for the buses. Greydog’s goal is to minimize the total downtime for all drivers. How should Greydog assign crews to trips? Note: It is permissible for a driver’s “day” to overlap midnight. For example, a Washington-based driver can be assigned to the Washington–Boston 3 P.M. trip and the Boston–Washington 6 A.M. trip.

Q:103Five male characters (Billie, John, Fish, Glen, and Larry) and five female characters (Ally, Georgia, Jane, Rene, and Nell) from Ally McBeal are marooned on a desert island. The problem is to determine what percentage of time each woman on the island should spend with each man. For example, Ally could spend 100% of her time with John or she could “play the field” by spending 20% of her time with each man. Table 55 shows a “happiness index” for each potential pairing of a man and woman. For example, if Larry and Rene spend all their time together, they earn 8 units of happiness for the island.

Q:104Play matchmaker and determine an allocation of each man and woman’s time that earns the maximum total happiness for the island. Assume that happiness earned by a couple is proportional to the amount of time they spend together. b Explain why the optimal solution to this problem will, for any matrix of “happiness indices,” always involve each woman spending all her time with one man.

Q:105Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert island with Jennifer Aniston, Courteney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility measures” in Table 52 indicate how much happiness each couple would experience if they spent all their time together. The happiness earned by a couple is proportional to the fraction of time they spend together. For example, if Freddie and Gwyneth spend half their time together, they earn happiness of  1 2 (9) 4.5. a Let xij be the fraction of time that the ith man spends with the jth woman. The goal of the eight people is to maximize the total happiness of the people on the island. Formulate an LP whose optimal solution will yield the optimal values of the xij’s

Q:106 Explain why the optimal solution in part (a) will have four xij 1 and twelve xij 0. The optimal solution requires that each person spend all his or her time with one person of the opposite sex, so this result is often referred to as the Marriage Theorem. c Determine the marriage partner for each person. d Do you think the Proportionality Assumption of linear programming is valid in this situation?

Q:107 Any transportation problem can be formulated as an assignment problem. To illustrate the idea, determine an assignment problem that could be used to find the optimal solution to the transportation problem in Table 56. (Hint: You will need five supply and five demand points).  The Chicago board of education is taking bids on the city’s four school bus routes. Four companies have made the bids in Table 57.  Suppose each bidder can be assigned only one route. Use the assignment method to minimize Chicago’s cost of running the four bus routes.

Q:108 Show that step 3 of the Hungarian method is equivalent to performing the following operations: (1) Add k to each cost that lies in a covered row. (2) Subtract k from each cost that lies in an uncovered column. 10 Suppose cij is the smallest cost in row i and column j of an assignment problem. Must xij 1 in any optimal assignment?

Q:109 General Ford produces cars at L.A. and Detroit and has a warehouse in Atlanta; the company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in Table 60 (“—” means that a shipment is not allowed). L.A. can produce as many as 1,100 cars, and Detroit can produce as many as 2,900 cars. Houston must receive 2,400 cars, and Tampa must receive 1,500 cars.

a Formulate a balanced transportation problem that can be used to minimize the shipping costs incurred in meeting demands at Houston and Tampa.

b Modify the answer to part (a) if shipments between L.A. and Detroit are not allowed.

c Modify the answer to part (a) if shipments between Houston and Tampa are allowed at a cost of \$5.

Q:110 Sunco Oil produces oil at two wells. Well 1 can produce as many as 150,000 barrels per day, and well 2 can produce as many as 200,000 barrels per day. It is possible to ship oil directly from the wells to Sunco’s customers in Los Angeles and New York. Alternatively, Sunco could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or Los Angeles. Los Angeles requires 160,000 barrels per day, and New York requires 140,000 barrels per day. The costs of shipping 1,000 barrels between two points are shown in Table 61. Formulate a transshipment model (and equivalent transportation model) that could be used to minimize the transport costs in meeting the oil demands of Los Angeles and New York.

Q:111 Rework Problem 3 under the assumption that Galveston has a refinery capacity of 150,000 barrels per day and Mobile has one of 180,000 barrels per day. (Hint: Modify the method used to determine the supply and demand at each transshipment point to incorporate the refinery capacity restrictions, but make sure to keep the problem balanced.) 5 General Ford has two plants, two warehouses, and three customers. The locations of these are as follows: Plants: Detroit and Atlanta Warehouses: Denver and New York Customers: Los Angeles, Chicago, and Philadelphia.

Q:112Cars are produced at plants, then shipped to warehouses, and finally shipped to customers. Detroit can produce 150 cars per week, and Atlanta can produce 100 cars per week. Los Angeles requires 80 cars per week; Chicago, 70; and Philadelphia, 60. It costs \$10,000 to produce a car at each plant, and the cost of shipping a car between two cities is given in Table 62. Determine how to meet General Ford’s weekly demands at minimum cost.

Q:113 Assume that payment of bills can be made after they are due, but a penalty of 5¢ per month is assessed for each dollar of cash demands that is postponed for one month. Assuming all bills must be paid by the end of month 6, develop a transshipment model that can be used to minimize the cost of paying the next six months’ bills. (Hint: Transshipment points are needed, in the form Ct cash available at beginning of month t after bonds for month t have been sold, but before month t demand is met. Shipments into Ct occur from bond sales and Ct 1. Shipments out of Ct occur to Ct 1 and demands for months 1, 2,…. t.)

Q:114 1 Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped to three customers. The profit earned per tube depends on the site where the tube was produced and on the customer who purchases the tube (see Table 64). Customer 1 is willing to purchase as many as 80 tubes per week; customer 2, as many as 90; and customer 3, as many as 100. Televco wants to find a shipping and production plan that will maximize profits. a Formulate a balanced transportation problem that can be used to maximize Televco’s profits. b Use the northwest corner method to find a bfs to the problem. c Use the transportation simplex to find an optimal solution to the problem.

Q:115 A company must meet the following demands for a product: January, 30 units; February, 30 units; March, 20 units. Demand may be backlogged at a cost of \$5/unit/month. All demand must be met by the end of March. Thus, if 1 unit of January demand is met during March, a backlogging cost of 5(2) \$10 is incurred. Monthly production capacity and unit production cost during each month are given in Table 66. A holding cost of \$20/unit is assessed on the inventory at the end of each month. a Formulate a balanced transportation problem that could be used to determine how to minimize the total cost (including backlogging, holding, and production costs) of meeting demand. b Use Vogel’s method to find a basic feasible solution. c Use the transportation simplex to determine how to meet each month’s demand. Make sure to give an interpretation of your optimal solution (for example, 20 units of month 2 demand is met from month 1 production).

Q:116 Currently, State University can store 200 files on hard disk, 100 files in computer memory, and 300 files on tape. Users want to store 300 word-processing files, 100 packaged-program files, and 100 data files. Each month a typical word-processing file is accessed eight times; a typical packaged-program file, four times; and a typical data file, two times. When a file is accessed, the time it takes for the file to be retrieved depends on the type of file and on the storage medium (see Table 68). a If the goal is to minimize the total time per month that users spend accessing their files, formulate a balanced transportation problem that can be used to determine where files should be stored.

b Use the minimum cost method to find a bfs. c Use the transportation simplex to find an optimal solution.

Q:117 The Gotham City police have just received three calls for police. Five cars are available. The distance (in city blocks) of each car from each call is given in Table 69. Gotham City wants to minimize the total distance cars must travel to respond to the three police calls. Use the Hungarian method to determine which car should respond to which call.

Q:118  There are three school districts in the town of Busville. The number of black and white students in each district are shown in Table 70. The Supreme Court requires the schools in Busville to be racially balanced. Thus, each school must have exactly 300 students, and each school must have the same number of black students. The distances between districts are shown.

Q:119 Find the optimal solution to the balanced transportation problem in Table 72 (minimization). 11 In Problem 10, suppose we increase si to 16 and d3 to 11. The problem is still balanced, and because 31 units (instead of 30 units) must be shipped, one would think that the total shipping costs would be increased. Show that the total shipping cost has actually decreased by \$2, however. This is called the “more for less” paradox. Explain why increasing both the supply and the demand has decreased cost. Using the theory of shadow prices, explain how one could have predicted that increasing s1 and d3 by 1 would decrease total cost by \$2.

Q:120 Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce 500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per day. Oil is sent from the fields to a refinery, either in Dallas or in Houston (assume that each refinery has unlimited capacity). It costs \$700 to refine 100,000 barrels of oil at Dallas and \$900 at Houston. Refined oil is shipped to customers in Chicago and New York. Chicago customers require 400,000 barrels per day of refined oil; New York customers require 300,000. The costs of shipping 100,000 barrels of oil (refined or unrefined) between cities are given in Table 74. Formulate a balanced transportation model of this situation.

Q:121 For the Powerco problem, find the range of values of c24 for which the current basis remains optimal. 16 For the Powerco problem, find the range of values of c23 for which the current basis remains optimal. 17 A company produces cars in Atlanta, Boston, Chicago, and Los Angeles. The cars are then shipped to warehouses in Memphis, Milwaukee, New York City, Denver, and San Francisco. The number of cars available at each plant is given in Table 75. Each warehouse needs to have available the number of cars given in Table 76. The distance (in miles) between the cities is given in Table 77.

Q:122 Assuming that the cost (in dollars) of shipping a car equals the distance between two cities, determine an optimal shipping schedule. b Assuming that the cost (in dollars) of shipping a car equals the square root of the distance between two cities, determine an optimal shipping schedule.

Q:123During the next three quarters, Airco faces the following demands for air conditioner compressors: quarter 1—200; quarter 2—300; quarter 3—100. As many as 240 air compressors can be produced during each quarter. Production costs/compressor during each quarter are given in Table 78. The cost of holding an air compressor in inventory is \$100/quarter. Demand may be backlogged (as long as it is met by the end of quarter 3) at a cost of \$60/compressor/quarter. Formulate the tableau for a balanced transportation problem whose solution tells Airco how to minimize the total cost of meeting the demands for quarters 1–3.

Q:124 During each of the next two months you can produce as many as 50 units/month of a product at a cost of \$12/unit during month 1 and \$15/unit during month 2. The customer is willing to buy as many as 60 units/month during each of the next two months. The customer will pay \$20/unit during month 1, and \$16/unit during month 2. It costs \$1/unit to hold a unit in inventory for a month. Formulate a balanced transportation problem whose solution will tell you how to maximize profit.

Q:125 A firm producing a single product has three plants and four customers. The three plants will produce 3,000, 5,000, and 5,000 units, respectively, during the next time period. The firm has made a commitment to sell 4,000 units to customer 1, 3,000 units to customer 2, and at least 3,000 units to customer 3. Both customers 3 and 4 also want to buy as many of the remaining units as possible. The profit associated with shipping a unit from plant i to customer j is given in Table 82. Formulate a balanced transportation problem that can be used to maximize the company’s profit.

Q:126 The Carter Caterer Company must have the following number of clean napkins available at the beginning of each of the next four days: day 1—15; day 2—12; day 3—18; day 4—6. After being used, a napkin can be cleaned by one of two methods: fast service or slow service. Fast service costs 10¢ per napkin, and a napkin cleaned via fast service is available for use the day after it is last used. Slow service costs 6¢ per napkin, and these napkins can be reused two days after they are last used. New napkins can be purchased for a cost of 20¢ per napkin. Formulate a balanced transportation problem to minimize the cost of meeting the demand for napkins during the next four days.

Q:127A firm producing a single product has three plants and four customers. The three plants will produce 3,000, 5,000, and 5,000 units, respectively, during the next time period. The firm has made a commitment to sell 4,000 units to customer 1, 3,000 units to customer 2, and at least 3,000 units to customer 3. Both customers 3 and 4 also want to buy as many of the remaining units as possible. The profit associated with shipping a unit from plant i to customer j is given in Table 82. Formulate a balanced transportation problem that can be used to maximize the company’s profit.

Q:128 A company can produce as many as 35 units/month. The demands of its primary customers must be met on time each month; if it wishes, the company may also sell units to secondary customers each month. A \$1/unit holding cost is assessed against each month’s ending inventory. The relevant data are shown in Table 83. Formulate a balanced transportation problem that can be used to maximize profits earned during the next three months.

Q:129Powerhouse produces capacitors at three locations: Los Angeles, Chicago, and New York. Capacitors are shipped from these locations to public utilities in five regions of the country: northeast (NE), northwest (NW), midwest (MW), southeast (SE), and southwest (SW). The cost of producing and shipping a capacitor from each plant to each region of the country is given in Table 85. Each plant has an annual production capacity of 100,000 capacitors. Each year, each region of the country must receive the following number of capacitors: NE, 55,000; NW, 50,000; MW, 60,000; SE, 60,000; SW, 45,000. Powerhouse feels shipping costs are too high, and the company is therefore considering building one or two more production plants. Possible sites are Atlanta and Houston. The costs of producing a capacitor and shipping it to each region of the country are given in Table 86. It costs \$3 million (in current dollars) to build a new plant, and operating each plant incurs a fixed cost (in addition to variable shipping and production costs) of \$50,000 per year. A plant at Atlanta or Houston will have the capacity to produce 100,000 capacitors per year. Assume that future demand patterns and production costs will remain unchanged. If costs are discounted at a rate of 11 1 9 % per year, how can Powerhouse minimize the present value of all costs associated with meeting current and future demands?

Q:130 During the month of July, Pittsburgh resident B. Fly must make four round-trip flights between Pittsburgh and Chicago. The dates of the trips are as shown in Table 87. B. Fly must purchase four round-trip tickets. Without a discounted fare, a round-trip ticket between Pittsburgh and Chicago costs \$500. If Fly’s stay in a city includes a weekend, then he gets a 20% discount on the round-trip fare. If his stay in a city is at least 21 days, then he receives a 35% discount; and if his stay is more than 10 days, then he receives a 30% discount. Of course, only one discount can be applied toward the purchase of any ticket. Formulate and solve an assignment problem that minimizes the total cost of purchasing the four round-trip tickets. (Hint: Let xij 1 if a round-trip ticket is purchased for use on the ith flight out of Pittsburgh and the jth flight out of Chicago. Also think about where Fly should buy a ticket if, for example, x21 1.)

Q:131Three professors must be assigned to teach six sections of finance. Each professor must teach two sections of finance, and each has ranked the six time periods during which finance is taught, as shown in Table 88. A ranking of 10 means that the professor wants to teach that time, and a ranking of 1 means that he or she does not want to teach at that time. Determine an assignment of professors to sections that will maximize the total satisfaction of the professors.

Q:132 It costs \$40 to buy a telephone from the department store. Assume that I can keep a telephone for at most five years and that the estimated maintenance cost each year of operation is as follows: year 1, \$20; year 2, \$30; year 3, \$40; year 4, \$60; year 5, \$70. I have just purchased a new telephone. Assuming that a telephone has no salvage value, determine how to minimize the total cost of purchasing and operating a telephone for the next six years.

Q:133 At the beginning of year 1, a new machine must be purchased. The cost of maintaining a machine i years old is given in Table 5. The cost of purchasing a machine at the beginning of each year is given in Table 6. There is no trade-in value when a machine is replaced. Your goal is to minimize the total cost (purchase plus maintenance) of having a machine for five years. Determine the years in which a new machine should be purchased.

Q:134 A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box is given in Table 7. The variable cost (in dollars) of producing each box is equal to the box’s volume. A fixed cost of \$1,000 is incurred to produce any of a particular box. If the company desires, demand for a box may be satisfied by a box of larger size. Formulate and solve a shortest-path problem whose solution will minimize the cost of meeting the demand for boxes.

Q:135Four workers are available to perform jobs 1–4. Unfortunately, three workers can do only certain jobs: worker 1, only job 1; worker 2, only jobs 1 and 2; worker 3, only job 2; worker 4, any job. Draw the network for the maximum-flow problem that can be used to determine whether all jobs can be assigned to a suitable worker.

Q:136 The Hatfields, Montagues, McCoys, and Capulets are going on their annual family picnic. Four cars are available to transport the families to the picnic. The cars can carry the following number of people: car 1, four; car 2, three; car 3, three; and car 4, four. There are four people in each family, and no car can carry more than two people from any one family. Formulate the problem of transporting the maximum possible number of people to the picnic as a maximum-flow problem.

Q:137 Suppose a network contains a finite number of arcs and the capacity of each arc is an integer. Explain why the Ford–Fulkerson method will find the maximum flow in the finite number of steps. Also show that the maximum flow from source to sink will be an integer. 12 Consider a network flow problem with several sources and several sinks in which the goal is to maximize the total flow into the sinks. Show how such a problem can be converted into a maximum-flow problem having only a single source and a single sink.

Q:138 For the networks in Figures 21 and 22, find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 6 Seven types of packages are to be delivered by five trucks. There are three packages of each type, and the capacities of the five trucks are 6, 4, 5, 4, and 3 packages, respectively. Set up a maximum-flow problem.

Q:139 For the networks in Figures 21 and 22, find the maximum flow from source to sink. Also find a cut whose capacity equals the maximum flow in the network. 6 Seven types of packages are to be delivered by five trucks. There are three packages of each type, and the capacities of the five trucks are 6, 4, 5, 4, and 3 packages, respectively. Set up a maximum-flow problem.

Q:140 Fly-by-Night Airlines is considering flying three flights. The revenue from each flight and the airports used by each flight are shown in Table 11. When Fly-by-Night uses an airport, the company must pay the following landing fees (independent of the number of flights using the airport): airport 1, \$300; airport 2, \$700; airport 3, \$500. Thus, if flights 1 and 3 are flown, a profit of 900 800  300  700  500 \$200 will be earned. Show that for the network in Figure 25 (maximum profit) (total revenue from all flights)  (capacity of minimal cut). Explain how this result can be used to help Fly-by-Night maximize profit (even if it has hundreds of possible flights). (Hint: Consider any set of flights F (say, flights 1 and 3).

Q:141During the next four months, a construction firm must complete three projects. Project 1 must be completed within three months and requires 8 months of labor. Project 2 must be completed within four months and requires 10 months of labor. Project 3 must be completed at the end of two months and requires 12 months of labor. Each month, 8 workers are available. During a given month, no more than 6 workers can work on a single job. Formulate a maximum-flow problem that could be used to determine whether all three projects can be completed on time. (Hint: If the maximum flow in the network is 30, then all projects can be completed on time.)

Q:142 A company is planning to manufacture a product that consists of three parts (A, B, and C). The company anticipates that it will take 5 weeks to design the three parts and to determine the way in which these parts must be assembled to make the final product. Then the company estimates that it will take 4 weeks to make part A, 5 weeks to make part B, and 3 weeks to make part C. The company must test part A after it is completed (this takes 2 weeks). The assembly line process will then proceed as follows: assemble parts A and B (2 weeks) and then attach part C (1 week). Then the final product must undergo 1 week of 1 2 4 3 FIGURE 39 Network for Problem 1 testing. Draw the project network and find the critical path, total float, and free float for each activity. Also set up the LP that could be used to find the critical path.

Q:143Consider the project network in Figure 40. For each activity, you are given the estimates of a, b, and m in Table 18. Determine the critical path for this network, the total float for each activity, the free float for each activity, and the probability that the project is completed within 40 days. Also set up the LP that could be used to find the critical path. 4 The promoter of a rock concert in Indianapolis must perform the tasks shown in Table 19 before the concert can be held (all durations are in days).

a Draw the project network.

b Determine the critical path.

c If the advance promoter wants to have a 99% chance of completing all preparations by June 30, when should work begin on finding a concert site?

d Set up the LP that could be used to find the project’s critical path.

Q:144 Horizon Cable is about to expand its cable TV offerings in Smalltown by adding MTV and other exciting stations. The activities in Table 22 must be completed before the service expansion is completed.

a Draw the project network and determine the critical path for the network, the total float for each activity, and the free float for each activity.

b Set up the LP that can be used to find the project’s critical path.

Q:145When an accounting firm audits a corporation, the first phase of the audit involves obtaining “knowledge of the business.” This phase of the audit requires the activities in Table 23. a Draw the project network and determine the critical path for the network, the total float for each activity, and the free float for each activity. Also set up the LP that can be used to find the project’s critical path.

Q:146 Explain why an activity’s free float can never exceed the activity’s total float.  A project is complete when activities A–E are completed. The predecessors of each activity are shown in Table 25. Draw the appropriate project diagram. (Hint: Don’t violate rule 4.)  Determine the probabilities that 1–2–4 and 1–3–4 are critical paths for Figure 37.  Given the information in Table 26, (a) draw the appropriate project network, and (b) find the critical path.

Q:147 Write a LINGO program that can be used to crash the project network of Example 6 with the crashing costs given in Table 14.  Consider the project diagram in Figure 42. This project must be completed in 90 days. The time required to complete each activity can be reduced by up to five days at the costs given in Table 27. Formulate an LP whose solution will enable us to minimize the cost of completing the project in 90 days.

Q:148 Formulate the problem of finding the shortest path from node 1 to node 6 in Figure 2 as an MCNFP. (Hint: Think of finding the shortest path as the problem of minimizing the total cost of sending 1 unit of flow from node 1 to node. Find the dual of the LP that was used to find the length of the critical path for Example 6 of Section 8.4. b Show that the answer in part (a) is an MCNFP. c Explain why the optimal objective function value for the LP found in part (a) is the longest path in the project network from node 1 to node 6. Why does this justify our earlier claim that the critical path in a project network is the longest path from the start node to the finish node?

Q:149 Fordco produces cars in Detroit and Dallas. The Detroit plant can produce as many as 6,500 cars, and the Dallas plant can produce as many as 6,000 cars. Producing a car costs \$2,000 in Detroit and \$1,800 in Dallas. Cars must be shipped to three cities. City 1 must receive 5,000 cars, city 2 must receive 4,000 cars, and city 3 must receive 3,000 cars. The cost of shipping a car from each plant to each city is given in Table 33. At most, 2,200 cars may be sent from a given plant to a given city. Formulate an MCNFP that can be used to minimize the cost of meeting demand.

Q:150 Each year, Data Corporal produces as many as 400 computers in Boston and 300 computers in Raleigh. Los Angeles customers must receive 400 computers, and 300 computers must be supplied to Austin customers. Producing a computer costs \$800 in Boston and \$900 in Raleigh. Computers are transported by plane and may be sent through Chicago. The costs of sending a computer between pairs of cities are shown in Table 34. a Formulate an MCNFP that can be used to minimize the total (production distribution) cost of meeting Data Corporal’s annual demand.

Q:151 Oilco has oil fields in San Diego and Los Angeles. The San Diego field can produce 500,000 barrels per day, and the Los Angeles field can produce 400,000 barrels per day. Oil is sent from the fields to a refinery, in either Dallas or Houston (assume each refinery has unlimited capacity). To refine 100,000 barrels costs \$700 at Dallas and \$900 at Houston. Refined oil is shipped to customers in Chicago and New York. Chicago customers require 400,000 barrels per day, and New York customers require 300,000 barrels per day. The costs of shipping 100,000 barrels of oil (refined or unrefined) between cities are shown in Table 35. a Formulate an MCNFP that can be used to determine how to minimize the total cost of meeting all demands. b If each refinery had a capacity of 500,000 barrels per day, how would the part (a) answer be modified?

Q:152 Braneast Airlines must determine how many airplanes should serve the Boston–New York–Washington air corridor and which flights to fly. Braneast may fly any of the daily flights shown in Table 36. The fixed cost of operating an airplane is \$800/day. Formulate an MCNFP that can be used to maximize Braneast’s daily profits. (Hint: Each node in the network represents a city and a time. In addition to arcs representing flights, we must allow for the possibility that an airplane will stay put for an hour or more. We must ensure that the model includes the fixed cost of operating a plane. To include this cost, the following three arcs might be included in the network: from Boston 7 P.M. to Boston 9 A.M.; from New York 7 P.M. to New York 9 A.M.; and from Washington 7 P.M. to Washington 9 A.M.)

Q:153 Daisymay Van Line moves people between New York, Philadelphia, and Washington, D.C. It takes a van one day to travel between any two of these cities. The company incurs costs of \$1,000 per day for a van that is fully loaded and traveling, \$800 per day for an empty van that travels, \$700 per day for a fully loaded van that stays in a city, and \$400 per day for an empty van that remains in a city. Each day of the week, the loads described in Table 37 must be shipped. On Monday, for example, two trucks must be sent from Philadelphia to New York (arriving on Tuesday). Also, two trucks must be sent from Philadelphia to Washington on Friday (assume that Friday shipments must arrive on Monday). Formulate an MCNFP that can be used to minimize the cost of meeting weekly requirements. To simplify the formulation, assume that the requirements repeat each week. Then it seems plausible to assume that any of the company’s trucks will begin each week in the same city in which it began the previous week.

Q:154 Consider the problem of finding the shortest path from node 1 to node 6 in Figure 2.

a Formulate this problem as an MCNFP.

b Find a bfs in which x12, x24, and x46 are positive. (Hint: A degenerate bfs will be obtained.)

c Use the network simplex to find the shortest path from node 1 to node 6. 2 For the MCNFP in Figure 62, find a bfs

Q:155 A truck must travel from New York to Los Angeles. As shown in Figure 68, a variety of routes are available. The number associated with each arc is the number of gallons of fuel required by the truck to traverse the arc.  Use Dijkstra’s algorithm to find the route from New York to Los Angeles that uses the minimum amount of gas. Formulate a balanced transportation problem that could be used to find the route from New York to Los Angeles that uses the minimum amount of gas. Formulate as an MCNFP the problem of finding the New York to Los Angeles route that uses the minimum amount of gas.

Q:156 Telephone calls from New York to Los Angeles are transported as follows: The call is sent first to either Chicago or Memphis, then routed through either Denver or Dallas, and finally sent to Los Angeles. The number of phone lines joining each pair of cities is shown in Table 39.  Formulate an LP that can be used to determine the maximum number of calls that can be sent from New York to Los Angeles at any given time. b Use the Ford–Fulkerson method to determine the maximum number of calls that can be sent from New York to Los Angeles at any given time.

Q:157 Before a new product can be introduced, the activities in Table 40 must be completed (all times are in weeks). a Draw the project diagram. b Determine all critical paths and critical activities. c Determine the total float and free float for each activity. d Set up an LP that can be used to determine the critical path. e Formulate an MCNFP that can be used to find the critical path. f It is now 12 weeks before Christmas. What is the probability that the product will be in the stores before Christmas?

Q:158 The duration of each activity can be reduced by up to 2 weeks at the following cost per week: A, \$80; B, \$60; C, \$30; D, \$60; E, \$40; F, \$30; G, \$20. Assuming that the duration of each activity is known with certainty, formulate an LP that will minimize the cost of getting the product into the stores by Christmas.

Q:159 Find a minimum spanning tree for the network in Figure 68.  A company produces a product at two plants, 1 and 2. The unit production cost and production capacity during each period are given in Table 41. The product is instantaneously shipped to the company’s only customer according to the unit shipping costs given in Table 42. If a unit is produced and shipped during period 1, it can still be used to meet a period 2 demand, but a holding cost of \$13 per unit in inventory is assessed. At the end of period 1, at most six units may be held in inventory. Demands are as follows: period 1, 9; period 2, 11. Formulate an MCNFP that can be used to minimize the cost of meeting all demands on time. Draw the network and determine the net outflow at each node, the arc capacities, and shipping costs.

Q:160 A project is considered completed when activities A–F have all been completed. The duration and predecessors of each activity are given in Table 43. The LINDO output in Figure 69 can be used to determine the critical path for this project. a Use the LINDO output to draw the project network. Indicate the activity represented by each arc. b Determine a critical path in the network. What is the earliest the project can be completed?

Q;161 During the next two months, Machineco must meet (on time) the demands for three types of products shown in Table 45. Two machines are available to produce these † Based on Mulvey (1979). TAB LE 43 Immediate Activity Duration Predecessors A 2— B 3— C 1A D 5 A, B E 7B, C F 5D, E † This problem is based on Brown, Geoffrion, and Bradley (1981). products. Machine 1 can only produce products 1 and 2, and machine 2 can only produce products 2 and 3. Each machine can be used for up to 40 hours per month. Table 46 shows the time required to produce one unit of each product (independent of the type of machine); the cost of producing one unit of each product on each type of machine; and the cost of holding one unit of each product in inventory for one month. Formulate an MCNFP that could be used to minimize the total cost of meeting all demands on time.

Q:162 Coach Night is trying to choose the starting lineup for the basketball team. The team consists of seven players who have been rated (on a scale of 1 poor to 3 excellent) according to their ball-handling, shooting, rebounding, and defensive abilities. The positions that each player is allowed to play and the player’s abilities are listed in Table 9. The five-player starting lineup must satisfy the following restrictions:  At least 4 members must be able to play guard, at least 2 members must be able to play forward, and at least 1 member must be able to play center.  The average ball-handling, shooting, and rebounding level of the starting lineup must be at least 2.  If player 3 starts, then player 6 cannot start.  If player 1 starts, then players 4 and 5 must both start. 5 Either player 2 or player 3 must start.

Q:163Suppose we add the following restriction to Example 1 (Stockco): If investments 2 and 3 are chosen, then investment 4 must be chosen. What constraints would be added to the formulation given in the text? 5 How would the following restrictions modify the formulation of Example 6 (Dorian car sizes)? (Do each part separately.)  If midsize cars are produced, then compacts must also be produced.  Either compacts or large cars must be manufactured

Q:164 To graduate from Basketweavers University with a major in operations research, a student must complete at least two math courses, at least two OR courses, and at least two computer courses. Some courses can be used to fulfill more than one requirement: Calculus can fulfill the math requirement; operations research, math and OR requirements; data structures, computer and math requirements; business statistics, math and OR requirements; computer simulation, OR and computer requirements; introduction to computer programming, computer requirement; and forecasting, OR and math requirements. Some courses are prerequisites for others: Calculus is a prerequisite for business statistics; introduction to computer programming is a prerequisite for computer simulation and for data structures; and business statistics is a prerequisite for forecasting. Formulate an IP that minimizes the number of courses needed to satisfy the major requirements.

Q:165 Glueco produces three types of glue on two different production lines. Each line can be utilized by up to seven workers at a time. Workers are paid \$500 per week on production line 1, and \$900 per week on production line 2. A week of production costs \$1,000 to set up production line 1 and \$2,000 to set up production line 2. During a week on a production line, each worker produces the number of units of glue shown in Table 12. Each week, at least 120 units of glue 1, at least 150 units of glue 2, and at least 200 units of glue 3 must be produced. Formulate an IP to minimize the total cost of meeting weekly demands.

Q:166 In Example 7 (Euing Gas), suppose that x 300. What would be the values of y1, y2, y3, z1, z2, z3, and z4? How about if x 1,200? 8 Formulate an IP to solve the Dorian Auto problem for the advertising data that exhibit increasing returns as more ads are placed in a magazine (pages 495–496). 9 How can integer programming be used to ensure that the variable x can assume only the values 1, 2, 3, and 4?

Q:167 The Lotus Point Condo Project will contain both homes and apartments. The site can accommodate up to 10,000 dwelling units. The project must contain a recreation project: either a swimming–tennis complex or a sailboat marina, but not both. If a marina is built, then the number of homes in the project must be at least triple the number of apartments in the project. A marina will cost \$1.2 million, and a swimming–tennis complex will cost \$2.8 million. The developers believe that each apartment will yield revenues with an NPV of \$48,000, and each home will yield revenues with an NPV of \$46,000. Each home (or apartment) costs \$40,000 to build. Formulate an IP to help Lotus Point maximize profits

Q:168 Use LINDO, LINGO, or Excel Solver to find the optimal solution to the following IP: Bookco Publishers is considering publishing five textbooks. The maximum number of copies of each textbook that can be sold, the variable cost of producing each textbook, the sales price of each textbook, and the fixed cost of a production run for each book are given in Table 16. Thus, for example, producing 2,000 copies of book 1 brings in a revenue of 2,000(50) \$100,000 but costs 80,000 25(2,000) \$130,000. Bookco can produce at most 10,000 books if it wants to maximize profit.

Q:169 Comquat owns four production plants at which personal computers are produced. Comquat can sell up to 20,000 computers per year at a price of \$3,500 per computer. For each plant the production capacity, the production cost per computer, and the fixed cost of operating a plant for a year are given in Table 17. Determine how Comquat can maximize its yearly profit from computer production.

Q:170 The Lotus Point Condo Project will contain both homes and apartments. The site can accommodate up to 10,000 dwelling units. The project must contain a recreation project: either a swimming–tennis complex or a sailboat marina, but not both. If a marina is built, then the number of homes in the project must be at least triple the number of apartments in the project. A marina will cost \$1.2 million, and a swimming–tennis complex will cost \$2.8 million. The developers believe that each apartment will yield revenues with an NPV of \$48,000, and each home will yield revenues with an NPV of \$46,000. Each home (or apartment) costs \$40,000 to build. Formulate an IP to help Lotus Point maximize profits.

Q:171 Eastinghouse sells air conditioners. The annual demand for air conditioners in each region of the country is as follows: East, 100,000; South, 150,000; Midwest, 110,000; West, 90,000. Eastinghouse is considering building the air conditioners in four different cities: New York, Atlanta, Chicago, and Los Angeles. The cost of producing an air conditioner in a city and shipping it to a region of the country is given in Table 18. Any factory can produce as many as 150,000 air conditioners per year. The annual fixed cost of operating a factory in each city is given in Table 19. At least 50,000 units of the Midwest demand for air conditioners must come from New York, or at least 50,000 units of the Midwest demand must come from Atlanta. Formulate an IP whose solution will tell Eastinghouse how to minimize the annual cost of meeting demand for air conditioners

Q:172 2 Consider the following puzzle. You are to pick out 4 three-letter “words” from the following list: DBA DEG ADI FFD GHI BCD FDF BAI For each word, you earn a score equal to the position that the word’s third letter appears in the alphabet. For example, DBA earns a score of 1, DEG earns a score of 7, and so on. Your goal is to choose the four words that maximize your total score, subject to the following constraint: The sum of the positions in the alphabet for the first letter of each word chosen must be at least as large as the sum of the positions in the alphabet for the second letter of each word chosen. Formulate an IP to solve this problem.

Q:173 † Breadco Bakeries is a new bakery chain that sells bread to customers throughout the state of Indiana. Breadco is considering building bakeries in three locations: Evansville, Indianapolis, and South Bend. Each bakery can bake as many as 900,000 loaves of bread each year. The cost of building a bakery at each site is \$5 million in Evansville, \$4 million in Indianapolis, and \$4.5 million in South Bend. To simplify the problem, we assume that Breadco has only three customers, whose demands each year are 700,000 loaves (customer 1); 400,000 loaves (customer 2); and 300,000 loaves (customer 3). The total cost of baking and shipping a loaf of bread to a customer is given in Table 21. Assume that future shipping and production costs are discounted at a rate of 11 1 9 % per year. Assume that once built, a bakery lasts forever. Formulate an IP to minimize Breadco’s total cost of meeting demand (present and future). (Hint: You will need the fact that for x 1, a ax ax2 ax3

a/(1 x).) How would you modify the formulation if either Evansville or South Bend must produce at least 800,000 loaves per year?

Q;174Governor Blue of the state of Berry is attempting to get the state legislature to gerrymander Berry’s congressional districts. The state consists of 10 cities, and the numbers of registered Republicans and Democrats (in thousands) in each city are shown in Table 23. Berry has five congressional representatives. To form congressional districts, cities must be grouped according to the following restrictions:

1 All voters in a city must be in the same district.

2 Each district must contain between 150,000 and 250,000 voters (there are no independent voters). Governor Blue is a Democrat. Assume that each voter always votes a straight party ticket. Formulate an IP to help Governor Blue maximize the number of Democrats who will win congressional seats.

Q:175 You have been assigned to arrange the songs on the cassette version of Madonna’s latest album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette must total between 14 and 16 minutes in length. The length and type of each song are given in Table 28. The assignment of songs to the tape must satisfy the following conditions: 1 Each side must have exactly two ballads. 2 Side 1 must have at least three hit songs. 3 Either song 5 or song 6 must be on side 1. 4 If songs 2 and 4 are on side 1, then song 5 must be on side 2. Explain how you could use an integer programming formulation to determine whether there is an arrangement of songs satisfying these restrictions.

Q:176A Sunco oil delivery truck contains five compartments, holding up to 2,700, 2,800, 1,100, 1,800, and 3,400 gallons of fuel, respectively. The company must deliver three types of fuel (super, regular, and unleaded) to a customer. The demands, penalty per gallon short, and the maximum allowed shortage are given in Table 29. Each compartment of the truck can carry only one type of gasoline. Formulate an IP whose solution will tell Sunco how to load the truck in a way that minimizes shortage costs.

Q:177 Simon’s Mall has 10,000 sq ft of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and maximum number of each type of store (along with the square footage of each type) is given in Table 30. The annual profit made by each type of store will, of course, depend on how many stores of that type are in the mall. This dependence is given in Table 31 (all profits are in units of \$10,000). Thus, if there are two department stores in the mall, each department store earns \$210,000 profit per year. Each store pays 5% of its annual profit as rent to Simon’s. Formulate an IP whose solution will tell Simon’s how to maximize rental income from the mall.

Q:178 The Smalltown Fire Department currently has seven conventional ladder companies and seven alarm boxes. The two closest ladder companies to each alarm box are given in Table 33. The city fathers want to maximize the number of conventional ladder companies that can be replaced with tower ladder companies. Unfortunately, political considerations dictate that a conventional company can be replaced only if, after replacement, at least one of the two closest companies to each alarm box is still a conventional company.

a Formulate an IP that can be used to maximize the number of conventional companies that can be replaced by tower companies.

b Suppose yk 1 if conventional company k is replaced. Show that if we let zk 1 yk, the answer in part (a) is equivalent to a set-covering problem.

Q:179 A power plant has three boilers. If a given boiler is operated, it can be used to produce a quantity of steam (in tons) between the minimum and maximum given in Table 34. The cost of producing a ton of steam on each boiler is also given. Steam from the boilers is used to produce power on three turbines. If operated, each turbine can process an amount of steam (in tons) between the minimum and maximum given in Table 35. The cost of processing a ton of steam and the power produced by each turbine is also given. Formulate an IP that can be used to minimize the cost of producing 8,000 kwh of power

Q:180 An Ohio company, Clevcinn, consists of three subsidiaries. Each has the respective average payroll, unemployment reserve fund, and estimated payroll given in Table 36. (All figures are in millions of dollars.) Any employer in the state of Ohio whose reserve/average payroll ratio is less than 1 must pay 20% of its estimated payroll in unemployment insurance premiums or 10% if the ratio is at least 1. Clevcinn can aggregate its subsidiaries and label them as separate employers. For instance, if subsidiaries 2 and 3 are aggregated, they must pay 20% of their combined payroll in unemployment insurance premiums. Formulate an IP that can be used to determine which subsidiaries should be aggregated.

Q:181 The Indiana University Business School has two rooms that each seat 50 students, one room that seats 100 students, and one room that seats 150 students. Classes are held five hours a day. The four types of requests for rooms are listed in Table 37. The business school must decide how many requests of each type should be assigned to each type of room. Penalties for each type of assignment are given in Table 38. An X means that a request must be satisfied by a room of adequate size. Formulate an IP whose solution will tell the business school how to assign classes to rooms in a way that minimizes total penalties.

Q:182 A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box are given in Table 39. The variable cost (in dollars) of producing each box is equal to the box’s volume. A fixed cost of \$1,000 is incurred to produce any of a particular box. If the company desires, demand for a box may be satisfied by a box of larger size. Formulate and solve (with LINDO, LINGO, or Excel Solver) an IP whose solution will minimize the cost of meeting the demand for boxes.

Q:183 A large drug company must determine how many sales representatives to assign to each of four sales districts. The cost of having n representatives in a district is (\$88,000 \$80,000n) per year. If a rep is based in a given district, the time it takes to complete a call on a doctor is given in Table 48 (times are in hours). Each sales rep can work up to 160 hours per month. Each month the number of calls given in Table 49 must be made in each district. A fractional number of representatives in a district is not permissible. Determine how many representatives should be assigned to each district.

Q:184 Venture capital firm JD is trying to determine in which of 10 projects it should invest. It knows how much money is available for investment each of the next N years, the NPV of each project, and the cash required by each project during each of the next N years (see Table 52). a Write a LINGO program to determine the projects in which JD should invest. b Use your LINGO program to determine which of the 10 projects should be selected. Each project requires cash investment during the next three years. During year 1, \$80 million is available for investment. During year 2, \$60 million is available for investment. During year 3, \$70 million is available for investment. (All figures are in millions of dollars.)

Q:185 I am moving from New Jersey to Indiana and have rented a truck that can haul up to 1,100 cu ft of furniture. The volume and value of each item I am considering moving on the truck are given in Table 61. Which items should I bring to Indiana? To solve this problem as a knapsack problem, what unrealistic assumptions must we make?

Q:186 Each day, Sunco manufactures four types of gasoline: lead-free premium (LFP), lead-free regular (LFR), leaded premium (LP), and leaded regular (LR). Because of cleaning and resetting of machinery, the time required to produce a batch of gasoline depends on the type of gasoline last produced. For example, it takes longer to switch between a lead-free gasoline and a leaded gasoline than it does to switch between two lead-free gasolines. The time (in minutes) required to manufacture each day’s gasoline requirements are shown in Table 75. Use a branch-andbound approach to determine the order in which the gasolines should be produced each day.

Q:187 There are four pins on a printed circuit. The distance between each pair of pins (in inches) is given in Table 76. a Suppose we want to place three wires between the pins in a way that connects all the wires and uses the minimum amount of wire. Solve this problem by using one of the techniques discussed in Chapter 8. b Now suppose that we again want to place three wires between the pins in a way that connects all the wires and uses the minimum amount of wire. Also suppose that if more than two wires touch a pin, a short circuit will occur. Now set up a traveling salesperson problem that can be used to solve this problem. (Hint: Add a pin 0 such that the distance between pin 0 and any other pin is 0.)

Q:188 A manufacturer of printed circuit boards uses programmable drill machines to drill six holes in each board. The x and y coordinates of each hole are given in Table 79. The time (in seconds) it takes the drill machine to move from one hole to the next is equal to the distance between the points. What drilling order minimizes the total time that the drill machine spends moving between holes?

Q:189 A court decision has stated that the enrollment of each high school in Metropolis must be at least 20 percent black. The numbers of black and white high school students in each of the city’s five school districts are shown in Table 90. The distance (in miles) that a student in each district must travel to each high school is shown in Table 91. School board policy requires that all the students in a given district attend the same school. Assuming that each school must have an enrollment of at least 150 students, formulate an IP that will minimize the total distance that Metropolis students must travel to high school.

Q:190 The Cubs are trying to determine which of the following free agent pitchers should be signed: Rick Sutcliffe (RS), Bruce Sutter (BS), Dennis Eckersley (DE), Steve Trout (ST), Tim Stoddard (TS). The cost of signing each pitcher and the number of victories each pitcher will add to the Cubs are shown in Table 92. Subject to the following restrictions, the Cubs want to sign the pitchers who will add the most victories to the team.

Q:191Consider a country where there are 1¢, 5¢, 10¢, 20¢, 25¢, and 50¢ pieces. You work at the Two-Twelve Convenience Store and must give a customer 91¢ in change. Formulate an IP that can be used to minimize the number of coins needed to give the correct change. Use what you know about knapsack problems to solve the IP by the branchand-bound method. (Hint: We need only solve a 90¢ problem.)

Q: 192 A soda delivery truck starts at location 1 and must deliver soda to locations 2, 3, 4, and 5 before returning to location 1. The distance between these locations is given in Table 94. The soda truck wants to minimize the total distance traveled. In what order should the delivery truck make its deliveries?

Q:193 Eastinghouse ships 12,000 capacitors per month to their customers. The capacitors may be produced at three different plants. The production capacity, fixed monthly cost of operation, and variable cost of producing a capacitor at each plant are given in Table 96. The fixed cost for a plant is incurred only if the plant is used to make any capacitors. Formulate an integer programming model whose solution will tell Eastinghouse how to minimize their monthly costs of meeting their customers’ demands.

Q:194 Monsanto annually produces 359 million lb of the chemical maleic anhydride. A total of four reactors are available to produce maleic anhydride. Each reactor can be run on one of three settings. The cost (in thousands of dollars) and pounds produced (in millions) annually for each reactor and each setting are given in Table 99. A reactor can only be run on one setting for the entire year. Set up an IP whose solution will tell Monsanto the minimum-cost method to meet its annual demand for maleic anhydride.

Q:195 Hallco runs a day shift and a night shift. No matter how many units are produced, the only production cost during a shift is a setup cost. It costs \$8,000 to run the day shift and \$4,500 to run the night shift. Demand for the next two days is as follows: day 1, 2,000; night 1, 3,000; day 2, 2,000; night 2, 3,000. It costs \$1 per unit to hold a unit in inventory for a shift. Determine a production schedule that minimizes the sum of setup and inventory costs. All demand must be met on time.

Q:196 Gotham City has been divided into eight districts. The time (in minutes) it takes an ambulance to travel from one district to another is shown in Table 100. The population of each district (in thousands) is as follows: district 1, 40; district 2, 30; district 3, 35; district 4, 20; district 5, 15; district 6, 50; district 7, 45; district 8, 60. The city has only two ambulances and wants to locate them to maximize the number of people who live within 2 minutes of an ambulance. Formulate an IP to accomplish this goal.

Q:197 Arthur Ross, Inc., must complete many corporate tax returns during the period February 15–April 15. This year the company must begin and complete the five jobs shown in Table 102 during this eight-week period. Arthur Ross employs four full-time accountants who normally work 40 hours per week. If necessary, however, they will work up to 20 hours of overtime per week for which they are paid \$100 per hour. Use integer programming to determine how Arthur Ross can minimize the overtime cost incurred in completing all jobs by April 15.

Q:198 PSI believes it will need the amounts of generating capacity shown in Table 103 during the next five years. The company has a choice of building (and then operating) power plants with the specifications shown in Table 104. Formulate an IP to minimize the total costs of meeting the generating capacity requirements of the next five years.

Q:199You are the sales manager for Eli Lilly. You want to have sales headquarters located in four of the cities in Table 113. The number of sales calls (in thousands) that must be made in each city are given in Table 113. For example, San Antonio requires 2,000 calls and is 602 miles from Phoenix. The distance between each pair of cities is given in Table 114 and in file Test1.xls. Where should the headquarters be located to minimize the total distance that must be traveled to make the needed calls?

Q:200 If a company has m hours of machine time and w hours of labor, it can produce 3m1/3w2/3 units of a product. Currently, the company has 216 hours of machine time and 1,000 hours of labor. An extra hour of machine time costs \$100, and an extra hour of labor costs \$50. If the company has \$100 to invest in purchasing additional labor and machine time, would it be better off buying 1 hour of machine time or 2 hours of labor?