7-14 The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available, and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach.

7-15 Electrocomp’s management realizes that it forgot to include two critical constraints (see Problem 7-14). In particular, management decides that there should be a minimum number of air conditioners produced in order to fulfill a contract. Also, due to an oversupply of fans in the preceding period, a limit should be placed on the total number of fans produced.

1. If Electrocomp decides that at least 20 air conditioners should be produced but no more than 80 fans should be produced, what would be the optimal solution? How much slack or surplus is there for each of the four constraints?

2. If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, what would be the optimal solution? How much slack or surplus is there for each of the four constraints at the optimal solution?

7-16 A candidate for mayor in a small town has allocated $40,000 for last-minute advertising in the days preceding the election. Two types of ads will be used: radio and television. Each radio ad costs $200 and reaches an estimated 3,000 people. Each television ad costs $500 and reaches an estimated 7,000 people. In planning the advertising campaign, the campaign manager would like to reach as many people as possible, but she has stipulated that at least 10 ads of each type must be used. Also, the number of radio ads must be at least as great as the number of television ads. How many ads of each type should be used? How many people will this reach?

7-17 The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 board feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 board feet of redwood; each picnic table takes 6 labor hours and 35 board feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoor Furniture produce to obtain the largest possible profit? Use the graphical LP approach.

7-18 The dean of the Western College of Business must plan the school’s course offerings for the fall semester. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum?

7-19 MSA Computer Corporation manufactures two models of smartphones, the Alpha 4 and the Beta 5. The firm employs five technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e., all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield $1,800 each. Determine the most profitable number of each model of smartphone to produce during the coming month.

7-20 A winner of the Texas Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical firm and a public utility. Although a long-range goal is to get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1–10 (with 10 being the most risky) is assigned to each of the two stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock.

The following table provides a summary of the return and risk:

The investor would like to maximize the return on the investment, but the average risk index of the investment should not be higher than 6. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment?

7-21 Referring to the Texas Lotto situation in Problem 7-20, suppose the investor has changed his attitude about the investment and wishes to give greater emphasis to the risk of the investment. Now the investor wishes to minimize the risk of the investment as long as a return of at least 8% is generated. Formulate this as an LP problem, and find the optimal solution. How much should be invested in each stock? What is the average risk for this investment? What is the estimated return for this investment?

7-22 Solve the following LP problem using the corner point graphical method. At the optimal solution, calculate the slack for each constraint:

7-23 Consider this LP formulation:

Graphically illustrate the feasible region, and apply the isocost line procedure to indicate which corner point produces the optimal solution. What is the cost of this solution?

7-24 The stock brokerage firm of Blank, Leibowitz, and Weinberger has analyzed and recommended two stocks to an investors’ club of college professors. The professors were interested in factors such as short-term growth, intermediate growth, and dividend rates. The data for each stock are as follows:

Each member of the club has an investment goal of (1) an appreciation of no less than $720 in the short term, (2) an appreciation of at least $5,000 in the next three years, and (3) a dividend income of at least $200 per year. What is the smallest investment that a professor can make to meet these three goals?

7-25 Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the dog food. How many pounds of beef and grain should be included in each pound of dog food? What are the cost and vitamin content of the final product?

7-26 The seasonal yield of olives in a Piraeus, Greece, vineyard is greatly influenced by a process of branch pruning. If olive trees are pruned every two weeks, output is increased. The pruning process, however, requires considerably more labor than permitting the olives to grow on their own and results in a smaller size olive. It also permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labor and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labor hours but takes 2 acres of land. An olive grower has 250 hours of labor available and a total of 150 acres for growing. Because of the olive size difference, a barrel of olives produced on pruned trees sells for $20, whereas a barrel of regular olives has a market price of $30. The grower has determined that because of uncertain demand, no more than 40 barrels of pruned olives should be produced. Use graphical LP to find

1. the maximum possible profit.

2. the best combination of barrels of pruned and regular olives.

3. the number of acres that the olive grower should devote to each growing process.

7-27 Consider the following four LP formulations. Using a graphical approach, determine

1. which formulation has more than one optimal solution.

2. which formulation is unbounded.

3. which formulation has no feasible solution.

4. which formulation is correct as is.

Formulation 1

Formulation 2

Formulation 3

Formulation 4

7-28 Graph the following LP problem and indicate the optimal solution point:

1. Does the optimal solution change if the profit per unit of X changes to $4.50?

2. What happens if the profit function should have been $\$3X+\$3Y?$

7-29 Graphically analyze the following problem:

1. What is the optimal solution?

2. If the first constraint is altered to $X+3Y\le 8,$ does the feasible region or optimal solution change?

7-30 Examine the LP formulation in Problem 7-29. The problem’s second constraint reads

If the firm decides that 36 hours of time can be made available on machine 2 (namely, an additional 12 hours) at an additional cost of $10, should it add the hours?

7-31 Consider the following LP problem:

1. What is the optimal solution to this problem? Solve it graphically.

2. If a technical breakthrough occurred that raised the profit per unit of X to $8, would this affect the optimal solution?

3. Instead of an increase in the profit coefficient X to $8, suppose that profit was overestimated and should only have been $3. Does this change the optimal solution?

7-32 Consider the LP formulation given in Problem 7-31. If the second constraint is changed from $2X+3Y\le 240$ to $2X+4Y\le 240,$ what effect will this have on the optimal solution?

7-33 The computer output given below is for Problem 7-31. Use this to answer the following questions.

1. How much could the profit on X increase or decrease without changing the values of X and Y in the optimal solution?

2. If the right-hand side of constraint 1 were increased by 1 unit, how much would the profit increase?

3. If the right-hand side of constraint 1 were increased by 10 units, how much would the profit increase?

7-34 The computer output on the next page is for a product mix problem in which there are two products and three resource constraints. Use the output to help you answer the following questions. Assume that you wish to maximize profit in each case.

1. How many units of product 1 and product 2 should be produced?

2. How much of each of the three resources is being used? How much slack is there for each constraint? Which of the constraints are binding, and which are nonbinding?

3. What are the dual prices for each resource?

4. If you could obtain more of one of the resources, which one should you obtain? How much should you be willing to pay for this?

5. What would happen to profit if, with the original output, management decided to produce one more unit of product 2?

7-35 Graphically solve the following problem:

1. What is the optimal solution?

2. Change the right-hand side of constraint 1 to 11 (instead of 10), and resolve the problem. How much did the profit increase as a result of this?

   

   7.5-14 Full Alternative Text

   

   7.5-15 Full Alternative Text

3. Change the right-hand side of constraint 1 to 6 (instead of 10), and resolve the problem. How much did the profit decrease as a result of this? Looking at the graph, what would happen if the right-hand-side value were to go below 6?

4. Change the right-hand-side value of constraint 1 to 5 (instead of 10), and resolve the problem. How much did the profit decrease from the original profit as a result of this?

5. Using the computer output on this page, what is the dual price of constraint 1? What is the lower bound on this?

6. What conclusions can you draw from this regarding the bounds of the right-hand-side values and the dual price?

7-36 Serendipity5

• The three princes of Serendip

• Went on a little trip.

• They could not carry too much weight;

• More than 300 pounds made them hesitate.

• They planned to the ounce. When they returned

• to Ceylon

• They discovered that their supplies were just

• about gone

• When, what to their joy, Prince William found

• A pile of coconuts on the ground.

• “Each will bring 60 rupees,” said Prince Richard with a grin

• As he almost tripped over a lion skin.

• “Look out!” cried Prince Robert with glee

• As he spied some more lion skins under a tree.

• “These are worth even more—300 rupees each

• If we can just carry them all down to the beach.”

• Each skin weighed fifteen pounds and each coconut, five,

• But they carried them all and made it alive.

• The boat back to the island was very small

• 15 cubic feet baggage capacity—that was all.

• Each lion skin took up one cubic foot

• While eight coconuts the same space took.

• With everything stowed they headed to sea

• And on the way calculated what their new wealth might be.

• “Eureka!” cried Prince Robert, “Our worth is so great

• That there’s no other way we could return in this state.

• Any other skins or nut that we might have brought

• Would now have us poorer. And now I know what—

• I’ll write my friend Horace in England, for surely

• Only he can appreciate our serendipity.”

• Formulate and solve Serendipity by graphical LP in order to calculate “what their new wealth might be.”

7-37 Bhavika Investments, a group of financial advisors and retirement planners, has been requested to provide advice on how to invest $200,000 for one of its clients. The client has stipulated that the money must be put into either a stock fund or a money market fund and the annual return should be at least $14,000. Other conditions related to risk have also been specified, and the following linear program was developed to help with this investment decision:

where

$S=\text{dollars}$ invested in stock fund

$M=\text{dollars}$ invested in money market fund

The QM for Windows output is shown below.

1. How much money should be invested in the money market fund and the stock fund? What is the total risk?

2. What is the total return? What rate of return is this?

3. Would the solution change if the risk measure for each dollar in the stock fund were 14 instead of 12?

4. For each additional dollar that is available, how much does the risk change?

5. Would the solution change if the amount that must be invested in the money market fund were changed from $40,000 to $50,000?

7-38 Refer to the Bhavika Investments (Problem 7-37) situation once again. It has been decided that, rather than minimizing risk, the objective should be to maximize return while placing a restriction on the amount of risk. The average risk should be no more than 11 (with a total risk of 2,200,000 for the $200,000 invested). The linear program was reformulated, and the QM for Windows output is shown above.

7-39 The Feed ’N Ship Ranch fattens cattle for local farmers and ships them to meat markets in Kansas City and Omaha. The owners of the ranch seek to determine the amounts of cattle feed to buy so that minimum nutritional standards are satisfied and at the same time total feed costs are minimized. The feed mix can be made up of the three grains that contain the following ingredients per pound of feed:

The cost per pound of stocks X, Y, and Z is $2, $4, and $2.50, respectively. The minimum requirement per cow per month is 4 pounds of ingredient A, 5 pounds of ingredient B, 1 pound of ingredient C, and 8 pounds of ingredient D.

The ranch faces one additional restriction: it can obtain only 500 pounds of stock Z per month from the feed supplier, regardless of its need. Because there are usually 100 cows at the Feed ’N Ship Ranch at any given time, this means that no more than 5 pounds of stock Z can be counted on for use in the feed of each cow per month.

1. Formulate this as an LP problem.

2. Solve using LP software.

7-40 The Weinberger Electronics Corporation manufactures four highly technical products that it supplies to aerospace firms that hold NASA contracts. Each of the products must pass through the following departments before being shipped: wiring, drilling, assembly, and inspection. The time requirement in hours for each unit produced and its corresponding profit value are summarized in the following table:

The production available in each department each month and the minimum monthly production requirement to fulfill contracts are as follows:

The production manager has the responsibility of specifying production levels for each product for the coming month. Help him by formulating (that is, setting up the constraints and objective function) Weinberger’s problem using LP.

7-41 Outdoor Inn, a camping equipment manufacturer in southern Utah, is developing a production schedule for a popular type of tent, the Double Inn. Orders have been received for 180 of these to be delivered at the end of this month, 220 to be delivered at the end of next month, and 240 to be delivered at the end of the month after that. This tent may be produced at a cost of $120, and the maximum number of tents that can be produced in a month is 230. The company may produce some extra tents in one month and keep them in storage until the next month. The cost for keeping these in inventory for one month is estimated to be $6 per tent for each tent left at the end of the month. Formulate this as an LP problem to minimize cost while meeting demand and not exceeding the monthly production capacity. Solve it using any computer software. (Hint: Define variables to represent the number of tents left over at the end of each month.)

7-42 Outdoor Inn (see Problem 7-41) expanded its tent-making operations later in the year. While still making the Double Inn tent, it is also making a larger tent, the Family Rolls, which has four rooms. The company can produce up to a combined total of 280 tents per month. The following table provides the demand that must be met and the production costs for the next three months. Note that the costs will increase in month 2. The holding cost for keeping a tent in inventory at the end of the month for use in the next month is estimated to be $6 per tent for the Double Inn and $8 per tent for the Family Rolls. Develop a linear program to minimize the total cost. Solve it using any computer software.

7-43 Modem Corporation of America (MCA) is the world’s largest producer of modem communication devices. MCA sold 9,000 of the regular model and 10,400 of the smart (“intelligent”) model this September. Its income statement for the month is shown in the Table for Problem 7-43. Costs presented are typical of prior months and are expected to remain at the same levels in the near future.

The firm is facing several constraints as it prepares its November production plan. First, it has experienced a tremendous demand and has been unable to keep any significant inventory in stock. This situation is not expected to change. Second, the firm is located in a small Iowa town from which additional labor is not readily available. Workers can be shifted from production of one modem to another, however. To produce the 9,000 regular modems in September required 5,000 direct labor hours. The 10,400 intelligent modems absorbed 10,400 direct labor hours.

Third, MCA is experiencing a problem affecting the intelligent modems model. Its component supplier is able to guarantee only 8,000 microprocessors for November delivery. Each intelligent modem requires one of these specially made microprocessors. Alternative suppliers are not available on short notice.

TABLE FOR PROBLEM 7-43

MCA wants to plan the optimal mix of the two modem models to produce in November to maximize profits for MCA.

1. Formulate, using September’s data, MCA’s problem as a linear program.

2. Solve the problem graphically.

3. Discuss the implications of your recommended solution.

7-44 Working with chemists at Virginia Tech and George Washington Universities, landscape contractor Kenneth Golding blended his own fertilizer, called “Golding-Grow.” It consists of four chemical compounds: C-30, C-92, D-21, and E-11. The cost per pound for each compound is indicated as follows:

The specifications for Golding-Grow are as follows: (1) E-11 must constitute at least 15% of the blend; (2) C-92 and C-30 must together constitute at least 45% of the blend; (3) D-21 and C-92 can together constitute no more than 30% of the blend; and (4) Golding-Grow is packaged and sold in 50-pound bags.

1. Formulate an LP problem to determine what blend of the four chemicals will allow Golding to minimize the cost of a 50-pound bag of the fertilizer.

2. Solve using a computer to find the best solution.

7-45 Raptor Fuels produces three grades of gasoline—Regular, Premium, and Super. All of these are produced by blending two types of crude oil—Crude A and Crude B. The two types of crude contain specific ingredients that help in determining the octane rating of gasoline. The important ingredients and the costs are contained in the following table:

In order to achieve the desired octane ratings, at least 41% of Regular gasoline should be ingredient 1; at least 44% of Premium gasoline must be ingredient 1, and at least 48% of Super gasoline must be Ingredient 1. Due to current contract commitments, Raptor Fuels must produce as least 20,000 gallons of Regular, at least 15,000 gallons of Premium, and at least 10,000 gallons of Super. Formulate a linear program that could be used to determine how much of Crude A and Crude B should be used in each of the gasolines to meet the demands at the minimum cost. What is the minimum cost? How much of Crude A and Crude B are used in each gallon of the different types of gasoline?