LINEAR PROGRAMMING FORMULATION PROBLEMS AND SOLUTIONS

 

 

 

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 maybe 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 to ensure an adequate supply of air conditioners for a contract, at least 20 air conditioners should be manufactured. Because Electrocomp incurred an oversupply of fans in the preceding period, management also insists that no more than 80 fans be produced during this production period. Resolve this product mix problem to find the new 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 3500 feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood; each picnic table takes 6 labor hours and 35 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 minicomputers, 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 minicomputer 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?

 

 

 

7-24     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. These data on 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 is the cost and vitamin content of the final product?

 

 

 

Chapter 8 

 

8-1       (Production problem) Winkler Furniture manufactures two different types of china cabinets: a French Provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

 

(a) Formulate as an LP problem.

 

(b) Solve using an LP software program or spreadsheet.

 

 

 

Cabinet Style	Carpentry (Hours/Cabinet)	Painting (Hours/Cabinet)	Finishing (Hours/Cabinet)	Net Revenue per Cabinet ($)
French Provincial	3	1.5	.75	28
Danish Modern	2	1	.75	25
Dept. capacity (hrs)	360	200	125

 

 

 

 

 

8-2       (Investment decision problem) The Heinlein and Krarnpf Brokerage firm has just been instructed by one of its clients to invest $250,000 for her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and bonds they believe are well rated, but within the following guidelines:

 

(a) Municipal bonds should constitute at least 20% of the investment.

 

(b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers.

 

(c) No more than 50% of the amount invested in municipal bonds should be placed in a high-risk, high-yield nursing home stock.

 

Subject to these restraints, the client’s goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return.

 

 

 

 

 

 

(a) Formulate this portfolio selection problem using LP. (b) Solve this problem.

 

 

 

8-3       (Restaurant work scheduling problem). The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day’s operation. (Hint: Let X_i equal the number of waiters and busboys beginning work in time period i, where i = 1, 2,3,4,5,6.)

 

 

 

 

 

 

8-4       (Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. 

 

                      The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.

 

          In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds.

 

 

 

 

 

8-6       (Media selection problem) The advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicago’s North Side, is considering two media possibilities. One plan is for a series of half- page ads in the Sunday Chicago Tribune newspaper, and the other is for advertising time on Chicago TV. The stores are expanding their lines of do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city’s neighborhoods and 60% in northwest suburban areas.

 

        The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in the northwest suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad. The cost of a half-page Tribune advertisement is $925; a television spot costs $2,000.

 

        Diversey Paint would like to select the least costly advertising strategy that would meet desired exposure levels.

 

(a) Formulate using LP.

 

(b) Solve the problem.

 

 

 

8-11     (College meal selection problem) Kathy Roniger, campus dietician for a small Idaho college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal-content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates. On a particular day, Roniger’s food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and the contribution to each of the five nutritional requirements are given in the accompanying table:

 

 

 

	Table of Food Values and Costs
Food Item	Calories/ Pound	Iron (mg/lb)	Fat (gm/lb)	Protein (gm/lb)	Carbs. (gm/lb)	Cost/ Pound ($)
Milk	295	0.2	16	16	22	0.60
Ground Meat	1216	0.2	96	81	0	2.35
Chicken	394	4.3	9	74	0	1.15
Fish	358	3.2	0.5	83	0	2.25
Beans	128	3.2	0.8	7	28	0.58
Spinach	118	14.1	1.4	14	19	1.17
Potatoes	279	2.2	0.5	8	63	0.33

 

 

 

What combination and amounts of food items will provide the nutrition Roniger requires at the least total food cost?

 

 

 

                    

8-12     (High tech production problem) Quitmeyer Electronics Incorporated manufactures the following six microcomputer peripheral devices: internal modems, external modems, graphics circuit boards, CD drives, hard disk drives, and memory expansion boards. Each of these technical products requires time, in minutes, on three types of electronic testing equipment, as shown in the table the following table:

 

 

 

	Internal Modem	External Modem	Circuit Board	CD Drive	Hard Drive	Memory Board
Test device 1	7	3	12	6	18	17
Test device 2	2	5	3	2	15	17
Test device 3	5	1	3	2	9	2

 

 

 

The first two test devices are available 120 hours per week. The third (device 3) requires more preventive maintenance and may be used only 100 hours each week. The market for all six computer components is vast, and Quitmeyer Electronics believes that it can sell as many units of each product as it can manufacture. The table that follows summarizes the revenues and material costs for each product:

 

 

 

 

 

 

In addition, variable labor costs are $15 per hour for test device 1, $12 per hour for test device 2. and $18 per hour for test device 3. Quitmeyer Electronics wants to maximize its profits.

 

(a) Formulate this problem as an LP model.

 

(b) Solve the problem by computer. What is the best product mix?

 

(c) What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3? Should Quitmeyer Electronics add more test device time? If so, on which equipment?

 

 

 

8-15     (Material blending problem)  Amalgamated Products has just received a contract to construct steel body frames for automobiles that are to be produced at the new Japanese factory in Tennessee. The Japanese auto manufacturer has strict quality control standards for all of its component subcontractors and has informed Amalgamated that each frame must have the following steel content:

 

 

 

 

 

 

Amalgamated mixes batches of eight different available materials to produce one ton of steel used in the body frames. The table below details these materials. Formulate and solve the LP model that will indicate how much of each of the eight materials should be blended into a 1-ton load of steel so that Amalgamated meets its requirements while minimizing cost.