Part 1 of 5
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.
Now, Electrocomp’s management also realizes 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, the optimal production mix of air conditioners and fans is _____________ (The first number is for air conditioners and the second is for fans
a. (50, 30)
b. (20, 80)
c. (40, 60)
d. (26.67, 80)
2. In the optimal solution for the above question (Question 1), the slack for the four constraints – Total hours of wiring, Total hours of drilling, Number of air conditioners, Number of fans are _________, respectively. (Hint: A negative number shows that the constraint is binding downward, i.e. it sets a lower rather than an upper limit. Take the negative sign out when interpreting the results.)
a. 0, 0, 20, 20
b. 0, 0, 10, 20
c. 0, 0, 20, 10
d. 10, 10, 10, 10
3. If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, the optimal production of air conditioners is 45 and the optimal productions of fans is 50 (Only enter an integer and include no units.)
4. In the optimal solution for the above question (Question 3), the slack for the four constraints – Total hours of wiring, Total hours of drilling, Number of air conditioners, Number of fans are __________, respectively. (Hint: A negative number shows that the constraint is binding downward, i.e. it sets a lower rather than an upper limit. You may take the negative sign out when interpreting the results)
a. 0, 0, 15, 10
b. 5, 0, 15, 0
c. 0, 5, 15, 0
d. 0, 0, 10, 15
Part 2 of 5
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 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.
5. The optimal production of benches is262 and the optimal production of tables is 25.(Please only include the integer and no units. Hint: Because of the resource constraints, you may not be able to produce 4 tables even if the solution is 3.8; you may have to just produce 3 tables)
Part 3 of 5
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 attached 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.
Stock
Estimated Return
Risk Index
Petrochemical
12%
9
Utility
6%
4
6. The optimal dollar amount invested in Petrochemical stock is 20000 dollars, and the optimal dollaramount invested in Utility stock is 30000 dollars. (Only use an integer and include no units. Hint: The total risk on each stock is the risk index times the investment in dollars. When you use Excel QM, you need to end the total risk in the right hand side of the constraint not simply the risk index.)
a. 20,000, 30,000
b. 30,000, 20,000
c. 10,000, 15,000
d. 15,000, 10,000
7. The average risk index (not in dollar amount) for the optimal investment is 6, and the estimated best/maximum return for this investment is 4200. (Please only use an integer and include no units).
Part 4 of 5
The Heinlein and Krampf Brokerage firm has just been instructed by one of its clients to invest $250,000 of 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 the attached list of high-quality stocks and bonds and their corresponding rates of return.
Hint: You will need to rearrange one of the constraints in this problem so that all variables will be at the left hand side of the constraint and the right hand side of the constraint will be zero. You may adjust he sign of the constraint accordingly. For example, rearranging x < y + 2 to x – y < 2.
Investment
Projected Rate of Return (%)
Los Angeles municipal bonds
5.3
Thompson Electronics, Inc
6.8
United Aerospace Corp.
4.9
Palmer Drugs
8.4
Happy Days Nursing Homes
11.8
8. The optimal money invested in the five stocks and bonds – Municipal bonds, Electronics, Aerospace, Drugs, and Nursing homes are ________ dollars, respectively.
a. 50,000, 50,000, 50,000, 50,000, 50,000
b. 50,000, 0, 0, 175,000, 25,000
c. 25,000, 0, 0, 175,000, 50,000
d. 50,000, 25,000, 25,000, 125,000, 25,000
Part 5 of 5
Mt. Sinai Hospital in New Orleans is a large, private, 600-bed facility, complete with laboratories, operating rooms, and x-ray equipment. In seeking to increase revenues, Mt. Sinai’s administration has decided to make a 90-bed addition on a portion of adjacent land currently used for staff parking. The administrators feel that the labs, operating rooms, and x-ray department are not being fully utilized at present and do not need to be expanded to handle additional patients. The addition of 90 beds, however, involves deciding how many beds should be allocated to the medical staff for medical patients and how many to the surgical staff for surgical patients.
The hospital’s accounting and medical records departments have provided the following pertinent information. The average hospital stay for a medical patient is 8 days, and the average medical patient generates $2,280 in revenues. The average surgical patient is in the hospital 5 days and receives a $1,515 bill. The laboratory is capable of handling 15,000 tests per year more than it was handling. The average medical patient requires 3.1 lab tests and the average surgical patient takes 2.6 lab tests. Furthermore, the average medical patient uses one x-ray, whereas the average surgical patient requires two x-rays. If the hospital was expanded by 90 beds, the x-ray department could handle up to 7,000 x-rays without significant additional cost. Finally, the administration estimates that up to 2,800 additional operations could be performed in existing operating room facilities. Medical patients, of course, do not require surgery, whereas each surgical patient generally has one surgery performed.
Assume that the hospital is open 365 days a year. Round off your answers to the nearest integer.
9. The optimal number of medical patients per year is 2791, and the optimal number of surgical patients per year is 2104. The maximum annual profit is9551040 dollars. (Please round to the closest integer and include no units)
10. Among the 90 additional beds, _____ beds should be used for medical patients and _________ beds used for surgical patients.
a. 45, 45
b. 60, 30
c. 61, 29
d. 29, 61
11. At the optimal, there are 2 empty beds, 877.5lab tests of unused capacity, 1x-rays of unused capacity, and 696unused operation rooms available. (Please round to the closest integer and include no units.)
