Problem 1
The Kleenglass Corporation makes a dishwasher that has excellent cleaning power. This dishwasher uses less water than most competitors, and it is extremely quiet. Orders have been received from several retails stores for delivery at the end of each of the next 3 months, as shown below:
Month Number of Units
June 195
July 215
August 205
Due to limited capacity, only 200 of these can be made each month on regular time, and the cost is $300 each. However, an extra 15 units per month can be produced if overtime is used, but the cost goes up to $325 each. Also, if there are any dishwashers produced in a month that are not sold in that month, there is a $20 cost to carry this item to the next month. Use linear programming to determine how many units to produce in each month on regular time and on overtime to minimize the total cost while meeting the demands.
Problem 2
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.
Formulate this problem to determine how many medical beds and how many surgical beds should be added to maximize revenues. Assume that the hospital is open 365 days a year.
Problem 3
Prepare a written report to the CEO of Mt. Sinai Hospital in Problem 8-18 on the expansion of the hospital. Round off your answers to the nearest integer. The format of presentation of results is important. The CEO is a busy person and wants to be able to find your optimal solution quickly in your report. Cover all the areas given in the following sections, but do not mention any variables or shadow prices.
(a) What is the maximum revenue per year, how many medical patients/year are there, and how many surgical patients/year are there? How many medical beds and how many surgical beds of the 90-bed addition should be added?
(b) Are there any empty beds with this optimal solution? If so, how many empty beds are there? Discuss the effect of acquiring more beds if needed.
(c) Are the laboratories being used to their capacity? Is it possible to perform more lab tests/year? If so, how many more? Discuss the effect of acquiring more lab space if needed.
(d) Is the x-ray facility being used to its maximum? Is it possible to do more x-rays/year? If so, how many more? Discuss the effect of acquiring more x-ray facilities if needed.
Is the operating room being used to capacity? Is it possible to do more operations/year? If so, how many more? Discuss the effect of acquiring more operating room facilities if needed.
Problem 4
Coast-to-Coast Airlines is investigating the possibility of reducing the cost of fuel purchases by taking advantage of lower fuel costs in certain cities. Since fuel purchases represent a substantial portion of operating expenses for an airline, it is important that these costs be carefully monitored. However, fuel adds weight to an airplane, and consequently, excess fuel raises the cost of getting from one city to another. In evaluating one particular flight rotation, a plane begins in Atlanta, flies from Atlanta to Los Angeles, from Los Angeles to Houston, from Houston to New Orleans, and from New Orleans to Atlanta. When the plane arrives in Atlanta, the flight rotation is said to have been completed, and then it starts again. Thus, the fuel on board when the flight arrived in Atlanta must be taken into consideration when the flight begins.
|
Leg
|
Minimum Fuel Required (1,000 gal.)
|
Maximum Fuel Allowed
(1,000 gal.)
|
Regular Fuel Consumption (1,000 gal.)
|
Fuel Price per Gallon
|
|
Atlanta-Los Angeles
|
24
|
36
|
12
|
4.15
|
|
Los Angeles-Houston
|
15
|
23
|
7
|
4.25
|
|
Houston-New Orleans
|
9
|
17
|
3
|
4.10
|
|
New Orleans-Atlanta
|
11
|
20
|
5
|
4.18
|
Along each leg of this route, there is a minimum and a maximum amount of fuel that may be carried. This and additional information is provided in the table above.
The regular fuel consumption is based on the plane carrying the minimum amount of fuel. If more than this is carried, the amount of fuel consumed is higher. Specifically, for each 1,000 gallons of fuel above the minimum, 5% (or 50 gallons per 1,000 gallons of extra fuel) is lost due to excess fuel consumption. For example, if 25,000 gallons of fuel were on board when the plane takes off from Atlanta, the fuel consumed on this route would be 12 + 0.05 = 12.05 thousand gallons. If 26 thousand gallons were on board, the fuel consumed would be increased by another 0.05 thousand, for a total of 12.1 thousand gallons.
Formulate this as a LP problem to minimize the cost.
How many gallons should be purchased in each city?
What is the total cost of his?
Problem 5
Quitmeyer Electronics Incorporated manufactures six microcomputer peripheral devices: internal modems, external modems, graphics circuit boards, floppy disk drives, hard disk drives, and memory expansion boards. Each of these technical products require time in minutes, on three type of electronic testing equipment, as shown in the table below.
| |
Internal Modem |
External Modem |
Circuit Board |
CD Drives |
Hard Drives |
Memory Boards |
| 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. Test 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 follow table summarizes the revenue and cost for each product.
Device Revenue Per Unit Material Cost per Unit
Internal Modem 200 35
External Modem 120 25
Graphics circuit board180 40
Floppy disk drive 130 45
Hard disk drive 430 170
Memory expansion board 260 60
In addition, variable labor cost are $15 per hour for test device 1, $12 per hour for test device 2, and $18 per hour for test device 3. Quitmeyer wants maximize profits.
1. Formulate the problem as an LP model (excel).
2. What is the best product mix?
3. What is the value of an additional minute of time per week on test device 1? Test device 2? Test device 3?
4. Should Quitmeyer electronics add more test device time? If so on which equipment.
Problem 6
Agri-Chem's contract with Enerco specified a maximum of 90,000 cu.ft. X 103 per day for its complexes. However, curtailments are projected to be based on actual usage rather than contractual maximums. The current natural gas usage is 85,680 cu. ft. x 103 per day. Enerco projects curtailments in the range of 20 to 40 percent.
Question
Which of Agri-Chem's complexes would be least affected by a gas curtailment? (Using Excel Solver).
|
TABLE 1 Contribution to profit and overhead
|
|
Product
|
$/Ton
|
|
Ammonia
|
80
|
|
Ammonium phosphate
|
120
|
|
Ammonium nitrate
|
140
|
|
Urea
|
140
|
|
Hydrofluoric acid
|
90
|
|
Chlorine
|
70
|
|
Caustic soda
|
60
|
|
Vinyl chloride monomer
|
90
|
|
TABLE 2 Operational data
|
|
|
|
Product
|
Capacity (tons/day)
|
Production Rate (% of capacity
|
Natural Gas Consumption
|
|
(1,000 cu. ft./ton)
|
|
Ammonia
|
1,500
|
80
|
8.0
|
|
Ammonium phosphate
|
600
|
90
|
10.0
|
|
Ammonium nitrate
|
700
|
70
|
12.0
|
|
Urea
|
200
|
80
|
12.0
|
|
Hydrofluoric acid
|
800
|
70
|
7.0
|
|
Chlorine
|
1,500
|
80
|
18.0
|
|
Caustic soda
|
1,600
|
80
|
20.0
|
|
Vinyl chloride monomer
|
1,400
|
60
|
14.0
|