Notes on Problem Solutions (for all questions on assignments, exams):
Show the details of your formulation, including:
Summarize the results in terms of the original problem as though you were reporting to a manager. If the results are unusual in any way, state why this might be so.
Be succinct. Each solution should require no more than a couple of pages.
Your cereal company makes two types of cereal: Wheaties and Heartland, both consisting entirely of wheat, sucrose, and barley. You have in stock 100 tons of wheat, 20 tons of sucrose,and 30 tons of barley. Wheatiesis made from a mix that must contain at least 15% sucrose, while Heartland must be at least 50% wheat and at least 20% barley. Each ton of Wheaties sells for $830 and each ton of Heartland sells for $770. Formulate a linear program to maximize the revenue from the sales of the two cereals.
A software company conducts a job-training program for managers in its marketing department.
Trained managers are used as teachers in the program at a ratio of one for every ten trainees.
The training program lasts for one month. From past experience it has been found that out of ten trainees hired, only seven complete the program successfully (the unsuccessful trainees are released).
Trained managers are also needed for management, and the company’s requirements for trained managers to perform management for the next three months are as follows:
In addition, the company requires 250 trained managers by the beginning of April. There are 130 trained managers available at the beginning of the year. Payroll costs per month are:
Each trainee $4000
Each trained manager $7000 (Managing or teaching)
Each trained manager idle $5000 (Union contract forbids firing a trained manager)
The company plans to hire managers and trainees at the beginning of January, February, and March, but not April. Fractional hires are possible due to part-time arrangements. Find a schedule for hiring managers and trainees that will meet the company’s requirements while minimizing their staffing costs.
A houseware manufacturer produces two products: Drapes at a profit of $0.80 each and tablecloths at a profit of $1.00 each. Market limits dictate that tablecloths should be manufactured at a rate of no more than 12 per day. The manufacturer has a total of 16 hours of manufacturing labour available per day. Each tablecloth requires 1 hour of labour and each drape requires 0.5 hours of labour. Each drape requires 2 m2 of fabric to manufacture, and each tablecloth requires 1 m2 of fabric. The supply of fabric is limited to 40 m2 per day. At what rate should the manufacturer make each product to maximize the profit rate?
A company manufactures raspberry and blueberry sports drinks. The net profit per litre of raspberry drink is $3 and $6 per litre of blueberry drink. The company wishes to control the production rates of the two drinks within a certain band. The production rate of raspberry drink should not exceed the production rate of blueberry drink by more than 4 litres per minute, and the production rate of blueberry drink should not exceed the production rate of raspberry drink by more than 1 litre per minute. How many litres per minute of each drink should the company produce to maximize profits?