CSC 445/545 Notes 1.1: Linear Programming Applications

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 A manufacturing company produces two models of industrial microwave ovens: X and the Y. Components for both the products need to machined then assembled. It takes 4 hours to machine the components for X and 3 hours to machine the components for Y. A total of 100 machine hours are available per day. It takes 2 hours to assemble X and 3 hours to assemble Y. There are 48 hours of assembly time available per day. The market potential for X is 15 per day. The market potential for Y is 20 per day. The marketing department wants the company to produce at least 10 products per day, in any combination. Each X contributes $2000 to profit and each Y contributes $2500 to profit.

 How many X and Y products should be produced per day to maximize profit?

 Marketing Research Problem

 Market Survey, Inc. (MSI), specializes in evaluating consumer reaction to new products, services, and advertising campaigns. A client firm has requested assistance from MSI in ascertaining consumer reaction to a recently marketed product for household use. During meetings with the client it was agreed that door to door personal interviews would be used to obtain information from both households with children and without children. In addition, it was agreed that both day and evening interviews would be conducted to allow for a variety of household work schedules. Specifically, the client's contract called for MSI to conduct 1000 interviews with the following quota guidelines:

 1 At least 400 households with children would be interviewed.

2 At least 400 households without children would be interviewed.

3. The total number of households interviewed during the evening would be at least as great as the number of households interviewed during the day.
4. At least 40% of the interviews for households with children would be conducted during the evening.
5. At least 60% of the interviews without children will be conducted during the evening.

Since the interviews for households with children take additional interviewer time, and since evening interviewers are paid more than daytime interviewers, the cost of an interview varies with the type of interview. Based on the previous research studies, estimates of the interview costs are the following:

 Find the best interviewing plan.

 Financial Portfolio Selection Problem

 Welte Mutual Funds, Inc. has just obtained $100,000 by converting industrial bonds to cash and is now looking for other investment opportunities for these funds. The firm's top financial analyst recommends that all new investments should be made in the oil or steel industry, or in government bonds. Specifically, the analyst has identified five investment opportunities. The projected rate of return for these companies are shown in the attached table.

 Investment Projected Rate of Return (%)

 Atlantic Oil 7.3%

Pacific Oil 10.3%

Midwest Steel 6.4%

Huber Steel 7.5%

Government Bonds 4.5%

The management of Welte has imposed the following investment guidelines:

1. Neither oil nor steel industry should receive more that $50,000 of the total investment. 
2. Government bonds should be at least 25% of the steel industry investments.
3. The investment in Pacific Oil, the high return but high risk investment, cannot be more than 60% of the total oil industry investment.

What is the best investment plan for Welte?

 Transportation Problem

 Foster Generators has production facilities in Cleveland, Bedford and York. Production capabilities for these plants over the next 3 month planning period for one particular type of generator are as follows:

 The firm distributes its generators through four regional distribution centers located in Boston, Chicago, St. Louis, and Lexington; the 3 month forecast of demand for the distribution centers are as follows:

Management would like to know determine how much of its production should be shipped from each plant to each distribution center. The cost for each unit shipped via each route is given in the following table:

 Unable to hire new police officers because of budget limitations, the Gotham City Police commissioner is trying to utilize the force better. The minimum requirements for police patrols for weekdays are noted below:

The police officers can start their shifts at the starting time of any of the above time periods. However they have to patrol for 8 hours. What is the minimum number of officers required to satisfy the requirements?