Question 1: Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize profit. Each roll of Grade X carpet uses 50 units of synthetic fiber, requires 25 hours of production time, and needs 20 units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic fiber, requires 28 hours of production time, and needs 15 units of foam backing.
The profit per roll of Grade X carpet is $240 and the profit per roll of Grade Y carpet is $160. In the coming production period, Muir has 3000 units of synthetic fiber available for use. Workers have been scheduled to provide at least 1800 hours of production time (overtime is a possibility). The company has 1500 units of foam backing available for use.
(Assume that you can produce any fraction of carpet, which means your decision variables are not necessary integers). Please find the optimal solution and generate answer and sensitivity reports and then answer the following questions:
1. How many constraints are binding in this question?
2. Suppose the profit per roll of Grade X is increased to $1000 due to the shortage in the market, is your solution still optimal (please type Y or N)? The new profit will be ___ thousand dollars.
3. Given that synthetic fiber is a relevant cost and its initial unit cost is 5 dollars. What is the maximum amount Muir Manufacturing should pay for 50 units of synthetic fiber?
4. If a new policy requires that workers have been scheduled to provide at least 2200 hours of production time, is your solution still optimal?
Question 2: Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.5 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%.
The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $75,000. According to Innis's risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis's client also specifies that at least $300,000 be invested in the money market fund.
Please find the optimal solution and then use the sensitivity and answer reports to answer the following questions.
1. How many constraints are not binding in this question?
2. Suppose the risk index of each unit invested in the money market is increased to 7, is your solution still optimal (please type Y or N)?
3. Suppose the risk index of each unit invested in the stock fund is decreased to 4, is your solution still optimal (please type Y or N)?
4. Given that the maximum amount of money Innis has be authorized to invest in the two funds is increased to 2 million dollars, is your solution still optimal?
5. If the client requires that the annual income should be at least $75,100, how much would the total risk index change?