National Insurance Associates carries an investment portfolio of stocks, bonds, and other investment alternatives. Currently $200,000 of funds are available and must be considered for new investment opportunities. The four stock options National is considering and the relevant financial data are as follows:

The risk measure indicates the relative uncertainty associated with the stock in terms of its realizing the projected annual return; higher values indicate greater risk. The risk measures are provided by the firm's top financial advisor.
National's top management has stipulated the following investment guidelines: The annual rate of return for the portfolio must be at least 9% and no one stock can account for more than 50% of the total dollar investment.
a. Use linear programming to develop an investment portfolio that minimizes risk.
b. What is the objective coefficient ranges for the four variables? Interpret these ranges.
c. Suppose that the firm decides that the annual rate of return must be at least 10%. What does the dual value associated with this constraint indicate about the change in risk that would occur from this increased rate of return?

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Stock Price per share Annual rate of return $100 0.12 0.10 $50 0.08 $80 0.06 0.05 $40 0.10 0.08 Risk measure per dollar invested