Investment Advisors Inc is a brokerage firm that manages sto

Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:

The sensitivity report for this problem is shown in the figure below.

What are the shadow prices for the constraints? If required, round your answers to three decimal places.

Interpret each shadow price.

The shadow price is the value that the______________ will change by if you increase the constraint by one unit.

The funds available constraint has a shadow price of ________________ which means that, if we add one dollar to the funds available, the value of the objective function would _______________ .

Risk maximum has a shadow price of____________ which means that, if we add one point to the risk index, the value of the objective function would _____________ .

US Oil has a shadow price of______________ which means that if we increase the number of shares allowed to be purchased by one then the value of the objective function would______________ .

A constraint with a shadow price of 0 means that______________ .

Would it be beneficial to increase the maximum amount invested in U.S. Oil?

__________ because________________ .

Solution

1) U=800 H=1200 8400 is the return.

2) There is no slack on constraint 1 and 2, so these are the binding constraints. This means that if the value of the constraint is changed, the solution changes.

The binding constaints are

25U + 50H <= 80,000 Funds available

0.50U + 0.25H <= 700 Risk maximum

It is easy to verify that these are equalities at U=800 H=1200

3) The dual values are 0.093333 (clearly, really 7/75) for constraint 1 and 1.33333 (really, 4/3) for constraint 2 This means that a change of 1 in the constraints yields this change in the objective function. Note how 7/75*80000+4/3*700 = 8400

4) It would not be beneficial, as we are at the optimum solution. From 0.50U + 0.25H <= 700, an increase of 1 in U will mean that H will have to decrease by 2 (.5*1+(.25)*(-2)=0)

Then, the objective function will change by 3U + 5H = 3(1) +5(-2) = 3 - 10 = -7; this is a decrease of 7.