Linear Programming Case Study
Your instructor will assign a linear programming project for this assignment according to the following specifications.
It will be a problem with at least three (3) constraints and at least two (2) decision variables. The problem will be bounded and feasible. It will also have a single optimum solution (in other words, it won't have alternate optimal solutions). The problem will also include a component that involves sensitivity analysis and the use of the shadow price.
You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.
Writeup.
Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is. For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs.
After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints. Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean.
Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price.
Excel.
As previously noted, please set up your problem in Excel and find the solution using Solver. Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the setup of the model, and the results.
The investment firm of Reynolds and Atherton has $10 million to invest. The firm wishes to invest a significant fraction of the money in high-yield but relatively risky funds, while keeping much of the money in safer but lower-yield investments. The safe but low-yield possibilities are certificates of deposit (CDs), yielding a 4.9% annual return, and treasury bills that yield 6.3%. High-grade corporate bonds (8.5% annual return) and a real estate fund (10% annual return) are higher-yielding but still reasonably secure possibilities. The firm can also invest in a foreign stock fund and a high-tech stock fund, with anticipated yields of 11.2% and 12.5% per annum, respectively. At the high end of both risk and return is a venture capital fund. This fund has yielded 18% over the last few years and is expected to do so again in the coming year.
To achieve a reasonable balance between potential profit and risk, the firm has decided to restrict the ratio of high-yield investments (foreign stock fund, high-tech stock fund, and venture capital fund) to low-yield investments (CDs and treasury bills) to no more than 1.5. Further, the sum of the investments in corporate bonds and real estate should be no greater than the sum of all of the other investments combined. In order to promote high yield while preventing too much money from being invested in the risky venture capital fund, the firm decides to invest at least twice as much in the foreign and high-tech stocks combined as it does in the venture capital fund and CDs combined. Also, the money invested in the venture capital fund will be at most 30% of the money invested in the foreign and high-tech stock funds combined. As a final gesture toward diversification, it is decided that the combined investment in treasury bills and corporate bonds can be as much as but no more than 20% of the total investment. The firm wants to invest the entire $10 million. How should Reynolds and Atherton invest the money in order to maximize its annual profit?
Please do the following:
(a.) Formulate a linear programming model for this problem, clearly defining the decision variables and listing the objective function and all of the constraints.
(b.) Use Excel to solve the linear programming model and produce a sensitivity analysis report.
(c.) Describe the solution in detail, including any interesting or unusual features.
(d.) Discuss the meaning of the sensitivity ranges of the objective function coefficients.
(e.) What is the shadow price associated with the total investment? What does it mean?
(f.) If the investment firm were to eliminate one constraint inequality, which one would result in the greatest increase in profit? Why do you think that occurs? Would eliminating this constraint increase risk? Why or why not?
(g.) Comply with all of the specifications listed in the description of this assignment in the course shell.