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Optimization with a constraint: An LP problem

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Below problem is one that I'm drawing a blank on for even setting up the equation. Any help would be greatly appreciated.

Smith recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.

What is the value of the optimal solution, i.e., the value of the objective function once you have determined the values of decision variables?

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Solution Summary

The following outlines the process of determining the objective function and constraint then demonstrates the process of finding the constrained optimum.

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Optimization / LP Word Problem

A poet has inherited 90 ha (or acres) of woods from her late grandmother. The poet finds that talks in the woods makes her happy, although she has to rely on income from the woods to support her during times of writers' block. In all she has 90 ha, 40 ha of which are covered in red pine forest, the other 50 ha are covered in mixed hardwood forest. The poet wants to figure out how much land to manage under each forest type in order to maximize her revenue. She has calculated that she needs to spend 2 days/ha/year to manage the red pine and 3d/ha/year to manage the mixed hardwood. She makes $90/ha/year on the red pine and $120/ha/year on the hardwoods. In order to have enough time for her writing she wants to spend no more than 180/year working on the land.

A. Formulate this problem using algebraic notation

B. Graph the feasible region and find the optimal solution

C. User Solver in Excel to solve the problem and provide a printout of your spreadsheet. Does the optimal solution match your graphical solution?

D. Write out the dual to this problem, solve it in Excel and provide your spreadsheet.

E. What are the shadow prices on the constraints (of the primal)?

F. Say there is some uncertainty with respect to returns on hardwood management and instead of expecting $120/ha/year, the poet could expect up to $150/ha/year. How does this change the objective function value and optimal solution? Is this what you would expect? Why?

G. Assume instead that the poet is considering working 182 days in the woods instead of 180. How does this change the objective function? Explain this change in relation the shadow prices.

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