Problem Situation: Olive Growing With Additional Labor and Land Possibilities The seasonal yield of olives in Piraeus, Greece is greatly influenced by a process of branch pruning. If Olive trees are pruned every two weeks, output is increased. The pruning process, however, requires considerably more labor than permitting the olives to grow on their own and results in a smaller size olive. It also, though, permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labor and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labor hours but takes 2 acres of land. Assume an olive grower has a base amount of 250 hours of labor available and owns 150 acres of land for growing. Because of the olive size difference, a barrel of olives produced on pruned trees generates a profit of $20, whereas a barrel of regular olives generates a $30 profit. The grower has determined that because of uncertain demand, no more than 40 barrels of pruned olives should be produced. In addition, the grower can obtain up to 50 overtime hours of labor, for a premium (above the cost already factored into the per barrel profit margins) of $2.00 /hour. Also, the grower can rent up to 30 acres of land from the adjoining farm at a cost of $12.50 /acre. Tasks 1. LP Formulation (i.e. Algebraic Formulation). Write a mathematical formulation of the problem, which simultaneously determines how many barrels of each olive to produce, how many overtime hours of labor to use, and how many acres of land to rent from the adjoining farm (so we are trying to figure out 4 numbers in total). Clearly define your decision variables (letter denoting variable, complete description, with units), and then write the objective function and constraints algebraically. Hint: At first, forget about overtime and the additional land, and formulate the problem with only the two olive types, regular labor time, and land. Then start expanding your formulation to include overtime labor and the additional land possibilities (new variables? new terms in the objective? Any change in the existing constraints? Any new constraints?). Write your complete LP formulation for the full problem in the Word document in the spreadsheet. 2. Implement and Solve Model. Using the Excel file provided, on the Model sheet, implement your formulation from part (1) in Excel. Then report the optimal solution and total profit in the Word document. It is okay if the numbers don't turn out integer. Do not force Solver to find integer solutions, and do not format the resulting values in the spreadsheet to integer quantities. Make sure to label all variables and constraints properly. Make sure your Solver parameters are set properly. Someone familiar with the Olive problem and Linear Programming should be able to easily understand your model. (Hint/Check Value: The optimal (maximum) objective value is $2,412.50 profit. Remember this is the "what" of an optimization problem (i.e., measure of how good); the "how" is the values of the decision variables.)
