The seasonal yield of olives in a vineyard 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 requires considerably more labour than permitting the olives to grow on their own, and it results in smaller-sized olives. It also permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labour and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labour hours but takes 2 acres of land. An olive grower has 250 hours of labour available and a total of 150 acres for growing. Because of the olive size difference, a barrel of olives produced on pruned trees has a profit of £20, whereas a barrel of regular olives has a profit of £30. The grower has determined that, because of market preferences, the production of regular olives should be more than 1.5 times the production of pruned olives.
a) Ignoring any other considerations, develop a linear programming model to find the optimal combination of barrels of pruned and regular olives that will yield the maximum profit.
b) Show all the constraints and the feasible region graphically and find the optimum solution.
c) Classify each of the constraints as redundant, non-binding or binding and explain how the olive grower should interpret this information.
d) Determine the shadow prices of the different constraints.
e) Suppose the profit of regular olives is £45 per barrel. Determine the new optimal solution (profit and value of the decision variables) and use the graphical solution approach to clarify your reasoning.
f) In addition to maximising profit, the grower wants a solution that will also minimise the acres of land that will be used to grow olives. This acres requirement is of secondary importance compared to the original, maximising profit, requirement. Develop a ranked goal programming model with these two requirements. The grower expects not to use more than 120 acres but expects a profit of at least £1800. Only a formulation of the goal programming problem is expected for this part and you do need not to solve the problem.