Question: Suppose that in given Problem, the space consumed by each vegetable is not proportional to its cost. In particular, suppose that one pound of lettuce required 0.4 square foot of space and one pound of zucchini required 1 square foot of space. Determine upper and lower bounds on the optimal values of the order quantities in this case. Test different values of the Lagrange multiplier to find the optimal values of the order quantities. (A spreadsheet is ideally suited for this kind of calculation. If you solve the problem using a spreadsheet, place the Lagrange multiplier in a cell so that its value can be changed easily.)
Problem: A local outdoor vegetable stand has exactly 1,000 square feet of space to display three vegetables: tomatoes, lettuce, and zucchini. The appropriate data for these items are given in the following table.
Item
Tomatoes Lettuce Zucchini
Annual demands 850 1,280 630
(in pounds)
Cost per pound $0.29 $0.45 $0.25
The setup cost for replenishment of the vegetables is $100 in each case, and the space consumed by each vegetable is proportional to its costs, with tomatoes requiring 0.5 square foot per pound. The annual interest rate used for computing holding costs is 25 percent. What are the optimal quantities that should be purchased of these three vegetables?