1. Grain Distribution A grain distributor buys wheat from farms and then sells it directly to local markets or stores it for sale in subsequent periods. Relevant data affecting the distributor's decision of how much to buy, sell and maintain in inventory in each of three successive periods are shown in the following table. Periods 1 2 3 Purchase price/bushel ($) 2.00 3.00 2.50 Selling price/bushel ($) 3.10 4.20 3.50 Max bushels to purchase 1400 1800 1600 Max bushels to sell 1000 2000 2500 The indicated upper limits on the number of bushels that can be purchased and sold are based on projected availabilities and demands. The maximum quantity that can be stored in inventory from one period to the next is 800 bushels, and the distributor's storage cost is $0.40 per bushel. Finally, the distributor has 300 bushels already in inventory at the start of Period 1, and would like to have at least 200 bushels on hand at the end of Period 3 for safety reserve consideration. a. What optimization decisions are to be made by the distributor? b. What is the objective function to be optimized (maximized or minimized)? c. What constraints are to be satisfied? d. Write out an algebraic formulation for the addressed optimization problem: sets and parameters, decision variables, objective function and constraints, and elaborate it in detail. e. Solve the model using OPL Studio. What is the optimal production plan? What is the optimal objective value?

2. Grain Distribution with Fixed Setup Costs Now suppose that the distributor in Problem 1 will need to pay a fixed cost of $500 every time it makes a purchase. What new decision variables will you need? Present and elaborate your new algebraic formulation, and solve it using OPL Studio.