Question: Inventory Problem with Variable demand
A company knows that the demand for its product during each of the next four months will be as follows: month 1, 1 unit; month 2, 3 units; month 3, 2 units; month 4, 4 units. At the beginning of each month, the company must determine how many units should be produced. Producing any number of units incurs in a setup cost of $3. In addition, there is a variable cost of $1 for every unit produced. At the end of each month, a holding cost of 50¢per unit on hand is incurred. Capacity limitations allow a maximum of 5 units to be produced during each month.
The size of the company's warehouse restricts the ending inventory for each month to 4 units at most. The company wants to determine a production schedule that will meet all demands on time and will minimize the sum of setup, production and holding costs during the four months. Assume that 0 units are on hand at the beginning of the first month, and at the end of month four, inventory should be zero as well. Formulate this multi-period deterministic variable demand inventory problem as a Mixed Integer Problem (MIP). Assume the company only checks inventory at the end of the month (or at the beginning of the next month, before any ordering or demand occurs). Usesias the demand parameter for monthi.
Hint: Define I_i as inventory level at the end of month i-1 and x_i as the production size at the beginning of month i. What condition must I, x and s satisfy? Make sure to add variables to account for the setup cost.