Production planning

Suppose that you are the production manager of a manufacturing company that produces hiking-boots. For the next year, the demand for the hiking-boots pairs in months 1, 2, and 3 are 3000, 2000, and 5000, respectively. Each hiking-boots pair costs $30 to manufacture. At the beginning of month 1, there are 1,500 hiking-boots pairs available. As the production manager, you want to determine how many hiking-boots pairs to produce within each month so that the available inventory at the beginning of a month plus the number of hiking-boots pairs produced within the same month is at least sufficient to satisfy the demand. However, the production facility has manufacturing capacity such that it cannot produce more than 3,500 hiking-boots pairs in any of the months. Also, the ending inventory in months 1 and 2, i.e., the number of hiking-boot pairs remaining unsold at the end of month 1 and month 2, can be sold in the following months (i.e., in month 2 and month 3); but, there is a $8 unit cost of inventory for each pair remaining at the end of a month. There will be no remaining pairs at the end of month 3. As the production manager, you want to find the cost minimizing production plan for the hiking-boots pairs, where the total cost is equal to the production plus inventory costs. That is, you want to determine how many hiking-boots pairs to produce within each month so that you satisfy the demand in each month and minimize the total cost (assume that you can produce fractional number of hiking-boots pairs).

a) Represent the above problem as a network optimization problem. Particularly, you will need to formulate a minimum cost flow problem. Draw the network by defining the nodes, node values, and what they represent; and, the arcs, arc costs, arc capacities (if any), and what they represent. Then, state the problem as a minimum cost flow problem and give the mathematical formulation for this network optimization problem. (Hint: you will have 4 nodes, 1 node is the production facility, which will be the supply node, the other three nodes are the months, which will be demand nodes. Total supply will be equal to the total demand minus the available inventory at the beginning, and the demand at the node for month 1 will be month 1's demand minus the available inventory at the beginning. You will also have 5 arcs in total.)

b) Formulate the problem you had in part a in Excel and solve it using excel solver.