Consider the Pigskin example (Example 3.3) from Chapter 3. Find Pigskin’s optimal production policy if, in addition to the given production and holding costs, there is a fixed cost of $50,000 during any month in which there is positive production. Assume now that storage capacity is 20,000 footballs.
EXAMPLE 3.3 PRODUCING FOOTBALLS AT PIGSKIN
The Pigskin Company produces footballs. Pigskin must decide how many footballs to produce each month. The company has decided to use a six-month planning horizon. The forecasted monthly demands for the next six months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Pigskin wants to meet these demands on time, knowing that it currently has 5000 footballs in inventory and that it can use a given month’s production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next six months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95, respectively. The holding cost per football held in inventory at the end of any month is figured at 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all customer demand exactly when it occurs—at whatever the selling price is. Therefore, Pigskin wants to determine the production schedule that minimizes the total production and holding costs.
Objective To use LP to find the production schedule that meets demand on time and minimizes total production and inventory holding costs.
WHERE DO THE NUMBERS COME FROM?
The input values for this problem are not all easy to find. Here are some thoughts on where they might be obtained. (See Figure 3.25.)
■ The initial inventory in cell B4 should be available from the company’s database system or from a physical count.
■ The unit production costs in row 8 would probably be estimated in two steps. First, the company might ask its cost accountants to estimate the current unit production cost. Then it could examine historical trends in costs to estimate inflation factors for future months.
■ The holding cost percentage in cell B5 is typically difficult to determine. Depending on the type of inventory being held, this cost can include storage and handling, rent, property taxes, insurance, spoilage, and obsolescence. It can also include capital costs—the cost of money that could be used for other investments.
■ The demands in row 18 are probably forecasts made by the marketing and sales department. They might be “seat-of-the-pants” forecasts, or they might be the result of a formal quantitative forecasting procedure as discussed in Chapter 14. Of course, if there are already some orders on the books for future months, these are included in the demand figures.
■ The production and storage capacities in rows 14 and 22 are probably supplied by the production department. They are based on the size of the workforce, the available machinery, availability of raw materials, and physical space.