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Inventory Model.pdf

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Abe Igelfeld

4. An Inventory Model In this section we shall construct a simple quantitative model to describe the cost of maintaining an inventory. Suppose you must meet an annual demand of V units of a certain product for which the rate of demand is constant throughout the year. Suppose further that you replenish your stock periodically throughout the year by ordering x units of the product just when your stock is about depleted. In this case a graph of your inventory level versus time would look something like Figure 1. x x/2 t Figure 1 The decreasing lines are parallel since the rate of demand is constant throughout the year. We shall incorporate three costs into our model: storage costs, re-ordering costs, and purchasing costs. Storage Costs. Suppose the cost in dollars per annum of storing one unit is A . The average x inventory level (see Figure 1) is , so we shall take the annual storage cost to be 2 Ax 2 : Re-ordering Costs. Suppose the ▯xed cost in dollars of placing an order is B . If we order x units at a time, we must order V times per year, so the re-ordering cost is x BV x : Purchasing Costs. Let p(x) be the cost in dollars of purchasing x units. In practice this cost may involve discounts as an incentive to place large orders [see Example 2, Exercises 4, 5], 1 but for this model let us assume that the per unit cost of purchasing x units is ▯xed| that is, p(x) = kx | so that the total cost of purchasing V units is V ▯ kx = V k : x This is simply the cost of purchasing V units, and is a constant. If C(x) is the cost of maintaining the inventory, we then have C(x) = Ax + BV + kV : 2 x Mathematically we can use this model to minimize annual inventory costs. C (x) = A ▯ BV 2 x2 = 0 , x2 = 2BV A r 2BV ) x = A 2BV This value of x minimizes C since C (x) = is positive (since we can assume x is x3 positive), and is called the economic lot size. Note that the economic lot size is independent of purchasing cost, so that our simple model depends only on storage and re-ordering costs. Example 1. A department store sells 500 refrigerators per year. The annual storage and carrying cost per unit is $30 and the ▯xed reorder costs are $50. At present, lots of 100 are ordered. How much can be saved by an adjustment of the order size? Solution. Present annual costs are 50(500) C(100) = 1500 + = 1750 : 100 For a general lot size x , annual cost would be, in dollars, 25;000 C(x) = 15x + : x A minimum is reached if r x = 2(500)(50) 30 r 25;000 = 15 : = 40:8 2 Since x must be an integer, we calculate 25;000 C(41) = 615 + 41 = 615 + 609:76 = $1224:76 C(40) = 600 +25;000 = 600 + 625 = $1225 : 40 Judgment suggests that the round number of 40 units is worth the 26 cents per year of savings foregone. Hence the saving by a change from 100 to 40 units per order is 1750 ▯ 1225 = 625
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