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# G carlsson u of mn isye lecture 10 inventory models

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Unformatted text preview: e and set both to zero to find that √ √ √ √ 2aK p 2aK p+h ∗ ∗ · ; Q= · S= h p+h h p J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 13 / 55 Observations The optimal cycle length is Q∗ t∗ = = a √ √ The largest shortage we’ll ever have is √ ∗ ∗ Q −S = p+h p 2K · ah √ 2aK · p h p+h The fraction of time that no shortage exists is p S∗ /a = ∗ Q /a p+h which is independent of K and a When p becomes large, we have Q∗ − S∗ → 0; when h becomes large, we have S∗ → 0 (we don’t want positive inventory levels) J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 14 / 55 Lead times in ordering The EOQ model describes the optimal times when new orders should arrive at our facility However, there may be a lead time L between the time that we place the order and the time when the new order actually arrives It is helpful to define a reorder point, which is a specified inventory level at which we should order more units: “Order the quantity Q∗ whenever the inventory level drops to X units” The formula for the reorder point is straightforward: X = aL J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 15 / 55 Quantity discounts In the √ original EOQ model, we found that the optimal order quantity was Q∗ = 2aK/h, which does not depend on the unit cost c of the product This is because we assumed that the unit cost of the product was constant The EOQ model with quantity discounts replaces this assumption in the following way [1] : The unit cost of an item now depends on the quantity in the batch. In particular, an incentive is provided to place a large order by replacing the unit cost for a small quantity by a smaller unit cost for every item in a larger batch, and perhaps by even smaller unit costs for even larger batches. [1] F.S. Hillier and G.J. Lieberman. Introduction to Operations Research. Introduction to Operations Research. McGraw-Hill Higher Education, 2010. J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 16 / 55 Quantity discounts Consider a simple EOQ model: a= 8000, constant demand rate of the product K= h= \$12000, setup cost for ordering one batch \$0.3, holding cost per unit per unit of time in inventory Let’s now assume that there are different purchasing costs ci : c1 c2 c3 J. G. Carlsson, U of MN ISyE = = = \$11 if the order is less than 10000 \$10 if the order is between 10000 and 80000 \$9.50 if the order is greater than 80000 Lecture 10: Inventory Models November 7, 2013 17 / 55 Quantity discounts From previous analysis, we know that the cost as a function of Q is given by hQ aK + aci + , Q 2 as shown in the above plot (the three costs only differ by a constant!) From visual inspection, the optimal order quantity is 25, 298 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 18 / 55 Quantity discounts From previous analysis, we know that the cost as a function of Q is given by hQ aK + aci + , Q 2 as shown in...
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