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**Unformatted text preview: **Probabilistic Inventory Models All the inventory models discussed in Chapter 15 require that demand during any period of time be known with certainty. In this chapter, we consider inventory models in which demand over a given time period is uncertain, or random; single-period inventory models, where a prob- lem is ended once a single ordering decision has been made; single-period bidding models; versions of the EOQ model for uncertain demand that incorporate the important concepts of safety stock and service level; the periodic review ( R , S ) model; the ABC inventory classifica- tion system; and exchange curves. 16.1 Single-Period Decision Models In many situations, a decision maker is faced with the problem of determining the value q for a variable ( q may be the quantity ordered of an inventoried good, for example, or the bid on a contract). After q has been determined, the value d assumed by a random variable D is observed. Depending on the values of d and q , the decision maker incurs a cost c ( d , q ). We assume that the person is risk-neutral and wants to choose q to minimize his or her expected cost. Since the decision is made only once, we call a model of this type a single-period decision model. 16.2 The Concept of Marginal Analysis For the single-period model described in Section 16.1, we now assume that D is an integer-valued discrete random variable with P ( D d ) p ( d ). Let E ( q ) be the decision makers expected cost if q is chosen. Then E ( q ) d p ( d ) c ( d , q ) In most practical applications, E ( q ) is a convex function of q . Let q * be the value of q that minimizes E ( q ). If E ( q ) is a convex function, the graph of E ( q ) must look something like Figure 1. From the figure, we see that q * is the smallest value of q for which E ( q * 1) E ( q *) (1) Thus, if E ( q ) is a convex function of q , we can find the value of q minimizing expected cost by finding the smallest value of q that satisfies Inequality (1). Note that E ( q 1) E ( q ) is the change in expected cost that occurs if we increase the decision variable q to q 1. 1 6 . 3 The News Vendor Problem: Discrete Demand 881 To determine q *, we begin with q 0. If E (1) E (0) 0, we can benefit by in- creasing q from 0 to 1. Now we check to see whether E (2) E (1) 0. If this is true, then increasing q from 1 to 2 will reduce expected cost. Continuing in this fashion, we see that increasing q by 1 will reduce expected costs up to the point where we try to in- crease q from q * to q * 1. In this case, increasing q by 1 will increase expected cost. From Figure 1 (which is the appropriate picture if E ( q ) is a convex function), we see that if E ( q * 1) E ( q *) 0, then for q q *, E ( q 1) E ( q ) 0. Thus, q * must be the value of q that minimizes E ( q ). If E ( q ) is not convex, this argument may not work. (See Problem 1 at the end of this section.) Our approach determines q * by repeatedly computing the effect of adding a marginal unit to the value of q . For this reason, it is often called ....

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