A reply to Lau and Lau
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A reply to Lau and Lau In their thorough critique of our paper (see, Bradford and Sugrue1) on a Bayesian approach to the two-period newsvendor problem with replenishment, Lau and Lau2 presented two very useful and important points for discussion. The ®rst of these examines a required correction to our `original' cost model formulation, while the second outlines an improved, more ef®cient solution procedure to the same. For the most part we agree with their general modi®cations, recommendations and conclusions (particularly with regard to the mathematical corrections to the original formulation), although we also feel that some additional interpretation and discussion is warranted concerning their proposed improvement to the model's solution procedure. In this rejoinder, we address each of these interpretations, extensions and conclusions in turn. For the sake of consistency and brevity in the arguments to follow, where possible we will maintain Lau and Lau's mathematical notation. Required corrections in the cost model The two-period (henceforth, periods 1 and 2, respectively) nature of the original newsvendor problem with replenishment necessitated the inclusion of a residual or `carry-over' inventory acquistion cost into period 2 for those items which had unsold units remaining at the end of period 1, which are subsequently available for sale in period 2. Lau and Lau provided the appropriate correction to the original formulation of the cost model, which focuses on a modi®cation of the conditional expected pro®t levels in period 2 (that is, px ), as de®ned by their Equation (3): px b ÿ v
Hx ÿ H x;
x 0; 1; 2; . . . ; H;
1
where, b is the sum of the ®rst four terms from Lau and Lau's Equation (2), or conversely the ®rst four terms from Bradford and Sugrue's Equation (10); v; is the unit acquisition cost; Hx (x 0, 1, 2, . . . ) is the conditional (period 2) provisioning level for those items which had the unconditional demand rate of exactly x units in period 1 and H is the unconditional provisioning level in period 1. Of particular interest to the following discussion is the term (Hx 7 H x), which represents the net inventory acquisition level in period 2, and re¯ects the residual or carry-over of unsold units from period 1. The derivation and use of this quantity is illustrated by the following example, which essentially derives from the results and data as
presented in the original article. Suppose that at the beginning of period 1, each item is provisioned at the level of H 6 units. Suppose further that a particular item has the (unconditional) demand rate of exactly x 3 units during the same period. Then clearly, the residual carry-over inventory levels for this item at the beginning of period 2 would be (H 7 x) (6 7 3) 3 units. Assuming for discussion purposes that the optimal (conditional) provisioning policy for this item is H*x 5 units, the item's `net' (period 2) p
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