ACCEPTANCE SAMPLING: THE BUYER'S PROBLEM

A natural statistical problem arises from the following situation. Lots of varying quality are available from several manufacturers. A buyer is interested in purchasing some of these lots. The intrinsic problem for the buyer is finding a suitable strategy for making his purchases.;Three different sampling schemes are developed and illustrated. The simplest is the set of (m,n) procedures in which the buyer draws a random sample of size n from each of m lots and then makes a decision about which lot to purchase. More general but still elementary is the sequential fixed sample size procedures. Here the buyer is permitted to sample the lots sequentially according to a stopping rule, while still drawing a fixed number of observations from each. Most general is the class of fully sequential procedures in which both the choice of which lot to sample (or resample) next and the number of observations to be taken are determined sequentially.;To analyze the fully sequential scheme the optimal control model of Haggstrom is extended to include any (integrable) reward process {Z(,(alpha)), F(,(alpha)), (alpha)(ELEM)(DELTA)} index by a tree (DELTA) and to permit conditional branching at each node to any of a countable number of successor nodes. Starting from first principles a theory is developed which generalizes many of the theorems proved by Chow, Robbins, and Siegmund for (DELTA) linearly ordered and improves upon many derived by Haggstrom for his optimal control model. By martingale techniques the following major new results are obtained: (1) Sufficient and necessary conditions for a control variable, which is a generalization of a stopping variable, to be optimal when the reward process has finite value. (2) Characterization of the optimal return process as the minimal regular supermartingale dominating the reward process. (3) Sufficient and necessary conditions for an infinite horizon reward process to be equal in value to the limit of the sequence of horizon truncated reward processes. (4) Representation of a control variable as a stopping variable on a stochastic track. The model is then extended to include control variables which do not stop almost surely, by defining a class of limiting nodes and terminal rewards. It is proved that this will not increase the value of the control problem. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI.;In the literature on acceptance sampling the comparative nature of such "buyer's problems" has not been well understood. The author argues that in accepting a lot the buyer foregoes the opportunity of purchasing some other, potentially more favorable lot in its place. This essential feature distinguishes buyer problems from producer problems and leads to acceptance sampling procedures differing substantially from those currently in use.