A Bayes Sequential Sampling Inspection Plan

Abstract

Given a lot of size $N$ whose items are obtained from a statistically controlled process with an unknown probability $p, 0 < p < 1$, of an item being defective, a rectifying sampling inspection problem arises when the loss involved in sending out the lot with $d, 0 \leqq d \leqq N$, defectives in it is $kd$ where $k > 0$ is the loss involved in sending out a defective item instead of replacing it with a good one. The cost of inspecting $n$ items (defective items detected during inspection are replaced by good ones) is $cn, c > 0$. An inspection plan is to be devised that minimizes (in some suitable sense) the risk (expected loss plus inspection cost). Let $\psi$ denote any sequential inspection plan, which determines sequentially the (random) number $\mathscr{R}$ of items to be inspected. If the lot is sent out after inspecting $n$ items, the cost of inspection incurred is $nc$ and the expected loss due to the defectives remaining in the lot is $(N - n)pk$. Then the risk is $(N - n)pk + nc = Npk + nk(c/k - p),$ and hence the risk associated with the plan $\psi$ is given by \begin{equation*}\tag{1.1}R(p, \psi) = Npk + k(c/k - p)E_p(\mathscr{R}).\end{equation*} Since $E_p(\mathscr{R}) \geqq 0$ for every $p \varepsilon (0, 1)$, it follows from (1.1) that if $p$ were known then the optimal plan (which we denote by $\psi_p$) would be to inspect the whole lot or not to inspect at all according as $p >$ or $\leqq p_0 \equiv c/k$. The risk $R(p, \psi_p)$ of this plan is given by $R(p, \psi_p) = Npk,\quad\text{for} p \leqq p_0, \\ = Nc,\quad\text{for} p > p_0$ (see Figure 1). Let $d$ be the number of defectives observed in the first $n$ items inspected when $p$ is unknown. Since $d$ is a sufficient statistic for $p$ based on "the past history," the decision to stop or continue inspection may be made to depend on $d$. Thus we may restrict our attention to plans defined by two mutually exclusive and exhaustive sets in the $(n, d)$ plane called the continuation and stopping sets. If at any stage the current position $(n, d)$ belongs to the continuation set we inspect another item from the lot. If it belongs to the stopping set we stop inspection and send out the lot. Obviously all the points with $n \geqq N$ or with $d \geqq N$ are stopping points. We now define the boundary set as the set of stopping points which are accessible from some continuation point. Because of the monotonic nature of the two types of cost associated with a plan one would expect that for an admissible plan, the boundary set consists of points $\{(n, d_n)\}$ with $d_n$ monotonically nondecreasing in $n$ and the points $(n, d)$ are stopping or continuation points according as $d \leqq d_n$ or $d > d_n$. Such plans are thus defined by a set of boundary points $\{(n, d_n)\}$ or equivalently $\{(d, n_d)\}$. Anscombe [1] considers a class of linear plans where $n_d$ is defined by \begin{equation*}\tag{1.2}n_d = n_0 + d\omega,\quad d = 0, 1, 2, \cdots, \lbrack (N - n_0)/(\omega + 1)\rbrack,\end{equation*} ([ ] denoting the greatest integer part). He considers the problem of choosing the integers $n_0$ and $\omega$ so that the following conditions are satisfied: (1) Whatever the unknown number of defectives in the lot, there is at most a preassigned risk that after inspection the lot will contain more than a specified number of defectives. (2) The average number of items inspected is as small as possible for some range of values (or simply for a fixed value) of the unknown number of defectives. It is to be noted that Anscombe's problem as formulated above, is typical of the classical Neyman-Pearson theoretic formulation and as such differs from our decision theoretic formulation of minimizing (in some suitable sense) the risk associated with a plan. In fact, Wurtele, was first to consider this formulation in her thesis [7]. She gives a characterization of the Bayes sequential plan $\{(d, \tilde n_d)\}$ as a multiple sampling plan which is a consequence of the fact that $\tilde n_d$ is an increasing function of $d$. She presents algorithms for obtaining the large sample behavior of the optimal (Bayes) boundary in the limiting case appropriate for the Poisson approximation. We shall concern ourselves with the Bayes sequential plans for beta prior distributions. To study the large sample behavior of the optimal boundary and associated risk we formulate the following associated normal version of the sampling inspection problem. We observe $X_t$, a Wiener process with unknown drift $\mu$ per unit time and variance one per unit time. Let $T$ be the time at which observation is stopped. Subject to $T \leqq t_0$, it is desired to locate a stopping set which minimizes $E(T\mu)$ (where $E$ denotes the expectation operator over the marginal distribution of the data with a normal prior distribution for $\mu$). If $\mu$ were known, an optimal procedure would be to stop immediately if $\mu \geqq 0$ and to continue as long as possible if $\mu > 0$. The associated problem is then tackled with the help of techniques developed by Chernoff [3], [4], [5] in connection with the sequential testing for the sign of $\mu$. The results are then related to the original sampling inspection problem. Numerical computations are given comparing the boundary and the risk calculated using the asymptotic formulae with the exact ones. Lastly, the optimal (Bayes) plan is compared with the optimal plan among the class of linear plans (1.2). Here the optimum $n_0$ and $\omega$ are approximated by the solution of the corresponding Wiener process problem.