Stochastics and Statistics
Designing repetitive screening procedures with imperfect inspections: An empirical Bayes approach

https://doi.org/10.1016/j.ejor.2016.03.003Get rights and content

Highlights

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    A batch of complex items is inspected more than once to discover conforming items in repetitive screening.

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    The number of conforming items still remaining in the batch is often estimated by the maximum likelihood method.

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    We propose an empirical Bayes estimation method which considers the heterogeneity in batch quality.

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    We also deal with the problem of when to stop the screening process and salvage the remaining items.

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    In numerical study, we show that the empirical Bayes method is superior to the traditional estimation method.

Abstract

A batch of expensive items, such as IC chips, is often inspected multiple times in a sequential manner to further discover more conforming items. After several rounds of screening, we need to estimate the number of conforming items that still remain in the batch. We propose in this paper an empirical Bayes estimation method and compare its performance with that of the traditional maximum likelihood method. In the repetitive screening procedure, another important decision problem is when to stop the screening process and salvage the remaining items. We propose various types of stopping rules and illustrate their procedures with a simulated inspection data. Finally, we explore various extensions to our empirical Bayes estimation method in multiple inspection plans.

Introduction

In statistical quality control, “acceptance sampling” is an effective tool that improves and assures the quality of the items in a batch (Schilling and Neubauer 2012). A sample of items is picked at random from the batch and, on the basis of information that is yielded by the sample, we make a decision regarding the disposition of the remaining items in the batch.

In some situations, however, all the items in the batch are subjected to a series of rigorous tests, and only the items that pass the test successfully are taken out of the batch. Some authors call it “100% inspection” (Duffuaa, Al-Turki, & Kolus, 2009) or “100% attribute sampling” (Dhavale, 1987). Because no real sampling takes place in such a case, we use the term “screening” throughout the paper (Tang & Tang, 1994).

During a round of screening, not every conforming item is identified due to inevitable inspection errors. That is why some expensive items are screened more than once in a sequential manner. This “repetitive screening” is also known as a sequential defect removal sampling (Bonnett & Woodward, 1994), a repetitive testing (Greenberg & Stokes, 1995), a repeat inspection (Duffuaa and Khan, 2005, Elshafei et al., 2006), or a sequential review (Rallis & Lansdowne, 2001).

Consider, for example, the semiconductor industry (Greenberg & Stokes, 1995). The fabrication of an integrated circuit (IC) or “chip” is accomplished by depositing a series of film layers upon a silicon wafer, which forms the foundation of the chip. A single wafer may hold hundreds of chips, each of which will go on to power such electronic devices as MP3 players, cell phones, personal computers, and more. At the final stage, packaged chips are subjected to a functionality test, where hundreds of different input patterns are used to ensure that each chip works in accordance with manufacturing specifications.

It is possible for a non-defective chip to fail the screening procedure. (e.g., the electrical leads on the packaged device may be contaminated with a small amount of dust or not lined up perfectly with the test head.) Due to the stringency of the test design, however, it is not likely for any defective chips to pass the functionality test successfully. Because the market value of a chip is much higher than the cost of its functionality test, it is cost-effective to re-test any chips that have failed the functionality test.

In such a situation, an important managerial decision is how to estimate the number of non-defective chips still remaining in the batch after a given number of screening cycles (Chun, 2006). Conversely, we also want to determine the number of screening cycles that produce the maximum profit or the minimum cost (Chun, 2009). In this paper, we first propose a Bayesian method for estimating the total number of confirming items in a batch, and then we deal with the problem of when to stop the screening procedure.

One of the difficulties in Bayesian analysis is how to determine the “prior distribution” of the total number of conforming items in a batch and estimate its parameter values. In the paper, we use the so-called “empirical Bayes” estimation, where the parameter of a binomial prior distribution is estimated from the inspection data. Based on the empirical Bayesian estimate of the number of conforming items still remaining in a batch, we can decide whether to stop the screening procedure or to conduct another round of screening. The optimal stopping rule is the one that satisfies a pre-specified condition or maximizes the expected profit (Jaraiedi, Kochhar, & Jaisingh, 1987).

The remainder of the paper is organized as follows. In the next section, we contrast two types of inspection problems, formally introduce the notation that will be used throughout the paper, and briefly introduce the basic concept of the Bayesian approach. We consider the method of maximum likelihood in Section 3, and we propose the empirical Bayes method in Section 4. We then consider various types of stopping rules in the repetitive screening procedure in Section 5. In Section 6, we compare the performance of the empirical Bayes estimate with that of the maximum likelihood estimate, and illustrate the stopping rules with simulated inspection data. The concluding remarks on possible extensions to our empirical Bayes estimation method are given in Section 7.

Section snippets

Quality engineering terminology

A complex product is composed of a number of elements or items that jointly describe what the consumer or user thinks of as quality (Montgomery, 2012, p. 8). These parameters are often called quality characteristics. Quality characteristics are often evaluated relative to specifications. A products or component is called nonconforming if it fails to meet one or more of its specifications. A specific type of failure is called a nonconformity. A nonconforming product is considered defective if it

Maximum likelihood estimation

There are originally n conforming items in a batch of size N. After (j–1) rounds of repetitive screening procedures, there are (n–sj–1) conforming items that are still undiscovered and remain in the batch. Thus, the probability that exactly xj conforming items will be detected during the next round of screening is a binomial distribution as shown below: P[xj|n]=(n−sj−1)!(n−sj)!xj!qn−sj(1−q)xj,where q is the overlook probability of a conforming item during a screening process.

If we conduct k

Bayesian analysis

Let us consider a certain batch of N items, among of which only n items are conforming to manufacturer's specifications. To consider the batch-to-batch variability, we assume that the total number of conforming items n is a random variable which differs from batch to batch. As the prior distribution of n, we suggest using a binomial distribution which belongs to the same distributional family as the likelihood function f(x|n) in (5): g0(n|π)=(Nn)πn(1−π)N−n.Notice that the “prior mean” of the

Optimal screening procedure

Upon completing the kth round of screening, we need to decide if the batch of remaining items should be screened further to discover more conforming items still remaining in the batch. We can imagine various types of stopping rules for the repetitive screening procedure. Among them, some stopping rules depend on the inspection results at each stage, and other stopping rules have a fixed stopping point regardless of the batch size or inspection results.

Data

To illustrate the empirical Bayes estimation method and demonstrate various stopping rules, we simulated the inspection data for various batch sizes Ni as shown in Table 1. The batch size Ni for i = 1, 2, …, 9 was systematically changed from 800 to 1,200 by 200. The total number of conforming items ni in each batch was simulated using a binomial random number with the batch size Ni and the average conforming rate 0.9. For the Monte Carlo simulation, we used the Microsoft Excel functions,

Concluding remarks

In the paper, we considered the problem of screening a batch of items multiple times in a sequential manner. We proposed an empirical Bayes method to estimate the number of conforming items in a batch. We also proposed various stopping rules that determine the optimal number of screenings. In a modest simulation study, we evaluated the accuracy of the empirical Bayes method and showed its outstanding performance over the traditional maximum likelihood method. Although we may need more extensive

Acknowledgments

I'd like to express my gratitude for the financial support from Cherie H. Flores Endowed Chair of MBA Studies at Louisiana State University. I'd also like to express my sincere thanks to two anonymous referees for their thorough and knowledgeable reviews that helped me improve the original manuscript significantly.

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