Design of Screening Procedures

Design of Screening Procedures: A Review

KWEI TANG

Louisiana State University, Baton Rouge, LA 70803

JEN TANG

Purdue University, West Lafayette, IN 47907

It has been well accepted that dependence on inspection to correct quality problems is ineffective and costly, and hence screening (100% inspection) should not be used as a long-term solution for improving product quality. However, screening may be an attractive practice for removing nonconforming items from a population in the short term because of the advances in automatic inspection equipment and computer control in manufacturing. Important factors considered in designing screening procedures include the selection of screening variable, available information on the population being studied, cost of inspection, losses caused by decision errors, the variation in product quality, and inspection and manufacturing environments. This paper presents a sy stematic review of the literature on the design of screening procedures.

Introduction

lID ECENT advances in automation and computer l&. control in manufacturing are changing the fundamental role and functions of quality control/assurance. In particular, the use of automatic test equipment (ATE) has greatly increased inspection speed and accuracy (Kivenko and Oswald (1974)). Consequently, screening (100% inspection) is becoming an attractive practice for removing nonconforming items, and it has been suggested that inspection will essentially become an inherent part of modern manufacturing processes (Stile (1987)).

However, as pointed out by Deming (1986), dependence on inspection to correct quality problems is ineffective and costly, and hence screening should not be used as a long-term solution for improving product quality or reducing the costs incurred by nonconforming items. Instead, implementing successful process control and quality improvement programs is essential for a manufacturer to survive in the competitive business world.

Dr. K. Tang is Professor and Chairman in the Department of Quantitative Business Analysis. He is a Member of ASQC.

Dr. J. Tang is an Associate Professor at Krannert Graduate School of Management.

Vol. 26, No.3, July 1994 209

Several factors are usually considered in designing a screening procedure. These factors include the goal to be accomplished, the nature of the performance variables, screening methods and criteria, available information on the population, and economica and manufacturing environments. As a result, the complexity of the design issue is affected by these factors. For example, it can be as simple as designing a single screening operation, or as complicated as designing a system of screening operations for a multi-stage manufacturing process. Furthermore, the situation in which a training sample is required to estimate the population parameters is more difficult to deal with than when accurate information on the population is available. The basic factors considered in designing screening procedures are listed in Figure 1 and are described in what follows.

Objective. Two separate objectives have been commonly used to design screening procedures. One is to optimize the expected total profit associated with a screening procedure, and the other is to use screening to reach certain statistical goals, such as controlling the outgoing nonconforming rate of the product. The methods using these objectives are known as economic and statistical designs of screening procedures, respectively.

In an economic design three cost components are commonly considered: the cost of inspection, the cost

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210

Objective

Performance Variable

Screening Variable

Information on Population

Logistics

KWEI TANG AND JEN TANG

Statistical Goals

Economic Goals

Statistical and Economic Goals

Attribute Type

Variable

Single Variable Number

Multiple Variables

Single Grades Specifications

Multiple Grades

Inspection Error Performance Variable

No Inspection Error

Single Variable Correlated Variable

Multiple Variables

Parameters Known

Parameters Partially Known

Parameters Unknown

Manufacturing Systems

Inspection Methods

Corrective Actions

Process Conditions

Other

FIGURE 1. Basic Factors in Designing Screening Procedures.

L type

S type

N type

of rejection, and the cost of acceptance. The cost of inspection may include expenses of testing materials, labor, equipment, and so forth. The cost of rejection is incurred by corrective actions taken on re-

jected items, such as repairing, scrapping, or returning the items to the supplier. The cost of acceptance is caused by the items of imperfect quality that reach the customers. This may include damage caused by

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DESIGN OF SCREENING PROCEDU RES: A REVIEW 211

product failure, warranty cost, handling cost, loss in sales, loss in goodwill, and so forth (Hald (1960)).

When profit is used as the objective, it is computed by the difference between the revenue and the cost. The revenue depends on product quality and market structure. For example, items may be sold to several markets with different product specifications and prices.

The most commonly used statistical criterion is the outgoing conforming rate. Note that when inspection is error-free, the outgoing conforming rate should be 100% after screening. However, the outgoing conforming rate becomes a meaningful and important design criterion when nonconforming items may not be detected because of inspection error or for other reasons. Further note that economic factors are usually considered implicitly in selecting statistical goals. For example, the outgoing conforming rate should be set at a high level when the cost of accepting nonconforming items is large. In fact, it is also possible to incorporate both the economical and statistical criteria in designing a screening procedure. For example, one may want to minimize the total related cost and, at the same time, require the outgoing conforming rate to be above a given level.

Performance Variable. A performance variable is a measure of a product's ability to satisfy stated or implied expectations of the customers. A product may have one or more performance variables such as weight, color, and dimensions. A performance variable can be a continuous variable or an attribute (qualitative) variable. Continuous variables can be further divided into three types: the-nominal-thebest (N type), the-smaller-the-better (S type), or the-larger-the-better (L type) (Taguchi, Elsayed, and Hsiang (1989)). A product may have multiple grades with specifications for their performance variables being different.

Screening Variable. A screening variable is a variable used to develop screening criteria (rules). When the performance variable is used as the screening variable, all the nonconforming items will be identified if the inspection is error-free. However, since screening errors frequently occur because of inherent variability in testing materials, environment, and/or human inspectors, they should be taken into consideration if the inspection outcomes are significantly affected.

In some situations it is attractive to use a surrogate variable that is correlated with the performance vari-

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able as the screening variable when measuring the performance variable is expensive, time-consuming, or even destructive. This issue is interesting because the relationship between the performance variable and the surrogate variable is usually not perfect, and it is also possible to use more than one correlated variable.

Availability of Information on Population. Both the statistical and economic approaches require using the probability distributions of the performance and/or the screening variables to evaluate the objective functions. Most models for continuous variables assume univariate or bivariate normality. If the distribution parameters are unknown, sampling information and, possibly, prior distributions are used to estimate the distribution parameters. In general, the unknown-parameter cases are more complicated, especially when more than one parameter is unknown.

Logistics. Knowledge of the manufacturing environment is essential to designing a screening procedure. For example, to effectively control the related costs or conforming rate for a manufacturing system with multiple stages (operations), screening procedures used after the stages should be designed jointly. In addition, more efficient screening methods may be designed for some special testing techniques. For example, group testing is applicable when a single test can determine whether a pool of items is free of defect, and burn-in can be used to test all the outgoing items under normal or stress conditions for a certain period to screen out early failed items. The disposition of rejected items, such as scrapping or repairing, also affects the complexity of the problem.

The objective of this paper is to provide a systematic review of the area of screening. The paper is organized as follows. The next section identifies four representative models for single screening procedures. These are Deming's (1986) all-or-none rules, Taguchi's (1984) model for tolerance design, Tang's (1988a) economic model for using correlated variables, and statistical models for using correlated variables. A literature review of single screening procedures is given in the following section. Then, two special screening procedures, burn-in and group testing, are discussed. Finally, special topics on inspection planning, production process design, and selective assembly are discussed. The organization of the paper is shown in Figure 2.

Note that when the inspection is based on the performance variable, screening will identify all the non-


21 2 KWEI TANG AND JEN TANG

conforming items in the inspected population if inspection is error-free. However, if the distribution parameters are unknown, one may be interested in various statistical inferences on the population prior to screening, either for planning purposes or for determining whether screening should be performed. Much is written on this issue, including such topics as confidence intervals for mean and variance, tolerance intervals for a proportion of the population, and so forth. The unknown parameter case when correlated variables are used as screening variables is more complicated because the screening limits are typically functions of the sample means, variances, and correlation coefficient of both the performance and correlated variables. These statistical inference issues on univariate and bivariate normal distributions are not covered in this paper. Readers are referred to the reviews and references given by Hahn

Basic Models

Single Screening

Additional Models

Burn-In Special Screening Methods

Group Testing

(1970a, 1970b), Odeh and Owen (1980), Hutchinson and Lai (1990), Fountain and Chou (1991). Moreover, the collection of the equations for integrals of functions of univariate, bivariate, and multivariate normal density functions given by Owen (1980) is very useful in evaluation and optimization of screening models.

Basic Models for Single Screening

In a typical single screening procedure all the outgoing items are subject to acceptance inspection. If an item fails to meet the predetermined screening specifications, the item is rejected and subject to corrective actions. In this section the formulation methods and solutions of four basic models for single screening procedures are discussed. These models

Deming's All-or-None Rule

Taguchi's Tolerance Design

Economic Models Using Correlated Variables

Statistical Models Using Correlated Variables

Statistical Models

Univariate Economic Models

Multiple Performance Variables

Multiple Correlated Variables

Inspection Errors

Selection of Screening Variables

Inspection Effort Allocation Special Topics

Selection of Process Parameters

Selective Assembly

FIGURE 2. Organization of the Paper.

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DESIGN OF SCREENING PROCEDU RES: A REVIEW 213

provide the basis for understanding and discussing more specialized and more complex models.

Deming's All-or-None Rules

Deming (1986) showed that partial inspection to remove nonconforming items is not economical for a stable process (this is known as the all-or-none rules). Suppose a lot of N items produced by a stable process is subject to attribute inspection. If an item is inspected and found to be nonconforming, it costs r to rework or replace the item, and an unfound nonconforming item will result in an acceptance cost of a. Inspection is assumed to be error-free. Let s denote the cost of inspecting an item and f the proportion of the items that are inspected. Then the expected total cost for this lot is

ETC = Nsf + rNpf + aNp(l- f)

where p is the lot proportion nonconforming. The objective is to determine the value f* that minimizes ETC. One can express ETC as

ETC = N (pa -f [(a -r)p -s])

where pa is the expected cost of accepting an item without inspection, and (a - r)p - s is the per-item expected payoff of inspecting an item. Consequently,

f* = {OO% ifp>s/(a-r) otherwise.

This well-known result and its applications were discussed in detail by Deming (1986) and Papadakis (1985).

Taguchi's Model for Tolerance Design

Consider a screening procedure where each outgoing item is subject to inspection on the performance variable. Let Y denote an N-type performance variable with the target (ideal) value 7 and f(y) the probability density function (pdf) of Y. Taguchi (1984) suggested that the cost associated with an item with Y = y be determined by the following quadratic function

where k is a positive constant. For the purpose of screening let [7 -8, 7 + 8] be the acceptance region, and if the value y is outside this region, the item is rejected and excluded from shipment. Let r denote the cost associated with the disposition of a rejected item and Sy the per-item cost of inspection. Then

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the per-item expected total cost associated with the screening procedure is

+Sy

where the first and second terms are, respectively, the per-item expected costs of acceptance and rejection. The value 8* that minimizes ETCy is equal to ..;:rJk (Tang (1988a)), which is the point where (y,7) is equal to r. This is intuitive since it is economical to reject an item when the acceptance cost is higher than the rejection cost. Furthermore, 8* is independent of f(y). These results can be easily extended for the S-type and L-type performance variables (Taguchi, Elsayed, and Hsiang (1989)).

Economic Models Using Correlated Variables

Let the relationship between X and Y be described by a joint pdf, h(x, y), and [Lx, Ux] be the acceptance region for X, so that an item is rejected if its observed value x falls outside the acceptance region. Let the per-item cost of measuring X be s x and m( x) be the marginal pdf of X. Then the per-item expected total cost is

Ux 00

ETCx = J J (y, 7)h(x, y)dydx Lx -ex>

where the first and the second terms are the peritem expected costs of acceptance and rejection, respectively. Tang (1988b) gave the optimal screening limits L and U; when (y,7) is a step, linear, or quadratic function; and h(x, y) is a bivariate normal pdf with known parameters. The case of an L-type performance variable was discussed by Tang (1987).

Statistical Models Using Correlated Var'iables

A common objective of statistical models when using correlated variables is to determine screening limits that raise the conforming rate from the prescreening value 7 to a larger value A. Let p denote the correlation coefficient between X and Y. Since most studies dealt with the L-type performance variable and assumed a positive p, the discussion in the remainder of this paper is for that situation unless it is specified otherwise. Note that the screening


214 KWEI TANG AND JEN TANG

procedures for other situations (i.e., the S-type and N-type variables and/or a negative p) can be easily obtained by simple transformations when h(x, y) is symmetrical.

Suppose an item is conforming if its Y value is in the interval n. If the parameters of h(x, y) are known, the objective is to find the lower screening limit Lx for X, so that the conforming rate of the accepted items is at least >., that is

Pr [Y E nix 2: Lx] 2: >..

Owen, McIntire, and Seymour (1975) developed tables for finding Lx when h(x, y) is a bivariate normal pdf with known parameters.

If the distribution parameters are unknown, Lx is a function of the sample means and variances of X and Y and their sample correlation coefficient. As a result

Pr [Y E nix 2: Lx] becomes a random variable. In that case the objective is to find Lx so that

Pr {Pr [Y E nix 2: Lx] 2: >.} 2: 1 - a (1)

where 1 - a is the confidence level. This problem is difficult because it involves five unknown parameters. There have been several different approaches to address this problem. A literature review is given in the next section.

Additional Models for Single Screening In addition to the basic models discussed in the

last section, more specialized models have been developed based on these basic models.

Statistical Models Using Correlated Variables

Owen and Boddie (1976) considered the situation where the distribution parameters are partially known. In addition to equation (1), they also considered the expected tolerance interval suggested by Wilks (1941), which satisfies

E {Pr [Y E nix 2: Lx]} = >..

Owen and Su (1977) studied several situations where p and/or 'Yare unknown, which were not considered by Owen and Boddie (1976).

When all the distribution parameters are unknown, the standardized conditional distribution of Y given the sample t-statistic of X is called the normal conditioned on t-distribution (Owen and Hass


(1978) and Owen and Ju (1977)). Li and Owen (1979) considered the N-type performance variable and assumed all the distribution parameters are unknown. Their method uses>. and the lower tolerance limits of p and 'Y as the parameters of the normal conditioned on t-distribution to find the standardized lower screening limit. Odeh and Owen (1980, p. 12) gave the same method for the L-type performance variable. A brief review of the above methods was also given by Owen (1988).

Owen, Li, and Chou (1981) (OLC) studied a situation where items are inspected using correlated variable until a specified number n of items are accepted. The screening limit is determined so that at least lout of the n items are conforming with a specified confidence level. Both the known-parameter and unknown-parameter cases were included in their discussion. Note that the procedures proposed by Owen and his co-authors use the non-central t-distribution and the normal conditioned on t-distribution. These distributions and related references were discussed in detail by Odeh and Owen (1980).

Madsen (1982) proposed a selection procedure with a slightly different objective; that is, to determine the largest subset of accepted items from a finite inspection lot so that the conforming rate of the subset meets a pre-specified level with a given probability. Wong, Meeker, and Selwyn (1985) used a noninformative prior in a Bayesian model for the OLC procedure and showed, by a simulation study, that the screening limits obtained using their method were closer to the accurate values than those given by OLC. Mee (1990) provided a more efficient but less stringent method by directly approximating the conditional probability

Pr {Pr [Y E nix 2: Lx] 2: 'Y} .

Using simulation, the method was shown to yield fewer rejected items than the methods in Odeh and Owen (1980).

Boys and Dunsmore (1986) considered a predictive probability function approach using a prior distribution and a sampling distribution to find screening limits so that the probability an accepted item is conforming reaches a satisfactory level. Boys and Dunsmore (1987) tried to simplify the problem structure for the unknown parameter case by transforming the performance variable into an indicator variable T = 0 or 1. Two approaches were discussed. The first one is the diagnostic paradigm (Aitchison and

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DESIGN OF SCREENING PROCEDURES: A REVIEW 215

Dunsmore (1975)), which is based on the predictive probability function of T given X, and the second one is the sampling paradigm, which is based on the conditional distribution function of X given T.

Tsai and Moskowitz (1986) introduced the concept of individual unit misclassification error (IME) and developed a one-sided screening procedure to control both the IME and the outgoing conforming rate. They assumed bivariate normality with known parameters.

Univariate Economic Models

Menzefricke (1984) introduced a cost structure for the OLe screening procedure. Three costs were considered: cost of inspection, cost of not having l conforming items, and cost of accepting nonconforming items. Both known and unknown parameter cases were discussed. Boys and Dunsmore (1986) introduced a cost structure for considering the losses incurred by screening out conforming items and retaining nonconforming items. Moskowitz, Plante, and Tsai (1991) combined economic factors, IME, and average outgoing nonconforming rate as a basis for selecting a screening procedure. Bai, Kim, and Riew (1990) studied one-sided and two-sided procedures for the situations where all the parameters are known and where some of the parameters are known. Based on the same cost structure, Kim and Bai (1992a) studied the case where all the parameters are unknown. Kim and Bai (1992b) also considered screening procedures in which the performance variable is a dichotomous variable (similar to Boys and Dunsmore (1987)), and its relation with the correlated variable is described by the logistic model or normal model.

To reduce the possible screening errors caused by the imperfect relationship between the performance variable and the correlated variable, Tang (1988c) proposed a two-stage procedure where each item is inspected using a correlated variable at the first stage and inspected using the performance variable only when the result at the first stage is inconclusive. Hui (1991) assumed the process mean (the mean of the performance variable) may shift to another value (the out-of-control state). A model was developed to derive the screening limits as well as the control limits for monitoring the process. The performance variable is used as the screening variable, and the control limits are based only on the current (single) observation.

Tang (1989) considered a situation where the outgoing items are sorted into one of two grades or

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scrapped. The two product grades have different specifications and, thus, prices. A loss is incurred to the producer when an item is classified into a grade where quality does not meet the consumer's requirement for that grade. On the other hand, a loss in selling price is incurred when an item is classified into a lower product grade while it can meet the consumer's requirement for a higher grade. Models based on the performance variable and a correlated variable were developed. Bai and Hong (1992) developed a similar model for multiple markets and also discussed situations where some distribution parameters are unknown. Kim, Tang, and Peters (1992) extended the two-stage model of Tang (1988c) to a model with two product grades.

Park, Peters, and Tang (1991) proposed a sequential procedure for screening a lot with unknown nonconforming rate. A decision is to be made after inspecting each item on whether to inspect another item or to reject the remainder of the lot. An optimal stopping rule was developed using a Bayesian approach to maximize the expected difference between the payoff of finding conforming items and inspection cost.

Multiple Performance Variables

Tang and Tang (1989b) considered a product with several performance variables and formulated economic models for two screening procedures. In the first procedure all the outgoing items are subjected to acceptance (attribute) inspection using all the performance variables, and the disposition of each item is determined by whether this item conforms to the screening specifications of the variables. In the second procedure the exact values of all the item's variables are obtained for making a decision on this item. A joint decision rule based on an aggregation of the variables is developed. Lo and Tang (1990) extended this model to a product with two product grades. A bivariate model was discussed by Hui (1990), where acceptance cost is a linear combination of functions of two individual variables', either quadratic functions or absolute values of the quality deviations from the target values. The effects of inspection error were also discussed.

Tang (1990) considered a product with multiple performance variables produced by a serial production system, where a performance variable is determined at each stage of the system. If an unacceptable item is rejected and excluded from production in an early stage, the production and inspection costs that



would be invested on this item in the later stages can be saved. Therefore, the screening rule at any stage should be based on the quality of the product, the total investment already on the item, and the investment and the expected quality cost that would be incurred in the later stages.

Butler and Lieberman (1984) considered a product (system) with several components. The product fails if one or more components fails. When an item fails, its components are sequentially tested until a failed component is found. A heuristic procedure for sequencing the order of inspecting the components was proposed to identify a failed component after the product (system) fails.

Multiple Correlated Variables

Owen, McIntire, and Seymour (1975) suggested two decision rules that use two correlated variables in a screening procedure. The first method uses a screening rule that requires an accepted item to conform to the individual screening specifications of the two correlated variables. The second screening rule is based on a linear combination of correlated variables. Thomas, Owen, and Gunst (1977) provided tables and procedures for using two correlated variables based on the trivariate normal distribution. They also obtained a linear combination of the two correlated variables that maximizes the chance of obtaining conforming items.

Moskowitz and Tsai (1988) extended their previous work (Tsai and Moskowitz (1986)) based on the IME to a double (two-stage) screening procedure using two correlated variables. At the first stage an item is inspected using a correlated variable. When a decision cannot be reached at the first stage, the item is inspected using a second correlated variable. Recently, Moskowitz, Plante, and Tsai (1993) proposed a multistage (sequential) screening procedure that controls the maximum and average misclassification errors for detecting hypertension in a series of blood pressure measurements. Tang and Tang (1989a) developed a cost model on the basis of the second method suggested by Owen, McIntire, and Seymour (1975). Tang and Tang also showed that the optimal linear combination of the correlated variables should have the maximum correlation coefficient with the performance variable.

Inspection Error

It is well known that most inspection processes have inherent variability due to various factors such

Journal of Qualify Technology

as variations in testing materials and inspecton.. For attribute inspection there are two types of errors (Case, Benett, and Schmidt (1975)). A Type I error occurs when a conforming item is classified as nonconforming, and a Type II error occurs when a nonconforming item is classified as conforming. For variable inspection, inspection error is characterized in terms of bias and imprecision. Bias is the difference between the true value of the performance variable of an item and the average of a large number of repeated measurements of the same item, and imprecision is the dispersion among the measurements of the same item (Mei, Case, and Schmidt (1975)).

When inspection error is present, losses are incurred by rejecting conforming items and accepting nonconforming items. Raz and Thomas (1983) discussed sequencing several inspectors with different inspection precision levels to meet a predetermined outgoing conforming rate at minimum cost. Drury, Karwan, and Vanderwarker (1986) examined the performance of different methods of combining two inspectors for making inspection decisions.

Raouf, Jain, and Sathe (1983) considered a product with multiple performance variables and assumed that failure to meet the specifications of any one of the variables results in rejection of the product. Because of inspection errors, it may be necessary (or economical) to inspect an item on the same performance variable more than once. A mathematical model is formulated to determine the optimal sequence of measuring the performance variables and the optimal number of inspections to be performed on each item in order to minimize the total expected cost per accepted item. Duffuaa and Raouf (1990) provided the mathematical proof of the optimal sequencing rule given in Raouf, Jain, and Sathe (1983). Lee (1988) developed a simplified version of the Raouf, Jain, and Sathe (1983) model and derived an efficient solution procedure for finding the optimal number of inspections. Jaraiedi, Kochhar, and Jaisingh (1987) considered a similar problem and developed a method to determine the minimum number of inspections that must be performed on a lot to meet a desired lot outgoing conformance rate.

Tang and Schneider (1987) discussed how to determine screening limits when inspection error is present, and they investigated the economic effects of inspection imprecision on a screening procedure. It is assumed that the rejected items are reworked, and two rework conditions were considered. In the first situation the rejected items can be reworked so

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that the performance variable is exactly equal to the target value. In the second situation rework is based on the first inspection result; therefore, the value of the performance variable of the reworked items may not be exactly equal to the target value.

Tang and Schneider (1990) showed that when inspection error is present, the observed value of the performance variable can be treated as a correlated variable. Consequently, all the results associated with using correlated variables in screening are applicable to the inspection error situation.

The inspection precision level may actually be a decision variable in some situations. For example, the inspection precision can be improved by using the result of multiple tests on the same item. This practice has been used to test IC chips in the computer industry. Tang and Schneider (1988) discussed a method of determining the optimal inspection precision level based on the tradeoff of inspection cost and the costs incurred by inspection errors.

Raz and Thomas (1990) collected several related papers on human factors (errors) in inspection, including a review paper by Raz (1986) and another one by Dorris and Foote (1978) on statistical quality control methods.

Selection of Screening Variables

Searle (1965) used applications in genetics to study the effectiveness of using an indirect selection method based on a correlated variable. A measure of the relative selection efficiency for an indirect selection method relative to a direct selection method was introduced, and the sample standard error of this measure was derived. Then, conditions were given under which an indirect selection method should be used. Menzefricke (1984) used the screening procedure suggested by Owen, Li, and Chou (1981) to illustrate a method for deciding whether a correlated variable should be used in lieu of the performance variable. The basic tradeoff of using a correlated variable is between the saving in inspection cost and the loss caused by accepting nonconforming items and not having enough conforming items. Both known and unknown parameter cases were discussed.

Tang and Schneider (1990) showed that the benefit of using a correlated variable as the screening variable is dependent on the correlation between the correlated variable and the performance variable. Inspection error may "dilute" the correlation between the two variables, which, consequently, reduces the

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effectiveness of using the correlated variable. However, in practice it is often possible to find a correlated variable that requires a less complicated measuring process, so that the inspection error of using a correlated variable is relatively lower than that of using the performance variable. This may further support the use of a correlated variable. Tang and Schneider (1990) illustrated both theoretically and empirically when a correlated variable should be used as the screening variable.

Special Screening Procedures

In this section two special screening procedures are discussed. The first is burn-in, which is used to reduce early product failure by testing (operating) all the outgoing items under a normal or stress condition for a fixed amount of time before shipping to customers. The second is group testing, which is used when it is possible to use a single test to verify whether a group of items is free of nonconforming items.

Burn-In

Many industrial products have high failure rates in their early lives. "Burn-in" is a procedure that operates all the outgoing items for a fixed period under normal or stress conditions to reduce early failures before shipping a product to consumers. Recently, Tusin (1990) reported an interesting development and success in using environmental stress screening (burn-in under special stress conditions) to improve electronics reliability.

For many products there are three phases in their product life cycle. The early stage (with relatively high but decreasing failure rate) is called the infantmortality phase, the stage with a constant failure rate is called the normal phase, and the last stage (with an increasing failure rate) is called the wear-out phase. The point that separates the infant-mortality and normal phases is called the change-point.

A common practice is to test the product until it reaches its change-point. If the burn-in period is too long, then stress conditions are used to accelerate the "aging" process. In order to estimate the changepoint, several product life distributions have been used, including Wei bull , gamma, lognormal, nonhomogeneous Poisson, mixed Wei bull-exponential , empirical distributions, and others (Kuo and Kuo (1983), Boukai (1987), and Hjorth (1980)). A survey of change-point estimation was given by Zacks (1983). Since then, Yao (1986) has studied the prop-



erties of maximum likelihood estimation and Boukai (1987) proposed a Bayes sequential estimation procedure. Note that these studies assumed that all the distribution parameters except the change-point are known. If all the distribution parameters are unknown, it is a classical statistical estimation problem. However, since stress conditions are often used, the problem becomes much more complicated. This is known as the area of accelerated life testing. First, there should be a known relationship between the actual product life and the life under stress conditions. A well-known example is the Arrhenius model (Nelson (1971)), which describes degradation over time as a function of the operating temperature. Furthermore, the stress conditions may be applied in different manners, such as step-stress (Nelson (1980)) and progressive stress (Allen (1958) and Yin and Sheng (1987)). Wadsworth, Stephens, and Godfrey (1986, Chap. 18) provided a good introduction on how to design an accelerated life test, as well as useful references, such as Nelson and Meeker (1978), Nelson (1980, 1982), and Mann, Schafer, and Singpurwalla (1974).

The cost structure of designing a burn-in procedure is very similar to the basic model in Tang (1987). However, burn-in models are usually complicated because the burn-in (inspection) costs and the product life distribution after burn-in are functions of burn-in time. Moreover, the cost associated with failed items after burn-in may be difficult to calculate. Some examples of cost models are as follows.

Stewart and John (1972) developed a Bayesian model for determining the burn-in time and replacement schedule for non-repairable products. The cost components considered in the model are burn-in cost, manufacturing cost, and costs incurred by scheduled and unscheduled replacements. Canfield (1975) studied a similar problem with a known product life distribution. Plesser and Field (1975) considered a repairable product with the number of failures following a Poisson process. It was assumed that the product failure rate remained unchanged after repair. The optimal burn-in time is determined by minimizing the expected total cost of operating an item in both burn-in period and service periods. Cozzolino (1970) considered how long to continue the burn-in process for repairable products. Weiss and Dishon (1971) studied two situations where a specific number of items are required at the end of the burn-in process. In the first situation, failed items in the process are repaired, and in the second situation, failed


items are not repaired and the shortage at the end of the burn-in period is made up by new items with zero burn-in time. Nguyen and Murthy (1982) formulated a model for determining the burn-in time for a product sold with a warranty. Two types of warranty polices are considered. The first is the failurefree policy, where all the failed items are repaired or replaced in the warranty period. The second is a rebate policy, where the customer is refunded some portion of the sales price if the product fails during the warranty period. Chou and Tang (1992) considered Nguyen and Murthy's model for the failure-free policy, but used the Wei bull-exponential mixed distribution to describe the infant-mortality and normal phases. They also studied the situation where the change-point is unknown.

Much is written in the literature on burn-in procedures. Readers are referred to Leemis and Beneke (1990) and Kuo and Kuo (1983) for more detailed reviews of burn-in models.

Group Testing

Group testing is used when all the nonconforming items have to be removed from a population and it is possible that a (group) test on a pool of items can be used to detect whether the items in the pool are all conforming. The benefit of group testing is the savings in the cost of testing individual items when all the items are conforming. If the group test indicates that the items are not all conforming, the items are retested individually, and nonconforming items are identified and removed from the lot. This inspection procedure is called the two-stage group testing procedure.

A well-known example is the blood-testing problem considered by Dorfman (1964) where a large number of blood samples are to be tested for contamination. Portions of blood samples can be tested together. These samples are tested individually only when the group test is positive. Additional examples are leakage tests, flow tests (Sobel and Groll (1959) and Hwang (1984)), and group factor screening in experimental design (Watson (1961), Li (1962), and Gurnow (1965)).

Designing a group testing procedure consists of selecting an appropriate group size to minimize the total inspection effort. Using a large group size reduces the frequency of group tests but increases the chance of retests. It is evident that the distribution of the number of nonconforming items is an impor-

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DESIGN OF SCREEN I N G PROCEDU RES: A REVIEW 219

tant factor in making this decision. A common goal of group testing models is to minimize the expected total number of group and individual tests. An implicit assumption is that the costs of performing a group test and an individual test are the same. If the number of nonconforming items follows a binomial distribution, the per-item expected number of tests in Dorfman's problem is given by l/k + 1 - (1 _ p)k, where k is the group size, and p is a known nonconforming rate. Samuels (1978) gave an easy way to find the exact optimal group size. Many other researchers; including Nebenzahl and Sobel (1973), Nebenzahl (1975), Hwang (1972, 1975, 1978, 1980), Lin (1974), Kumar and Sobel (1971), and Hwang, Song, and Du (1981); studied the same problem under different distributional assumptions. Graff and Roeloffe (1972) incorporated inspection errors into Dorfman's model.

There are many other forms of group testing. Sterrett (1957) suggested a procedure to sequentially test the items in a group that failed the group test. Individual testing is done until the first nonconforming item is found. Then the remaining items are again tested in a group, and the procedure is repeated until all the nonconforming items are found. The efficiency of this method was studied by Sobel and Groll (1959). Gill and Gottlieb (1974) proposed a procedure to divide the group that was found to contain nonconforming items into two sub-groups, and this procedure is applied recursively to the sub-groups that fail the group test. Sobel and Groll (1959) considered a procedure that divides the group that failed the group test into successively smaller sub-groups. Recursion equations were developed to determine the sizes of the sub-groups. A detailed discussion of this procedure was given by Mundel (1984), and tables for using this procedure were provided by Snyder and Larson (1969). Mundel (1985) developed a cost model which assumes that the costs of the group test and individual test are different. The optimal group size is found by minimizing the total expected test cost. Li (1962) proposed a multi-cycle procedure for screening experimental factors in which the whole group of factors is divided into sub-groups at each cycle, and all the sub-groups that fail the group test (at least one factor is significant) are pooled into the group for the next cycle. Kumar (1965) considered a multiple-grade situation where three possible inspection outcomes are considered: good, mediocre, and defective. Sobel and Groll (1959) discussed the situation where a limited number of tests (group or individual) can be performed on an individual item.

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Sobel (1960) discussed the restriction that a group test can be applied only to adjacent items, not to any arbitrary subset of items.

When p is unknown, Sobel and Groll (1966) developed a Bayesian model using a beta prior distribution. They compared it with the procedure assuming p is known and another procedure based on continually updating estimates of p. They also discussed a different type of procedure that allows "mixing" the groups found to contain nonconforming items and the uninspected items in order to form a new group for further testing. Hwang (1984) proposed a robust procedure for the case where only the mean nonconforming rate is available (the form of the distribution is unknown). Schneider and Tang (1990) formulated a cost model using the Bayesian approach for the two-stage procedure and showed that using variable group sizes based on simple updating procedure (Bayes' rule) can substantially reduce the total inspection effort.

An interesting issue related to group testing is to minimize the number of tests required to find one nonconforming item from a population. However, the objective is not to screen all the nonconforming items from the population. This issue was discussed by Kumar and Sobel (1971), Hwang (1974), and Garey and Hwang (1974). Kotz and Johnson (1982) provided methods to compute the outgoing conforming rates and the expected total number of tests for group testing with inspection error.

Inspection Planning and Selection of Process Parameters

In this section three special topics are discussed. The first is how to allocate screening efforts in a multi-stage manufacturing system, the second is how to design the process paraeters to optimize the profit/cost, and the third is product variation reduction by selective assembly.

Inspection Effort Allocation

In a multiple-stage manufacturing system "where to inspect" and "how many to inspect" are important decisions for controlling manufacturing costs. The "topology" of manufacturing systems makes these two decisions difficult. Two types of manufacturing systems have been studied. The first one is a serial system, where each stage or operation (except for the beginning one) has only one immediate predecessor. The second type is a non-serial system, where



some stages may have multiple predecessors. In addition, various dispositions of nonconforming items found in the manufacturing process, such as repair and replacement need to be considered.

There have been several heuristic rules concerning inspection location (Moore (1973) and Peters and Williams (1984)). Some of these are:

1. Inspect after operations that are likely to produce nonconforming items

2. Inspect before costly operations 3. Inspect before operations where nonconforming

items may damage or jam machines 4. Inspect before operations that cover up noncon

forming items 5. Inspect before assembly operations where rework

is very costly.

The issue of inspection locations has been investigated using mathematical programming methods. For example, White (1966) considered a serial manufacturing system with m stages. A conforming item becomes nonconforming at each stage in the system with a known probability. A lot may be inspected partially or completely prior to entering any stage, and if a nonconforming item is found, it is replaced by a conforming item. The cost of inspection may be different in different stages, and the cost of replacing a nonconforming item increases as the item moves further through the system. The tradeoff is whether to inspect an item to avoid further wasteful investments in a nonconforming item. The cost structure at each stage is similar to that of Deming's model (1986). However, the decisions at all the stages are related. In particular, the decisions at earlier stages should consider the cost and probabilistic structure in the later stages, and the decisions made at earlier stages will affect entering nonconforming rates at the later stages.

A mathematical model is formulated to find the optimal inspection proportions It, h, . . . , and 1m in an m-stage manufacturing system. Note that if Ii is greater than 0, the ith stage becomes an inspection location. White (1966) showed that the optimal inspection proportions should be either 0 or 100%, which enables the model to be solved by simple dynamic programming or integer programming methods. Britney (1972) considered non-serial systems and found that the all-or-none rules also apply. In fact, most of the researchers have either proved that a 0 or 100% inspection plan is optimal or have assumed it based on previous research results (Rabinowitz (1988)). Chakravarty and Shtub


(1987) extended White's (1966) model to a multiproduct situation where additional costs, setup, and inventory carrying costs are considered. Ballou and Pazer (1982) discussed the inspection allocation issue for a serial production system when inspection errors are present. Two detailed reviews on this issue were recently prepared by Raz (1986) and Rabinowitz (1988).

Selection of Process Parameters

In some manufacturing situations, such as a bottling process, material (production) cost is a function of the performance variable, and lower and/or upper product specification limits are specified. The process mean affects both the production cost and the chance of producing nonconforming items. If inspection is not destructive, nonconforming items can be identified by screening. Depending on the sales and production situations, nonconforming items may be scrapped, reworked, or sold at reduced prices. Consequently, the decision on setting a process mean should be based on the tradeoffs among material cost, payoff of conforming items, and the costs incurred due to nonconforming items. Based on the process condition, the studies in this area can be classified into two categories. In the first, the process mean is assumed to be stable over time, and in the second, the process mean decreases/increases over time (or with the number of items produced). The latter category is known as the "tool wearing process".

Stable Process. Springer (1951) considered a manufacturing situation where upper and lower specification limits are both present and the performance variable follows a gamma distribution. The per-item cost associated with nonconforming items above the upper specification limit (overfilled items) can be different from those below the lower specification limit (underfilled items). However, these costs are assumed to be constants (independent of the value of the performance variable). The process mean that minimizes the total cost associated with nonconforming items is obtained. Nelson (1979) gave a nomograph for Springer's solution. Bettes (1962) studied a similar situation with a given lower specification limit and an arbitrary upper limit. Underfilled and overfilled items are reprocessed at a fixed cost. The optimal process mean and upper specification limit are determined simultaneously.

Hunter and Kartha (1977) discussed the situation where underfilled items can be sold at a (constant)

Vol. 26, No.3, July J 994


reduced price and a penalty (give-away cost) is incurred by conforming items with excess quality (the difference of the performance variable and the lower limit). They derived a procedure for calculating the optimal process mean. Nelson (1978) also provided approximate solutions to this problem. Bisgaard, Hunter, and Pallesen (1984) modified Hunter and Kartha's model by assuming that the selling price of nonconforming items is a linear function of the performance variable. Carlsson (1984) discussed a more general sales situation where the selling prices of the conforming and nonconforming items are linear functions of excess ( "give-away") quality and "deficit in quality" , respectively.

Golhar (1987) assumed that only the regular market (fixed selling price) is available for the conforming items and that the underfilled items are reprocessed and sold in the regular market. Golhar and Pollock (1988) extended this model to include an upper limit to reduce the cost associated with excess quality by reprocessing the items above this limit. Solution procedures for the optimal process mean and the upper limit were also given in Golhar (1988). Carlsson (1989) discussed a situation in which the lots produced by a production process are subjected to lot-by-lot acceptance sampling by variables, and Boucher and Jafari (1991) studied the same problem except that an attributes sampling plan is used to decide whether a lot is accepted.

An implicit assumption in Golhar and Pollock's model is that the process has an unlimited capacity that can be used to reprocess items above the upper limit. Schmidt and Pfeifer (1991) considered the situation where the process capacity is fixed. Melloy (1991) considered products that are subject to regulatory auditing (compliance tests) schemes. The performance variable is the weight of the package, which is determined by the weights of the product and the tare (e.g., boxes). The process mean and two-sided screening limits are used to minimize the "give-away" product weight, subject to an acceptable level of risk of failing the compliance tests. Tang and Lo (1993) developed a model for jointly determining the optimal process mean and screening limits when a correlated variable is used in inspection. Note that since a correlated variable is not perfectly correlated with the quality characteristic, acceptance cost may be incurred by accepting nonconforming items for shipment.

Tool Wearing Process. A tool wearing process is a production process that exhibits decreasing (or in-

Vol. 26, No.3, July 1994

creasing) patterns in the process mean and/or variance during the course of production. Typical examples are machining, stamping, and molding operations (Gibra (1967)). To control the loss incurred by nonconforming items, it may be necessary to periodically stop the process in order to reset the process by, for example, changing the tools or cleaning the molds. If the process mean has a decreasing pattern, then setting a higher initial process mean can reduce the frequency of resetting and, thus, the loss of production time. Nevertheless, it can also result in a higher production cost. Consequently, the initial process condition and the run size (time for resetting) are jointly determined to minimize the total related cost.

Gibra (1967) considered a process where the process mean decreases constantly (linearly) over time and the process variance remains constant. The optimal process mean and run size are obtained by minimizing the sum of resetting cost and the loss due to nonconforming items. Arcelus, Banerjee, and Chandra (1982) considered a situation with both upper and lower specification limits and where the process mean and variance increase linearly or nonlinearly over time. Optimal solutions that minimize the average production cost per conforming item were obtained for both infinite (continuous) and finite horizon production situation. Arcelus and Banerjee (1985) incorporated the profit/cost structure given by Bisgaard, Hunter, and Pallesen (1984) into Gibra's model. Schneider, O'Cinneide, and Tang (1988) used an AOQL constraint and a more general assumption about tool wearing that allows the deterioration (decrease) in the process mean to be a random variable. The economic model for this problem was developed by Schneider, Tang, and O'Cinneide (1990). Note that screening was explicitly assumed by Arcelus, Banerjee, and Chandra (1982) but not by Gibra (1967) and Schneider, Tang, and O'Cinneide (1990).

Selective Assembly. The quality variation of a product is affected by the variations of its components and the assembly method. Random assembly is a method in which components are chosen randomly for assembly. Suppose that weight is the performance variable, and it is determined by the sum of the weights of two components. Using random assembly, the variance of the performance variable is equal to the sum of the variances of the two components.

In contrast, selective assembly matches components according to certain rules. For example, the



units of each component can be first sorted (screening) into several groups, and then the units in a group are assembled only with the units in a selected group of another component. Using this approach, nonconforming rate and product variation are reduced. Malmquist (1990) reviewed the literature and discussed various approaches to reduce product variation and nonconforming rate by using selective assembly.

Discussion

In this paper, we reviewed the literature in the area of screening. The structure of the area of screening is provided by the four basic models: Deming's allor-none rules, Taguchi's model for tolerance design, Tang's economic model for using correlated variables, and statistical models for using correlated variables. Then, various more detailed models based on these four models were presented for different inspection and manufacturing environments.

Most of the existing studies assume a stable process or a deteriorating process with a known pattern. However, little has been done on using screening data (especially on correlated variables) in process control and improvement. Furthermore, in practice, production decisions and the design of a quality control/ assurance system should be considered jointly. For example, it was shown that a vendor decision may be changed due to implementing a screening procedure (Tang (1988d)). Shih (1980) modified the simple inventory economic lot sizing (EOQ) model by assuming that all the orders are screened. Kalro and Gohil (1982) considered a lot size model with backlogging where the number of items received may be different from the order quantity. The difference is described by a normal random variable. Lee and Rosenblatt (1985) derived optimal order quantities under two inspection policies. In the first policy, a lot is accepted without inspection, and is partially inspected before it is sold to customers. In the second policy, all the items are screened before purchase, and thus are free of nonconforming items. Porteus (1986) considered the production-lot sizing problem when there is a possibility that the production process may be out of control.

It should be pointed out again that a manufacturer has to use process control and improvement programs to improve product quality. This will, in turn, enhance ones ability to survive in this competitive business world. Screening should be considered


only as a short-term method to remove nonconforming items from a population, and dependence on inspection to solve quality problems is ineffective and costly.

Acknowledgments

Dr. Kwei Tang's research was supported in part by National Science Foundation Grant #DDM-8857557 and Southern Scrap Material Company, Baton Rouge, LA.

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Key Words: Burn-In, Economic Design, Group Testing, Multi-Stage Manufacturing System, Screening.

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