Research Article  Open Access
A Study on Overestimating a Given Fraction Defective by an Imperfect Inspector
Abstract
It has been believed that even an imperfect inspector with nonzero inspection errors could either overestimate or underestimate a given FD (fraction defective) with a 50 : 50 chance. What happens to the existing inspection plans, if an imperfect inspector overestimates a known FD, when it is very low? We deal with this fundamental question, by constructing four mathematical models, under the assumptions that an infinite sequence of items with a known FD is given to an imperfect inspector with nonzero inspection errors, which can be constant and/or randomly distributed with a uniform distribution. We derive four analytical formulas for computing the probability of overestimation (POE) and prove that an imperfect inspector overestimates a given FD with more than 50%, if the FD is less than a value termed as a critical FD. Our mathematical proof indicates that the POE approaches one when FD approaches zero under our assumptions. Hence, if a given FD is very low, commercial inspection plans should be revised with the POE concept in the near future, for the fairness of commercial trades.
1. Introduction
Our research started from a BLU (backlight unit) company in Korea. Inspections of a BLU, which is one of the major components attached to the back of a thin film transistor liquid crystal display unit, can be divided into several functional inspections and external appearance inspections. In the Korean BLU industry, inspection operation is usually done, like other production lines, at the end of a line, due to related costs, and the reworkability of a BLU. The inspection decision is made only by a single attribute: conforming (or C) or nonconforming (or NC). In reality, items will be misclassified, even if only a few. A C (or NC) item may be classified as NC (or C), and this probability is typically termed as type I (or type II) error. Correctly or falsely accepted items at the end of each BLU line are packaged into lots and transported to a clean storage area, where an acceptance sampling plan by attribute is performed, by a source inspector affiliated to a buyer. Even if only one NC item is found in a lot, the lot is rejected by the source inspector. The rejected items, even if there are very few NC items, must be 100% reinspected later, in another clean room (Yang and Cho [1]).
As the fraction defective (FD) of BLU items waiting for source inspections has been gradually lowered to either the thousands or hundreds PPM level, most of the quality control managers have continually raised the possibility that the FD judged by a source inspector has “always” been overestimated, because his inspection could not be perfect, but also his inspection severity would be advantageous to his company. In addition, various questions about unknown dependencies between FD and inspection errors have been raised. In fact, the FD judged by an inspector should be underestimated or overestimated with a 50 : 50 chance, regardless of a given FD and/or inspections errors. In order to verify their presumptions, after having formed lots with a very low FD comprised of hundreds of BLU items randomly mixed up with C and NC items, they carried out significant field experiments controlled by an expert, where the lots were tested by an inspector, with type I error 0.86% and type II error 4.50%, as estimated. They concluded that the possibility of overestimation by the source inspector seemed to be at least significantly larger than 50% but they could not prove it mathematically and that there might exist some relationships between overestimation, very low FD, and inspection errors. From the above facts, many basic questions may be raised, but above all, here we are interested only in the following fundamental and theoretical questions:
“Does an imperfect inspector overestimate a given fraction defective, when it is very low?”
In other words, “Could the FDA (fraction defective after inspection) always be larger than the FDB (fraction defective before inspection), if the FDB is very or extremely low?” In order to answer the above question, we need to find a way to compute the probability of overestimation (POE), when an FD and inspection errors are given.
As far as we know, there have been no papers directly related to our problem. However, some studies dealing with nonzero inspection errors have appeared in the literature, since the 1970s. Collins et al. [2] considered the effects of inspection error on the probability of a lot of acceptance, average outgoing quality, and average total inspection, under both replacement and nonreplacement assumptions, and suggested that an acceptance sampling plan may be designed, based on inspection error. Dorris and Foote [3] surveyed the state of knowledge on inspection and measurement errors and suggested future lines of investigation about inspection errors. Raz and Thomas [4] presented a branchandbound method for determining an optimum sequencing inspection plan, for a group of inspectors operating at different skill and cost levels. Tang [5] provided a rule for determining the optimal sequence of multiple quality characteristics, for minimizing the cost of inspection within each inspection stage. Lee [6] developed the stop rule, for seeking the optimal number of inspection stages. Sylla and Drury [7] dealt with the apparent fraction nonconforming , where is an FD, is the probability of rejecting a C item, and is the probability of accepting an NC item. They found , the sample size and , the cutoff value for single sampling by attributes, considering fraction defective, types I and II errors, and errorrelated payoffs, and proposed the concept of liability which is an inspector’s ability to respond to information, like payoffs, fraction defectives, and discriminability between noise and signal distributions. Burk et al. [8] derived a relation and showed a table about the relation. They noted that as the type I error approaches , approaches zero and that for very good process, is actually a type I error. They suggested a procedure for estimating the types I and II errors and gave an industrial example. These were shown as major variables, in the last contribution related to our current research.
Several models that are partially related to our problem have appeared in the literature. In order to attain a prespecified quality rate at the end of an assembly line, Yang [9] suggested a Kstage inspectionrework (KIR) system, which was composed of a series of K stages, each of which included an inspection process and a rework process. He suspected the effectiveness of the KIR system and proved mathematically that FDA is always larger than FDB if FDB is less than a value that depends on a FD of rework and inspection errors. Based on his assumptions, he suggested a necessary condition for inspection effectiveness that the sum of two errors must be less than one. However, the necessary condition is so rough that it cannot be used practically.
In this paper, we deal with the above fundamental question. In Section 2, we describe our problem in detail. Assuming that an imperfect inspector classifies an infinite sequence of items with a known FD and that each inspection error of type I or type II is either a constant or a uniform random variable on an interval, we provide four mathematical models: Model I (C, C) with both type I error and type II error being constant, Model II (R, C) with type I error and type II error being random and constant, respectively, Model III (C, R) with type I error and type II error being constant and random, respectively, and Model IV (R, R) with type I error and type II error being random and random, respectively. In Sections 3 through 6, for each model, we derive formulas for computing the probability of overestimation (POE) and a critical FD satisfying POE = 50%. In Section 7, in order to extract some relation between the results of the previous sections, we make a reasonable assumption and prove a theorem that answers our question.
2. Problem Statement
Suppose that an imperfect inspector with nonzero inspection errors classifies onebyone an infinite sequence of items with a known fraction defective , but which is unknown to the inspector. Then the sequence can be considered as an infinite Bernoulli process such that, for each , the value of is either zero, representing a C item, or one, representing an NC item; for all values of , the probability that , , is the same number . Let be the probability (the type I error) that the th C item is misclassified as NC and falsely rejected by the inspector; let be the probability (the type II error) that the th NC item is misclassified as C and falsely accepted by the inspector. Let be zero, if the th item is judged as a C item by the inspector, and one, otherwise. That is, and . Then, and can be obtained as
Since the expected number of rejected items after the th inspections is , the FD of an infinite number of items judged by the inspector, denoted by , becomes and can be reduced to
The above equation implies that the value of becomes if all and are zeros since the value of the limit term becomes zero. We assume that two types of inspection errors are nonzero and less than or equal to one unless specially mentioned. That is, for all . Hence, the value of the limit terms can be positive, zero, or negative, corresponding to overestimation, correct estimation, or underestimation, respectively. Either overestimation or underestimation does not raise any problems by themselves, as long as their probabilities are exactly the same. In fact, we are likely to believe that all inspectors are expected to either overestimate or underestimate a given FD with a 50 : 50 chance. Otherwise, either buyer or supplier must face economic loss due to an unfair inspection game. However, unfortunately, it turns out in this paper that the 50 : 50 chance is not always true and that it depends upon and . Let POE be the probability of overestimation by the inspector, which can be reduced to
If is an input variable, can be expressed as . If there exists a unique FD such that , then is termed as “the critical fraction defective” or CFD. This definition implies that ; that is, is an conditional unbiased estimator of CFD when is a random variable. In the case that such a CFD does not exist, will be called as CFD only if an inspector estimate correctly, that is, , and is defined to be 0.5.
Since inspection error as well as can be assumed to be either a constant or a random variable, we need four kinds of analyses of POE and CFD, as shown in Table 1. Note that the subscript “” (or “”) under the right sides of POE and CFD in the table represents that the type I or type II error is assumed to be constant (or random). In order to obtain some fundamental properties, we assume that each random variable follows a uniform distribution with an interval on (0, an upper value]. That is, , and where is an indicator function with one for , and zero otherwise. This uniformity assumption with zero/an upper value may be justified, since it has been hoped that inspection errors would become smaller and smaller, and there has been almost no information on the distribution of inspection error, up to now. It is well known that uniform distribution gives maximum uncertainty. Our problem can be summarized as follows: derive both and CFD for each model and answer the fundamental question:
“Does an imperfect inspector overestimate a given FD when it is very low?”

Throughout this paper, the first and second derivatives of a function will be expressed as and , respectively, and the expectation of a random variable will be denoted by .
3. Analysis of Model I (C, C)
Suppose that and , for all , and both and are realvalued constants. Then, since and , the process becomes an infinite Bernoulli process with being the same number . The following proposition indicates that depends on and where and that CFD_{cc} depends only on , not . Let and assume that is a rational number.
Proposition 1. Under the assumption that and , for all , and both and are realvalued constants with , (1)(2),(3)an inspector with always overestimates for , estimates correctly for with , and always underestimates for .
Proof. Equations (2) and (3) can be derived, respectively, as
For , since the inequality is “always” true, (i.e., an inspector with “always” overestimates ), can be defined to be one. For , since an inspector with “always” underestimates , can be defined to be zero. In the case of , since , that is, an inspector estimate correctly, by our definition in Section 2, it follows that and . However, is a rational number (note that can be expressed as , which can be expressed as the fraction of two integers, with ), while can be an irrational number depending on the values of and . Hence, cannot be equal to if (or ) is an irrational number. Since we assume that (or ) is a rational number, we have, if and only if and Proposition 1 holds true.
From Proposition 1, can be drawn as shown in Figure 1(a). Suppose that both and are input variables. Then, since an inspector with overestimates if and only if , every point satisfying gives overestimation. For , every point satisfying gives overestimation. For , every point for gives overestimation. Let be the overestimation region as shown in Figure 1(b). Then, an inspector with overestimates if , estimates correctly if , and underestimates otherwise. Note that the point () cannot be included in since due to the assumption that .
(a) The graph of
(b) The region of overestimation in plane
(c) The region of overestimation in plane when
Suppose that both and are input variables and that is given as a constant. Then, since an inspector with overestimates if and only if , every point satisfying gives overestimation. For , every point satisfying gives overestimation where . For , every point satisfying gives overestimation. Let be the overestimation region bounded by , , and as shown in Figure 1(c) where . That is, an inspector with overestimates if , estimates correctly if , and underestimates otherwise.
If , then from (2) and the inspector estimates correctly if and only if . Hence, has two values either 0.5 (when ) or one (when ) depending on . If , then from (2), and the inspector estimates correctly if and only if . Hence, has two values either 0.5 (when ) or zero (when ) depending on . These special cases of and will be discussed only if necessary.
Since CFD_{cc} depends only on , CFD_{cc} remains the same even if both and are multiplied by the same number. For example, the CFD_{cc} of is 0.25, the same as that of . Typical CFD_{cc} values for are summarized in Table 2. As shown in the table, the value of CFD_{cc} is 50% on the diagonal, more than 50% above the diagonal, and less than 50% below the diagonal. It can be proved that is zero if and only if and is one if and only if except the case that , and for a given , CFD_{cc} increases strictly and forms a concave shape as increases, while for a given , CFD_{cc} decreases strictly and forms a convex shape as increases.

Example 2. Suppose that % and in order to estimate the FD correctly, we must select one inspector among three inspectors with for as shown in Table 3. In order to reflect actual inspection errors of a backlight unit manufacturer, the value of is set to be smaller than that of . At first glance, the inspector with either or is more likely to be selected than the inspector with since and . However, it is a wrong decision since inspector 1 underestimates as 4.995% and inspector 2 overestimates as 5.065%, while the right decision is to select the worst inspector 3 since only inspector 3 can estimate correctly. This decision seems to be very strange at first time but gives a new concept to quality control managers, who could utilize intentionally this perspective in special situations.

4. Analysis of Model II (R, C)
Suppose that and for all where is a constant with and is a random variable distributed with the probability density function for and . Then, since and , the process becomes an infinite “randomBernoulli” process with being not a constant but the random variable . From (2), can be obtained as . From (3), we have
For , (6) can be further reduced to where . For , becomes zero since is always smaller than . If and , then since , every inspector with always estimates correctly regardless of any distribution of , and by our definition of POE, . Hence, has two values either 0.5 (when ) or zero (when ) depending on . Suppose that . Then, we have the following proposition indicating that can be expressed as a function of two input variables , and where .
Proposition 3. Under the assumptions that and for all , where is a constant with and is a random variable distributed with for and ,(1) where ,(2) is a strictly decreasing concave function of for ,(3),(4)the inspector with overestimates with for , estimates correctly with for , and underestimates with for .
Proof. From (7), since for , can be derived as above. Since and , is a strictly decreasing concave function of for . Since for , solving for gives . Hence, holds true.
Note that and CFD_{rc} for any density function can be generally derived as shown in Proposition A.1 in Appendix. Also note that where .
A representative graph of is shown in Figure 2(a). Suppose that both and are input variables. Then, solving for gives and Proposition 3 implies that every point in the shaded region as shown in Figure 2(b) gives overestimation with , where can be represented by . That is, an inspector with overestimates if , estimates with if , and underestimates otherwise. Note that if , then every point for on the line gives overestimation with since is always greater than .
(a) The graph of
(b) The region of overestimation in plane
(c) The region of overestimation in plane given when
Suppose that both and are input variables and is given as a constant. Then, solving for gives and Proposition 3 implies that every point in the shaded region as shown in Figure 2(c) gives overestimation with where . That is, an inspector with overestimates with if , estimates with if , and underestimates with otherwise.
Example 4. For and three inspectors given in Table 4, and CFD_{rc} for each inspector can be computed and summarized in the table. If we would like to select the inspector satisfying , then inspector 3 will be selected again.

5. Analysis of Model III (C, R)
Suppose that and for all where is a constant with and is a random variable distributed with the probability density function for and . Then, since , , the process becomes an infinite randomBernoulli process with being the random variable . From (2), can be obtained as . From (3), we have
For , (9) can be further reduced to where .
For , becomes one since is always greater than from (9). If , then since is always equal to , every inspector with always estimates correctly regardless of any distribution of , and, by our definition of POE, is not one but 50% in this case. This result implies that can be either one or 50% depending on the value of . Since we assume that , we have . Suppose that . Then, we have the following proposition indicating that can be expressed as a function of two input variables and where .
Proposition 5. Under the assumption that and for all where is a constant with and is a random variable distributed with for and , (1) where ,(2) is a strictly decreasing convex function of for ,(3) ,(4) the inspector with overestimates with for , estimates with for , and underestimates with for .
Proof. From (10), since for , can be derived as above. Since and for , is a strictly decreasing convex function of for . Since for , solving for gives . Hence, holds true.
Note that generally and CFD_{cr} for any density function can be derived as shown in Proposition A.2 in Appendix. Also note that where .
A representative graph of is shown in Figure 3(a). Suppose that both and are input variables. Then, solving for gives and Proposition 5 implies that every point in the shaded region as shown in Figure 3(b) gives overestimation with where . That is, an inspector with overestimates if , estimates with if , and underestimates with otherwise. Note that if , then every point for gives underestimation since .
(a) The graph of
(b) The region of overestimation in plane
(c) The region of overestimation in plane given when
Suppose that both and are input variables and is given as a constant. Then, solving for gives and Proposition 5 implies that every point in the shaded region as shown in Figure 1(c) which gives overestimation with where . That is, an inspector with overestimates with if , estimates with if , and underestimates with otherwise.
Example 6. For and three inspectors given in Table 5, and CFD_{cr} for each inspector can be computed and summarized in the table. If we would like to select the inspector satisfying , then inspector 3 will be selected again.

6. Analysis of Model IV (R, R)
Suppose that and for all where and are random variables distributed with the probability density functions and , respectively, for , and . Then, since and , the process becomes an infinite randomBernoulli process with being the random variable . From (2), can be obtained as . From (3), POE can be expressed as
If , then since ; that is, even perfect inspector always underestimates regardless of any distribution of . On the other hand, if , then since ; that is, even perfect inspector always overestimates regardless of any distribution of . For , (12) can be reduced to where .
Suppose that and . Then, we have the following proposition indicating that can be expressed as a function of two input variables and where .
Proposition 7. Under the assumption that and for all where and are random variables distributed with and , respectively, for , , and , (1) where ,(2) is a strictly decreasing concave function of with , and is a strictly decreasing convex function of with ,(3),(4)the inspector with overestimates with for , estimates with for , and underestimates with for .
Proof. It is proved from (12) that and . For , since the shape of depends upon and , for as shown in Figure 4(a), (13) can be reduced to
and for as shown in Figure 4(b), (13) can be reduced to
Equation (15) holds when (equivalently, ) since replacing in (15) with zero gives and (16) holds when (equivalently, ) since replacing in (16) with one gives . For , since and , is a strictly decreasing concave function of . For , since and , is a strictly decreasing convex function of . Note that is differentiable at . Hence, is a strictly decreasing function of with and . Since , CFD_{rr} is . Since 0.5 for and 0.5 for , holds true.
(a) For (or )
(b) For (or )
Since and are uniform random variables, respectively, (15) and (16) can be derived by a different method as follows. Equation (13) can be expressed as
Since the value of the integral is equivalent to the size of the area as shown in Figure 4, can be interpreted as the ratio of the size of to the size of the rectangle . The shape of varies depending on the straight line , which cuts the rectangle into two parts, that is, an upper part and a lower part as shown in Figure 4. The lower part of the rectangle under the line corresponds to , a rightangled triangle when , and a trapezoid when . Since the size of the trapezoid and the triangle can be derived as and , respectively, can be easily obtained.
It seems to be mathematically hard to derive and CFD_{rr} generally for any density functions. From Propositions A.1 and A.2 in Appendix, our conjecture is that . Note that .
From Proposition 7, a graph of can be drawn as shown in Figure 5(a). Suppose that and are input variables. Then, since 0.5, solving for gives and Proposition 7 implies that every point in the shaded region as shown in Figure 5(b) gives overestimation with where . That is, an inspector with overestimates with if , estimates with if , and underestimates with otherwise.
(a) The graph of
(b) The region of overestimation in plane
(c) The region of overestimation in plane for given when
Suppose that is a given constant and both and are input variables. Then, similarly solving for gives and Proposition 7 implies that every point in the shaded region as shown in Figure 5(c) gives overestimation where . That is, an inspector with overestimates with if , estimates with if , and underestimates with otherwise.
Example 8. For and three inspectors given in Table 6, and CFD_{rr} for each inspector can be computed and summarized in the table. If we would like to select the inspector satisfying %, then inspector 3 will be selected again as before.

7. Summary
In order to find relations between four propositions, the constants and can be assumed to be and , respectively. This assumption may be justified since a constant can be interpreted as a representative value. Since and are uniform random variables on and , respectively, we have, and , respectively. Thus, the following theorem holds true.
Theorem 9. Assuming an infinite sequence of items with a known FD and nonzero inspection errors with and , (1)an imperfect inspector with nonzero inspection errors has his/her own POE curve and CFD,(2)POE is a function of two variables and , denoted by ,(3)POE is a decreasing function of with and ,(4)POE is a decreasing function of with and ,(5)there always exists a unique , which depends only on inspection errors and not ,(6)the inspector overestimates with for , estimates with for , and underestimates with for .
Proof. If and , then we have and , where . Let . Then, from the results of four propositions, the theorem except holds true and those results are summarized in Table 7. By using the similar method used for proofs of propositions, can be proved.
