Single Sampling Plans for Variables Indexed by AQL and AOQL with Measurement Error

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1 Journal of Modern Alied Statistical Methods Volume 11 Issue 2 Article Single Samling Plans for Variables Indexed by AQL and AOQL with Measurement Error R. Sankle Vikram University, India, Ujjain (M. P.) J.R. Singh Vikram University, India, Ujjain (M. P.) Follow this and additional works at: htt://digitalcommons.wayne.edu/jmasm Part of the Alied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Sankle, R. and Singh, J.R. (2012) "Single Samling Plans for Variables Indexed by AQL and AOQL with Measurement Error," Journal of Modern Alied Statistical Methods: Vol. 11: Iss. Article 12. Available at: htt://digitalcommons.wayne.edu/jmasm/vol11/iss2/12 This Regular Article is brought to you for free and oen access by the Oen Access Journals at DigitalCommons@WayneState. It has been acceted for inclusion in Journal of Modern Alied Statistical Methods by an authorized administrator of DigitalCommons@WayneState.

2 Journal of Modern Alied Statistical Methods Coyright 2012 JMASM, Inc. November 201 Vol. 11, No /12/$95.00 Single Samling Plans for Variables Indexed by AQL and AOQL with Measurement Error R. Sankle J. R. Singh Vikram University, India, Ujjain (M. P.) Single samling lans are investigated for variables indexed by accetable quality level (AQL) and average outgoing quality limit (AOQL) under measurement error. Procedures and tables are rovided for selection of single samling lans for variables for given AQL and AOQL when rejected lots are 100% insected for relacement of a nonconforming unit. For a articular samling lan in oeration for an observed measurement, a method for determining true oerating characteristic (OC) functions and average outgoing quality (AOQ) is described for various error sizes. Key words: Measurement error, AQL, AOQL. Introduction One difficulty with roduction rocesses is achieving a desired quality level of manufactured roduct while maintaining economy in roduction cost. Statistical techniques have been successfully alied to address this roblem; to emloy statistical techniques, insections are conducted on intermediate and finished roducts. In every insection system, there is always a ossibility for error in acceting a non-conforming unit and rejecting a conforming unit. These errors, which are mainly due to chance, are termed insection errors and they can be estimated. This is imortant because corrective action must be taken if the number of insection errors is large. Jackson (1957) studied the effect of insection errors on waste and on the quality of outgoing roduct assuming 100% insection. Considering that error is a substantial art of observed variation, Diviney and David (1963) R. Sankle is on the faculty in the School of Studies in Statistics. her at: rajshreesankle@gmail.com. J. R. Singh is a Professor in the School of Studies in Statistics. him at: jrsinghstat@gmail.com. investigated the relationshi between measurement error and roduct accetance. The requirement that the measurement of an individual item does not exceed some secified limit is sometimes more imortant than the requirement that the mean and variability for the items be at or near some re-determined value. An accetance samling lan in which a secified number of units is samled from each lot, with the lot being acceted if less than a fixed number of non-conformance roducts are found in the samle, is one of the traditional statistical tools used for quality control. Lots that are not acceted can either be discarded or rectified. Rectification, that is, relacing or discarding all non-conforming units after 100% insection of rejected lots, is frequently used when manufacturing costs are high. Several authors have roosed redictors for estimating the number or rate of non-conformances in lots subjected to accetance samling (Hahn, 1986; Zaslavsky, 1988; Brush, et al. 1990; Martz & Zimmer, 1990). Greeberg and Stokes (1992) used the information obtained in rectification to devise a more efficient redictor than those reviously roosed. Greenberg and Stockes (1995) also considered an alication of quality control in which the test rocedure is imerfect. Two roblems may exist in accetance samling. Devices that are classified as non-conforming may be conforming (false ositive) and devices 396

3 R. SANKLE & J. R. SINGH that are classified as conforming may be nonconforming (false negative). Johnson, et al. (1991) rovided exressions and tables of the average outgoing quality for many tyes of samling lans when the false ositive and false negative rates are known. Lindsay (1985) described methods for estimating the robability of false ositives and false negatives and the rates and numbers of non-conformances when a samle is reeatedly insected. However, these authors do not consider lans with rectification. A lot-by-lot rectification insection scheme for a series of lots calls for 100% insection of rejected lots under the alication of a samling lan. If it is referable to use a single samling lan for variables under a rectification insection scheme, the index for the selection of the samling lan will be the average outgoing quality limit (AOQL), which is the worst average quality the consumer will receive in the long run, regardless of the incoming quality. Rejected lots are often a nuisance to the roducers because they result in extra work and extra cost. If too many lots are rejected the reutation of the roducer or sulier may be damaged. From the roducer s oint of view, it is referable to fix an accetable quality level (AQL) by designing a samling lan such that, if the incoming roduct quality is maintained at AQL most of the lots, for examle 95%, will be acceted during the samling insection stage. Thus, designing samling insection lans indexed by AQL and AOQL satisfies both the roducer and consumer whenever rectifying insection is necessary. The redictors are generally assumed to be measured without error, but this is often not the case. To identify the arameter in the model, the following assumtions are made concerning measurement errors. First, it is assumed that the true values and the measurement errors are uncorrelated and that the mean of the measurement errors is zero. Second, the measurement errors are assumed to be normally distributed with zero mean and a constant known variance. Third, the true values are assumed to be normally distributed with a mean estimated by the mean of the observed values and a variance estimated using the reliability of the observed values. The reliability of a variable measured with error is the ratio of the variance of the true values to the variance of the observed values; the closer this ratio is to 1 the more reliable the measurement. The reliability can be rovided by reliability coefficients (Hand, 2004). Alternatively, a range of lausible reliabilities can be exlored to carry out a sensitivity analysis of the results to estimate the severity of the unobserved measurement error. This study examines single samling lans for variables indexed by AQL and AOQL under measurement error. Procedures and tables are rovided for selecting single samling lans for variables for given AQL and AOQL when rejected lots are 100% insected for relacement of nonconforming units. For a articular samling lan in oeration for observed measurement, a method of determining the OC function and AOQ curves is described for various errors sizes. Model descrition for Variable Single Samling Plan indexed by AQL and AOQL under Measurement Errors: Consider the distribution of the true quality characteristics x to be normal with mean μ and known standard deviation. The density function is: f 1 ( x) = Φ x μ, (2.1) where Φ(x) is the standardized normal robability density function given by ( ) Φ x x = e 2. (2.2) 2π The mean and standard deviation of the observed measurement (X = x + e) can be written as E(X) = E(x) + E(e) = μ where μ is the mean of x and e is the random error at measurement and is indeendent of x, and V(X) = V(x) + V(e) = e = X. 397

4 SAMPLING PLANS FOR AQL AND AOQL VARIABLES WITH MEASUREMENT ERROR The correlation coefficient ρ between the true and observed measurement is given by {( )( )} E x μ X μ ρ = X {( ) 2 + ( ) } E x μ x μ e = Noting that x and e are indeendent, E(e) = 0 and E(x) = μ, it can be shown that X. z. The accetance criterion for the single samling lan is: For the uer secification limit, accet the lot if, where ~ N( 0,1) x+ k U, (2.6) and, for the lower secification limit, accet the lot if P x k L. (2.7) The fraction nonconforming in a given lot is P ρ = = X 2. X (2.3) with ( K ), Φ = (2.8) K U μ = (2.9) The relation between the size of measurement error r and correlation coefficient ρ is: where r ρ = (2.4) 2 1+ r r = In referencing a single samling variable lan when is known, the following symbols are used: L: Lower secification limit; U: Uer secification limit; N: Samle size; k: Accetance arameter; and x : Samle mean y Φ ( y) = ex z dz, 2π 2 (2.5) e. where K is the ercent oint of the standard normal distribution. If is the roortion defective in the lot, then U = μ + K (2.10) and its robability of accetance under measurement error will be with a ( ) ( ), P =Φ w (2.11) n w= ( K k). (2.12) ρ If the quality of the acceted lot is and all nonconforming units found in the rejected lots are relaced by conforming units in a rectification insection scheme, the AOQ can be aroximated as AOQ =.P a (). (2.13) If m is the roortion non-conforming at which AOQ is maximum, then AOQL = m P a ( m ). (2.14) 398

5 R. SANKLE & J. R. SINGH If AQL ( 1 ) is rescribed, then the corresonding value of K AQL or K 1 will be fixed, and if P a ( 1 ) is fixed at 95%, then, w AQL = w 1 = 1.645; hence, = (K 1 k) n, (2.15) ρ so that for a given AQL, k is determined by the samle size n. Results Table 1 is used for selecting a single samling variables lan under measurement errors for known case. For examle, if the AQL is fixed at 1%, the AOQL is fixed at 1.25% and r = 4, 6 and, Table 1 yields n = 39, 27, 26 and 25, and k = 1.989, 1.990, and 1.998, resectively. It shows that, when the size of the error increases, the value of n increases and, due to measurement errors, the samle sizes are affected but there is a very minor change in accetance arameter k. Further, suose that it is decided to use, an accetance criterion where is known to be 2.0. Let there exist an uer secific limit U = 10.0 and a unit for which the quality characteristic x > U is considered as nonconforming. Table 2 shows the erformance characteristics of a samling lan with n = 25 and k = 2.0 under a rectifying insection scheme. If the true rocess average quality is oerating at AQL (μ = 5.346) and r =, then 95% of the lots submitted will be acceted during the samling insection stage itself and only 5% of the rejected lots will be rectified by relacing non-conforming units with conforming units. In such a case, the AOQ will be only about 1%. If the submitted quality deteriorates to 1.79 % (error free case, that is, r = ), then only about 70% of the lots will be acceted by the samling lan and aroximately one out of every three lots will be rejected and rectified. The AOQ in such a case will not exceed the AOQL of 1.25% fixed, meaning that, irresective of the roduct quality submitted by the roducer, the consumer will receive an average quality not worse than 1.25% under the rectification scheme. The worst case is when r = 2; the AOQ in such a case will just exceed the AOQL of 1.25% fixed for different errors sizes. The values of P a ( m ) for known case are resented in Table and a visual comarison of AOQ curves for different error sizes is shown in Figure 1. As Figure 1 illustrates, the effect of measurement error is serious on the AOQ curves. When using Table 1 to select samling lans, limitations of lans indexed by AOQL under measurement error must be taken into account. Samling with rectification of rejected lots reduces the average ercentage of nonconforming items in the lots; however, it also introduces non-homogeneity in the series of lots finally acceted. That is, any articular lot will have a quality of % or 0% non-conforming deending on whether the lot is acceted or rectified. Thus, the assumtion underlying the AOQL rincile is that the homogeneity in the qualities of individual lots is unimortant and only average quality matters. Table 3 gives P a ( m ) values for the lans given in Table 1. If AQL is 0.25%, AOQL is 1.25% and r = 4, 6 and, then P a ( m ) is 0.354, and 0.338, resectively, and m = AOQL/P a ( m ) for r = 4, 6 and, is 3.53%,3.65%, 3.67%, 3.69%. Thus, if the lot quality is 3.69% then, on average, among every three lots assed on to the consumer two will be free from non-conforming items while the third lot will contain 3.69% non-conforming items: this is about 15 times the AQL secified. In order to avoid such error, the roducer should maintain the rocess quality aroximately at the set AQL because a high rate of rejecting lots at = m will also indirectly ut ressure on the roducer to imrove the submitted quality. References Brush, G. G., Hoadley, B., & Saerstein B. (1990). Estimating outgoing quality using the quality measurement lan. Technometrics, Diviney, T. E., & David, N. A. (1963). A note on the relationshi between measurement error and roduct accetance. Journal of Industrial Engineering, 14,

6 SAMPLING PLANS FOR AQL AND AOQL VARIABLES WITH MEASUREMENT ERROR r 2 4 Table 1: Single Samling Plans for Variables Indexed by AQL and AOQL under Measurement Error AOQL (%) AQL (%) , , 75, , 19, , 10, 17, , 6, 9, 14, 47, , 5, 8, 14, 49, , 5, 7, 1 47, , 6, 1 39, , 6, 10, 25, , 7, 26, , 7, 24, , 21, , 51, , 17, 45, , 9, 15, , 6, 8, 1 34, , 5, 7, 1 34, , 4, 7, 11, , 6, 10, 27, , , , , , , , , , , , ,

7 R. SANKLE & J. R. SINGH r 6 Table 1 (continued): Single Samling Plans for Variables Indexed by AQL and AOQL under Measurement Error AOQL (%) AQL (%) , , 48, , 16, , 9, 14, , 6, 8, , 5, 7, , 4, 6, 11, 30, , 6, 10, 26, , 5, 8, 18, , 7, 18, , 6, 16, , 14, , , 46, , 16, 40, , 9, 14, 30, , 6, 8, 1 31, , 5, 7, 1 31, , 4, 6, 11, 29, , 6, 10, 25, , , , , , , , , , ,

8 SAMPLING PLANS FOR AQL AND AOQL VARIABLES WITH MEASUREMENT ERROR Table 2: Performance Characteristics of the Variables Plan under Measurement Error for AQL = 0.01, AOQL = , U = 10, SD = 2 r μ v' (%) w Pa AOQ

9 R. SANKLE & J. R. SINGH r AOQL (%) Table 3: Pa(m) Values of Known Sigma Plans Under Measurement Error AQL (%)

10 SAMPLING PLANS FOR AQL AND AOQL VARIABLES WITH MEASUREMENT ERROR Table 3 (continued): Pa(m) Values of Known Sigma Plans Under Measurement Error r AOQL (%) AQL (%)

11 R. SANKLE & J. R. SINGH Figure 1: Average Outgoing Quality Curves under Measurement Error Average Outgoing Quality r=2 r=4 r=6 r= Proortion Defective Greenberg, B. S., & Stokes, S. L. (1992). Estimating nonconformance rates after zero defect samling with rectification. Technometrics, 34, Greenberg, B. S., & Stokes, S. L. (1995). Reetitive testing in the resence of insection errors. Technometrics, 37, Hahn, G. J. (1986). Estimating the ercent nonconforming in the acceted roduct after zero defect samling. Journal of Quality Technology, 18, Hand, D. J. (2004). Measurement theory and ractice. London, UK: Arnold. 405

12 SAMPLING PLANS FOR AQL AND AOQL VARIABLES WITH MEASUREMENT ERROR Jackson, E. J. (1957). Effect of insection errors on waste and on quality of outgoing roduct. Industrial Quality Control, VIX(6), 5-8. Johnson, N. L., Kotz, S., & Wu, X. (1991). Insection errors for attributes in quality control. London, UK: Chaman & Hall. Martz, H. F., & Zimmer, W.J. (1990). A nonarametric Bayes emirical Bayes rocedure for estimating the ercent nonconforming in acceted lots. Journal of Quality Technology, Zaslavsky, A. (1988). Estimating nonconformity rates in C-defect samling. Journal of Quality Technology, 20,