Electron. J. Math. Phys. Sci., 2002, Sem. 1, 15-21

Electronic Journal of Mathematical

and Physical Sciences

 

 

ISSN 1538-3318
© 2002 by EJMAPS

www.ejmaps.org

 

A Technical Note on 2. The Effects of Inspection Errors on the Average Outgoing Quality in an Industrial Process

Abraham F. Jalbout ^1*$, Hadi Y. Alkahby ², Fouad N. Jalbout ³ , Abdulla Darwish ³

¹Department of Chemistry, University of New Orleans, New Orleans, LA 70148-2820 USA, E-mail: Ajalbout@ejmaps.org

²Department of Mathematics, Dillard University, New Orleans, LA 70112 USA

³Department of Physics and Engineering, Dillard University, New Orleans, LA 70112 USA

 

*Author to whom correspondence should be addressed.

^$Based on the A.F. Jalbout Lectures at the Dillard University seminar series

Received: 14 December 2001 / Accepted: 5 January 2002/Pubished: 15 January 2002

 

Abstract: In recent years researchers in various quality control procedures consider the possibility of inspection errors as an important issue. The presence of these errors leads to changes in the so-called operational characteristic (O.C.) control curve, and as a result the average outgoing quality of an industrial process. We present a new mathematical model that can be applied to calculate such quantities as the expected number of defective items replaced in an accepted lot, and other functions of this process.

Keywords: Inspection Errors, industrial process, Bayesian methods, statistics

 

 

© 2002 by EJMAPS (http://www.ejmaps.org). Reproduction for noncommercial purposes permitted

1.      Introduction

 

In attribute sampling plans the errors are generally of two kinds:

 

Type I error: a good item is classified as bad, with a probability e₁

Type II error: a bad item is classified as good, with a probability e₂

Collins and Case [1] derived an expression for the probability of acceptance under inspection errors. An expression was later derived for the marginal distribution of the observed defectives. (Hald [2-5] Case, Bennett and Schmidt [6-7] developed formulas for calculating the average outgoing quality (AOQ) when attribute inspection is subject to Type I and Type II inspection errors. Nine different rectification inspection policies are considered. These policies were first introduced by Wortham and Mogg [8].  Beainy and Case [9] later generalized these models, and they developed nine different sample/rest-of-lot disposition policies for single and double sampling along explicitly developed AOQ models.  In this work both the attribute variable plans are considered [10] and a Bayesian technique is developed to estimate different parameters. Appendix A lists all of the notations used in this work. Although more the more recent work of Johnson, Kotz and Wu [11] describes some more modern industrial approaches to inspection errors with attributes in quality control, they still forget to include the Bayesian methods, a novel approach which is presented in this work.

2.                  Mathematical Development

 

Hald [2-5] has derived the following form of the marginal distribution of :

 

                                               (2.1)

 

under the assumption that the number of defectivesin a lot sizeis binomially

 

 

distributed, with a p.d.f:

                          (2.2)

whereis the process fraction defective.

The second assumption of equation (2.1) is that the number of defectivesin a sample size given is hypergeometric:

                                                ,                                                 (2.3)

thus this proves that the Hald’s [2-5] derivations of the binomial distribution is reproduced by hypergeometric sampling.  Thus for the Bayesian operating characteristic (BOC) curve the probability of lot acceptance is derived from the above equations as:

                         (2.4)

where is the acceptance number. For the inspection error analysis the observed defectives from a sample is replaced by observed number of defectives. The probability of lot acceptance given in (2.4), will be reduced:

                                                 (2.5)

where:

                        (2.6)

Equation (2.4) gives the probability of lot acceptance for perfect inspection. The probability of lot acceptance when inspection errors are presented as in equation (2.5), and using equation (2.6) we can derive:

                                       (2.7)

 

We can now deduce an expression for the average outgoing quality (AOQ).

            The AOQ can be defined as:

   (2.8)

An expression fro AOQ can be derived by introducing the following terms: , the number of defectives in the uninspected portion of an accepted , the number of defective of defective items classified as being good in the screened portion of the rejected lot. is the number of defective items classified as good in the sample, is the number of defective items introduced through replacement into the lot. For an accepted lot, the expected number of defective items replaced in the lot is:

.                                                  (2.9)

The probability that an item is classified as being good is then:

                                                            .                                 (2.10)

A set of  items are selected at random, tested and classified as good or bad. A total of items were needed to replace the defective items in the accepted lot. This procedure of sampling defines a negative binomial process. The expected number of items tested to obtain items, which are good, is then:

.                                                       (2.11)

The expected number of defective items replaced in an accepted lot is then:

                                          (2.12)

The expected number of defective items replaced in a rejected lot, which is screened, is:

.                                  (2.13)

The expected number of items to be replaced is:

                            (2.14)

The expression AOQ is then:

                (2.15)

Expression (14) can be reduced to the form:

                 (2.16)

Similarly the AOQ expression for sampling with no replacement can be derived as:

                 (2.17)

This is true so no defectives are introduced through the replacement process.

3.      Conclusions

 

In this work expressions for the average outgoing quality were derived for both the model involving replacement of defective items in the lot and when the items are not replaced. The potential application of this work lies in the ability of industrial researchers to calculate both of these quantities and decide the loss of such events, which are so very common in real life inspections. This simple model, is novel, and will be fruitful for any industrial process involving a constant inspection process. (see Jalbout, Alkahby [12])

REFERENCES

 

1. R.D. Collins, K.E. Case, AIIE Transactions, 1981, 8(3), pp. 375-378

 

2. A. Hald, Statistical Theory of Sampling Inspection by Attributes Part 1, 1976, University of Copenhagen, Institute of Mathematical Statistics, Denmark

 

3. A. Hald, Statistical Tables for Sampling Inspection by Attributes, 1976, University of Copenhagen, Institute of Mathematical Statistics, Denmark

 

4. A. Hald, Statistical Theory of Sampling Inspection by Attributes Part 2, 1978, University of Copenhagen, Institute of Mathematical Statistics, Denmark

 

5. J.L. Herbert, Communications in Statistics. Theory and Methods, 1994, 23, pp. 1173-1180.

 

6. K.E. Case, J.W. Schmidt, G.K. Bennett, AIIE Transactions, 1977, 7(4), pp. 363-369.

 

7. R.D. Collins, K.E. Case, G.K. Bennett, International Journal of Production Research , 1978, 10(1), pp. 2-9

 

8. A.W. Wortham, J. Mogg,  Journal of Quality Technology, 1970, 2(1), pp. 30-31

 

9. I. Beainy,  K.E. Case, Journal of Quality Technology, 1981, 13(1), pp. 30-40

 

10. J.W. Schmidt, K.E. Bennett, Journal of Quality Technology ,1980, 12(1), pp. 10-17.

 

11. N.L. Johnson, S. Kotz, X. Wi, Inspection Errors for attributes in    quality control, 1991, Chapman and Hall Ltd (London; New York)

 

12. F.N. Jalbout, H.Y. Alkahby, Elect. J. Diff. Eqns. , 1999, Conf. 02, pp. 105-11

 

Appendix A: Notations Used in text