Electron. J. Math. Phys. Sci., 2002, Sem. 1, 1-14
Electronic Journal of
Mathematical
and Physical Sciences
ISSN 1538-3318
© 2002 by EJMAPS
www.ejmaps.org
A Technical Note on 1. Classical verses
Bayesian Economic Cost Plans
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.
$ Invited speaker in the Natural Sciences Division for
the Dillard University Seminar Series
Received: 12 December
2001 / Accepted: 2 January 2002/Published: 15 January 2002
Abstract: The
primary focus of this work is to specify the basic parameters in terms of prior
distributions and finding the appropriate conditional posterior distributions
to affect the transformation. The principle parameters include the upper and
lower limit for the process’s quality characteristic X, its mean μ, and
the standard deviation, the materials fraction defective p, the sample size n,
and the lot size N. The Bayesian model’s posterior distributions are derived
using known priors as functions of these parameters.
Keywords: Bayesian, cost model, comparison, lot size,
fraction defective
© 2002 by EJMAPS (http://www.ejmaps.org).
Reproduction for noncommercial purposes permitted
authors tried to identify an optimal solution for various sample sizes. This was done in terms of a minimax criterion for large lot size. The procedure seems to work in the case some prior distributions when either rejection or acceptance of a lot leads to the same expected cost.
1.3 Model Comparisons
Equivalent classical attribute and variable sampling acceptance plans for fraction defective can be used on the same process. This is done by specifying the average outgoing quality (AOQ) and the average total inspection parameters for both types of plans. In addition, the variance for the process with variable sampling must be a known. The underlying sampling distribution governing the variable plan is assumed to be a normal. The binomial is assumed for the attribute plan. In both plans the fraction defective is given in terms of the AOQ and ATI. To make the transition from the classical attribute plan to the variable one, Romig supplies a table to obtain the number in the sample, s, and the acceptance number, c, for this variable plan.
The approach presented above by Duncan and Romig laid the groundwork for a more sophisticated equivalence methodology put forward by Hamaker (1979) Hamaker's approach calls for additional criteria of constructing equivalent OC curves for both plans. Here, the equivalent OC curves possess the same slope at the fraction defective level p, corresponding to the 0.5 probability of acceptance value. Unlike the Duncan and Romig result, however, Hamaker provides approximation techniques to consider an unknown process variance for the variable acceptance plan. In summary, the classical acceptance sampling plans prescribe a procedure to specify the risk of accepting lots of a certain quality.
An outgrowth of the above equivalence methodologies for classical plans is to develop a similar set of criteria and framework for Bayesian type attribute and variable sampling plan. This technical note details the parameter requirements to go from a Bayesian attribute plan to the equivalent variable plan. The implication of the work is being currently developed for practical use. It should be emphasized that the procedures outlined above for the classical plans do not consider the economic impact of producing an item. Bayesian attribute and variable sampling plans fall into two major modeling categories:
(1) Not Economic
(2) Economic
For this note, the approach considered is a Bayesian
economic attribute and variable sampling plan. The transformation procedure of
going from a Bayesian variable plan to an attribute plan requires the
specification distributions for the process characteristic X and its mean value
to be found. In this case, the distribution for the fraction defective parameter,
p, this ω(p), and its corresponding conditional expectations E(p|μ)
and E(p|x) can be derived. However, if
the distribution for p is beta as in the case for attribute sampling, the
distribution for μ and E (p|
) are derived. The distributions, conditional expectations,
and the economic parameters are employed to define the economic profit and cost
and evaluate a set of decision points for a variable sampling plan for fraction
defective. The migration from classical plans to the Bayesian plan is
summarized in table 1. A complete list of all the parameters and functional
definitions used in table 1, are in Appendix A. The example, integrates
the total cost and profit with Bayesian sampling into the transformation
procedure.
1.4 Conclusions
We have thus compared the three models of Bayesian noneconomic and economic models with the classical results. This provides the starting model for further industrial advancements in which Bayesian analysis is to be used for quality control testing.
REFERENCES
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5. Easterling, R.G., 9 th Conference on Reliability and Maintainability, Detroit, July 20-22, 1970, pp. 31-36
6. Fertig, K. W. and N.R. Mann, Journal of Quality Technology, 1983, 2(3), pp. 139-148.
7. Hald, A. Statistical Tables for Sampling Inspection by Attributes, University of Copenhagen, Institute of Mathematical Statistics, Demnark
8. Hamaker, H., Journal of Quality Technology, 1983, 11(3), pp. 139-148
9. Schafer, R.E., Bayes, Naval Research Logistic Quarterly, 1967, 24(1), pp. 81-88
Appendix
A: Statistical and Economic Terms
