Bayesian Repetitive Deferred Sampling Plan Indexed Through Relative Slopes

This paper deals with designing of Bayesian Repetitive Deferred Sampling Plan (BRDS) indexed through incoming and outgoing quality levels with their relative slopes on the OC curve. The Repetitive Deferred Sampling (RDS) Plan has been developed by Shankar and Mohapatra (1991) and this plan is an extension of the Multiple Deferred Sampling Plan MDS (c 1 , c 2 ) , which was proposed by Rambert Vaerst (1981).


INTRODUCTION
A cceptance sampling uses sampling procedure to determine whether to accept or reject a product or process.It has been a common quality control technique that used in industry and particularly in military for contracts and procurement of products.It is usually done as products that leave the factory, or in some cases even within the factory.Most often a producer supplies number of items to consumer and decision to accept or reject the lot is made through determining the number of defective items in a sample from that lot.The lot is accepted, if the number of defectives falls below the acceptance number or otherwise, the lot is rejected.Acceptance sampling by attributes, each item is tested and classified as conforming or non-conforming.A sample is taken and contains too many non-conforming items, then the batch is rejected, otherwise it is accepted.For this method to be effective, batches containing some non-conforming items must be acceptable.If the only acceptable percentage of non-conforming items is zero, this can only be achieved through examing every item and removing the item which are non-conforming.This is known as 100% inspection.Effective acceptance sampling involves effective selection and the application of specific rules for lot inspection.The acceptance-sampling plan applied on a lot-by-lot basis becomes an element in the overall approach to maximize quality at minimum cost.Since different sampling plans may be statistically valid at different times during the process, therefore all sampling plans should be periodically reviewed.
The general problem for process control is one towards maintaining a production process in such a state that the output from the process confirms to design specifications.As the process operates it will be subject to change, which causes the quality of the output to deteriorate.Some amount of deterioration can be tolerated but at some point it becomes less costly to stop and overhaul the process.The problem of establishing control procedures to minimize long-run expected costs has been approached by several researchers through Bayesian decision theory.
Classical analysis is directed towards the use of sample information.In addition to the sample information, two other types of information are typically relevant.The first is the knowledge of the possible consequences of the decision and the second source of non sample information is prior information.Suppose a process a series of lots is supplying product.Due to random fluctuations these lots will be differing quality, even though the process is stable and incontrol.These fluctuations can be separated into within lot variation of individual units and between lot variations.

BAYESIAN ACCEPTANCE SAMPLING
Bayesian Acceptance Sampling approach is associated with utilization of prior process history for the selection of distributions (viz., Gamma Poisson, Beta Binomial) to describe the random fluctuations involved in Acceptance Sampling.Bayesian sampling plans requires the user to specify explicitly the distribution of defectives from lot to lot.The prior distribution is the expected distribution of a lot quality on which the sampling plan is going to operate.The distribution is called prior because it is formulated prior to the taking of samples.The combination of prior knowledge, represented with the prior distribution, and the empirical knowledge based on the sample which leads to the decision on the lot.
A complete statistical model for Bayesian sampling inspection contains three components: 1.The prior distribution (i.e.) the expected distribution of submitted lots according to quality.2. The cost of sampling inspection, acceptance and rejection.3. A class of sampling plans that usually defined by means of a restriction designed to give a protection against accepting lots of poor quality.
Risk-based sampling plans are traditional in nature, drawing upon producer and consumer type of risks as depicted by the OC curve.Economically based sampling plans explicitly consider certain factors as cost of inspection, accepting a non-conforming unit and rejection a conforming unit, in an attempt to design a cost-effective plan.Bayesian plan design procedures take into account the past history of similar lots submitted previously for the inspection purposes.Non-Bayesian plan design methodology is not explicitly based upon the past history.Case and Keats (1982) have examined the relationship between defectives in the sample and defectives in the remaining lot for each of the five prior distributions.They observe that the use of a binomial prior renders sampling useless and inappropriate.These serve to make the designers and users of Bayesian sampling plans more aware of the consequence associated with selection of particular prior distribution.Calvin (1984) has provided procedures and tables for implementing Bayesian Sampling Plans.A set of tables presented by Oliver and Springer(1972)which are based on assumption of Beta prior distribution with specific posterior risk to achieve minimum sample size, which avoids the problem of estimating cost parameters.It is generally true that Bayesian Plan requires a smaller sample size than a conventional sampling plan with the same producer and consumer risk.Scafer (1967) discusses single sampling by attributes using three prior distributions for lot quality.Hald (1965) has given a rather complete tabulation and discussed the properties of a system of single sampling attribute plans obtained by minimizing average costs, under the assumptions that the costs are linear with fraction defective p and that the distribution of the quality is a double binomial distribution.The optimum sampling plan (n, c) depends on six parameters namely N,p r ,p s ,p 1 ,p 2 and w 2 where N is the lot size, p r , p s are normalized cost parameters and p 1 ,p 2 ,w 2 are the parameters of prior distribution.It may be shown, however that the weights combine with the p's is such a way that only five independent parameters are left out.
The Repetitive Deferred Sampling Plan has been developed by Shankar and Mohapatra (1991) and this plan is essentially an extension of the Multiple Deferred Sampling Plan MDS -(c 1 , c 2 ) which was proposed by Rambert Vaerst (1981).In this plan the acceptance or rejection of a lot in deferred state is dependent on the inspection results of the preceding or succeeding lots under Repetitive Group Sampling (RGS) inspection.RGS is a particular case of RDS plan.Vaerst (1981) has modified the operating procedure of the MDS plan of Wortham and Baker (1976) and designed as MDS-1.Wortham and Baker (1976) have developed Multiple Deferred State Sampling (MDS) Plans and also provided tables for construction of plans.Suresh (1993) has proposed procedures to select Multiple Deferred State Plan of type MDS and MDS-1 indexed through producer and consumer quality levels considering filter and incentive effects.Vedaldi (1986) has studied the two principal effects of sampling inspection which are filter and incentive effect for attribute Single Sampling Plan and also proposed a new criterion based on the (AQL, 1−α ) point of the OC curve and an incentive index.Lilly Christina (1995) has given the procedure for the selection of RDS plan with given acceptable quality levels and also compared RDS plan with RGS plan with respect to operating ratio (OR) and ASN curve.Suresh and Pradeepa Veerakumari (2007) have studied the construction and evaluation of performance measures for Bayesian Chain Sampling Plan (BChSP-1).Suresh and Saminathan (2010) have studied the selection of Repetitive Deferred Sampling Plan through acceptable and limiting quality levels.Suresh and Latha (2001) have studied Bayesian Single Sampling Plan through Average Probability of Acceptance involving Gamma-Poisson model.
The operating ratio was first proposed by Peach (1947) for measuring quantitatively the relative discrimination power of sampling plans.Hamaker (1950) has studied the selection of Single Sampling Plan assuming that the quality characteristics follow Poisson model such that the OC curve passes through indifference quality level and the relative slope of OC curve at that quality level.
This paper related to Bayesian Repetitive Deferred Sampling Plan for Average Probability of Acceptance function for consumer's and producer's quality levels and relative slopes.

CONDITIONS FOR APPLICATION OF RDS PLAN
1. Production is steady so that result of past, current and future lots are broadly indicative of a continuing process.
2. Lots are submitted substantially in the order of their production.
3. A fixed sample size, n from each lot is assumed.
4 Inspection is by attributes with quality defined as fraction nonconforming.

OPERATING PROCEDURE FOR RDS PLAN
1. Draw a random sample of size n from the lot and determine the number of defectives (d) found therein.Here c 1 and c 2 are acceptance numbers such that c 1 < c 2. When i=1 this plan reduces to RGS plan.

Accept the lot if
The operating characteristic function P a (p) for Repetitive Deferred Sampling Plan is derived by Shankar and Mohapatra (1991) In particular, the average probability of acceptance for c 1 = 0, c 2 =1 is obtained as follows:
2. Find the value in Table 1(b) under the column for the appropriate α and β, which is closest to the desired ratio.
3. Corresponding to the located value of μ 2 /μ 1 the value of s, i can be obtained.
4. The sample size can be obtained as nμ 1 /μ 1 where nμ 1 can be obtained against the located value μ 2 /μ 1 .

EXTAMPLE
Suppose the value for μ 1 is assumed as 0.004 and value for μ 2 is assumed as 0.065 then the operating ratio is calculated as 16.25.Now the integer approximately equal to this calculated operating ratio and their corresponding parametric values are observed from the table 1(b).The actual nμ 1 = 0.2731 and nμ 2 = 2.9667 at (α = 0.05 and β = 0.10), defects per unit in the submitted lots p can be modeled with Gamma distribution having parameters α and β.Let p has a prior distribution with density function given as Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 2, March 2013

Table 1 (b):
Values of μ 2 /μ 1 tabulated against s and i for given α and β for Bayesian Repetitive Deferred Sampling Plan Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 2, March 2013 s i 48 Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 2, March 2013 Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 2, March 2013

Table 2 (
a) is used to select the parameters for Bayesian Repetitive Deferred Sampling Plan indexed with μ 1 and h 1 .For example, for given μ 1 = 0.01 and h 1 = 0.07 from Table 2(a) under the column headed h 1, locate the value is equal to or just greater than the desired value h 1 .Corresponding to this h 1, the values of parameters associated with the relative slopes are nμ 1 = 0.1457, s =1 and i = 5.From this one can obtain the sample size as n = nμ 1 /μ 1 ≈ 14.57.Thus the parameters are n = 15, s =1 and i = 5 6.2.

Selection of Parameters with Relative Slope h 2 at The Limiting Quality LevelTable 2 (
a) is used to select the parameters for Bayesian Repetitive Deferred Sampling Plan indexed with μ 2 and h 2 .For example, for given μ 2 = 0.2 and h 2 = 1.7 from Table 2(a) under the column headed h 2 , locate the value is equal to or just greater than the desired value h 2 .Corresponding to this h 2, the values of parameters associated with the relative slopes are nμ 2 = 3.5673, s =3 and i = 4. From this one can obtain the sample size as n = nμ 2 /μ 2 ≈ 17.8365.Thus the parameters are n = 18, s = 3 and i = 4.

Selection of Parameters with Relative Slope h 0 at The Inflection PointTable 2 (
a) is used to select the parameters for Bayesian Repetitive Deferred Sampling Plan indexed with μ 0 and h 0 .For example, for given μ 0 = 0.05 and h 0 = 0.86 from Table 2(a) under the column headed h 0, locate the value is equal to or just greater than the desired value h 0 .Corresponding to this h 0, the values of parameters associated with the relative slopes are nμ 0 = 0.8373, s = 5 and i = 5.From this one can obtain the sample size as n = nμ 0 /μ 0 ≈ 16.746 .Thus the parameters are n = 17, s = 5 and i = 5.

Table 2 (
a): Relative slopes for Acceptable, Indifference and Limiting Quality Levels