Selection of multiple deferred sampling (0, 1) plan with zero – inflated poisson distribution based on quality region

Introduction

Statistical Quality Control is the term used to describe the set of statistical tools used to describe the set of statistical tools used by quality engineers professionals. One of the major areas of Statistical Quality Control is Acceptance sampling. Acceptance sampling is a methodology that deals with procedures, by which decision to accept or reject are based on the inspection of sample. Acceptance sampling is "the middle of the road" approach between no inspection and 100% inspection. There are two major classifications of acceptance plan: by attributes and variables. Therefore, there is a change of rejection a good lot (Producer's risk) and acceptance a bad lot(consumer's risk). Acceptance sampling by attributes each item is tested and classified as conforming and non-conforming. A sampling is taken and contains two many non -conforming items, then the batch is reject, otherwise it is accepted. To ensure a good quality of the final product from the factory, inspection of the raw material and the product material should be done. The acceptance sampling plans are well known to reduce the above risks. In all the type of sampling plans namely single sampling plan, double sampling, multiple sampling plan and chain sampling plan, the basic assumption is the lot or fraction of defective goods is constant. Even though the process is stable, in practical situations, the goods produced from a process will slightly differ in their quality due to random fluctuations. The concept of the Multiple Dependent (or deferred) State (MDS) sampling plan was introduced by Wortham and Baker (1976) [12] . MDS sampling plan belongs to the group of conditional sampling procedures. Subramani. and Govindaraju (1990) [8] have presented tables for the selection of multiple deferred state MDS-1 sampling plan for given acceptable and limiting quality levels using Poisson distribution. Suresh (1993) has proposed procedures to select Multiple Deferred State Sampling plan of type MDS and MDS-1 index through producer and consumer quality levels considering filter and incentive effect. The Zero-Inflated Poisson (ZIP) distribution can be used as the appropriate probability distribution to data which are consisting of many over dispersed zeros. ZIP distribution has been used in a wide range of disciplines such as agriculture, epidemiology, econometrics, public health, process control, medicine and manufacturing, etc. Some of the applications of ZIP distribution can be found in Bohning et al. (1999) [1] , Lambert (1992) [3] , and Yang et al. (2011) [13] . Construction of control charts using ZIP distribution are discussed in Sim and Lim (2008). Some theoretical aspects of ZIP distributions are mentioned in McLachlan and Peel (2000) [7] . Single sampling plans by attributes under the condition of ZIP distribution are determined by Loganathan and Shalini (2013) [4] , Suresh and Sangeetha (2010) [10] have given a procedure for selection of Repetitive Deferred Sampling plan through quality region. Latha and Palanisamy (2018) [5] have discussed the Selection of Bayesian Single Sampling Plan with Zero -Inflated Poisson Distribution Based on Quality Region. Latha and Palanisamy (2019) [6] have discussed the Selection of Multiple Deferred Sampling (0,1) Plan with Zero -Inflated Poisson Distribution The main aim of this paper is to account for determining Acceptance Quality Level (AQL), Limiting Quality Level (LQL), Producer's Risk (α), Consumer's Risk (β) and Indifference Quality Level (IQL). This article also leads to final the Probabilistic Quality Region (PQR) and Indifference Quality Region (IQR) for specified parameters. Procedures and tables given in this article provides applications in industrial ground

Condition for application of MDS-1(C1, C2)

Rembert Vaerst (1980) has developed Multiple Deferred State MDS-1(C 1 , C 2 ) Sampling Plans in which the acceptance or rejection of a lot is based in not only on the results from the current lot but also on sample results of the past or future lots.  Interest centers on an individual quality characteristic that involves destructive or costly tests such that normally only a small number of tests per lot can be justified.  The product to be inspected comprises a series of successive lots or batches (or material or of individual units) produced by an essentially continuing process.  Under normal conditions the lots are expected to be essentially of the same quality.  The product comes from a source in which the consumer has confidence.

The Operating procedure of the MDS (C1, C2)

The operating procedure for this plan is stated as:  From each lot, select a random sample of n units and observe the number of nonconforming units d. The OC function of the MDS (0.1) plan is given by the equation

Rembert Vaerst has presented certain tables giving minimum MDS-1(C 1 ,C 2 ) plans indexed by AQL and LQL and observes the following properties.  MDS-1(C 1 ,C 2 ) Plans are natural extension of ChSP-1

Plans of Dodge (1955) [2] .  MDS-1 (C 1 ,C 2 ) plans allows significant reduction in sample size as compared to single sampling plans.  The use of acceptance number C 2 increases the chances of acceptance in the region of principal interest. Where the product percent defective is very low.  When m= 0, the plan becomes a single sampling plan with sample size n, and acceptance number C 2 .  When m = ∞, the plan becomes a single sampling plan with sample size n, and acceptance number C 1

Operating characteristic function for multiple deferred sampling mds (0, 1) plan with zip model

The OC function is defined as

Where "p" is the probability of fraction defective

The numbers of defects are zero for many samples there may consider Zeroinflated Poisson probability distribution. The probability mass function of the ZIP (𝜑, 𝜆) distribution is given by Lambert (1992) [3] and McLachlam and peel (2000) [7] 𝑃(𝑋 = 𝑥 |φ, λ ) = 𝑓(𝑥) + (1 -𝜑) 𝑃(𝑋 = 𝑥|λ)

The probability mass function can also be expressed as

x np np In this distribution, φ may be called as the mixing proportion. φ and (np) are the parameters of the ZIP distribution. According to McLachlan and Peel (2000) [7] , a Zip distribution is a special kind of mixture distribution.

6. Designing Plans for given AQL, LQL, α and β Tables 1 and2 are used for selecting a Modified chain sampling plan with zero inflated possion for specified AQL and LQL, α, β and i by the following steps.  and m are increased, the average product quality is decreased.

The steps utilized for selecting

Example 2: Suppose the value for p 1 is assumed as 0.02and value for p 2 is assumed as 0.037 then the operating ratio is calculated as 18.5. Now the integer approximately equal to this calculated operating ratio and their corresponding parametric values are observed from the table 2. The actual np 1 =0.1265 and np 2 =2.3337at (α=0.05 and β=0.01).

Designing of quality interval Multiple Deferred State Sampling MDS (0,1) Plan with Zero -Inflated Poisson 8.1 Probabilistic Quality Region (PQR)

PQR is an interval of quality (p 1 < p< p 2 ) in which product is accepted with a minimum probability 0.10 and maximum probability 0.95. The probabilistic quality range, denoted by d 2 = (p 2p 1 ), is derived through the probability of acceptance.

Indifference Quality Region (IQR)

The IQR is an interval of quality (p 1 < p< p 0 ) in which product is accepted with a minimum probability 0.50 and maximum probability 0.95. The indifference quality range, denoted as d 0 = (p 0p 1 ), is derived through the probability of acceptance. In the similar way, the above equations are equated to the average probability of acceptance 0.95 and 0.10; AQL (np 1 ) and IQL (np 2 ) are obtained in Table 3.

Selection of the Sampling plan

Conclusion

Acceptance Sampling is the best technique, which deals with the procedure in which decision to accept or reject lots or process based on their examination of past history or knowledge of samples. The ZIP distribution has been shown to be useful for modeling outcomes of manufacturing process producing numerous defect-free products. The work presented in this paper mainly related to procedure for designing multiple deferred state sampling plan for probability of acceptable, producer's and consumer's risk and limiting, indifference for quality levels and quality regions. The quality level and quality interval sampling plan possesses wider potential applicable in industry ensuring higher standard of quality attainment for product or process. Thus quality interval and quality level are good measure for defining and designing for acceptance sampling plan which are readymade use to industrial shop-floor situations

References

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