Designing STDS plan for zero-inflated Poisson distribution through various quality level

Introduction

The Poisson distribution is the standard model for the analysis of count data. However, in many situations this type of observations exhibits a substantially larger proportion of zeros than what is expected for the Poisson model. For instance, this is often the case with count data coming from medical and public health research (see Bohning et al. [1] ). This phenomenon usually arises when the distribution generating the data is a mixture of two populations, the first of which yields Poisson-distributed counts whereas the second one always contributes with a zero. Often, a Poisson or Negative Binomial model cannot accommodate the numbers of zeros in the sample properly. Both models would under predict them. There is said to be an "excess zeros" problem. New models are needed to deal with these type of data. Zero-inflated Poisson models are designed to deal with situations where there is an "excessive" number of individuals with a count of 0. In most count data sets, the conditional variance is greater than the conditional mean, often much greater, a phenomenon known as over-dispersion. If data consist of non-negative, highly skewed sequence counts with a large proportion of zeros, Zeroinflated models are useful for analyzing such data. In the present scenario of industrial field, the technological development results the production processes are well designed in such way that the products are in perfect state, so that the number of zero defects will be more in those cases. However, random fluctuations in the production processes may lead some products to an imperfect state. The appropriate probability distribution to describe such situations is a zero-inflated Poisson (ZIP) distribution. The ZIP distribution can be viewed as a mixture of a distribution which degenerates at zero and a Poisson distribution. ZIP distribution has been used in a wide range of disciplines such as agriculture, epidemiology, econometrics, public health, process control, medicine, manufacturing, etc. Some of the applications of ZIP distribution can be found in Bohning et al. [1] , Lambert [5] , Naya et al. [8] and Ridout et al. [9] . Loganathan and Shalini [6] designed single sampling plans by attributes under the condition of Zero-inflated Poisson distribution.

Construction of control charts using ZIP distribution are discussed inXie et al. [11] . Some theoretical aspects of ZIP distributions are mentioned in McLachlan and Peel [7] . The probability mass function (p.m.f) of the ZIP (ω, λ) distribution is given by Lambert [5] and McLachlan and peel [7] 𝑝(𝐷 = 𝑑|𝜔, 𝜆) = 𝜔𝑓(𝑑) + (1 -𝜔)𝑃(𝐷 = 𝑑|𝜆) Where 𝑓(𝑑) = { 1, 𝑖𝑓 𝑑 = 0 0, 𝑖𝑓 𝑑 ≠ 0

And 𝑃(𝐷 = 𝑑|𝜆) = 𝑒 -𝑛𝑝 𝜆 𝑑 𝑑! , 𝑤ℎ𝑒𝑛 𝑑 = 0,1,2, …

The above probability mass function can also be indicated as 𝑃(𝐷 = 𝑑|𝜔, 𝜆) = 𝑓(𝑥) = { 𝜔 + (1 -𝜔)𝑒 -𝜆 , 𝑤ℎ𝑒𝑛 𝑑 = 0

(1 -𝜔) 𝑒 -𝜆 𝜆 𝑑 𝑑! , 𝑤ℎ𝑒𝑛 𝑑 = 1,2, … , 0 < 𝜔 < 1, 𝜆 > 0

In this distribution, ω may be named as the mixing proportion which is alsoalso is known as the probability of extra zeros. ω and λ are the parameters of the ZIP distribution. According to mc Lachlan and Peel [10] , a ZIP distribution is a special kind of mixture distribution.

The mean of the ZIP (ω, λ) is (1 -𝜔)𝜆 and the variance is 𝜆(1 -𝜔)(1 + 𝜆𝜔) . When sampling plans are construct for product characteristics that concerning costly or destructive (harmful) testing by attributes, it is usual practice to use a single sampling plan with acceptance number such as Ac=0 and Ac=l Dodge [2] . However, the OC curves of single sampling plans with Ac=0 and Ac=l, lead to inconsistent interest between the producer and the consumer. The Ac=0 plan performs favorably to consumer while the Ac=l plan favors to the producer. For example if the AQL is set at 0.01 for a SSP with sample size 10, the producer's risk for Ac=0 plan is 0.095 while for the Ac=l plan it is 0.005. Similarly if the LQL is set at 0.30, one may observe that the consumer's risk for the Ac=0 plan becomes 0.05 and for the Ac=l plan it becomes 0.199. While fixing a plan for the above situation, the consumer will favor Ac=0 plan and the producer will favor the Ac=l plan. Such struggle can be overcome if one is able to design a suitable plan having an OC curve lying between the 0C curves of Ae=0 and Ac=l plans. Special type of double sampling plan (STDSP), invented by Govindaraju [3] , is a lot-by-lot sampling plan by attributes in which requirements are made to utilize only small acceptance numbers such as 0 or 1, and to inspect the submitted lot by drawing a second random sample even if the first random sample comprises zero defective items. A special feature, which led to the development of STDSP, is that the operating characteristics of the plan lies between those of c = 0 and c = 1 single sampling plans, and thus provides better discrimination over the single sampling plans with wide range of operating ratios. (See, Govindaraju [3] ). Kaviyarasuand Deepa [4] designed Special Type Double Sampling Plan using Fuzzy Poisson distribution. Vinothand Kaviyarasu [10] studied on Quick Switching System with Special Type Double Sampling Plan as Reference Plan.

2. The operating procedure of the STDS plan is as follows 1. From a lot, select a random sample of size 𝑛 1 units and count the number of defectives𝑑 Cannot accept Although this plan is valid under general conditions for application of attributes sampling inspection, this will be specially useful to product characteristics involving costly or destructive tests.

Expression for Operating Characteristic function

Let us consider the number of defects in the lot follows the Zero-Inflated Poisson model having probability mass function is given by

The probability of getting no defects from a sample of size 𝑛 1 will be

The probability of getting one defect from a sample of size 𝑛 2 will be

The probability of getting one or less defects from a sample of size 𝑛 2 will be

An STDS plan accepts a lot only when no defects are found in a sample of size 𝑛 1 and one or less defects are found in a sample of size𝑛 2 . Therefore 𝑃 𝑎 (𝑝)of STDS plan will given by

Here n is the combined sample size.

and 𝑥 = 𝑛𝑝, equation (2) becomes

Here  is the ratio of the size of the second sample to that of the combined sample. When𝑛 2 = 0, equation (2) becomes

Which is the OC function of single sampling plan with Ac=0. When 𝑛 2 = 𝑛 equation ( 2) [or when =1, equation 3] becomes

Which is the OC function of single sampling plan with Ac=1 The second sample size will be 𝑛 2 = 𝑛 -𝑛 1 = 35 -9 = 26. According to Govindraju [6] , the STDS plan under the condition of Poisson distribution is also𝑛 1 = 9 𝑎𝑛𝑑 𝑛 2 = 26 for the same𝑝 1 = 0.005, 𝛼 = 0.05, 𝑝 2 = 0.1, 𝛽 = 0.1 𝑎𝑛𝑑 𝜔 = .0001. It indicates that the sampling plan, STDS under the conditions of Poisson distribution can be obtained under the conditions of ZIP distributions. It may also be noted that observing non-defects would not be an event occurring frequently when ω is small. In such case, the STDS under conditions of ZIP distribution would be same as the STDS under the conditions of Poisson distribution. Hence, the STDSP's designed under the conditions of Poisson distribution will be a special case of the STDS under ZIP distribution.

Illustration 2

Tofind an STDS plan for 𝑝 1 = 0.004, 𝛼 = 0.05, 𝑝 2 = 0.1, 𝛽 = 0.1 𝑎𝑛𝑑 𝜔 = .0001, compute tabulated against  for given 𝑃 𝑎 (𝑝)for different values of ω of a STDS plan are given in following pages.

Plotting the OC curve of Given STDS Plan

An OC curve for an STDS plan characterized by𝑛 1 𝑎𝑛𝑑 𝑛 2 , can be constructed using table 4 and5 by the value of 𝑛 = 𝑛 1 + 𝑛 2 . The result of each division is the value of p, whose probability of acceptance 𝑃 𝑎 (𝑝) is given in the column heading. The tables of np values provide some points to plot the OC curves for different ω values. For example, for the STDS plan 𝑛 1 = 6, 𝑛 2 = 14, division of the entries opposite = As described early the operating procedure of STDS plan is as follows:

1. From a lot, select a random sample of size 𝑛 1 units and count the number of defectives𝑑 1 . If 𝑑 1 ≥ 1 reject the lot. If 𝑑 1 = 0, select a second random sample of size 𝑛 2 and count the number of defectives𝑑 2 . 2. If 𝑑 2 ≤ 1, accept the lot; otherwise, that is, if𝑑 2 ≥ 2, reject the lot.

For fixed 𝑛 2 and p, equation (2) becomes a function of n denoted as 𝑓(𝑛) = [𝜔 + 𝑘 ′ 𝑒 -𝑛𝑝 ]𝑘", where k' and k" are constants independent of n. it can be easily shown that 𝑓(𝑛) > 𝑓(𝑛 + 1) for0 < 𝑝 < 1. For, 𝑓(𝑛) > 𝑓(𝑛 + 1)implies,1 > 𝑒 -𝑝 , which is always true. That is, 𝑃 𝑎 (𝑝) is a decreasing function of n for fixed 𝑛 2 and p. For fixed n and p, equation (2) becomes 𝑓(𝑛 2 ) = (𝜔 + 𝑘′𝑒 -𝑛 2 p )(𝜔 + 𝑘"𝑒 -𝑛 2 p (1 + 𝑛 2 p)) Where k' and k" are constants independent of𝑛 2 . For 0 < 𝑝 < 1, it can be shown that 𝑓(𝑛 2 ) < 𝑓(𝑛 2 + 1). For 𝑓(𝑛 2 ) < 𝑓(𝑛 2 + 1) implies 0 < 𝑘". That is, 𝑃 𝑎 (𝑝) is an increasing function of 𝑛 2 for fixed n and p.

Expression for ASN function

Assuming no curtailment of inspection, the expression for ASN will be derived. A first sample of size 𝑛 1 will always taken. A second sample of size 𝑛 2 will be taken only when no defects are found in the first sample of 𝑛 1 units. Therefore,

The following properties of ASN can be observed a) ASN (p) is decreasing function of p when 𝑛 1 and 𝑛 2 are fixed. b) At p=0, ASN (p) occurs it is maximum value, because for fixed 𝑛 1 and𝑛 2 , ASN (p) is maximum only when 𝑒 -𝑛 1 𝑝 is maximum. The maximum value of 𝑒 -𝑛 1 𝑝 for 0≤p≤1 is 1 when p=0. Therefore ASN (p) will decrease from 𝑛 = 𝑛 1 + 𝑛 2 to 𝑛 1 as p increases from 0 to 1.

From equation (3.6), one gets,

Equation (7) gives the ratio of average sample size of STDS plan to the combined sample size as function of  and x.

Plotting of ASN curve of a given STDS Plan

Assuming no curtailment of inspection in both the first and second stages of sampling, an ASN curve of a given STDS plan can be constructed using Table 7 or 8 by multiplying each entry in the row for the given  by the value of n. The result of each multiplication is the ASN at p that has a probability of acceptance 𝑃 𝑎 (𝑝) shown in the column heading. The value of p that has a known 𝑃 𝑎 (𝑝)can be found using For very small proportion defective, the ASN value is higher because a second sample must be taken more frequently to make a decision. For very large values of p, the value of ASN approaches the first sample size𝑛 1 .

Conclusion

This paper deals with Zero-Inflated Poisson distribution which is the one of the important discreet probability distribution in a manufacturing sector. In a production process being well-monitored, occurrence of non-defects would be more frequent and a ZIP distribution is the appropriate probability distribution to the number of defects per unit. The aim is to development a sampling plan which meets the requirements with respect to the producer's and consumer's risk can be obtained that is optimal with respect to the sample size. The OC function of the STDS plan under the conditions of ZIP distribution has been derived. The procedure for designing STDS using unity values under the conditions of ZIP distribution is presented. Tables providing the SSPs are presented for some specified strength.

A comparison is also made with OC function of SSP and STDS plan under the condition of Zero-Inflated Poisson distribution.

The STDS plans under Poisson distribution become special cases of the STDS plans under ZIP distribution.

References

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