Designing of Special Type of Double Sampling Plan for Compliance Testing through Generalized Poisson Distribution

Transcription

1 Volume 117 No , 7-17 ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Designing of Special Type of Double Sampling Plan for Compliance Testing through Generalized Poisson Distribution V.Kaviayarasu 1 and V.Devika 2 1,2 Department of Statistics, Bharathiar University, Coimbatore , Tamil Nadu, India. 1 kaviyarasu@buc.edu.in 2 devumol20@gmail.com Abstract Acceptance sampling plans plays an important role to ensure the quality of the product in compliance testing by giving a guarantee to the manufacturing industrialists. In this article, a Special Type of Double Sampling (STDS) plan by attributes has proposed towards emphasizing the significance of the STDS plan through Generalized Poisson Distribution by achieving through a smaller number of acceptance over a single sampling plan under similar conditions. A search procedure has carried out for designing this method by considering two points on the OC curves namely Acceptable Quality Level and Limiting Quality Level. The procedures for designing the plan are illustrated through ongoing applications for industrialists to secure both producer and consumer at same level. Comparison study was made with Poisson Distribution over the Generalized Poisson Distribution for the sampling plan and few illustration are also given. AMS Subject Classification:62pxx. Key Words: Single Sampling Plan, Special Type of Double Sampling Plan, Acceptable Quality Level, Limiting Quality Level, Generalized Poisson Distribution, Operating Characteristics Curve. 1 Introduction In most of the acceptance sampling methods, the product quality characteristics are tested and it becomes the most important issue for ensuring the overall quality of the product. These product characteristics may involve in testing the destructive and safety related items by attributes which ensures to protect both the producer and consumer simultaneously. In compliance testing, the quality characteristic of 7

2 a sampling plan are tested for multiple items simultaneously to save the time and cost of the inspection process and it is achieved through Single Sampling Plan with smaller acceptance numbers. The sampling plan under this method may provide only one chance to decide about the lot. Hahn [2] has establishes that a Single Sampling Plan with zero acceptance number is a minimum size plan and it gives protection to the consumer only. According to Govindaraju [4, 5], the Operating Characteristics (OC) curves of Single Sampling Plans with smaller acceptance numbers results in conflicting interest between the producer and consumer. While Single Sampling Plan with acceptance number c=0, which pays more attention towards the consumer but on the other hand acceptance number c=1 may favor to producer. This conflicting nature can be nullified through this proposed plan since its OC curve is lying between the acceptance number c=0 and c=1 of Single Sampling Plan. The proposed sampling plan is an extension of Single Sampling Plan (SSP) through Generalized Poisson Distribution (GPD). Guenther [8] has developed various sampling plans using traditional distributions like Hyper-Geometric, Binomial and Poisson distributions. Here, the STDS plan parameters are considered for the GPD and various values of acceptance numbers, λ 1, λ 2 and φ. In the proposed plan, inspection procedure consists of two phases. The lot acceptance is not decided in the first phase, and another chance is given through the second phase. The probability α of rejecting a good lot is called the Producer s Risk and the probability β of accepting a bad lot is known as Consumer s Risk. The parameters of the proposed plan is determined for ensuring both the risks involved in it. Therefore, a sampling plan is designed through the underlying distribution called as GPD. The GPD is also known as Lagrangian Poisson distribution. According to Consol and Jain [6, 7], this distribution is an extension of Poisson distribution with two parameters λ 1 and λ 2. When λ 2 = 0, this distribution may reduce to Poisson distribution. The density function of GPD is given as follows: f(x, λ 1 ; λ 2 ) = λ 1 (λ 1 + xλ 2 ) x 1 e (λ 1+xλ 2 ) ; x = 0, 1, 2,... (1) x! Where, λ 1 is the scale parameter and λ 2 is the shape parameter which determines the shape of the performance measures. Here we consider, the case λ 2 > 0, when establishing the tables, GPD is considered and the sampling plan is developed for this distribution. The GPD possess the twin properties of over-dispersion as well as under-dispersion according to λ 2 > 0 or λ 2 < 0 respectively. It ensures the protection for producer and consumer especially, the STDS plan is applicable in the area of safety related testing like medicine, army weapons, chemicals, etc. 2 Special Type of Double Sampling Plan In acceptance sampling plan, the acceptance number which may ensures to satisfies the consumer and producer at certain level. According to Schilling and Neubauer [1], a SSP with zero acceptance number are very useful in the area of compliance and safety related testing. To overcome the pitfalls in SSP with smaller acceptance 8

3 number, a lot by lot STDS sampling plan is used as an alternative. In the first phase of the sampling inspection the acceptance number are considered as Ac = 0 and in second phase the acceptance number is relaxed for the large sample size which may have Ac = 1. The operating procedure of STDS plan under GPD is very simple as given below: 1. A random sample of size n 1 units are taken from a lot and observe the number of defectives d 1. If d 1 >= 1, reject the entire lot. 2. If d 1 = 0, select a second random sample of size n 2 and observe the number of defectives d 2. If d 2 <= 1, accept the lot. Otherwise reject the lot. The OC function and ASN function for STDS plan are designed and expressed as follows; P a (p) = P (d 1 = 0/n 1 ; p) P (d 2 <= 1/n 2 ; p) (2) ASN = n 1 + n 2 P (d 1 = 0/n 1 ; p) (3) Where d 1 is the number of defectives found in the random sample of size n 1 and d 2 is the number of defectives found in the random sample of size n 2 for the lot quality p. The application of GPD provides the flexibility of dealing the under dispersion and over dispersion data with the help of interim parameter. When the interim parameter plays a positive role in improving the quality of products exponentially, the mean value of the event p gets changed. To overcome this situation, the Generalized Poisson model is used for designing the effective usage of a sampling plan. Thus the probability function for the number of non-conforming units in the sample modeled through the GPD is given by P a (p) = e x (1 + φxe λ 2 ) (4) Where, x = np, φ = n 2 (n 1 +n 2 ) and λ 2 is the interim parameter. Equation (4) gives the probabilities of acceptance for the submitted lot for various values of p, n 1, n 2, φ and λ 2. Table-1: Operating Ratio Values for SSP 9

4 λ c α=0.05 β=0.10 p 2 /p 1 for α=0.05 β=0.05 α=0.05 β=0.01 α=0.01 β=0.10 p 2 /p 1 for α=0.01 β=0.05 α=0.01 β= Significance of STDS Plan In a manufacturing firm, their ultimate aim is to produce quality products with zero defective items as a result to satisfy both producer and consumer. This proposed STDS plan using the conditions of GPD plays an alternative in reducing the proportion defectives than the PD-STDS plan through the involvement of interim parameter. This plan consists of various plan parameters as n 1, n 2, λ 1, λ 2, and φ. Based on the plan parameters one can draw a suitable OC curve for the desired discrimination. In order to find an efficiency of this sampling plan, it is a common practice to fix the OC curve of desired probability level. A ratio is obtained between limiting quality level and acceptable quality level as given in Table-1 which is called as operating ratio values of SSP under the conditions of GPD. It is difficult to design a sampling plan whose operating ratio values lies in between and Thus to avoid this conflicting nature occurred in the (0, 1) acceptance SSP, the proposed STDS Plan using GPD may protect both producer and consumer with the help of fixed interim parameter. The significance of the GPD-STDS plan over PD-STDS plan is given below in Figure-1. Any sampling inspection objective is to reduce the sample size as well as the lower number of proportion defectives. For this plan, the Average Sample Number curve is drawn in Figure-2. Accordingly, the efficiency of PD-STDS plan and GPD- STDS plan is given in Table-2, which may reduce the number of non-conforming items per lot significantly. Further it shows the efficiency of a sampling plan between the PD-STDS plan and GPD-STDS plan. 10

5 Figure 1: OC curves for STDS Plan Figure 2: ASN Curve of GPD-STDS Plan Table-2: Efficiency of GPD-STDS Plan over PD-STDS Plan in terms of Proportion Defectives 11

6 Distribution Fraction Defectives for ASN GPD PD Designing Methodology of Special Type of Double Sampling Plan The STDS plan is designed and developed through GPD plan parameters. The development of this plan is obtained by operating ratio and unity value approach elucidated by Cameron [3]. Using this method, the table values are calculated for different set of plan parameters. Table-3 provides the np 1 and np 2 values. Table-4 contains the Operating Ratio (OR) values for various combination of α and β, such as [(α=0.05, β = 0.10), (α=0.05, β = 0.05), (α=0.05, β = 0.01), (α=0.01, l = 0.10), (α=0.01, β = 0.05) and (α=0.01, β = 0.01)]. The plan parameters for designing the STDS Plan can be done by the following procedure: 1. Select the Acceptable Quality Level (AQL), Lot Tolerance Percent Defectives (LTPD) and the interim parameter λ Calculate the operating ratio (OR) by using the formula OR = p 2 p Choose the corresponding unity values np 1 and np 2 from T able 3 which is very close to the calculated OR value. 4. Determine the sample size as n = np 1 p 1 and n = np 2 p 2. Also select the corresponding parameters based on the OR value. 5. Then the plan parameters for the proposed GPD-STDS plan is formulated as (n 1, n 2, λ 2, φ) for the given interim parameter λ 2. 5 Numerical Illustration Assume that the proposed plan is adopted in production process when the production is continuous. The plan will fix a maximum of 5% producer s risk and a maximum of 10% consumer s risk as their quality limits for designing the plan parameters. For example, Let, p 1 = 0.003, p 2 = 0.1 then calculate the operating ratio by the formula OR = p 2 p 1 = The value is 33.33, which corresponds to the value 0.1 of λ 2 = 0.5 and φ = 0.80 from Table-2. Corresponding to these parameters, the calculated value of np 1 and np 2 are obtained from Table-1, which is np 1 = and np 2 = respectively. Solving for n 1 and n 2 gives the values as n 1 = 4; n 2 = Conclusion In this paper, Special Type of Double Sampling plan (STDS) is carried out to design the basic quality levels using the Generalized Poisson Distribution with their parameters λ 1 and λ 2. When the producer risk is fixed as α and the consumers risk is 12

7 fixed as β, the plan parameters are designed and given with suitable illustration. It is observed that, the STDS plan may reduce the size of the sample when it compared with SSP especially in compliance testing. Moreover, the GPD may yield more probability of acceptance over the Poisson distribution which ensures the quality of the product. Thus, GPD-STDS plan can be an be alternative method when a quality engineer wants to adopt this method in manufacturing industries for destructive or safety related testing. Clearly these test would be benefited in terms of test time and cost in a sampling inspection methods. 7 Acknowledgement The authors are happy to acknowledge the unknown referees and their parent University for providing necessary facilities in the Department through DST-FIST and UGC-SAP Programmes. 13

8 Table-3: Probability of Acceptance values of GPD-STDS Plan λ 2 Φ

9 Table-4: Operating Ratio Values of GPD-STDS Plan λ φ p 2 /p 1 for p 2 /p 1 for α=0.05 α=0.05 α=0.05 α=0.01 α=0.01 α=0.01 β=0.10 β=0.05 β=0.01 β=0.10 β=0.05 β= References [1] E. G. Schilling and D. V. Neubauer, Acceptance Sampling in Quality Control, CRC Press, Boca Raton, (2009). [2] G. J. Hahn, Minimum Size Sampling Plans, Journal of Quality Technology, 6: (1974), PP:

10 [3] J. M. Cameron, Tables for constructing and for computing the operating characteristics of single sampling plans, Industrial Quality Control, 9:, (1952), PP: [4] K. Govindaraju, Contributions to the Study of Certain Special Purpose Plans, Ph D., Thesis, (1984), Dept. of Statistics, Bharathiar University, Coimbatore. [5] K. Govindaraju, Fractional acceptance number single sampling plan, Communications in Statistics Simulation and Computation 20:, (1991), PP: [6] P. C. Consul and G. C. Jain, A Generalization of the Poisson Distribution, Technometrics, (1973), 15:, PP: [7] P. C. Consul, Generalized Poisson Distribution: Properties and Applications, Marcel Dekker, New York, (1989). [8] W. C. Guenther, Use of the Binomial, Hyper-geometric and Poisson Tables to obtain Sampling Plans, Journal of Quality Technology,2:, (1969), PP:

11 17

12 18