Modeling Optimal Double Sampling Plan using the Truncated Poisson Distribution

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Modeling Optimal Double Sampling Plan using the Truncated Poisson Distribution

Subramani Kandasamy¹, Ambika Subramanian²

1Department of Statistics, Government Arts College, Coimbatore, India

2Department of Mathematics and Statistics,Sri Krishna Arts and Science College, Coimbatore, India.

Abstract

We have made a work to construct optimal attribute double sampling plan using the truncated Poisson distribution. The producers and the consumers are known as the single party while taking the final decision. The risk of the sellers and the buyers are derived by means of acceptance of quality level and Limiting Quality Level values. Here Truncated Poisson distribution is used to improve the efficiency of risks for both the parties. The sampling plan used in this article is particularly useful for finding reliability of the completed goods in shop floor conditions. In this paper for various operating ratios we have found the total sum of risk using Truncated Poisson distribution.

Keywords: Double sampling attribute plan, Acceptance quality level, Limiting quality level, Minimum risk plan, Truncated Poisson distribution.

1. Introduction

An OC curve is used to visualize the sampling plan to show how it is good in picking up of good lots by disposing the bad lots.(p1, 1- ) and (p2,) are the two points on the OC curve used to select the sampling plan. Golub(1953) developed the binomial model with fixed sample size to find the Acceptance number of SSP with reduced sum of risk for known AQL and LQL values.

Soundararajan followed Golub procedure and developed the OC curve for the Poisson model. The idea of finding the parameters of the SSP with minimum sum of supplier and buyers risk for known AQL and LQL was given by Govindaraju and Subramani (1990a).For Romboski(1969) QSS-1, Govindaraju and Subramani(1991) have developed the similar table for Poisson distribution.

Govindaraju and Subramani (1990b) given Minimum risk multiple deferred state plan of type MDS–1 of Rombert Vaerst (1981). Double sampling plan with less quantity of risk was developed by Govindaraju (1992).

For designing various sampling plan Hamakar(1960) has given some common points to remember,

(1) Suppliers capacity, buyers needs and capacity of the inspectors must be in proper balanced manner

(2) Proper separation of good lots from bad.

(3) Economical conditions in the selection of number of observation (4) Increasing the lot size to reduce the risk of taking incorrect decision.

(5) Accumulated sample data is used as a valuable source of information.

(6) Quality of the lot with unreliable information exert pressure over buyer and seller (7) For reliable and satisfactory product the sampling can be reduced.

Hald(1981) have developed the table with fixed producer’s risk =0.05 and the consumer’s risk

=0.10 for the preference of single and double sampling plan.

Major ethic of acceptance sampling plan was reviewed by G.B.Wetherill and W.K. Chiu (1975).

In which semi-economic rooted extremity on the OC curve determines anassociation between the number of acceptance and the size of the size. For prefixed sample size the sum of risk for supplier and buyers were reduced by introducing acceptance sampling plan for salvageable lot by Dey(1970)

Guenther (1969) introduced the procedure to find the parameter of SSP by attributes on the OC curve by two points (p1, 1- ) and (p2,).

The working procedure of DSP is given as,

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1. From an assigned lot select n1 as a random sample size and identify the number of defective units, d1.

2. Accept the lot if d1≤c1. Reject the lot if d1 ≥R1

A Second sample of size n2 is taken if c1< d1< R1, and a number of defective units d2 is identified. Let D2 = d1+d2.

Lot is accepted if D2≤c2, otherwise the lot is rejected (D2 ≥R2).

Table I represents the above plan

Table I: Operating procedure of DSP for attributes Stage Sample

Size

Acceptance Number

Rejection Number 1

2

n1

n2

c1

c2

R1

R2 (= c2+1)

In this present study we have developed a tables and a procedure to optimize the risk of buyer and seller for double sampling attribute plan for known AQL and LQL values using Truncated Poisson distribution.

2. Truncated Poisson distribution

Fisher(1950) discussed the exact variance test of a complete Poisson distribution by considering the conditional distribution in which the sample total is fixed. Rao and Chakravarti (1956) generalized these results to the case where the zero class is truncated.

Neyman and Scott (1966), Potthoff and Whittinghill (1966), and Moran (1973) have investigated the conditions under which the variance test is optimal for testing homogeneity of the complete Poisson distribution. Most of the latter results carry over to the left truncated case. John J.Gart(1974) have modified and generalized the homogeneity for the zero-truncated case given by Rao and Chakravati(1956).Paul R. Rider (1953) estimated the parameter for the lower end truncated Poisson distribution.

D.J.Finney and G.C.Varley(1955) have recorded the frequency distribution of eggs and gall- cells of the knapweed gall-fly in flower-heads of black knapweed which have been used in illustration of the use of the Truncated Poisson distribution for representing biological observations. R.F.Tate and R.L.Goen (1958) have found the sum of n independent similar distribution for Truncated Poisson random variables, and then well-known properties of sufficient statistics to the obtained parameters.

M.A.Hamdan (1972) has estimated the statistical measures for Bivariate Poisson distribution by using the concept of two different kinds of truncation while taking the samples.J.C.Ahuja (1972) have shown left truncated unbiased estimators with less variance at zero and one for the statistical measures of population following Poisson distribution hinge on the sample of a size and its total .

John J. Gart (1974) has modified and generalized the homogeneity test for the zero-truncated case given by Rao and Chakravarti.J.G.Spain(1982) proposed the “Best Fit” to the Truncated Poisson Distribution with first coefficient is zero, and the second coefficient is positive. David P.M.Scollnik (1998) discussed the guesstimating the parameters by means Bayesian method for generalized

truncated Poisson distribution.

Charles J.Geyern made an attempt over the discrete variable Y which follows Poisson or negative binomial with the condition Y>k where k may be Truncated value of Poisson and negative binomial distribution.

Johan Springael (2006) has derived the frequency function and CDF of rv’s, which was given as the total of unconventional ZTP random variables. In operations management system for modeling common buying model to stochastic production storage model

J.Valero, M.Perez-Casany and J.Ginebra (2010) proposed the probability models obtained by Zero-Truncating Poisson (ZTMP) and Mixing Zero-truncated Poisson ((MZTP) distributions have been characterized by means of their PGF’s.

Subramani and Haridoss (2016) developed tables for the identification of double Sampling Plan by means of Weighted Poisson distribution which was also known as left truncated Poisson distribution.

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According to our best knowledge, so far, no venture has been tried to construct a Double sampling plan for attributes using ZTP in the area of minimizing sum of risks

Truncated Poisson distribution is one of the discrete distributions in the probability theory. If the random variable of Poisson distribution assigned with non-negative and the non-zero values, it is known to be Conditional Poisson distribution or Positive distribution.

The PMF g(k;) with the truncation k>0 from usually known Poisson distribution f(k,) is given as,

k = 1, 2, 3…

3. Anthology of double sampling plan using Truncated Poisson distribution The parameters of the double sampling plan are

n1 – Size of first sample n2 – Size of second sample

c1 – acceptance number of first sample c2 – acceptance number of second sample R1 – rejection number of first sample R2 – rejection number of second sample

In double sampling plan the size of the second sample is considered to be some constant value (k) which was the constant multiple of involved in first sample size. In MIL-STD-105D (1963) the constant value is fixed as 1, so that the sample size for the first and the second become equal. Duncan (1986), noted that the OC curve become easier function of product between sample size and the acceptance quality level by assuming the standard combination of n1 and n2, for a known combination of acceptance and rejection number.

Double sampling attribute sampling plan for known p1 and p2 are derived in Table III, which reduces the sum of risks of consumer and producers by means of truncated Poisson distribution when k=1,2 and 3. Almost 10 percent of consumer’s and seller’s risk are observed in Table III. For the assigned value of p2/p1, body of the table II used to find the parameters of the DSP c1, c2, R1,R2 which is related with seller and buyer’s risk against np1. The following methodology used for determining the plans for known p1, p2,  and.

(1) find operating ratio p2/p1

(2) For the calculated value of p2/p1, see the value in the table headed by which is just less than or equal to the calculated rate for the assigned value of k.

(3) The parameters c1, c2, R1, R2 can be identified value in step 2, such that the table value of buyers and sellers risk may be just less than or equal to the expected value.

(4) The first sample size is calculated by n1=n=np1/p2, the value of np1 are given as the column heading corresponding to the parameter found in step 3. The second sample size is found with n2= kn.

For example: Let we consider p1 = 0.12, p2 = 0.02, = 1% and =1%, the values for the parameters of double sampling plan when k=1 is found as

(1) p2/p1 = 0.12/0.02 = 6

(2) From Table I for Truncated Poisson distribution when k=1, take the value p2/p1= 6

(3) From the body table the parameters are obtained as c1=1, c2 =5, R1 = 4, and R2 = 6, the value of the risk  = 0.1% and  = 0.5% for the corresponding value of  = 1% and =1%

(4) n1= n=np1/p2 = 1/0.02 = 50, n2= kn = 1(20) = 20

4. Estimation of Likeness and difference with Subramani’s technique of choosing double sampling plan attribute plan

For the known parameters Duncan(1986) has created the table for choosing the DSP with n1=n2

and n2=2n1. The table created by Schilling and Johnson (1980) for fixed value of =0.05 and = 0.10 to the matched plan of single, double and multiple sampling plan. Further Govindaraj and Subramani

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(1992)given the table without fixing the value of  and  to double sampling plan using Poisson distributions with the rounded value of operating ratio. Here we have constructed the table to find acceptance and rejection number which gives the minimum sum of sellers and buyers risk for assumed value of operating ratio.

The range of the truncated Poisson distribution is 1, 2, 3 …which truncated the value of zero from the Poisson distribution. The truncation of the zero values is more useful in various industries that are very careful in their quality of the product with zero defective, which helps them to become one of the top most brands in the market throughout the year. If they found at least one defective in the major part of the manufacturing units of lot, they will shift for the second selection of the lot for checking the quality. In these circumstances, the application of truncated Poisson distribution is used for developing the sampling plan admissible over sampling plan created by Govindaraju and Subramani (1992) with the range from zero. The plan created in this paper using truncated Poisson distribution reduces the sum of buyer and sellers risk.

Example 1:

For the fixed value of the parameter p1=0.02, p2= 0.08 with the risk =4% and  = 9%, Using Poisson distribution by Govindaraju and Subramani (1992) with the operating ratio as 8 and 1 as np1

(k=1), one can get the values for this plan as n1=n2=50, c1=0, c2 =6, R1 =7, R2= 7. With the value of risk as = 3% and  = 6% gives the sum of risk as 9%.

For the same fixed values, one can get the plan using the Truncated Poisson distribution.n1=n2=50, c1=1, c2 =4, R1 =5, R2= 5. With the value of risk as = 3% and  = 5% gives the sum of risk as 8%.

Example 2:

For the fixed value of the parameter p1=0.01, p2= 0.05 with the risk =4% and  = 4%, Using Poisson distribution by Govindaraju and Subramani (1992) with the operating ratio as 8 and 1 as np1

(k=1), one can get the values for this plan as n1=n2=75, c1=0, c2 =5, R1 =6, R2= 6. With the value of risk as = 3% and  = 5% gives the sum of risk as 8%.

For the same fixed values, one can get the new plan using the Truncated Poisson distribution.

n1=n2=75, c1=1, c2 =6, R1 =7, R2= 7. With the value of risk as = 1% and  = 3% gives the sum of risk as 4%.

Comparison between the Truncated Poisson distribution and Poisson distribution is illustrated in Table II.

Table II: Comparison of Double sampling plan with k=1 Given Values Poisson Distribution

Govindaraju and Subramani (1992)

Truncated Poisson

Distribution

p1 p2   c1 c2 R1 R2   c1 c2 R1 R2  

0.01 0.07 0.02 0.02 0 6 7 7 1 1 1 5 4 6 0.1 0.5

0.01 0.07 0.04 0.05 0 4 5 5 2 5 1 3 3 4 1 2

0.01 0.08 0.04 0.04 0 6 7 7 4 2 1 2 3 3 1 2

0.01 0.08 0.02 0.02 0 10 9 11 0 0.3 1 5 6 6 1 1 5. Creation of Table

Table III is constructed, the OC curve using Truncated Poisson model. When the size of the sample is large and the proportion of the probability is small, in the situation of nonconformities, the truncated Poisson distribution is an identical good approach to the discrete distributions like binomial and Poisson. The production process is stable when the fraction of nonconformities in the lot is constant based on the general assumption of attributes sampling plan in acceptance sampling. Thus the OC functions retain the same value for various aggregation of n and p gives their product is similar for known acceptance and rejection values. So it is easy for a person to select the double sampling attribute plan by means of single parameter np, instead of considering two different parameters n and p. The table constructed using the truncated Poisson distribution considered the same parameter  =np but it optimizes the risk of buyer and the seller.

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The OC function of Doublesampling attribute plan with the parameter n1, n2 =kn1, c1,c2, R1,and R2is Pa(p)= F (c1/n) + ∑ 𝑓 (^𝑑¹

𝑛) 𝐹(𝑐₂−^𝑑¹

𝑘𝑛)

𝑅₁−1

𝑑=𝑐₁+1 (1)

Where, 𝐹(𝑐 𝑛) = ∑ ^𝑛𝑝^𝑑

(𝑒^𝑛𝑝−1)𝑑!

𝑐𝑑=1

⁄

and 𝑓(𝑑 𝑛⁄ ) = ^𝑛𝑝^𝑑

(𝑒^𝑛𝑝−1)𝑑!

The sum of buyers and sellers risk expression is given as

 +  = 1- Pa(p1) + Pa(p2) (2)

The expression for np2 can be found out by known operating ratio (p2/p1) and np1 is,

np2 = (p2/p1)(np1) (3)

Form equation (3) the value of np2 is calculated from the known value of np1, then it is used in equation 2. By analyzing the various combination of c1=0(1)10, R1-1= 1(1)25 and c2= 1(1)40, for the fixed value of k, the parameters c1, c2, R1, R2 are found out from the corresponding sum of risk calculated with equation 2.

Table III: Parameters of Double Sampling Plan for given p2/p1 and np1 when k=1

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Table III (Contd.) p2/p1

0.5 0.75 1 1.5 2 2.5 3 4 5 6 7 8 9 10

7 1,2 1,3 1,5 1,6 1,9 1,11 1,13 2,22 7,30 1,35 1,39 1,40 6,40 10,40

3,3 3,4 4,6 6,7 8,10 9,12 11,14 20,23 21,31 23,36 25,38 25,41 26,41 28,41

1,8 1,2 0.1,0.5 0,0.4 0.1,0.1 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0

6.5 1,3 1,5 1,7 1,9 1,10 1,12 2,22 6,30 1,35 1,39 1,40 6,40 10,40

2,4 4,6 8,8 7,10 9,11 12,13 18,23 18,31 23,36 25,38 23,41 26,41 28,41

1,2 0.1,0.3 0.2,0.3 0.2,0.2 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0

6 1,4 1,5 1,6 1,9 1,10 1,12 2,21 4,22 1,35 1,39 1,40 6,40 10,40

4,5 4,6 5,7 7,10 8,11 10,13 18,22 17,23 23,36 25,38 23,41 26,41 28,41

1,2 1,1 0.8,1 0.2,0.2 0.1,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0

5.5 1,4 1,5 1,6 1,8 1,10 1,11 2,21 3, 21 1,33 1,39 1,40 6,40 10,40

4,5 5,6 5,7 7,9 8,11 9,12 18,22 19,22 24,34 25,38 24,41 26,41 28,41

1,2 1,4 1,1 0.4,0.3 0.1,0.2 0.1,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0

5 1,4 1,4 1,6 1,8 1,9 1,11 1,18 2,18 1,30 1,35 1,40 6,40 10,40

4,5 3,5 7,7 8,9 9,10 9,12 17,19 18,19 23,31 25,36 24,41 26,41 28,41

1,6 2,4 1,3 1,1 0.2,0.2 0.1,0.2 0,0 0,0 0,0 0,0 0,0 0,0 0,0

4.75 1,3 1,4 1,6 1,7 1,9 1,10 1,13 2,17 1,30 1,32 1,40 6,40 10,40

4,5 4,5 7,7 6,8 10,10 8,11 11,14 12,18 23,31 23,33 24,41 26,41 28,41

2,8 4,4 1,3 1,1 0.8,0.9 0.2,0.2 0,0 0,0 0,0 0,0 0,0 0,0 0,0

4.5 1,4 1,4 1,6 1,7 1,9 1,10 1,13 1,16 1,20 1,25 1,39 6,40 10,40

5,6 5,5 7,7 7,7 10,10 8,11 11,14 15,17 16,21 20,26 24,40 26,41 28,41

2,8 3,5 3,2 1,2 0.8,1 0.3,0.3 0,0.1 0,0 0,0 0,0 0,0 0,0 0,0

4 1,2 1,4 1,5 1,7 1,8 1,9 1,12 1,15 1,16 1,22 1,39 2,38 10,40

5,6 5,5 6,6 8,8 8,9 8,10 10,13 13,16 15,17 20,23 23,40 27,39 28,41

2,3 3,4 2,2 2,2 2,1 1,1 0.2,0.1 0.1,0.1 0,0 0,0 0,0 0,0 0,0

3.9 1,5 1,7 1,8 1,9 1,13 1,14 1,16 1,23 1,26 5,35 10,40

5,6 8,8 9,9 9,10 11,14 12,15 15,17 22,23 23,27 28,36 28,41

3,4 1,2 1,1 1,1 0.2,0.4 0.1,0.1 0,0 0,0 0,0 0,0 0,0

3.8 1,5 1,7 1,8 1,8 1,13 1,14 1,15 1,19 1,24 7,36 10,40

5,6 8,8 9,9 9,9 11,14 12,15 13,16 18,20 23,25 28,37 28,41

3,5 1,3 1,1 1,1 0.2,0.5 0.2,0.2 0,0 0,0 0,0 0,0 0,0

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6. Conclusion

The major focus area of this quality control paper is to minimize the risk of the seller and the

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buyer. That is the overall risk of taking the wrong decision about the product of the item is taken as the optimum criteria. Here the sum of risk of both buyer and seller is found by minimizing the risk of selection of the item without fixing them. Table II displayed in this paper created with the help of Truncated Poisson distribution for double sampling plan. Through the various means of analysis by means of Truncated Poisson distribution, we have arrived the decision that the sum of risk is minimized compared with the already existing double sampling plan using Poisson distribution. We have also constructed the OC curve which shows the better shoulder compared to the Poisson distribution. (Figure 1)

Figure 1. Operating Characteristic Curve

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