Selection of Bayesian Single Sampling Plan with Weighted Poisson Distribution based on (AQL, LQL)

Transcription

1 Selection of Bayesian Single Sampling Plan with Weighted Poisson Distribution based on (AQL, LQL) K. Subbiah * and M. Latha ** * Government Arts College, Udumalpet, Tiruppur District, Tamilnadu, India ** Kamarajar Government Arts College, Surandai, Tirunelveli District, Tamilnadu, India. Abstract The attributes for the principles of acceptance sampling plans have been done through the intrinsic assumption that production process from the chosen lots and non-confirming is stable. The lots which we chose have quality variations that occur owing to the random fluctuations. As a result the units of non-confirming proportions will vary further so as far as i am concerned that instead of conventional plan Bayesian methodology is an alternative framework used for making decisions about the submitted lots which helps to know the prior information in the process variation, this is what we call the Bayesian acceptance sampling plans. This paper involves about the single sampling plans by attributes which are developed under the conditions of producer s and consumer s risk for specified Acceptable Quality Level and Limiting Quality Level using Weighted Poisson distribution for prior process information. Keywords: Bayesian Single Sampling Plan (BSSP), Weighted Poisson, Minimum Risks( AQL,LQL), Operating Ratio(OR). Introduction In order to obtain single sampling attribute plans while the sample sizes are fixed and smaller, a technique of minimizing sum of producer s (p 1, 1-α) risks and consumer s (p 2, β) risks. When there is minimized sum of risks the values of particular producer s risk and consumer s risk is not taken into consideration. Sometimes the resulting plan may be disadvantageous to one of them. If the interest emphasis only for the consumer s risk, the risk can be fixed and also a plan with the fixed parameter, that gives a risk equal or nearly equal to the fixed consumer s risk can be obtained. If the plan for the consumer s risk is much smaller than the fixed quality, in which case fixing consumer s risk and obtaining a plan is in practice. In accordance with repeated procedure by Gunther ( 1971) to determine the parameters of single sampling plans by attributes for specified namely (p 1, 1-α) and (p 2, β). p 1 is refered to AQL and p 2 refered to LQL, α and β which meant producer s and consumer s risk. In the proposal of Cameron (1952) tables of unity values is a must for the construction of single sampling plans by attributes based on conventional Poisson distribution. Hald (1981) and Schilling (1982 ) gave procedure for determining appropriate plan parameters for Poisson based model on the unity values. Vijayathilakan (1982) stressed the importance in his another method in order to minimize the sum of risks with different weights for the producer s risk and consumer s risk. When larger weight can be assigned to consumer s risk than producer s risk, the interest is in the consumer s risk, otherwise it is vice versa. For the manipulation of Bayesian plan, Calvin (1984) provides tables and construction procedure. Oliver and Springler (1972) listed a set of tables that are based on the implicit assumptions of Beta prior distribution with specific posterior risk in order to achieve minimum sample size. It paves the way for avoiding the problem of calculating cost parameters. It is better that smaller sample size is required for the Bayesian sampling plan than that of conventional sampling plan with the same consumer s and producer s risk. 122

2 The Bayesian operating characteristic curve by Schafer (1967) has considered for the selection of sampling plans. Suresh and Latha (2001) have studied Bayesian single sampling plan through Average Probability of Acceptance involving Gamma Poisson model. Suresh and Latha (2001) obtained the Procedures and Tables for Selection of Bayesian Single Sampling with Weighted Risks. Latha and Subbiah (2015) have studied the selection of Bayesian Multiple deferred state (BMDS-1) sampling plan based on quality regions. Lauer analyzed the influence of prior information in comparison with the acceptance probability of sampling plans where the proportion defective p, that follows a Beta distribution with the conventional Operating Characteristic (OC) values. Here in this paper, it presents tables of unity values and methodology for finding the parameters n and c of Bayesian Single Sampling Plan (BSSP) that are done under the conditions of application of weighted Poisson distribution and also using operations as a measure of discrimination. The OC function of single sampling plan using weighted Poisson distribution is given by c e np (np) x 1 P a = x=1, xε N (1) Γ(x) Where p is the defective proportion of the lot. From the history of inspection, it is known that p follows a Beta distribution which is approximated by a Gamma distribution with density function w (p) w(p) = e pt t s p s 1 Thus, the average probability of acceptance P is approximately obtained by = Γs s, t > 0 and p > 0 (2) 1 P = P (a) (p)w(p)dp 0 c 1 s s ( nμ) x 1 x=1 β(x,s 1) (s 1) (s+nμ) s+x 1 x = 1,2,3... P = c ( s+x 2 s 1 ) ( nμ s+nμ )x 1 ( s x=1 s+nμ )s (3) Where μ = s is the first moment of the Gamma distribution for the product quality and it is worth mentioning t here that the equation has a negative binomial form. Procedure for selection of weighted Poisson SSP for given (µ 1, α) and (µ 2, β) For the points (µ 1,1-α) and (µ 2,β) are specified on the OC curve, the weighted Poisson SSP by attributes is determined by the following procedure. Step 1. The value of s can be estimated from the prior information of the process and, given µ 1 and µ 2 the value of R can be determined from the formula R= nµ 2 nµ 1 Step 2: For specified values of (α, β), s and c Table 2 is constructed with the values of R calculated from Table 1. Choose the value of c corresponding to the operating ratio which is equal to or just less than R. 123

3 Step 3: Enter the values of s and c obtained from the step 2 in the table 1 and choose the values of nµ 1 and nµ 2 corresponding to the column of ( P =1- α) and ( P = β) respectively. Step 4: From the ratio either nµ 1 µ 1 and nµ 2 µ 2 n can be computed. Thus the Weighted Gamma Poisson Single Sampling Plan is specified by (s,n,c). Construction of OC curve procedure for obtaining µ values: In order to construct the OC curve of a given weighted gamma Poisson SSP, the values of nµ are calculated and tabulated in Table 1. In order to obtain to the points for constructing the OC curve of a given weighted gamma-poisson SSP the values of nµ given in the table 1 are used. The OC curve of a sampling plan is a curve for (µ, P (µ)). The procedure for calculating µ is given below Step 1: The values s, n and c of a weighted gamma-poisson SSP are specified. Step 2: Enter Table 1 with the specified values of s and c. corresponding to s and c, Table 1 gives set of values of nµ which are associated with the specified values of P. Step 3: For each value of P we obtain the value of µ = nμ. This table can be extended for any value of (s, c). n Table 1: Values of nµ for which the proportion of lots expected to be accepted is given as the column heading for BSSP. s c P Illustration 1: The illustration to determine the plan parameters (n,c) for the specified strength (µ 1, α, μ 2, β) based on the table of unity values (Table 1) and the table of operating ratios (Table 5) is demonstrated. Consider the 124

4 strength (µ 1, α, μ 2, β) to be (0.02, 0.05, 05, 00) and the estimated value of s be 10. To determine the plan parameters (n,c) corresponding to the strength, proceed as follows: Step 1: Obtain the value using R= µ 2 µ 1 = =7.5. Step 2: The values of α =0.05 and β = 00 in Table 5 with the value of s=10 and R=7.5 are used to write the column. Step 3: When s=10, the Operating Ratio is , which is just less than 7.5. Associated with this Operating Ratio, the value of c is chosen as 4. Step 4: Enter the Table 1 with the values of s=10, c=4, 1- α =0.95 and β = 00 and choose nµ 1 and nµ 2 values as and respectively. At µ 1 =0.02, the value of n is determined as: n = nµ 1 = = 63 µ At µ 2 =05, the value of n is determined as: n = nµ 2 = = 53 µ 2 05 Table 2: Comparison of conventional Poisson, gamma-poisson and Weighted Gamma Poisson single sampling plans for the given strength (µ 1 =0.02, α = 0.05, µ 2 =05, β = 00). s Model Parameters nµ c - Poisson Gamma Poisson Weighted Gamma Poisson Among n= 63 (at µ 1 ) and n=53 (at µ 2 ), the largest value is preferred in order to ensure more discrimination. Thus the Weighted gamma- Poisson SSP by attributes for the given strength (0.02, 0.05, 05, 00) and for fixed s = 10 is determined as (63,4). Illustration 2 The procedure for the construction of OC curve of weighted gamma-poisson SSP by attributes, using the table of unity values (Table 1) is given. For the weighted gamma-poisson SSP determined in Illustration 1, the points for the plot of OC curve are obtained as follows: 1. From Table 1 for given s = 10 and c = 4, the values of nµ associated with their respective P values are obtained as follows: 125

5 Table 3: Unity values (np). P nµ The values of µ corresponding to P are determined as µ = nμ and are given below: Table 4: Determination of lot fraction nonconforming (p). P µ Illustration 3 In the following illustration, for different value of s and under the conditions of weighted gamma-poisson, gamma-poisson and Poisson distributions the features of the SSPs obtained are discussed. If we specify the strength of the plan as (µ 1 =0.02, α = 0.05, µ 2 =05, β = 00). From Table 2, for growth value of product quality, the Single Sampling Plan based on Poisson, Gamma Poisson and Weighted Gamma Poisson are obtained as fallows. n Table 5: Operating ratios for the constructing weighted gamma Poisson distribution Single Sampling Plan S 1 C α=0.05 β=00 R= µ 2 µ 1 for α=0.01 β=00 α=0.25 β=

6 S C α=0.05 β=00 R= µ 2 µ 1 for α=0.01 β=00 α=0.25 β= Plans with Weighted Risks Table II is used to select a Bayesian Single Sampling Plan using Weighted Poisson Distribution for the given AQL (µ 1 ) and LQL (µ 2 ) which involves minimum sum of risks. For the plan of Table V, producer s and consumer s risk will be at most 10% each against fixed values of the operating ratio μ 2. Suppose that ν 1 and ν 2 μ 1 are the weights considered such that ν 1 + ν 2 =1, then ν 1α + ν 2β can be minimized for obtaining the parameters of the required plan. Instead of minimizing ν 1α + ν 2β the expression α + νβ can be minimized, where ν = ν 2 is ν 1 the index of relative importance given to the consumer s risk in comparison with the producer s risk. When ν >1, the plan obtained will be more favorable to the consumer compared to the equal weights plan. When ν <1, it will be more favorable to the producer than the equal weights plan. Fixed Sample Size Minimizing α + νβ = P (R) μ1 + (A) P μ2 is equivalent to minimizing νp (R) μ2 (A). P μ1 (4) The Acceptance Quality Level (AQL) and Limiting Quality Level (LQL) corresponding to APA curve are referred as µ 1and µ 2, respectively. The AQL and LQL are usual quality levels in OC curve corresponding to the probability acceptance 1-α = 0.95 and β = 00, respectively. When sample size n is fixed the minimum value of expression (is obtained with P (c) < P (c 1) and P (c) > P (c + 1) (5) This result with c < 1+( ln ν+s ln(s+ nμ2 s+ nμ1 ) ) < c + 1 ln( nμ 1 s+ nμ2 nμ2 s+ nμ1 ) The optimum value of c is considered as the integral part of s+ 3 ν+s ln( nμ2 s+ nμ1 ) +( ln 2 ln( nμ 1 nμ2 ) (6) s+ nμ2 s+ nμ1 ) Table 6 gives the optimum value of c for n = 5, ν = 0.5 and s = 1, 5,

7 Illustration 4 It is given that AQL = 5%, LQL = 20% and n = 50. To find out the optimum plan for which ν = 2, it is observed that When Gamma Poisson Weighted Gamma Poisson s = 1, c = 1 c = 2 s = 5, c = 3 c = 4 s = 9, c = 3 c = 4 But for Conventional Single Sampling Plan it is observed that for the same value of AQL, LQL, n and ν, the acceptance number c = 5. Hence when protection is needed for consumer, more protection is given by Bayesian Gamma Poisson plan than conventional plan and when protection is needed for producer, more protection is given by Weighted Gamma Poisson than the Gamma Poisson for small values of s. Fixed Acceptance Number When acceptance number c is fixed, expression (4) is considered as a function of n and is minimized using differential calculus. The value of the optimum sample size is approximated to the nearest integer by solving the equation. ν ( µ 2 µ 1 ) c ( s+nµ 2 s+nµ 1 ) s+c = 0 (7) Table 7 gives the optimum values of n for given AQL, LQL, c = 1, ν = 0.5 and s =1, 5, 9. Illustration 5 When AQL = 5%, LQL = 20% acceptance number c = 1, ν = 0.5, it is observed that Gamma Poisson Weighted Gamma Poisson for s = 1, n = 10 n= 4 for s = 5, n = 13 n= 5 and for s = 9, n = 13 n =5 But for Conventional Single Sampling Plan it is observed that for the same value of AQL, LQL, c and ν, n = 7. Hence when protection is needed for consumer, more protection is given by Bayesian Gamma Poisson plan than conventional plan and when protection is needed for producer, more protection is given by Weighted Gamma Poisson than the Gamma Poisson for small values of s. Conclusion The values of nµ, are calculated by Newton-Rapson method for fixed values of s, c andp. The computed Operating Ratio (OR) values are tabulated in Table-5. For the fixed sample size n, values of c subject to minimum risk are obtained and similarly, fixed value of acceptance number c provides to obtain the values of n based on minimum risk. The acceptable values of Weighted Gamma Poisson Single Sampling Plan requires small number of sample size which protect producer by increased acceptance number nearly for the same lot quality. The Table values are generated by solving the functions in MS-Excel. Comparing with Gamma Poisson Single Sampling weighted Gamma Poisson requires smaller sample size for large values of s and the acceptance number increases which is favorable for producer increase of the same lot quality. 128

8 Table 6.(i) Acceptance numbers minimizing (α+2β) when s=1, Fixed sample size n=50 µ 1% µ 2%

9 Table 6.(ii) Acceptance numbers minimizing (α+2β) when s=5, Fixed sample size n=50 µ1% µ2%

10 Table 6.(iii) Acceptance numbers minimizing (α+2β) when s=9, Fixed sample size n=50 µ1% µ2%

11 0.02 Table 7 (i) μ Sampling size Minimizing (α+0.5β) with s=1 fixed Acceptance Number c= μ Table 7 (ii) Sampling size Minimizing (α + 0.5β) with s=5 fixed Acceptance Number c=1 μ μ

12 Table 7 (iii) Sampling size Minimizing (α + 0.5β) with s=9 fixed Acceptance Number c=1 μ 1 μ References: 1. Calvin T.W.(1984) How and when to perform Bayesian Acceptance Sampling, Vol. 7, American Society for Quality Control. 2. Cameron J.M.(1952), Tables for constructing and for computing the Operating Characteristics of Single Sampling Plan, Industrial Quality Control, Vol. 9, No. 1, pp Gunther, W.C (1971), On the Determination of Single sampling Attribute Plans Based upon a Linear Cost Model and a Prior Distribution. Technometrics, Vol. 13, pp Hald.A (1981), A statistical Theory of Sampling Inspection by Attributes, Academic Press, New York. 5. Latha, M and Subbiah, K(2015), Selection of Bayesian multiple deferred state (BMDS-1) sampling plan based on quality regions, International Journal of recent scientific Research, Vol.6, Issue 4, pp Lauer N.G(1978), Acceptance Probabilities for Sampling Plans when the proportion Defective has a Beta Distribution, Journal of Quality Technology, Vol. 10, No. 2, pp Oliver L.R and Springer M.D (1972), A General Set of Bayesian Attribute Acceptance Plans, American Institute of Industrial Engineers, Norcross, G.A 8. Schafer R.E (1967), Bayesian Single Sampling Plans by Attributes Based on the Posterior Risks, Navel Research Logistics Quarterly Vol4, (A) No, pp

13 9. Suresh K.K and Latha M (2001), Bayesian Single Sampling Plans for a Gamma Prior, Economic Quality Control, Vol6, No, pp Suresh K.K and Latha M (2001), Procedures and Tables for selection of Bayesian Single Sampling Plans with weighted risks, Far East Journal of Theoretical Statistics, Vol.5, No.2, pp Schilling E.G (1982), Acceptance Sampling in Quality Control, Marcel Dekker, Inc., New York. 12. Vijayathilakan J.P (1982), Studies in Lot Acceptance Procedures. Unpublished Ph.D. Thesis Submitted to University of Madras, India. 134