Selection of Optimum Single Sampling Plans under Prior Binomial distribution by minimizing the Average acceptance Cost

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1 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Selection of Optimum Single Sampling Plans under Prior Binomial distribution by minimizing the Average acceptance Cost M.Nirmala, 2 R.Subramaniam and 3 G.Uma PhD Research scholar, PSG College of Arts and Science, Coimbatore Associate Professor and Head, PSG College of Arts and Science, Coimbatore Assistant Professor, PSG College of Arts and Science, Coimbatore Abstract- This per deals with the determination of Optimum Single Sampling Plans (n,c) based on the Bayesian Prior Binomial Distribution for various values of the Lot sizes N by minimizing the average acceptance cost K(N, n, c, p) subject to the condition, the cost associated with a defective item which is accepted (A 2 ) is very small, Producer s Risk and Consumer s Risk are minimized (or) power -β is maximized. The ideal Operating Characterizing Curve for a Bayesian Single Sampling Plan is also constructed by considering AQL and LQL for the optimum acceptance number c. Index Terms- Bayesian Sampling Plan, Acceptance Quality Level, Limiting Quality Level, Producer s Risks (α), Consumer s Risks (β), Prior Distribution. A I. INTRODUCTION cceptance sampling is an inspection procedure used to determine whether to accept (or) reject a specific quantity of material. Inspection of a raw material and final product is necessary to ensure the good quality. It is well known that the acceptance sampling plans are used to reduce the cost of inspection. Acceptance sampling procedures are necessarily defensive the measures, instituted as productive devices against the threat of deterioration in quality. As such, they should be set up with the aim of discontinuance in favour of process control procedures as soon as possible. Sampling inspection plans provide decision-making procedures for controlling and improving the average quality of incoming items by specifying acceptable quality and determine when to accept (or) reject a given lot. An acceptance sampling is most useful to contact when product testing is destructive, very expensive, very time consuming (or) when product liability risks are significant [Starbird, S.A. 994]. Because of sampling, it could happen that the lot which has a satisfactory pre determined level of quality will be rejected and that the lot which does not have a satisfactory pre determined level of quality will be accepted. The first situation is known as Producer s Risk (or) α and the second situation is known as Consumer s Risk (or) β. [Wadsworth et al., 22]. The aim of the model is to find the Bayesian Optimum Single Sampling Plan (n,c) the sample size and the acceptance number by minimizing the average acceptance cost function K(N,n,c,p), which comprises cost of inspection and cost of reiring (A 2 ) (or) replacement of defective units subject to the condition of minimizing consumer s risk and producer s risk. The quantity most difficult to determine is normally the cost of accepting a defective items. If for example, the items considered are to be further processed, then the cost of accepting a defective may consist of the cost of handling and identifying the defective item, cost of assembling and disassembling, damage done to other items, cost of rework and cost of renewed testing and inspection. If however, the items are finished goods, the cost of ssing a defective may involve service and replacement cost, loss of goodwill which may be difficult to measure. For various cost models are available depending on a lot size, sample sizes. In rectifying inspection plans, each item produced by a process is inspected according to some plan and then according to the number of defectives units in the sample, the lot is either accepted (or) rejected. Guenther (97) explained how to obtained the rameter (n,c) for single sampling plans for ordinary sampling and for Bayesian sampling, using the model of expected total quality control cost depends on cost of inspection and sampling and cost due to wrong decision, which is accepting bad lot and rejecting good one. To improve the quality for any product and services, its customary to modernize the quality practiced and simultaneously reduced the cost for inspection and quality improvement. As a result of increasing customer quality requirement and development for new product technology may existing quality assurance practices and techniques need to be modified.

2 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September The need for such statistical and analytical techniques in a quality assurance is rapidly increasing owing to stilt competition in industry towards product quality improvement. This per introduces the designed model to minimize, Average acceptance cost subject to minimize the producer s and consumer s risk. Hald (96) has derived optimal solutions for the cost function K(N,n,c,p) in the cases where the prior distribution is Rectangular, Polya and Binomial. Bayesian acceptance sampling approach is associated with the utilization of prior process history for the selection of distribution [Viz., Gamma Poisson, Beta Binomial] to describe the random fluctuation involved in acceptance sampling.. Prior Distribution: The prior distribution is the expected distribution of a lot quality on which the sampling plan is going to operates. The distribution is called prior, because it is formulated prior to the taking of samples. The combination of prior knowledge represented with the prior distribution and the empirical knowledge based on the sample leads to the decision on the lot. A complete statistical model for basic sampling inspection contains three components. a. The prior distribution. That is the expected distribution of submitted lots according to quality. b. The cost of sampling inspection, acceptance and rejection. c. A class of sampling plan that usually defined by means of the restriction designed to give a production against accepting lot of poor quality. In this per the average acceptance cost is introduced when the proportional defective is random variable follows prior binomial distribution. The ultimate aim of a sampling plan is to obtained the optimum single sampling plan (n,c) based on prior binomial distribution by minimizing the producer s and consumer s risk. To design the single sampling plan, two point OC curve approach is followed. The plan rameters are determined and satisfying the condition that the producer s risk and consumer s risk are minimized at optimum level of A 2, the cost associated with a defective item which is accepted. It is important to note that there may exist multiples of solutions, in this case the cost associated with a defective items which is accepted (A 2 ) and acceptance number to be optimized and then determined the minimum average acceptance cost k(n,n,c,p) by minimizing consumer s risk and producer s risk. II. METHODOLOGY OF RESEARCH Since the aim of research insist on finding the optimum single sampling plans (n,c) based on prior binomial distribution by minimizing the average acceptance cost K (N,n,c,p) subject to the condition A 2,the cost associated with a defective item is minimum PP aa (pp ) αα () PP aa (pp 2 ) ββ (oooo) PP aa (pp 2 ) ββ (PPPPPPPPPP) (2) Where, KK(NN, nn, cc, pp) = nn(ss + ss 2 pp) + (NN nn){[(aa RR ) + (AA 2 RR 2 )pp]pp(pp) + (RR + RR 2 pp)} KK SS (pp) = SS + SS 2 pp KK aa (pp) = AA + AA 2 pp KK rr (pp) = RR + RR 2 pp p = Quality level corresponding to the producer s risk which is called Acceptable Quality Level. p 2 = Quality level corresponding to the consumer s risk which is called Limiting Quality Level. To satisfy, the aim of the following notations are necessary to build a model these are N: Lot size of product. n: Sample Size. p: xx percentage of defective in nn sample n. x: no of defectives in sample. S : Cost per item of sampling and testing. S 2 : Reir cost for a defective item found in sampling. A : Cost per item associated with the handling the (N-n) items not inspected in an acceptance lot. ( frequently it is Zero). A 2 : Cost associated with a defective item which is accepted (may be quite large). R : Cost per item of inspecting the remaining (N-n) items in a rejected lot. R 2: Reir cost associated with a defective item in the (N-n) items in a lot. α : Producer s Risk. β : Consumer s Risk. P(p): of accepting the production with quality p and it is P(p)=P(XX CC).

3 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Ka(p): Unit cost of acceptance Kr(p): Unit cost of rejection Ks(p): Unit cost of sampling K(N,n,c,p): Average acceptance cost under Bayesian sampling By giving different values of proportion defective p for various lot sizes N, the probabilities of acceptance Pa(p) will be calculated with optimum acceptance number c=7 and then identified Pa(p) =.95, the corresponding p value will be take it as Average Quality Level. Likewise, for various lot sizes N, the AQL values are obtained and are presented from Table to. Also identified the point Pa(p) =., the corresponding p value will be take it as Limiting quality level and then various lot sizes N, the LQL values are obtained and are presented from Table to 2. The determination of Optimum Single Sampling Plan(n,c) based on Bayesian Prior Binomial Distribution for various values of N were carried out through excel and are presented from table to 2. Table : Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 2: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 3: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2)

4 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 4: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n p np A 2 P(p) Ks(p) Kr(p) Ka(p K(N,n,c,p) c Table 5: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 6: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 7: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2)

5 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 8: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 9: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table : Determination of Optimum Sample Size n,, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Acceptable Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table : Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2)

6 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 2: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 3: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) Ka(p) K(N,n,c,p) c Table 4: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) ka(p K(N,n,c,p) c Table 5: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) Ka(p) K(N,n,c,p) c

7 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 6: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) Table 7: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) Ka(p) K(N,n,c,p) c Table 8: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) Ka(p) K(N,n,c,p) c Table 9: Determination of Optimum Sample Size n, acceptance number (c) and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2)

8 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 2: Determination of Optimum Sample Size n, acceptance number (c), and minimum average acceptance cost K(N,n,c,p) based on Limiting Quality Level. (A =, R =., R 2 =2, S =S 2 =2) N n P np A 2 P(p) Ks(p) Kr(p) Ka(p) K(N,n,c,p) c For various lot sizes N, the Optimum Single Sampling Plan (n,c) by using Prior Binomial Distribution, the Operating Characteristics Curve are presented from figure Defective (p) α=.5 Figure : N=, n=3, c= β=.927 } defectives (hundreds) (p) Defective (p) α= Figure 2: N=2, n=9, c= β=.92 } defectives (hundreds) (p)

9 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Defective (p) α= Figure 3: N=3, n=27, c= β=.96 } defectives (hundreds) (p) Defective (p) α= Figure 4: N=4, n=27,c= β=.28 } defectives (hundreds) (p) Defective (p) α= β=.37 } defectives (hundreds) (p) Figure 5: N=5,n=29, c=7

10 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Defective (p) α= Figure 6: N=6,n=33, c= β=.98 } defectives (hundreds) (p) Defective (p) α = β =.9 } defectives (hundreds) (p) Figure 7: N=7, n=35, c=7 Defective (p) α = β =.93 } defectives (hundreds) (p) Figure 8: N=8, n = 37, c=7

11 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Defective (p) α = Figure 9: N=9, n = 39, c= β=.999 } defectives (hundreds) (p) Defective (p) α =.489 Figure : N=, n = 4, c= β =.9 } defectives (hundreds) (p)

12 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 2: The following table represents the Optimum Single sampling plan (n,c) based on Prior Binomial Distribution. Lot Size N Sample Size n Average Lot Quality p Average Acceptance Cost K(N,n,c,p) Acceptance Number c A 2

13 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September From the above table clearly shows that the average lot quality p is higher at c=7 and the corresponding average acceptance cost K(N,n,c,p) are very less for various lot sizes N at the minimum level of A 2, cost of handling defective items in assembling and disassembling. Table 22: The following table shows the consolidated AQL, LQL and Average Lot Quality values based on the optimum values of the sample size n and acceptance number c. N n AQL LQL Based on Producer s Risk Quality Level Based on Consumer s Risk Average Acceptance Cost k(n,n,c,p) Average Lot LQL AQL Quality Based Based p -α -β A 2 c The above table clearly indicates that, for different lot sizes N the selection of Optimum Bayesian Single Sampling Plans (n,c) based on prior binomial distribution consist of minimum average acceptance cost K (N,n,c,p) at c=7 by minimizing A 2, cost of handling defective items in assembling and disassembling and also producer s risk and consumer s risk are at minimum level. The average acceptance cost K (N,n,c,p) associated with the average lot quality p always lies between the cost associated with AQL and LQL. Therefore the determination of single sampling plan (n,c) based on Bayesian prior binomial distribution is optimum and this single sampling plans satisfies the above mentioned conditions () and (2).

14 International Journal of Scientific and Research Publications, Volume 6, Issue 9, September Table 23: The following table shows the estimated AQL, LQL, Pa(p) and Average Acceptance Cost K (N, n, c, p) values when α=5% and β= %. Lot Size (N) Sample Size(n) α α value in percentage β β value in percentage The values of AQL and LQL are converted into percentages and are presented in table 23. A better single sampling plan would have a lower producer s risk and a lower consumer s risk. When AQL and LQL values are lies within (α =.5, β =.) then the plan is better single sampling plan. Hence in our case, for various lot sizes N, the percentages values of AQL and LQL are lies within α = 5%, β = %. This indicates that the selection of single sampling plans based on prior binomial distribution is further strengthened to be optimum. III. CONCLUSION In this study, various lot sizes N, Bayesian single sampling plans based on prior Binomial distribution, the optimum design rameters (n,c) are determined using the two points on the Operating Characteristics approach. In the proposed plan, the average acceptance cost K (N,n,c,p) is minimized based on the condition that the producer s risk and consumer s risk are minimum and also A 2 the cost of handling the defective item in assembling and disassembling are minimum at the acceptance number c=7. REFERENCES [] W.C. Guenther, On the determination of Single Sampling Attribute Plans Based upon a Linear Cost Model and a Prior distribution. Technometrics, vol.3, No.3, 97, pp [2] A. Hald, The Compound Hypergeometric Distribution and a System of a Single Sampling Inspection Plans Based on Prior Distribution s and Costs. Technometrics, vol.2, 96, pp , Discussion, pp [3] S.A.Starbird, The effect of Acceptance Sampling and Risk Aversion on the Quality Delivered by Suppliers, The journal of the Operational Research Society, vol. 45, No. 3,994, pp [4] H.M, Wadsworth, K.S, Stephens, A.B. Godfrey, Modern Methods for Quality Control and Improvement, 2 nd Ed., New York: John Willey & Sons, 22. AUTHORS First Author M.Nirmala, PhD Research scholar, PSG College of Arts and Science, Coimbatore nirmalastat25@gmail.com Second Author R.Subramaniam, Associate Professor and Head, PSG College of Arts and Science, Coimbatore psgstat@yahoo.com Third Author G.Uma, Assistant Professor, PSG College of Arts and Science, Coimbatore amug72@gmail.com Correspondence Author M.Nirmala, nirmalastat25@gmail.com,