Bayesian Double Sampling Plan under Gamma-Poisson Distribution

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1 Research Journal of Applied Sciences, Engineering and Technology 4(8): , 202 ISSN: Maxwell Scientific Organization, 202 Subitted: October 23, 20 Accepted: Noveber 5, 20 Published: April 5, 202 Bayesian Double Sapling Plan under Gaa-Poisson Distribution Sainathan Balaurali, 2 Muhaad Asla and 3 Chi-Hyuck Jun Departents of Matheatics, Kalasalinga University, Krishnankoil 62690, TN, India 2 Departent of Statistics, Foran Christian College University, Lahore, Pakistan 3 Departent of Industrial and Manageent Engineering, POSTECH, Pohang , Republic of Korea Abstract: The present study proposes a Bayesian double sapling plan for the inspection of attribute quality characteristics under the gaa-poisson distribution. The design paraeters of the proposed plan such as the saple sizes and the acceptance nubers are deterined for specified two points on operating characteristics curve such as acceptable quality level and liiting quality level along with the corresponding producer s and the consuer s risks. The optial paraeters are deterined using the iniu average saple nuber criteria. Extensive tables are provided for selection of paraeters of the proposed plan for selected cobinations of two quality levels and the results are explained with exaples. Key words:acceptable quality level, double sapling, gaa-poisson distribution, liiting quality level, producer s risk and consuer s risk INTRODUCTION Acceptance sapling is one of the ajor areas of statistical quality control. Inspection of the raw aterial and final product is very necessary to ensure the good quality. It is well known that the acceptance sapling plans are used to reduce the cost of inspection. One of the applications of sapling plans in that situation where the product under inspection is destructive or the cost of the life test or inspection of coplete product is very high or the inspection error is too high and the product liability risks are serious. Attributes sapling constitutes one of the ajor areas of acceptance sapling sapling. There are several sapling plans available in the literature for the inspection attribute quality characteristics naely single sapling plan, double sapling plan, ultiple sapling plan, chain sapling plan etc. Single sapling plan is one of the siplest attributes sapling plans which involves two paraeters naely the saple size n and the acceptance nuber c. Double sapling plan can be used to iniize the producer s risk. In double sapling if the results of the first saple are not definitive in leading to acceptance or rejection of a lot, a second saple is taken which then leads to a decision on the disposition of the lot. This approach akes sense not only as a result of experience, but also in the atheatical properties of the procedure. It is also to be noted that the average saple of the double sapling plan will be lesser than the single sapling plan. In all the attributes sapling plans, the basic assuption is that the lot or process fraction defective is constant, which indicates that the production process is stable. However, in practical situations, the lots fored fro a process will have quality variations which are due to rando fluctuations. These variations are classified into two, one is within-lot variation and the other is between-lot variation. When the second type variation is ore than the first type variation, the fraction defective ites in the lots will vary continuously. In such cases, the decision on the subitted lots should be ade with the consideration of the second type variation so that the conventional sapling plans cannot be applied. It is very iportant to note that acceptance sapling plans based on Bayesian ethodology can be used when the experienter has the prior knowledge on the process variation to ake a decision about the disposition of the lot. For further details about the prior distribution of the lot fraction defective, one can refer Hald (98) and Calvin (990). According to Vijayaraghavan et al. (2008) in the Bayesian theory, it is found that the gaa distribution is a natural conjugate prior for the sapling fro a Poisson distribution. When the saple ites are drawn randoly fro a process, the nuber of defects in the saple is distributed according to Poisson law, and the gaa distribution is the conjugate prior to the average nuber of defects per ite. Under these situations, Hald (98) derived Operating Characteristic (OC) function of the single sapling plan based on gaapoisson distribution. Vijayaraghavan et al. (2008) developed the tables for the selection of paraeters of single sapling plan using the gaa- Poisson distribution. In the literature, there is not double sapling plan is available for gaa-poisson situation. So this study attepts to develop a Bayesian Corresponding Author: Sainathan Balaurali, Departents of Matheatics, Kalasalinga University, Krishnankoil 62690, TN, India 949

2 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 double sapling plan under gaa-poisson odel. The optial paraeters of the proposed plan can be deterined plan for gaa prior and Poisson distribution under the specified requireents. Lp ( ) d! d! c e d ( np) c2 e d ( np) c2d e d n p n p n p 2 2 d0 dc d2 0 ( n2 p) d! 2 () DOUBLE SAMPLING PLAN UNDER GAMMA- POISSON DISTRIBUTION As entioned earlier, the double sapling plan is an extension of single sapling plan. The double sapling plan was investigated by any authors under various situations. Asla et al. (2009) investigated the double sapling plan for reliability tests under Weibull odel. Asla and Jun (200) developed tables for the selection of a double sapling plan under generalized log logistic distribution. Balaurali and Kalyanasundara (999) developed a conditional double sapling procedure based on soe switching rules. The operating procedure of the double sapling plan is as follows. Step : Select a rando saple of size n fro the lot of size N and observe the nuber of nonconforing ites in the saple, say d. If d #c, then accept the lot. If d >c 2, then reject the lot, If c <d #c 2, then go for second sapling as per step (2). Step 2: Select a rando saple of size n 2 fro the sae lot and observe the nuber of nonconforing ites in the saple, say d 2. If the cuulative nuber of nonconforing ites d +d 2 #c 2, then accept the lot. Otherwise reject the lot. Thus a double sapling plan is copletely specified by the paraeters n, n 2, c and c 2. Gaa-Poisson distribution, which is also known as the negative binoial distribution, is one of the ost widely used odels in reliability analysis and also in the acceptance sapling plans. Recently, Vijayaraghavan et al. (2008) developed the selection of single sapling plan using this distribution. When the production process produces output in a continuous strea and observed nuber of defects (or nonconforities) in the saple drawn fro this process is distributed as Poisson with paraeter np, where n is the saple size and p is the average nuber of defects per unit (Hald, 98). According to Schilling (982), the Poisson distribution is an appropriate odel for the nuber of nonconforing ites in the saple when the ratio of saple size to the population size ( ) is less than 0%, n is large and p<0.0 is sall such that np<5. Under the conditions of applications of Poisson odel, the probability of acceptance of the double sapling plan is given by: n N where c and c 2 are the acceptance nubers of the double sapling plan. sapling fro the Poisson distribution, the density function of prior distribution of p is given by (Vijayaraghavan When p varies fro lot-to-lot at rando and is distributed as gaa distribution which is the natural conjugate prior for et al., 2008): ap f p a e ( /, ) a p where a, 0 p, a, 0 (2) is the scale paraeter and is the shape paraeter. If E( p) p is given then the scale paraeter is obtained by a / p. Here is either specified or estiated using the Maxiu Likelihood Estiate (MLE) using prior inforation about the production process. The posterior distribution of the nuber of nonconforities is reduced to the gaa-poisson distribution. When the production is unstable, both the nuber of nonconforing ites in the saple, d and the average nuber of defects p are independently distributed. So, according to Hald (98), the sapling distribution of d, under the conditions p that the process average p 0., 02. is given by: ( d )! np Pdnp ( ;, ) d!( p)! np d np, d = 0,, 2,...(3) Based on this, the probability of acceptance of the double sapling plan under gaa-poisson odel is given by: P p a c2 dc c2d d2 0 d 0 d np d!! n p n p c d d! np d!! n p d2! n2 p d!! n 2 d n p d2 n p 2 2 DESIGNING OF GAMMA-POISSON DOUBLE SAMPLING PLAN The ultiate ai of a sapling plan is to iniize the producer s and consuer s risks. To design the double (4) 950

3 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Table : Optial gaa-poisson double sapling plan for given p, p 2, a = 5% and = 0% for = 5 p p ; ; ; ; 0. 36; 0. 32; 0. (96.453) (80.688) (69.028) (46.579) (4.822) (37.263) ; ; ; ; ; ; 0.2 (75.842) (23.937) (87.656) (76.585) (68.77) (48.227) ; 0.5 5; ; ; ; 0.4 5; 0.4 (456.66) (239.93) (45.608) (0.699) (82.930) (74.362) *** 366; ; 0.3 3; ; ; 0.5 (63.732) ( ) (79.722) (28.998) (88.377) *** *** 532; ; ; 0.2 9; 0.9 ( ) ( ) ( ) (44.320) Table 2: Optial gaa-poisson double sapling plan for given p, p 2, a = 5% and = 0% for =0 p p ; ; 0.2 4; 0. 36; 0. 32; 0. 29; 0. (88.359) (73.633) (48.333) (42.406) (37.695) (34.09) ; ; ; 0.3 4; ; ; 0.2 (38.23) (93.957) (80.789) (54.999) (49.388) (44.80) ; ; ; ; ; ; 0.3 ( ) (59.36) (7.622) (86.506) (63.228) (56.374) ; ; 0. 05; ; ; ; 0.4 ( ) ( ) (74.386) (20.03) (92.069) (69.55) *** 333; ; 0.4 0; ; ; 0.6 ( ) (289.73) (86.325) (2.005) (96.0) sapling plan, we use two points on the OC curve approach. The plan paraeters are deterined satisfying the following inequalities: Pa( p) a pa( p2 ) Here p is the quality level corresponding to the producer s risk, which is called Acceptable Quality Level (AQL). On the other hand, p 2 is the quality level corresponding to the consuer s risk which is also called Liiting Quality Level (LQL). It is iportant to note that there ay exist ultiple solutions, so we ay deterine these paraeters to iniize the Average Saple Nuber (ASN) at LQL. The ASN of the double sapling plan is given by: ASN ( p) np ( nn2)( p) (5) (6) (7) where P is the probability that the decision was ade by the first saple and based on gaa-poisson odel which is given by: c d 2 d np P ( )! d n p n p d c!( )! Hence to develop tables for designing an optial double sapling plan, we use the following nonlinear prograing proble: Miniize: Subject to: pa ( p2 ) ASN( p2 ) P ( p ) a a n ; n ; c 0; c c 2 2 (8a) (8b) (8c) (8d) The design paraeters such as the saple sizes n, n 2 and the acceptance nubers c and c 2 are deterined for various values of p, p 2, a, and. To reduce the nuber of paraeters, in this paper we consider n 2 =n. The optial paraeters of the double sapling plan under gaa-poisson odel are tabulated in Table -6 for the shape paraeters =5, 0, 25, 50, 00 and 50, respectively. In this study, we have assued that the shape paraeter of the gaa-poisson distribution is known. The proposed plan can also be used for the situation where the shape paraeter is unknown. Norally, producers keep the record of the estiated shape paraeter value for their product or it can be estiated fro the available data. Fro these tables we can observe interesting trends in the paraeter values. For 95

4 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Table 3: Optial gaa-poisson double sapling plan for given p, p 2, a = 5% and = 0% for = 25 P P ; 0.2 5; ; 0. 33; 0. 30; 0. 27; 0. (84.396) (69.744) (45.294) (39.406) (35.629) (32.066) ; 0.4 6; ; ; ; 0.2 3; 0.2 (32.976) (90.229) (77.090) (52.966) (46.496) (42.98) ; ; 0.5 6; ; 0.4 4; ; 0.3 (20.900) (32.68) (95.232) (83.867) (60.44) (54.483) ; ; ; ; ; ; 0.4 ( ) (29.298) (32.668) (99.945) (74.549) (66.922) ; ; 0.5 7; ; ; ; 0.5 ( ) ( ) ( ) (48.480) (02.990) (79.433) 0.03 *** 324; ; 0.6 0; 0. 84; ; 0.7 ( ) (37.824) (97.224) (46.837) (05.450) *** *** 278; ; ; 0.282; 0.0 (536.00) ( ) (89.884) (45.65) 0.04 *** *** *** 243; ; 0.80; 0.3 ( ) ( ) (84.09) *** *** *** 443; ; ; 0.9 ( ) (46.649) (26.559) 0.05 *** *** *** *** 367; ; 0.29 (72.93) (388.6) Table 4: Optial gaa-poisson double sapling plan for given p, p 2, a = 5% and = 0% for = 50 p p ; ; ; 0. 32; 0. 29; 0. 26; 0. (82.525) (68.77) (44.293) (38.49) (34.650) (3.095) ; ; 0.3 5; ; ; ; 0.2 (3.326) (89.324) (76.95) (52.04) (46.426) (4.263) ; ; ; ; ; ; 0.3 (208.98) (3.48) (94.425) (66.993) (59.549) (53.594) ; 0.2 2; ; ; ; ; 0.4 (337.93) (96.379) (3.24) (98.36) (73.726) (66.097) ; ; ; ; 0.7 6; ; 0.5 ( ) (303.28) (86.924) (30.924) (02.372) (78.689) ; ; ; ; ; 0.8 6; 0.7 ( ) ( ) ( ) (79.067) (3.237) (04.924) *** 460; ; ; ; 0. 73; 0.9 (93.999) (42.27) ( ) (73.638) (30.344) 0.04 *** *** 352; ; ;0.5 92; 0.2 ( ) ( ) ( ) (69.525) *** *** *** 286; ; ; 0.6 ( ) ( ) ( ) *** *** *** 497; ; ; 0.22 (99.880) (490.93) ( ) the sae values of p, p 2, ", $ as the value of shape paraeter increases, there is a drastic reduction in the saple size when copared to the single sapling plan based on gaa-poisson distribution. Also when p 2 increases, the saple size decreases for other fixed values and at the sae tie, as p increases, the saple size also increases. Exaples: Suppose that an experienter wants to run an experient to ake a decision on a product whether to accept or reject it. Assuing that the life tie of the product follows the gaa-poisson distribution with shape paraeter =0. Suppose one wants to find the paraeters of a double sapling schee for specified AQL (p ) = 0.05, LQL (p 2 ) = 0.07 with " = 5% and $ = 0%. Then fro Table 2, we can find the values of the paraeters as n = n 2 = 75, c = 0 and c = 5 with ASN = This double sapling plan is operated as follows: Step : Select a rando saple 75 fro the lot and observe the nuber of nonconforing ites. If there is no nonconforing ite is observed, then accept the lot. If ore than 5 nonconforing ites are observed, then 952

5 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Table 5: Optial gaa-poisson double sapling plan for given p, p 2, a = 5% and = 0% for = 00 p p ; ; ; 0. 32; 0. 28; 0. 26; 0. (82.468) (68.723) (43.346) (38.377) (33.73) (3.059) ; ; 0.3 5; ; ; ; 0.2 (3.392) (88.453) (76.83) (5.08) (45.526) (4.234) ; ; ; ; ; ; 0.3 (82.053) (30.408) (93.604) (66.24) (58.680) (53.58) ; 0. 0; ; ; ; ; 0.4 (3.640) (94.982) (30.558) (97.583) (72.900) (66.2) ; 0,20 5; ; ; ; ; 0.5 ( ) (28.37) (85.5) (30.284) (0.645) (78.770) ; ; ; ; ; ; 0.6 ( ) (447.79) (259.47) (78.683) (30.643) (9.48) *** 374; ; ; ; ; 0.8 (744.0) ( ) ( ) (58.708) (6.989) 0.04 *** *** 296; ; ; ; 0. ( ) (335.09) (25.659) (56.384) *** *** 499; ; ; ; 0.5 ( ) ( ) ( ) (207.28) *** *** *** 380; ; ; 0.20 ( ) (47.866) (268.63) ***: Plan does not exis (Value given in bracket is the ASN of the plan at LQL) Table 6: Optial gaa-poisson double sapling plan for given p, p 2, a =5% and = 0% for = 50 p p ; ; ; 0. 32; 0. 28; 0. 26; 0. (8.58) (67.838) (43.33) (38.363) (33.702) (3.047) ; ; ; ; ; ; 0.2 (30.550) (88.450) (58.47) (5.097) (45.56) (4.224) ; ; ; ; ; ; 0.3 (82.66) (30.460) (93.62) (66.23) (58.680) (52.726) ; 0, 0; ; ; ; 0.4 4; 0.4 (30.964) (95.82) (30.636) (97.623) (72.92) (65.275) ; ; ; ; ; ; 0.5 (50.203) ( ) (85.735) (30.390) (0.705) (78.797) ; ; ; ; ; ; 0.6 ( ) ( ) ( ) (78.927) (29.789) (9.535) *** 362; ; ; ; ; 0.8 ( ) ( ) ( ) (58.925) (7.09) 0.04 *** *** 286; ; 0.9 3; ; 0. ( ) (39.262) (24.738) (55.482) *** *** 463; ; ; ; 0.5 ( ) ( ) (298.5) ( ) *** *** *** 349; ; ; 0.9 ( ) ( ) (255.0) iediately reject the lot. If the nuber of nonconforing ites is in between 0 and 6 (i.e., to 5) then go for second sapling as per step (2). Step 2: Select another rando saple of size 75 fro the sae lot and observe the nuber of nonconforing ites. If the cuulative nuber nonconforing tes fro the first and second saple is less than or equal to 5 then the lot is accepted, otherwise the lot is rejected. Coparison: In this section, we will copare the results of the proposed plan with the gaa-poisson single sapling plan of Vijayaraghavan et al. (2008) by considering only two values of the shape paraeter naely 5 and 50 for different cobinations of AQL and LQL values. We have tabulated the saple sizes and the acceptance nubers along with the average saple nuber of the proposed plan as well as of the existing single sapling plan based on Gaa- Poisson distribution in Table 7. Fro this table, it is easily observed that the average saple nuber of the 953

6 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Table 7: Average saple nuber of the proposed plan and gaa-poisson single sapling plan for specified p, p 2, a = 5% and -= 0% Sapling Plan with ASN at p Gaa-Poisson ( = 5) Gaa-Poisson ( = 50) p p 2 SSP DSP SSP DSP (24) 62; 0.2 (80.687) 66. (66) 49; 0.2 (67.838) (66) 40; 0. (46.579) 50. (50) 32; 0. (38.363) (9) 55; 0.3 (76.585) 68.2 (68) 37; 0.2 (5.097) (06) 49; 0.3 (68.77) 60.2 (60) 33; 0.2 (45.56) (76) 57; 0.4 (82.930) 76.3 (76) 39; 0.3 (58.680) (38) 5; 0.4 (74.362) 54.2 (54) 35; 0.3 (52.726) (45) 83; 0.7(28.998) 9.4 (9) 46; 0.4 (72.92) (262) 59; 0.5(88.377) 82.4 (82) 4; 0.4 (65.275) *** 28;0.2( ) 9.6 (9) 60;0.6(0.705) *** 9;0.9(44.320) 95.5 (95) 48; 0.5 (78.797) Fig. : Operating characteristic curves of Poisson and gaa-poisson doublesapling plans with n = n 2 = 50, c = and c 2 = 3 for different shape paraeters Fig. 2: Operating characteristic curves of Poisson double sapling plan, gaa-poisson single sapling plan and gaa Poisson double sapling plans corresponding to p = 0.0 ("= 0.05) and p 2 =0.06 ($= 0.0) 954

7 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Fig. 3: Operating characteristic curves of Poisson double sapling plan, gaa-poisson single sapling plan and gaa Poisson double plans with sae saple size proposed plan is lesser than the ASN of the gaa- Poisson single sapling plan for all the cobinations of the AQL and LQL. Table 7 is around here. For further coparison, three OC curves are presented. Figure gives the OC curve of gaa-poisson double sapling plan for different shape paraeters. Fro this figure, it is understood that for sall fraction nonconforing, alost all the curves have equal probability of acceptance but at high fraction non-conforing, the OC curve of = 5 has ore probability of acceptable but when the values of increases the probability of acceptance reduces. Figure 2 provides the OC curves of gaa-poisson double sapling plan along with the gaa Poisson single sapling plan and conventional Poisson double sapling plans, all having sae AQL and LQL. Fro this figure, it can be observed that the OC curve of gaa-poisson double sapling plan has desirable shape as a coposite OC curve. For good quality, i.e. for saller values of fraction nonconforing, the OC curve of the gaa-poisson double sapling plan coincides with the OC curve of the conventional Poisson double sapling plan. As quality deteriorates, the OC curve of the gaa-poisson double sapling plan oves toward that for the gaa-poisson single sapling plan and coes close to it beyond the indifference quality level. It indicates that all the gaa-poisson double sapling schees, in general, protect the producer s interest against good quality levels and at the sae tie safeguard the consuer s interest against poor quality levels. In general, gaa-poisson double sapling plan has ore probability of acceptance than the other existing gaa-poisson single sapling plan and the conventional double sapling plan for fixed saple size and this can be easily observed fro Fig. 3. CONCLUSION In this study, we have developed a Bayesian double sapling plan under the gaa-poisson distribution. The optial design paraeters of the proposed plan are deterined using the two points on the OC curve approach. The proposed plan is better than the single sapling plan for the gaa-poisson distribution in ters of iniu ASN. The proposed plan provides the lesser ASN than the existing sapling plans. So, the double sapling plan perfors better than the conventional single, double sapling plan and gaa- Poisson single sapling plan and the proposed plan can be easily applied for the industrial use. REFERENCES Asla, M., C.H. Jun and M. Ahad, A double acceptance sapling plans based on life tests in the weibull odel. J. Stat. Theory Appli., 8(2): Asla, M. and C.H. Jun, 200. A double acceptance sapling plan for generalized log-logistic distributions with known shape paraeters. J. Appl. Stat., 37(3): Balaurali, S. and M. Kalyanasundara, 999. Deterination of conditional double sapling schee. J. Appl. Stat., 26(8): Calvin, T.W., 990. How to Perfor Bayesian Acceptance Sapling. Aerican Society for Quality Control, Wisconsin, Vol

8 Res. J. Appl. Sci. Eng. Technol., 4(8): , 202 Hald, A., 98. Statistical Theory of Sapling Inspection by Attributes. Acadeic Press, New York. Schilling, E.G., 982. Acceptance Sapling in Quality Control, Marcel Dekker, New York. Vijayaraghavan, R., K. Rajagopal and A. Loganathan, A procedure for selection of gaa-poisson single sapling plan by attribute. J. Appl. Stat., 35(2):