Design of Single Acceptance Sampling Plan Using Maximum Allowable Proportion Defective (MAPD) and Producer s Allowable Risk (PAR)

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1 Desig of Sigle Acceptace Samplig Pla Usig Maximum Allowable Proportio Defective (MAPD) ad Producer s Allowable Risk (PAR) Edokpa Idemudia Waziri 1* ad Oduayo J. Braimah 2 1 Departmet of Mathematics/Statistics, Ambrose Alli Uiversity, Ekpoma, Nigeria, 2 Departmet of Statistics, Abdul-Raheem College of Advaced Studies, Kwara State, Nigeria Abstract This paper focuses o sigle acceptace samplig desigs o icomig quality with Maximum Allowable percet Defective (MAPD) alog with a specified probability of acceptace referred to Producer s Allowable Risk (PAR) which is defied as the miimum probability of acceptace of the lot with a maximum allowable proportio defective. A statistical software (R) is used to obtai the table values ad graphs. These tables ad graphs are preseted i order to compare the efficiecy of this Sigle Samplig Pla so as to protect Acceptace Quality Level. Researchers thereafter suggest optimum criterio of sample size ad acceptace umber for a fixed ratio of PAR to MAPD withi a reasoable sample size regio. Keywords: probability of acceptace (PA), operatig characteristic, iflectio poit, Poisso distributio, optimum samplig pla, steepess agle. 1. Itroductio Maximum Allowable Percet Defective (MAPD) was suggested by Mayer (1956) ad later developed by Madelso (1962) below which the proportio of acceptace of the lot was expected to declie severely. Followig Poisso distributio by the umber of defectives, the ratio c is efficiet to divide the good ad bad lots for the idustrialist. There are some derivatios o some basic operatig procedure to locate sigle samplig pla o MAPD ad its properties, (Norma (1953), Soudararaja (1975) Ramkumar ad Suresh (1996)). Ramkumar (2010) also formed a criterio for developig Sigle Samplig Pla (SSP) o iterval quality desig i terms of MAPD i the p axis alike PAR i the Pa(p) axis. Also Ramkumar (2009) had suggested a samplig pla idexed through MAPD ad discrimiat distace, idicatig the effectiveess of the Operatig Characteristics (OC) curve o tagetial distace ad cocept of optimum samplig pla uder same operatig ratio. PAR is the miimum probability of acceptace of a lot of quality havig a maximum allowable proportio defective. I particular for a product with a icomig quality of MAPD <10%, more tha 60% (PAR) lots are always accepted savig the iterest of producer. Iflectio poit is the turig poit of OC curve with steepest decliatio taget idicatig the sharpess of OC curve esurig protectio to the cosumers. The steepess agle q is a sesitive measure to defie the discrimiatio of the required OC curve. This agle will be wider for more striget OC curve. Thus the parameters were capable of protectig both producer ad cosumer. So this pla is favorable i customer friedly, moderate costly, ispectio orieted products like daily usig items. The aim ad idea of this paper ca be used practically whe the producer had little iformatio about the resultat quality, but he is ready to ispect a rage of sample size for fial decisio. 2. Materials ad Method 2.1 Maximum Allowable Percet Defective (MAPD) MAPD is a key measure assessig to what degree the iflectio poit empowers the OC curve to discrimiate betwee good ad bad lots. MAPD is therefore defied as the proportio of defective beyod which cosumer wo t be willig to accept the lot. The desirability for developig a system of samplig pla idexed by MAPD is p = c 2.2 Operatig Characteristics (OC) Curve A acceptace samplig pla is best described i graphical terms o a operatig characteristic curve (OC curve), Baklizi, (2003). A OC curve is a plot of the actual umber of ocoformig uits i a lot (expressed as a percetage) agaist the probability that the lot will be accepted whe sampled accordig to the pla. The shape of a OC curve is determied primarily by sample size,, ad acceptace umber, c, although there is a small effect of lot size, N. A importat feature of a samplig pla is how it discrimiates betwee lots of high ad low quality. The ability of a samplig pla to discrimiate is described by its operatig characteristic (OC) curve 119

2 .2.3 Acceptace Quality Level Acceptable quality level (AQL) is a quality measure fixig producers risk at 5, 10 ad 1% i usual practice. But i a assemblig of compoets, it is ot good to prefix oly oe level of AQL at a fixed producer s risk. Also keepig at a level may exert pressure o the cosumers as well as the producer because differet items require differet levels of acceptace. Thus fixed AQL samplig plas declie the cofidece of both the producer ad the customer. This is the limitatio of all probability based idices with fixed levels as they are iadequate to specify the quality aspiratio of the customer. 2.4 Selectio ad costructio of the pla Fix p* ad Pa(p*) at suitable quality level i a OC curve. The Pa(p) = c r! where the umbers of defectives follow Poisso distributio. e c c r r=0 (1) Therefore, d = 1 P a (p) = 1 c r=0 (2) r! which is a fuctio of c oly, mootoically icreasig, so that uique pla holds for each d. Fid c matchig with the give d ( greater tha or equal to the earest d) But c = p. So that = c (3) p Also oe ca fid d from the steepess agle q opposite to distace d ta θ = d p (4) The d = p ta θ. This value ca be foud for give MAPD ad q. d p = d c = ta θ Therefore, from Pa(p ) or d ad θ usig equatio (5) The value of ca the be equivaletly be obtaied as ta θ e c c r is determied ad substitute for ta θ. = ( c ) ta θ (6) d 3. Result ad Data aalysis 3.1 Samplig Pla Techique Fixig various level of probability of acceptace at MAPD o the OC curve is called the producers allowable risk (PAR). PAR is probability liberalizatio to 70, 85 ad 93% etc, istead of 95 ad 100%. MAPD is a egieer s quality level with steepest decliatio beyod p*, so that the utmost quality of acceptace is prefixed by MAPD. Thus MAPD ad PAR give a balaced desig for both cosumer ad producer keepig iformatio o steepess of OC. This is the oly oe poit OC pla ad may be flexible for various compoets. The steepess agle produced by d is a direct measure of efficiecy of OC curve so the desigs are possible i terms of required steepess. Also i the composite productio uits or i the compoets, differet MAPD ad PAR ca be fixed. Thus MAPD-PAR samplig pla is a flexible strategy of ispectio of differet compoets i a composite product with oe poit OC curves. Also agle q ad PAR have a strog power of discrimiatio of lot as good ad bad. Cosiderig a example oe ca show the sigificace of this samplig pla. Fix the PAR=0.635, (d=0.365) ad agle of steepess 80 (1). The c = 4 ad = 62 with MAPD=0.065 ad AQL= For keepig the same AQL, with steepess agle 75, PAR will be , (d=0.3233) (2) ad MAPD satisfyig a samplig pla (24.2) (Figure 2). Thus fixig AQL it is better to adjust PAR ad the steepess agle so as to get required quality. 3.2 Costructio of table Table 1 is costructed by substitutig c = 1,2,..,10 i equatios (1) ad (2). Table 2 represets Sigle Samplig Plas (SSPs) for various combiatios of p* ad d. Fid c for each d from Table 1 ad the is determied by (3). Table 3 is the decliatio agle for various values p* ad PAR. Fidig d from (2), c is fixed from Table 1 ad ta θ is foud. It is helpful to idetify (,c) for give (p*, θ ). Table 4 is the suitable samplig plas at various steepess agles ad PA. For PAR ad θ, ca the be evaluated by the relatio d c For a defied PAR the decisive distace d ca be calculated (Equatio 2). Ispect the rage i which calculated d or PAR falls uiquely ad locates c (greater tha or equal to d) from Table 1. Hece for a prefixed MAPD, = c The steepess agle θ subteded by d o the OC curve with the p axis at Pa(p*) ad give MAPD ca also be p obtai usig equatio 4. (5) ta θ = from Table

3 Table 1: Values of PAR for specified values of c =1 to 20. Table 2: Cofidet SSP for give PAR (or d) ad MAPD. Table 3: Cofidet parametric combiatios of PAR, MAPD ad Agle θ 121

4 Table 4: Values of for give PAR (or d) ad agle θ. Figure 1: Operatig curve (OC) curve showig maximum allowable proportio defective (MAPD) ad producer s allowable risk (PAR). 122

5 Figure 2: Operatig characteristics (OC) curves for fixed acceptable quality level (AQL) defied o producer s allowable risk (PAR) ad steepess agle. Figure 3: Feasible acceptace umbers for a fixed OR=8. 4. Discussio ad Practical Examples Figure 1 shows the operatig characteristics (OC) curves striget or moderate at the same quality of AQL which ca be fialized o PAR ad steepess agle. Whe PAR is less ad agle is more there exists a more discrimiatig OC curve tha higher PAR ad lower agle. Also for such parameters MAPD icreases o PAR icrease. Figure 2 is the feasible acceptace umbers withi a sample size 50 to 100 keepig operatig ratio d p a costat, from which maximum ad miimum efficiet OC curves ca be derived. Figure 3 shows the OC curves satisfyig the defied OR=8.0 so that switchig rules could be implemeted. 4.1 Practical Examples Practically this pla ca be put to use whe the producer had little prior kowledge about the resultig quality, but he is ready to ispect a rage of sample size for fial decisio. Pa(p*) or d is a good measure of efficiecy of OC curve. The MAPD is a appropriate quality measure so that a ratio is fixed as costat ad this samplig pla is developed o these criteria. Example 1 The quality of a electroic is fixed at maximum allowable proportio defective 8% ad PAR = The decisive distace d = = From Table 1, d< for c=2. Samplig pla (, c) is (25. 2). Ad the steepess agle θ =

6 Example 2 A wheel-barrow producig compay fixes the MAPD =10% ad steepess agle of OC curve as 74 ad 68 respectively. The decisive distaces d 1 =p*.taθ 1 = = ad d 2 =p*.ta θ 2 =0.1x2.475 = o OC curve with samplig plas from Table 2 will be (30. 3) ad (100.1). Example 3 If the same wheel-barrow producig compay fixes a quality idex OR = 8 based o MAPD ad decisive distace, provided they are ready to bear a ispectio cost of 50 to 100 uits. The what will be the optimum samplig pla? Sice the quality idex is fixed at R=8, for a rage of sample = 50 to 100 ad c = 1 to 20. The the various samplig plas obtaied were (50.2), (67.3) ad (85.4), (Fig. 4). Amog these samplig plas, optimum is reached at (85.4), ad they ca start with (50.2) ad switch over to (67.3) o successive completio of fixed umber of lots. From the OC curves (85. 4), has a optimum probability of acceptace at maximum allowable proportio defective (Figure 4). 5.0 Coclusio This samplig pla therefore suggested to be efficiet to cotai a Maximum Allowable Percet Defective (MAPD) orieted Acceptace Quality Level (AQL) protectio satisfactory for both cosumer ad producer (i.e reducig both producer ad cosumer s risk). Decisive distace ad decliig agle is a cocept to exhibit the quality of Operatig Characteristics (OC) curve. Adoptig these samplig plas will ot highly affect the OC curve, but ca also reduce ispectio cost. Refereces Baklizi A., (2003). Acceptace samplig based o trucated life tests i the Pareto distributio of the secod kid. Adv. Appl. Stat., Vol. 3 Issue 1, pp Mayer PL (1956). Some Properties of iflectio Poit of a OC curve for Sigle Samplig Pla. Proceedigs of 2d Statistical Egieerig Symposium, Published i Chemical Corps, Egieerig Commad. Madelso J (1962). The Statisticia, Egieer ad Samplig Plas. Id. Qual. Cotrol, Vol.10 pp Norma B (1953). A method of discrimiatio for Sigle ad Double samplig OC Curves utilizig of the poit of iflectio. ENASR Report No.P.R.7 Egieerig Agecy. Ramkumar TB, Suresh KK 1996). Selectio of samplig pla idexed with Maximum Allowable Average Outgoig Quality. J. Appl. Stat., Vol. 23 Issue 6, pp Ramkumar TB (2009). Desig of Sigle Samplig Pla by discrimiat at MAPD.Probstat Forum, Vol. 2, pp Ramkumar TB (2010). Desigig a Quality Iterval Sigle Samplig Pla. It. J. Oper. Res. Optim., Vol. 1 Issue 1, pp Soudararaja V (1975). Maximum Allowable Percet Defective (MAPD) sigle samplig ispectio by attributes pla. J. Qual. Techol., Vol. 7 Issue 4, pp