Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire lot o Accept nd Reject Lots (does not chieve qulity improvement) o Lot sentencing o Audit tool Three pproches to lot sentencing: o Accept with no inspection o 100% inspection o Acceptnce smpling Resons for Acceptnce Smpling, not 100% inspection o Testing is destructive o Cost of 100% inspection is high o 100% inspection is not fesible (require too much time) o If vendor hs excellent qulity history 113
Advntges of Smpling o Less expensive o Reduced dmge o Reduces the mount of inspection error Disdvntges of Smpling o Risk of ccepting bd lots, rejecting good lots. o Less informtion generted o Requires plnning nd documenttion Types of Smpling Plns (ttribute smpling plns) o Single smpling pln o Double-smpling pln o Multiple-smpling pln o Sequentil-smpling Lot Formtion Considertions before inspection o Lots should be homogeneous o Lrger lots more preferble thn smller lots o Lots should be conformble to the mterils-hndling systems used in both the vendor nd consumer fcilities. 114
Rndom Smpling The units selected for inspection should be chosen t rndom. Rndom smples re not used, bis cn be introduced. If ny judgment methods re used to select the smple, the sttisticl bsis of the cceptnce-smpling procedure is lost. Definition of Single-Smpling Pln A single smpling pln is defined by smple size, n, nd the cceptnce number c. o N = lot size o n = smple size o c = cceptnce number o d = observed number of defectives N totl items in lot. Choose n of the items t rndom. If c or less number of items re defective, ccept the lot. The cceptnce or rejection of the lot is bsed on the results from single smple - thus single-smpling pln 115
The OC Curves The operting-chrcteristic (OC) curve mesures the performnce of n cceptnce-smpling pln. The OC curve plots the probbility of ccepting the lot versus the lot frction defective. P P{ d c} c d 0 n! p d!( n d)! d (1 p) n d Exmple. If the lot frction defective is p 0. 01, n=89 nd c=2, then P P{ d 2} 89! (0.01) 0!89! 0.9397 0 2 d 0 (0.99) 89! (0.01) d!(89 d)! 89 89! 1!88! (0.01) 1 d (0.99) (0.99) 88 89 d 89! (0.01) 2!87! 2 (0.99) 87 The OC curve shows the probbility tht lot submitted with certin frction defective will be either ccepted or rejected. 116
Effect of OC curves 117
Type-A OC curves bsed on Hypergeometric distribution Type-B OC curves bsed on binomil distribution Other spects of OC Curve Behvior: 118
AQL nd LTPD Acceptble qulity level (AQL, p 1 ) - poorest level of qulity for the vendor s process tht the consumer would consider to be cceptble s process verge. The probbility of such process not being ccepted is the Producer s risk. Lot tolernce percent defective (LTPD, p 2 ) poorest level of qulity tht the consumer is willing to ccept in n individul lot. The probbility tht lot with lower qulity level is ccepted is the Consumer s risk. Also clled rejectble qulity level (RQL) or limiting qulity level (LQL) AQL nd LTPD cn be used for the design of smpling plns Designing Single-Smpling Pln with Specified OC Curve Let the probbility of cceptnce be 1 for lots with frction defective p 1 Let the probbility of cceptnce be for lots with frction defective p 2. Assume binomil smpling (with type-b OC curves). The smple size n nd cceptnce number c re the solution to n! c d n d 1 p1 (1 p1) d 0 d!( n d)! n! c d n d p (1 p d d!( n d)! ) 2 2 0 119
Exmple. For p 1 = 0.01, =0.05 (or 1 0. 95 ), p 2 = 0.06, = 0.10, use computer softwre or grphicl pproch, it cn be shown tht the necessry vlues of n nd c re 89 nd 2, respectively. 120
Rectifying inspection Following prticulr smpling pln Accepted lots re pssed with non-conforming units replced Rejected lots re screened with 100% inspection, non-conforming units re replced The number of defective items pssing this rectifying inspection is: P p( N n) (1 P )0 P p( N n) The verge outgoing qulity (AOQ) of the lots pssing the inspection cn be clculted by: or simply use: P p( N n) AOQ N AOQ P where P is the cceptnce probbility, p is the frction defective, N is the btch size nd n is the smple size p 121
Exmple. N=10,000, n=89, c=2 nd p =0.01. From binomil distribution or the OC curve, we found tht P = 0.9397 Then: or simply: AOQ P p( N n) (0.9397)(0.01)(10,000 89) N 10,000 AOQ P p ( 0.9397)(0.01) 0.0094 0.0093 Averge outgoing qulity limit (AOQL) The worst possible verge qulity tht cn be resulted from the rectifying inspection progrm. Exmple. For rectifying inspection pln with n=89, c=2, we hve: p 0.0005 0.010 0.020 0.030 0.040 P 0.9897 0.9397 0.7366 0.4985 0.3042 AOQ 0.000495 0.009397 0.014732 0.014955 0.012168 p 0.050 0.06 0.070 0.080 0.090 P 0.1721 0.0919 0.0468 0.0230 0.0109 AOQ 0.008605 0.005514 0.00327 0.00184 0.000981 122
AOQL is the mximum point on the curve Averge totl inspection (ATI) per lot ATI n ( 1 P )( N n) Exmple. For N=10,000, n = 89, c = 2 nd p =0.01, we hve P =0.9397. Then n ( 1 P )( N n) 89 (1 0.9397)(10000 89) 687 ATI 123
Double Smpling Plns Procedure o n 1 = smple size of the first smple o c 1 = cceptnce number of the first smple o n 2 = smple size of the second smple o c 2 = cceptnce number for both smples 124
Exmple. For the pln with n 1 =50, c 1 =1, n 2 =100, c 2 =3, rndom smple of 50 will be tken from the lot. If d 1 1, the lot will be ccepted. If d, the lot will be rejected. If d 2 or d 3, the second smple of 1 3 1 100 will be tken. If d d 3, the lot will be ccepted. If d d 3, the lot will be rejected. 1 2 1 1 2 Remrks o Possible less inspection o Second chnce o More complicted o Less inspection my not be relized 125
Multiple Smpling Plns Similr to double smpling Possible less inspection More complicted My be further discussed lter 126
Militry Stndrd 105E (ANSI/ASQC Z1.4, ISO 2859) Developed during World Wr II Widely used cceptnce-smpling system for ttributes Gone through four revisions since 1950. collection of smpling schemes to mke smpling system Bsed on AQL Description of the Stndrd Three types of smpling re provided for: 1. Single 2. Double 3. Multiple Provisions for ech type of smpling pln include 1. Norml inspection 2. Tightened inspection 3. Reduced inspection The AQL is generlly specified in the contrct or by the uthority responsible for smpling Different AQLs my be designted for different types of defects Defects include criticl defects, mjor defects, nd minor defects Tbles re used to determine the pproprite smpling scheme 127
Switching Rules 1. Norml to tightened 2 out of five lots re rejected 2. Tightened to norml 5 lots re ccepted 3. Norml to reduced 10 lots hve been ccepted under norml inspection totl number of defectives of the 10 lots is less thn given limit stble production uthorized 4. Reduced to norml lot is rejected procedure termintes without meeting cceptnce or rejection criteri. Accept the lot nd chnge to norml production is not stble other conditions 5. Discontinunce of inspection 10 consecutive lots remin on tightened inspection. Discontinue nd tke ctions. 128
Procedure 1. Choose the AQL 2. Choose the inspection level 3. Determine the lot size 4. Find the pproprite smple size code letter from Tble 15-4 5. Determine the pproprite type of smpling pln to use (single, double, multiple) 6. Enter the pproprite tble to find the type of pln to be used. 7. Determine the corresponding norml nd reduced inspection plns to be used when required 129
Exmple Suppose product is submitted in lots of size N = 2000. The AQL is 0.65%. Assume tht we wnt to generte norml singlesmpling plns. For lots of size 2000 nd generl inspection level II, Tble 15-4 indictes the pproprite smple size code letter is K. From Tble 15-5 for single-smpling plns under norml inspection, the norml inspection pln is n = 125, c = 2. This mens tht we ccept the lot if there re 2 or less defective units in rndom smple of 125. We reject the lot if there re 3 or more defective units. If tightened inspection is to be used fter inspecting 5 lots with norml inspection, then Tble 15-6 shows tht n = 125, c =1 for tightened inspection. This mens tht we ccept the lot if there is 1 or 0 defective units in rndom smple of 125. We reject the lot if there re 2 or more defective units. If reduced inspection cn be used fter ccepting 10 consecutive lots with norml inspection, nd ll other conditions stisfied, then Tble 15-7 shows tht in the reduced inspection, the smple size is n =50 Ac=1 nd Re=3. This mens tht: If there re 1 or less defectives in the smple, we will ccept the lot If there re 3 or more defective units, we will reject the lot nd use norml inspection for inspecting the next lot. If there re 2 defective units, we will ccept the lot nd use norml inspection for inspecting the next lot. 130
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Discussion Severl points to be emphsized: MIL STD 105E is AQL-oriented The smple sizes selected for use in MIL STD 105E re limited The smple sizes re relted to the lot sizes. Switching rules from norml to tightened nd from tightened to norml re subject to some criticism. A common buse of the stndrd is filure to use the switching rules t ll. ANSI/ASQC Z1.4 or ISO 2859 is the civilin stndrd counterprt of MIL STD 105E. Differences include: 1. Terminology nonconformity, nonconformnce, nd percent nonconforming is used 2. Switching rules were chnged slightly to provide n option for reduced inspection without the use of limit numbers 3. Severl tbles tht show mesures of scheme performnce were introduced 4. A section ws dded describing proper use of individul smpling plns when extrcted from the system 5. A figure illustrting switching rules ws dded 133
Dodge-Romig Smpling Plns Bsed on AOQL or LTPD Use developed tbles AOQL plns o Minimize verge totl inspection o Rejected lots will be 100% inspected o Frction nonconforming is known or cn be estimted o The pln lso presents the LTPD vlues corresponding to P 0.10 on the OC curve of the pln. Or 90% of the lots will be rejected if its percent defective is higher thn the corresponding LTPD vlue. Exmple. N=5000, p=1%. From Tble 15-8, we find tht the pln with AOQL=3% will be n=65, c=3. The corresponding LTPD vlue t P 0.10 is 10.3%. 134
If the incoming qulity is indeed t the level of p=1%, we cn clculte or check the corresponding OC curve to see tht the cceptnce probbility is P 0. 9957 for p=1%. Then the verge totl inspection will be: n ( 1 P )( N n) 65 (1 0.9957)(5000 65) 86. 22 ATI LTPD plns o Provide plns for different LTPD vlues with lot cceptnce probbility of 10%. Exmple. N=5000 nd incoming percent defective is p=0.25%. We cn find from Tble 15-9 tht the pln with LTPD=1% will be n=770, c=4. The corresponding AOQL vlue for this pln is 0.28%. 135
Similrly, we cn find tht if indeed tht the incoming percent defective is p=0.25%, we cn clculte or check the corresponding OC curve to see tht the cceptnce probbility is P 0. 9541 for p=0.25%. Then the verge totl inspection will be: ATI n ( 1 P )( N n) 770 (1 0.9547)(5000 770) 961.62 Item-by-Item Sequentil Smpling Plns The procedure is to tke one unit from the lot to test t one time nd continue for number of items. Bsed on the test results, the entire lot will be ccepted or rejected. In doing so, it my reduce the totl number of items to be tested. This procedure is lso indexed on p 1, p 2 nd. It clcultes the following 2 lines for ech smple: X A X R h1 sn (cceptnce line) h sn 2 (rejection line) 136
If the totl number of nonconforming units is less thn or equl to the integer prt of X, we ccept the lot. A If the totl number of nonconforming units is equl to or greter thn the integer bove X, we reject the lot. R These prmeters re clculted by: 1 1 h 1 log / k, h 2 log / k 1 p 1 s log / k 1 p 2, k p2(1 p1) log nd p (1 p ) 1 2 Exmple. Assume tht p 1 =0.01, =0.05, p 2 =0.06 nd =0.10, then: p (1 p1) 0.06 0.99 k log log p (1 p ) 0.01 0.94 2 1 2 0.80066 1 0.95 h1 log / k log / 0.80066 1.22 0.10 1 0.90 h2 log / k log / 0.80066 1.57 0.05 1 p 1 0.99 s log / log / 0.80066 0.028 1 k p2 0.94 Then we hve: X A X R h1 sn = 1.22 0.028n h 2 sn 1.57 0.028n (cceptnce line) (rejection line) When n = 1, we hve: X A X R h1 sn = 1.22 0.028n = 1.22 0.028 1. 192 h 2 sn 1.57 0.028n 1.57 0.028 1. 598 137
As we just tke one smple, these results tell us nothing. When n = 2, we hve: h1 sn = 1.22 0.028 2 1.22 0.056 1. 164 X A X R h 2 sn 1.57 0.028 2 1.57 0.056 1. 626 So it sys nothing bout ccepting the lot but the lot will be rejected if both items re bd. For this pln, the process continues for X A until n = 44 when X A becomes positive. On the other hnd, the rejection numbers re shown in Tble 15-3. As the process continues, if there re int[ X R ] 1 bd ones, reject the lot. If the first 44 re ll good ones, ccept the lot. Otherwise, continue nd stop smpling when you rech 89 3 267 items. The number of 89 corresponds to the single smpling pln for p 1 =0.01, =0.05, p 2 =0.06 nd =0.10. 138
A similr exmple is to ssume tht p 1 =0.01, =0.05, p 2 =0.10 nd =0.10. In this cse, we hve: X A X R h1 sn = 0.939 0.04n h 2 sn 1.205 0.04n (cceptnce line) (rejection line) The results for n =1, 2, 3... 26 re tbulted below. n ccept reject n ccept reject 1 x x 14 x 2 2 x 2 15 x 2 3 x 2 16 x 3 4 x 2 17 x 3 5 x 2 18 x 3 6 x 2 19 x 3 7 x 2 20 x 3 8 x 2 21 x 3 9 x 2 22 x 3 10 x 2 23 x 3 11 x 2 24 0 3 12 x 2 25 0 3 13 x 2 26 0 3 If it continues, we hve the following vlues: n 49 58 74 83 100 109 Acceptnce 1 1 2 2 3 3 Rejection 3 4 4 5 5 6 The corresponding single smpling pln is (52, 2) nd the smpling my stop fter 156 smple items re tken. 139