UNIT 6 RECTIFYING SAMPLING PLANS

Transcription

1 UNIT 6 RECTIFYING SAMPLING PLANS Structure 6.1 Introduction Objectives 6.2 Rectifying Smpling Pln Implementtion of Rectifying Smpling Pln for Attributes 6.3 Averge Outgoing Qulity (AOQ) 6.4 Operting Chrcteristic (OC) Curve 6.5 Averge Smple Number (ASN) 6.6 Averge Totl Inspection (ATI) 6.7 Summry 6.8 Solutions/Answers 6.1 INTRODUCTION In Unit 5, you hve lernt bout the cceptnce smpling plns nd how these re implemented in industry. You hve lso lernt bout different terms relted to it such s cceptnce qulity level (AQL), lot tolernce percent defective (LTPD), producer s risk nd consumer s risk. In n cceptnce smpling pln, if the consumer rejects the lot on the bsis of the informtion provided by the inspected smple nd the producer is not ble to sell the rejected lot, he/she suffers loss. If this continues, the producer might fce huge loss nd not be ble to continue with production. He/she my even hve to shut it down. To void such situtions, the concept of the rectifying smpling inspection pln ws introduced. In this unit, you will lern bout the rectifying smpling plns for ttributes. In Sec. 6.2, we discuss wht is the rectifying smpling pln nd how it implement in industry. In Secs. 6.3 to 6.6, we introduce some more prmeters such s verge outgoing qulity (AOQ), operting chrcteristic (OC) curve, verge smple number (ASN) nd verge totl inspection (ATI). In the next unit, you will study single smpling plns. Objectives After studying this unit, you should be ble to: explin why rectifying smpling plns re needed; describe rectifying smpling pln; define verge outgoing qulity (AOQ) nd the verge outgoing qulity limit (AOQL); describe the operting chrcteristic (OC) curve; nd define verge smple number (ASN) nd verge totl inspection (ATI). 6.2 RECTIFYING SAMPLING PLAN In n cceptnce smpling pln, the consumer ccepts the lot if the cceptnce criteri re stisfied. Otherwise, he/she rejects the lot. This increses the

2 Process Control producer s risk. Rectifying smpling plns re used to reduce the producer s risk. In such pln, the entire lot is not rejected. Insted, ech nd every unit/item of the lot is inspected. It mens tht 100% inspection of the rejected lot is crried out nd the defective units found in the lot re replced by non-defective units. This procedure is known s rectifying or screening or detiling the rejected lots. The smpling pln in which 100% inspection is crried out for rejected lots is clled the rectifying smpling inspection pln. It is lso clled the rectifying smpling pln or rectifying inspection pln. Let us explin the procedure for implementing this pln Implementtion of Rectifying Smpling Pln for Attributes Suppose tht lots of the sme size (N) re received from the supplier or the finl ssembly line nd submitted for inspection one t time. The procedure for implementing the rectifying smpling pln to rrive t decision bout the lot is described in the following steps: Step 1: We drw rndom smple of size n from the lot received from the supplier or the finl ssembly. Step 2: We inspect ech nd every unit of the smple nd clssify it s defective or non-defective on the bsis of certin criteri. At the end of the inspection, we count the number of defective units found in the smple. Step 3: We compre the number of defective units found in the smple with the cceptnce criteri. Step 4: If cceptnce criteri re stisfied, we ccept the entire lot by replcing ll defective units in the smple by non-defective units. If the criteri re not stisfied, we ccept the lot by inspecting the entire lot nd replcing ll defective units in it by non-defective units. The steps described bove re shown in Fig Fig. 6.1: Procedure for implementtion of the rectifying smpling pln. Let us explin these steps further with the help of n exmple. Exmple 1: Suppose cricket bll mnufcturing compny supplies lots of 200 blls. To check the qulity of the lots, the buyer nd the mnufcturing

3 compny decide tht smple of size 20 will be drwn from ech lot. The lot would be ccepted if the inspected smple contins t most one defective bll. Otherwise, the lot would be rejected. If they use the rectifying smpling pln, explin the procedure for implementing it. Solution: For implementing the rectifying smpling pln, the buyer drws 20 blls from ech lot, inspects ech nd every bll of the smple nd clssifies it s defective or non-defective on the bsis of certin criteri. At the end of the inspection, he/she counts the number of defective blls found in the smple nd compres the number of defective blls with the cceptnce criteri. If the number of defective blls in the smple is less thn or equl to the cceptnce number, he/she ccepts the lot by replcing ll defective blls in the smple by non-defective blls. If the number of defective blls is greter thn the cceptnce number, he/she crries out 100% inspection for this lot insted of rejecting the lot. The defective blls found in the lot re replced by nondefective blls nd the lot is ccepted. You my like to explin the procedure for implementing rectifying smpling pln yourself. Try the following exercise. Rectifying Smpling Plns E1) A hospitl receives disposble injection syringes in lots of A single smpling pln with n = 25 nd c = 2 is being used for ccepting the lots. Explin the procedure for implementing the rectifying smpling pln in this cse. So fr you hve lernt bout the rectifying smpling pln. In Secs. 6.3 to 6.6, we explin vrious terms ssocited with the rectifying smpling plns. 6.3 AVERAGE OUTGOING QUALITY (AOQ) In smpling inspection pln, the items or units produced/mnufctured by the producer re formed in lots. The verge qulity level of the lots is set by the producer nd the consumer through negotition nd the producer sends the lots to the consumer for inspection. The qulity of the lots before the inspection is known s incoming qulity nd the qulity of the lots which hve been ccepted fter the inspection is known s outgoing qulity. In n cceptnce smpling pln, the lots re either ccepted or rejected. So the outgoing qulity is the sme s the incoming qulity. However, in rectifying smpling inspection pln, the rejected lots re rectified or screened. So the outgoing qulity will differ from the incoming qulity. Therefore, the concept of verge outgoing qulity (AOQ) is prticulrly useful for the evlution of rectifying smpling pln. You hve lernt in Sec.6.2 tht the consumer tkes smple from ech lot to check its qulity. He/she inspects ech item or unit of the smple for defects. If the number of defective units in the smple is less thn the cceptnce number, the lot is ccepted by replcing ll defective units found in the smple by nondefective units. If the number of defective units is greter thn the cceptnce number, ech nd every unit of the lot is inspected. It mens tht 100% inspection is crried out for ech rejected lot nd ll defective units found in the lot re replced by non-defective units. Therefore, these lots re ccepted fter 100% inspection with zero percent defective. As result, the ccepted stores will consist of lots of vrying qulity level, rnging from qulity levels 3

4 Process Control lower thn cceptnce qulity level to lots with zero defective. Therefore, we need to define the concept verge outgoing qulity. When ll lots re considered together, their verge qulity level my be considerbly different from the incoming qulity. The verge outgoing qulity (AQL) is defined s follows: The expected qulity of the lots fter the ppliction of smpling inspection is clled the verge outgoing qulity. It is clculted s follows: Number of defective units in the lot fter inspection AOQ Totl number of units in the lot (1) But we do not know the number of defective units in the entire lot fter the inspection of smples. So we hve n lterntive wy of clculting AOQ. If N is the size of ech lot, n is the smple size inspected in ech lot, p is the incoming qulity of the lots nd P is the probbility of ccepting the lot of incoming qulity p, AOQ for single smpling pln is given by p(n n)p AOQ N We shll derive this formul in Unit 7. The concept of AOQ is shown in Fig (2) Fig. 6.2: Concept of verge outgoing qulity. For the cceptnce smpling pln in which rectifiction is not done, the AOQ is the sme s the incoming qulity. If we drw grph of AOQ versus the incoming qulity, the curve so obtined is known s the AOQ curve. A typicl AOQ curve is shown in Fig

5 Rectifying Smpling Plns Fig. 6.3: The AOQ curve. If we nlyse the AOQ curve (shown in Fig. 6.3), we observe tht for p = 0, the incoming lots hve no defective units nd hence there re no defective units in the outgoing lots. The consumer ccepts ll lots. So in this cse, AOQ = 0. When the incoming qulity is good, lrge proportion of the lots will be ccepted by the rectifying smpling pln nd only smller frction will be screened. Hence, the outgoing qulity will be good. However, when the incoming qulity is not good, lrge proportion of the lots will be screened. In such cses, the outgoing qulity will lso be good becuse defective units or items will either be replced or rectified. Between these extremes, the AOQ increses up to mximum nd then decreses. The mximum vlue of AOQ represents the worst possible verge for the outgoing qulity. It is known s the verge outgoing qulity limit (AOQL). The AOQL tells us tht no mtter how poor the incoming qulity is, on n verge, the outgoing qulity will never be poorer thn AOQL. Let us tke up n exmple bsed on AOQ. Exmple 2: Suppose cricket bll mnufcturing compny formed lots of 500 blls. To check the qulity of the lots, the buyer drws 20 blls from ech lot nd ccepts the lot if the smple contins t most one defective bll. Otherwise, he/she rejects it. Suppose the qulity of the submitted lots is Clculte the AOQ for this smpling pln. If rejected lots re screened nd ll defective blls re replced by non-defective ones, clculte the AOQ gin. Solution: It is given tht N 500, n 20, c 1nd p 0.03 We know tht in n cceptnce smpling pln, the lots re either ccepted or rejected nd the verge outgoing qulity is the sme s the incoming qulity. Therefore, in the first cse, when the lot is either ccepted or rejected, the AOQ is the sme s the qulity of the submitted lot, i.e., AOQ = In the second cse, when the rejected lots re screened nd ll defective blls re replced by non-defective ones, we clculte the AOQ using eqution (2). For clculting the AOQ in this cse, we hve to clculte the probbility of ccepting lot (P ). Recll the method of fining it described in Sec. 5.5 of Unit 5. 5

6 Process Control If X represents the number of defective blls in the smple, the buyer ccepts the lot if X c. Here c = 1. Therefore, the probbility of ccepting the lot is given by 1 P P X c 1 P X x x0 Since N 10n, we cn use the binomil distribution nd Tble I given t the end of this block to obtin P. From Tble I, for n = 20, x = c = 1 nd p = 0.03, we hve 1 n x x0 x n x P P X 1 C p 1p On putting the vlues of N, n, p nd P in eqution (2), we get p N n P AOQ N 500 You my like to clculte the AOQ in the following exercises. E2) Suppose, in Exmple 2, the qulity of the submitted lots is 0.01, clculte AOQ for this smpling pln. E3) Suppose, in E1, the qulity of the submitted lot is 0.05, clculte AOQ for this smpling pln. We now describe nother feture of the smpling pln known s the operting chrcteristic curve. 6.4 OPERATING CHARACTERISTIC (OC) CURVE 6 You hve lernt tht in cceptnce smpling, the lot is either rejected or ccepted on the bsis of conclusions drwn from the smple. Sometimes, good lot my be rejected nd bd lot my be ccepted becuse we infer the qulity of ll units in the lot on the bsis of smple (smll prt of the lot). So we require n cceptnce smpling pln which ensures tht good lots re lwys ccepted nd the bd lots re lwys rejected. It mens tht the pln should perfectly discriminte between good nd bd lots. Therefore, we introduce tool clled the operting chrcteristic curve, which mesures the discriminting bility of pln. The operting chrcteristic (OC) curve of n cceptnce smpling pln reflects the bility of the pln to distinguish between good nd bd lots. The OC curve for smpling pln is grph of the probbility of ccepting the lot versus the proportion defective (proportion of defective units) in the lot. It is plotted by tking the proportion defective (p) long the X-xis nd the probbility of ccepting the lot long the Y-xis. When the cceptnce smpling pln is used, there is conflict of interest between the producer nd the consumer. The producer wnts tht ll cceptble lots to be ccepted nd the consumer wnts tht ll unstisfctory lots to be rejected. A smpling pln which perfectly discrimintes between good nd bd lots is clled n idel smpling pln. The OC curve for n idel smpling pln is clled n idel OC curve nd is shown in Fig. 6.4.

7 Rectifying Smpling Plns Fig. 6.4: Idel OC curve. The idel OC curve runs horizontlly t probbility of cceptnce P = 1.0 until level of lot qulity tht is considered bd is reched. At tht point, the curve drops verticlly to probbility of cceptnce P = 0. Then the curve runs horizontlly gin for ll lot proportion defectives greter thn the undesirble level. If such smpling pln could be employed, ll lots of bd qulity would be rejected nd ll lots of good qulity would be ccepted. But, the idel OC curve cn lmost never be obtined in prctice. In theory, it could be relised by 100% inspection, if the inspections were error free. But for 100% inspection, the cost is high nd the time of inspection is much more. So we use smpling plns. The idel OC curve shpe cn be pproched by incresing smple size. A typicl OC curve is shown in Fig The OC curve shows the cceptnce probbility of lot of certin qulity. It represents the reltionship between the probbility of cceptnce nd the qulity of the lot. If the lot hs no defective units, it is certin to be ccepted nd if it hs ll defective units then it is eqully certin to be rejected. Hence, the OC curve hs to strt with P = 1 when p = 0. As the proportion defective (p) of the lot increses, the probbility P of lot cceptnce decreses until it reches 0. Thus, P = 0 when p = 1. The OC curve for prticulr combintion of n nd c shows how well the pln discrimintes between good nd bd lots. In Fig. 6.5, we hve shown n OC curve for n = 100 nd c = 2. In this pln, if 0, 1, or 2 defective units re found in the smple of size n = 100, the lot would be considered cceptble nd if more thn 2 defective units re found, it would be rejected. Suppose, the ctul lot qulity is 0.01, i.e., there re 1 percent defective units in the lot, the OC curve shows tht the probbility of ccepting the lot is pproximtely It mens tht this smpling pln ccepts the lot bout 99% of the time nd rejects the lot bout 1% of the time. However, if the ctul qulity of lot is 0.05, i.e., there re 5 percent defective units, the OC curve shows tht the probbility of ccepting the lot is pproximtely

8 Process Control Fig. 6.5: the OC curve. Hence, if the ctul qulity of lot is good, the pln provides high probbility of cceptnce nd if its qulity is poor, the probbility of cceptnce of the lot is low. Thus, the OC curve shows how well given pln discrimintes between good nd bd qulity. The OC curve of smpling pln psses through two points greed upon by the producer nd the consumer. These points re (AQL, 1 α) nd (LTPD, β) where the verge qulity level (AQL) nd the lot tolernce percent defective (LTPD) re decided by the producer nd the consumer fter negotition with ech other. You know from Unit 5 tht α nd β re the producer s risk nd the consumer s risk, respectively. Since the OC curve represents the probbility of ccepting lot, insted of tking α (probbility of rejecting lot of qulity AQL) we tke 1 α (probbility of ccepting lot of qulity AQL). In prctice, the performnce of cceptnce smpling pln for discriminting good nd bd lots depends minly on the smple size (n) tken from the lot nd the cceptnce number (c) tht cn be permitted in the smple. Since n nd c re different in different smpling plns, the OC curve will be different for different smpling plns. Let us see how the smple size (n) nd the cceptnce smpling number (c) ffect the OC curve. Effect of incresing the smple size n on OC curve Study Fig It shows the effect of incresing smple size on OC curve. Here we hve three OC curves for three smpling plns with n = 100, c = 1; n = 200, c = 2 nd n = 300, c = 3. From Fig. 6.6, we note tht even though ech pln uses the sme percent defective which cn be llowed for n cceptnce lot (1/100, 2/200, 3/300 = 0.01, i.e.,1 percent), the OC curve becomes steeper nd lies closer to the origin s the smple size increses. 8

9 Rectifying Smpling Plns The greter the slope of the OC curve, greter the discrimintory power. Fig. 6.6: The OC curves for n = 100, c = 1; n = 200, c = 2 nd n = 300, c = 3. If we compre the discrimintion power of the three plns, we see tht ll three plns would ccept lots with bout 0.017, i.e., 1.7 percent defective units, bout 77 percent of the time. However, if the ctul qulity flls to 5 percent defective units, the pln n = 100, c = 1 ccepts the lot bout 19 percent of the time, n = 200, c = 2 bout 8 percent of the time nd n = 300, c = 3 bout 2 percent of the times. Therefore, the plns with lrger smple sizes hve definitely more discrimintion power. It mens tht the plns with lrger smple sizes re more effective in protecting the consumer ginst the cceptnce of reltively bd lots nd lso give protection to the producer ginst rejection of reltively good lots. Effect of decresing the cceptnce number (c) on OC curve In Fig. 6.6, you hve seen tht the OC curve chnges s the smple size chnges. Generlly, chnging the cceptnce number does not drmticlly chnge the slope of the OC curve. As the cceptnce number is decresed, the OC curve shifts to the left. For the sme n, plns with smller vlues of c provide discrimintion t lower levels of the lot proportion defective thn the plns with lrger vlues of c. You cn see this in Fig. 6.7, which shows the effect of cceptnce number on the OC curve. It hs three OC curves for n = 200 nd c = 1, 2 nd 3. 9

10 Process Control Fig. 6.7 Note tht in Fig. 6.7, the effect of decresing c is tht the OC curve becomes steeper. Thus, when we decrese the cceptnce number (c), the smpling pln becomes tighter or stricter. It mens tht the pln with smller cceptnce number hs more discrimintion power. Now tht you know wht the OC curve of smpling pln is nd why it is used, let us summrize its properties. Properties of the Operting Chrcteristic Curve The OC curve possesses the following properties: i) The OC curve of n cceptnce smpling pln shows the bility of the pln to distinguish between good nd bd lots. ii) iii) iv) The OC curve with lrger smple size is steeper. It mens tht the pln with lrger smple size hs more discrimintion power. The pln with lrger smple size protects the consumer ginst the cceptnce of reltively bd lots nd lso gives protection to the producer ginst rejection of reltively good lots. The OC curve for fixed smple size nd smller cceptnce smple number is steeper. It mens tht the pln with smller cceptnce number hs more discrimintion power. The OC curves with fixed smple sizes give constnt qulity protection, wheres the OC curves with reltive smple sizes give different qulity protections. We now discuss nother useful concept of smpling plns clled the verge smple number (ASN). 6.5 AVERAGE SAMPLE NUMBER (ASN) 10

11 In cceptnce smpling pln, the decision of cceptnce or rejection of lot is bsed on the informtion provided by the smple (s) drwn from the lot. So the verge smple number cn be defined s follows: The verge smple number (ASN) is defined s the verge (expected) number of smple units per lot, which is required to rrive t decision bout the cceptnce or rejection of the lot under the cceptnce smpling pln. The curve drwn between the ASN nd the lot qulity (p) is known s the ASN curve. In single smpling pln, we tke the decision of cceptnce or rejection of the lot on the bsis of only single smple of size n. Hence, the ASN in single smpling pln is simply the smple size n, which mens tht ASN is constnt. Therefore, the ASN curve for single smpling pln is stright line s shown in Fig Rectifying Smpling Plns Fig. 6.8: The ASN curve for single smpling pln. ASN for double smpling pln vries with the qulity of the lot nd is function of the ctul proportion defective (p) in the lot. We shll discuss the method of finding ASN nd ASN curve for double smpling pln in Unit 8. The concept of verge smple number is considered for the cceptnce smpling pln. The verge totl inspection (ATI) plys the sme role in rectifying smpling pln. 6.6 AVERAGE TOTAL INSPECTION (ATI) The verge totl inspection (ATI) for rectifying smpling pln is defined s follows: The verge number of units inspected per lot under the rectifying smpling pln is clled the verge totl inspection (ATI). In other words, The verge totl inspection (ATI) is the number of units inspected per lot to tke the decision for cceptnce or rejection of the lot under rectifying smpling pln clling for 100% inspection of the rejected lots. If the lot qulity is p, the verge totl inspection (ATI) for single smpling pln is given by 11

12 Process Control ATI n 1 P N n (3) We shll derive this formul in Unit 7. Therefore, ATI = ASN + (verge number of units inspected in the rejected lots) Thus, if the lot is ccepted on the bsis of the rectifying smpling pln, then Otherwise, ATI = ASN ATI > ASN If lot contins no defective unit, it will obviously be ccepted by the smpling pln nd only n items will be inspected. Therefore, in this cse ATI will be equl to the smple size n. If ll units of the lot re defective, the lot will be rejected nd 100% inspection of the lot will be clled. Therefore, in this cse the ATI will be equl to the lot size N. If the lot qulity lies between 0 nd 1, i.e., 0 < p < 1, the ATI will lie between the smple size n nd the lot size N. This mens tht ATI is function of the lot qulity p. The curve drwn between ATI nd the lot qulity (p) is known s ATI curve. A typicl ATI curve for single smple pln is shown in Fig. 6.9 for N = 2000, n = 30 nd c = Fig. 6.9: The ATI curve. We shll discuss the methods of obtining ATI nd ATI curves for single nd double smpling plns in Units 7 nd 8. Let us now compute ASN nd ATI for rel life sitution. Exmple 3: A hospitl receives disposble injection syringes in lots of A single smpling pln with n = 25 nd c = 2 is being used for inspection. Clculte the verge smple number (ASN). Suppose the incoming lots

13 contin 5% defective units. Clculte the verge totl inspection (ATI) for this pln, if the rejected lots re 100% inspected nd ll defective syringes re replced by non-defective syringes. Solution: It is given tht N 2000, n 25, c 2, p 0.05 In the first cse, the decision of cceptnce or rejection of the lot is tken only on single smple of size n = 25. Therefore, the ASN for this pln is simply the smple size, i.e., ASN = n = 25. We clculte ATI using eqution (3). We first clculte the probbility of ccepting lot of qulity p = If X represents the number of defective injection syringes in the smple, the hospitl ccepts the lot if X c = 2. Therefore, the probbility of ccepting the lot is given by 2 P P X 2 P X x x0 Since N 10n, we cn obtin this probbility by using Tble I. From Tble I, for n = 25, x = c = 2 nd p = 0.05, we hve Rectifying Smpling Plns 2 n x x 0 x n x P P X 2 C p 1p On putting the vlues of N, n nd P in eqution (3), we get ATI n 1 P N n It is time for you to do the following exercises bsed on ASN nd ATI. E4) Explin the following terms: i) Averge smple number (ASN), nd ii) Averge totl inspection (ATI). E5) A computer mnufcturer purchses computer chips from compny in lots of 400. Twenty computer chips re smpled from ech lot t rndom nd inspected for defects. The computer mnufcturer ccepts the lot if inspected smple contins t most one defective chip. Otherwise, he rejects it. Suppose the incoming lots contin 3% defective chips. Clculte ASN for this pln. If rejected lots re screened nd ll defective computer chips re replced by non-defective ones, clculte AOQ for this pln. You hve studied four curves, nmely, AOQ, OC, ASN nd ATI in Secs. 6.3 to 6.6. Their min purpose is to ssess the efficiency nd compre the costs of smpling plns. The AOQ nd OC curves serve to ssess protection of the consumer nd the producer in terms of their risks given by the smpling pln nd lso give n ide of the efficiency of the smpling pln. The ASN nd ATI curves give n indiction of the cost of inspection. With the help of these curves, we rrive t blnce between the costs of inspection nd the degree of 13

14 Process Control protection. If inspection (smpling) costs re more thn the costs of wrong decisions (without the id of such inspection), then tht prticulr inspection pln cnnot be justified on economic considertions. We now end this unit by giving summry of wht we hve covered in it. 6.7 SUMMARY 1. An cceptnce smpling pln in which rejected lots re 100% inspected is clled rectifying smpling pln. 2. The qulity of the lots before inspection is known s incoming qulity nd the qulity of the lots which hve been ccepted fter the inspection is known s outgoing qulity. 3. The expected qulity of the lots fter the ppliction of smpling inspection is clled verge outgoing qulity (AOQ) nd is given by Number of defective units in the lot fter inspection AOQ Totl number of units in the lot 4. The mximum vlue of AOQ represents the worst possible verge for the outgoing qulity nd is known s the verge outgoing qulity limit (AOQL). 5. The OC curve for smpling pln is grph of the probbility of ccepting the lot versus the proportion or frction defective in the lot. It shows the bility of the pln to distinguish between good nd bd lots. 6. A smpling pln which perfectly discrimintes between good nd bd lots is clled n idel smpling pln nd the OC curve for this smpling pln is clled n idel OC curve. 7. The precision with which smpling pln differentites between good nd bd lots increses with the size of the smple: The greter the slope of the OC curve, the greter is the discrimintion power of the pln. The pln with smller cceptnce number hs more discrimintion power. 8. The verge smple number (ASN) is defined s the verge (expected) number of smple units per lot, which is required to rrive t decision bout the cceptnce or rejection of the lot under cceptnce smpling pln. The curve drwn between ASN nd lot qulity (p) is known s the ASN curve. 9. The verge number of units inspected per lot under the rectifying smpling pln is clled the verge totl inspection. The curve drwn between ATI nd lot qulity (p) is known s the ATI curve. 6.8 SOLUTIONS/ANSWERS E1) To check the qulity of the lot, the qulity inspector of the hospitl drws 25 syringes rndomly from ech lot of disposble injection syringes nd inspects ech nd every syringe of the smple. He/she clssifies ech syringe s defective or non-defective on the bsis of certin defects. At the end of the inspection, he/she counts the number of defective syringes found in the smple nd compres the number of defective syringes found in the smple with the cceptnce number. If the number of defective syringes in the smple is greter thn c = 2, 14

15 sy, 3, insted of rejecting the lot, he/she clls for 100% inspection of this lot nd replces ll defective syringes found in the lot by nondefective syringes. Then he/she ccepts the lot. If the number of defective syringes is less thn or equl to c = 2, sy, 1, he/she ccepts the lot by replcing ll defective syringes in the smple by nondefective syringes. E2) It is given tht N 500, n 20, c 1nd p 0.01 We clculte AOQ for the single smpling pln using eqution (2). We first clculte P. Since N 10n, we cn obtin this probbility by using Tble I. From Tble I, for n = 20, x = c = 1 nd p = 0.01, we hve Rectifying Smpling Plns 1 n x x0 x n x P P X 1 C p 1p On putting the vlues of N, n, p nd P in eqution (2), we get AOQ E3) It is given tht p N n P N N 2000, n 25, c 2 nd p 0.05 We clculte AOQ for the single smpling pln using eqution (2). Since here N 10n, the probbility of ccepting the lot cn be obtined s in E2. 2 n x x0 x nx P P X 2 C p 1p On putting the vlues of N, n, p nd P in eqution (2), we get AOQ p N n P N E4) Refer to Sec. 6.5 for (i) nd Sec. 6.6 for (ii). E5) In this smpling pln, the decision of cceptnce or rejection of the lot is tken only using single smple of size n = 20. Therefore, the ASN for this pln is simply the smple size n = 20. It is given tht N 400, n 20, c 1, p 3% 0.03 We clculte ATI for the single smpling pln using eqution (3). Since N 10n, we cn obtin the probbility of ccepting lot of qulity p = 3% = 0.03 using Tble I. 15

16 Process Control From Tble I, for n = 20, x = c = 1 nd p = 0.03, we hve n x x 1 x0 n x P p P X 1 C p 1p On putting the vlues of N, n nd P in eqution (3) we get ATI n 1 P N n