UNIT 5 ACCEPTANCE SAMPLING PLANS 5.1 INTRODUCTION. Structure. 5.1 Introduction. 5.2 Inspection 5.3 Acceptance Sampling Plan

Transcription

1 UNIT 5 ACCEPTANCE SAMPLING PLANS Struture 5.1 Introdution Objetives 5.2 Inspetion 5.3 Aeptne Smpling Pln Advntges nd Limittions of Aeptne Smpling Types of Aeptne Smpling Plns 5.4 Implementtion of Aeptne Smpling Pln for Attributes 5.5 Terms Used in Aeptne Smpling Plns 5.6 Produer s Risk nd Consumer s Risk 5.7 Summry 5.8 Solutions/Answers 5.1 INTRODUCTION In mny situtions, the produt is so omplex tht ll omponents/prts of the produt re not mde by mnufturer. In suh ses, one or more omponents of the produt re purhsed from n outside gent or supplier nd the mnufturer does not hve diret ontrol over the qulity of suh omponents. Sine the finl produt is produed by the mnufturer, he/she fes problems suh s: How to ontrol the qulity of omponents reeived from others? How to ensure tht the lots produed do not ontin n exessively lrge proportion of defetive produts? Do the produts meet the desired speifitions? Suh problems belong to the tegory of produt ontrol. Produt ontrol refers to ontrol the produts in suh wy tht these re free from defets nd onform to speifitions. You hve lernt in Unit 1 tht produt ontrol n be done by 100% inspetion, i.e., eh nd every unit produed or reeived from the outside suppliers is inspeted. This type of inspetion hs the dvntge of ensuring tht ll defetive units re eliminted. However, it is very time onsuming nd ostly. Also, if the unit is destroyed under investigtion, e.g., light bulb, rker, mmunition, et., 100% inspetion is not prtible. In 1920, Hrold F. Dodge nd Hrry G. Roming developed sttistil methods for produt ontrol known s eptne smpling or smpling inspetion s n lterntive of 100% inspetion. Now--dys, produt ontrol is hieved through eptne smpling. In this unit, we introdue the eptne smpling plns nd the relted terminology. In Se. 5.2, we explin the mening of the term inspetion s used in industry. We disuss the onept of eptne smpling pln, its dvntges nd limittions nd the types of eptne smpling plns in Se In Se 5.4, we disuss the implementtion of the eptne smpling pln. In Ses. 5.5 nd 5.6, we explin the bsi terminology relted to eptne 1

2 Produt Control smpling pln, suh s lot, probbility of epting lot, eptne qulity level (AQL), lot tolerne perent defetive (LTPD), produer s risk nd onsumer s risk. In the next unit, we shll disuss the retifying smpling plns. Objetives After studying this unit, you should be ble to: distinguish between 100% inspetion nd smpling inspetion; define eptne smpling pln nd disuss the proedure of its implementtion; ompute the probbility of epting or rejeting lot; define eptne qulity level (AQL) nd lot tolerne perent defetive (LTPD) of the lot; nd ompute the produer s risk nd onsumer s risk for n eptne smpling pln. 5.2 INSPECTION A Go-No-Go guge is n inspetion tool whih is used to hek n item or unit or piee ginst its llowed tolernes. The nme Go-No-Go derives from its use. It mens tht we hek the item nd if the item is eptble (fulfil the speifitions), we sy Go nd if uneptble, we sy No-Go. 2 You hve lernt briefly bout 100% inspetion nd smpling inspetion in Se of Unit 1. In this setion, we disuss these onepts in some more detil. When mnufturer produes produt or buys some prts of the produt from outside gents or suppliers, he/she would like to ensure tht the finl produt is s per speifitions. For this purpose, he/she inspets the lot t every strtegi point. The method of heking, mesuring or testing one or more qulity hrteristis of the produt or the prts nd determining whether it stisfies the required speifitions or not, is lled inspetion. Inspetion is of two types: i) Inspetion of Vribles In this type of inspetion, the qulity hrteristi (s) of n item/unit is (re) mesured nd ompred with the required speifitions. For exmple, the desired speifition for the dimeter of bll bering is 50 mm, length of the refill of bll pen is 14 m, weight of riket bll is 162 grm, nd so on. ii) Inspetion of Attributes In inspetion of ttributes, tul mesurements re not tken. Insted, the item/units re tegorised s defetive or non-defetive on the bsis of Go-No-Go guges. It mens tht if unit fulfils ll required qulity hrteristis, it is tegorised s non-defetive nd if not, it is tegorised s defetive. In suh type of inspetion, the number of defetive units (or defets) is ounted. Methods of Inspetion There re two methods of inspetion: i) 100% Inspetion In this method of inspetion, eh nd every item/unit of ny given lot is inspeted. A deision regrding the qulity of the entire lot is tken on the bsis of ll inspeted units of the lot. This proedure needs huge expenditure of time,

3 money, lbour nd resoures. Also, if the produt is suh tht it is ompletely destroyed under the proess of inspetion (e.g., rker, mmunition, et.) then in suh ses, 100% inspetion is neither prtible nor eonomil. As n lterntive, we use smpling inspetion. ii) Smpling Inspetion In smpling inspetion method, some items/units (lled smple) re rndomly seleted from lot in suh wy tht the seleted smple is true representtive of the entire lot. Then eh nd every unit of the seleted smple is inspeted. A deision regrding the qulity of the entire lot is tken on the bsis of the informtion obtined from the smpled units. The question now is: How do we tke ny deision bout the qulity of lot on the bsis of smpling inspetion? For this, we need to lern bout the eptne smpling pln. 5.3 ACCEPTANCE SAMPLING PLAN Aeptne Smpling Plns Let us first define eptne smpling. A smpling inspetion in whih deision bout eptne or rejetion of lot is bsed on one or more smples tht hve been inspeted is known s eptne smpling. In other words: Aeptne smpling is tehnique in whih smll prt or frtion of the units/items is seleted rndomly from lot nd the seleted units re inspeted to deide whether the lot should be epted or rejeted on the bsis of the informtion provided by the smple inspetion. For exmple, suppose mnufturer of riket blls supplies them in lots of 500. A buyer wnts to inspet 20 blls from eh lot before epting the lot nd tkes deision bout eptne or rejetion of the lot on the bsis of the informtion provided by this smple. This is n exmple of eptne smpling beuse the deision bout the lot is tken on the bsis of smple. Consider nother exmple. Suppose mobile phone ompny pks the mobile phones it produes in lots of 100 units. To hek the qulity of the lots, the qulity inspetor of the ompny drws rndom smple of size 10 from eh lot. He/she tkes deision bout eptne or rejetion of the lot on the bsis of the informtion provided by this smple. When lot is rejeted on the bsis of the eptne smpling, it does not men tht ll units of the lot re of no use or ll units re defetive. It mens tht ll units of the lot hve been refully exmined nd the defetive units hve been either repired or removed. Thus, the remining units in the lot onform to prtiulr qulity level. The rejeted units my be sold t lower prie or the defets in the units my be removed so tht it n be retined. For instne, the lerne sles of, e.g., redy-mde lothes or shoes, et. re rried out to sell those items whih re not up to the speifitions or hve been rejeted. This is loss to the produer or the mnufturer, whih n be prevented if he/she ould mintin the desired level of qulity. Therefore, mnufturers should be very reful bout the qulity of goods, whih they supply to the ustomers. We n now nswer the question: Wht is n eptne smpling pln? 3

4 Produt Control An eptne smpling pln is speifi pln tht lerly sttes the rules for smpling nd the ssoited riteri for eptne or rejetion of lot. Aeptne smpling plns n be used for the inspetion of: 1. Mnuftured units/items, 2. Components, 3. Rw mterils, 4. Opertions, 5. Mterils in proess, 6. Supplies in storge, 7. Mintenne opertions, 8. Dt or reords, nd 9. Administrtive proedures. Aeptne smpling hs mny dvntges nd some limittions, whih we now desribe Advntges nd Limittions of Aeptne Smpling The min dvntges of eptne smpling re s follows: i) It is less expensive in terms of money, time nd lbour in omprison to 100% inspetion. ii) iii) iv) For items, whih nnot be used fter single inspetion, suh s rkers, bulbs, tube lights, food, et., 100% inspetion is not prtible. Smpling inspetion is the only wy for inspeting suh items. In eptne smpling, smple of smll number of items or units is inspeted nd hene smller inspetion stff is required. In mny ses, eptne smpling provides better outgoing qulity. In generl, it is seen nd greed tht good 100% inspetion removes only 85% to 90% of the defetive items, wheres very good 100% inspetion removes only 99% of the defetive items. However, due to humn error, it usully does not reh the 100% mrk. In other words, we n sy tht 100% inspetion is not lwys relible beuse it involves too muh routine work for the persons inspeting eh nd every item. Due to this, defetive items my lso be lbelled s stisftory nd my lso be epted t times when these persons re distrted. Hene, n pproprite smpling pln is preferble. v) Due to quik inspetion through the eptne smpling, the sheduling nd delivery times re sved. Aeptne smpling hs some limittions whih re given below: i) Sine, in eptne smpling, the entire lot is epted or rejeted on the bsis of onlusions drwn from one or more smples, there is lwys some risk of mking wrong inferene bout the qulity of the lot. These risks re termed the produer s risk nd onsumer s risk. You will lern more bout these in Se ii) The suess of eptne smpling depends on the rndomness of the smple, qulity hrteristis to be tested, lot size, eptne riteri, 4

5 et. Therefore, it is speilised job requiring reful plnning nd exeution nd every one nnot undertke it. So fr you hve lernt bout the eptne smpling pln nd its dvntges nd limittions. It is lso importnt for you to know bout different types of eptne smpling plns used in industries. Aeptne Smpling Plns Types of Aeptne Smpling Plns There re severl types of eptne smpling plns bsed on different pprohes tht n influene the deision bout lot. These re tegorised s follows: 1. Aeptne smpling plns for ttributes, nd 2. Aeptne smpling plns for vribles. 1. Aeptne Smpling Plns for Attributes Aeptne smpling plns in whih tul mesurements of the qulity hrteristis re not mde, but the units/items re tegorised s defetive or non-defetive on the bsis of Go-No-Go guges re lled eptne smpling plns for ttributes. These plns re of two types: A. Lot-By-Lot Aeptne Smpling Plns for Attributes Lot-by-lot eptne smpling plns for ttributes re the most ommonly used smpling plns nd, therefore, re simply lled eptne smpling plns for ttributes. Suh plns re used whenever the units to be inspeted n be onveniently grouped into bthes or lots. In lot-by-lot eptne smpling pln for ttributes, generlly, we use the following plns: i) Single smpling plns In the single smpling pln, the deision bout the eptne or rejetion of lot is bsed on single smple tht hs been inspeted. This is the simplest type of smpling pln. ii) Double smpling plns In the double smpling pln, the deision bout the eptne or rejetion of lot requires the evidene of two smples drwn from the lot. If the lot qulity is good (or bd), the lot is epted (or rejeted) on the bsis of the first smple. If the first smple shows n intermedite qulity, the deision bout the lot is tken on the evidene of the first nd seond smple ombined. This is more omplited thn the single smpling pln. iii) Multiple smpling plns A multiple smpling pln is n extension of the double smpling pln. This smpling pln my require more thn two smples to reh deision bout the eptne or rejetion of lot. iv) Sequentil smpling plns In the sequentil smpling pln, the smple size is not pre-fixed s in the se of single, double nd multiple smpling plns. The units/items re drwn from the lot, one t time nd inspeted. The deision bout epting or rejeting of the lot or ontinuing with the inspetion by tking one more unit from the lot, is mde on the bsis of informtion vilble up to tht stge. 5

6 Produt Control The multiple nd sequentil smpling plns re beyond the sope of this ourse. However, in Units 7 nd 8, we shll disuss the single smpling plns nd double smpling plns in detil. B. Continuous Prodution Aeptne Smpling Plns for Attributes Continuous prodution eptne smpling plns for ttributes re used when the units/items to be inspeted nnot be grouped into lots or bthes. Mny mnufturing opertions do not rete lots beuse in these opertions, the units re produed in ontinuous proess on onveyor belt or other stright-line systems. Continuous prodution eptne smpling plns for ttributes re beyond the sope of this ourse. 2. Aeptne Smpling Plns for Vribles The eptne smpling plns in whih tul mesurements of the qulity hrteristis re tken re lled eptne smpling plns for vribles. The eptne smpling plns for vribles re beyond the sope of this ourse. So we now fous on the eptne smpling plns for ttributes nd desribe the generl proedure for implementing it. 5.4 IMPLEMENTATION OF ACCEPTANCE SAMPLING PLAN FOR ATTRIBUTES Suppose tht lots of the sme size, sy, N, re reeived from the supplier or the finl ssembly line nd submitted for inspetion one t time. The proedure for implementing the eptne smpling pln to rrive t deision bout the lot is desribed in the following steps: Step 1: We drw rndom smple of size n from the lot reeived from the supplier or the finl ssembly. Step 2: We inspet eh nd every unit of the smple nd lssify it s defetive or non-defetive on the bsis of ertin riteri. At the end of the inspetion, we ount the number of defetive units found in the smple. Step 3: We ompre the number of defetive units found in the smple with the eptne riteri. Step 4: If the eptne riteri re stisfied, we ept the entire lot. Otherwise, we rejet the entire lot. The steps desribed bove re shown in Fig

7 Aeptne Smpling Plns Fig. 5.1: Proedure for implementtion of the eptne smpling pln. Let us explin these steps further with the help of n exmple. Suppose buyer of riket blls wnts to inspet smple of 20 blls from eh lot to hek their qulity. The eptne riterion is tht if the smple ontins t most one defetive bll, the lot would be epted. Otherwise, it would be rejeted. The buyer drws 20 blls from eh lot nd inspets eh nd every bll of the smple nd lssifies eh bll s defetive or nondefetive on the bsis of ertin defets. At the end of the inspetion, he/she ounts the number of defetive blls found in the smple nd ompres the number of defetive blls with the eptne riterion. If the number of defetive blls in the smple is more thn 1, he/she rejets the lot. If the number of defetive blls is zero or 1, he/she epts the lot. We shll disuss different types of smpling inspetion plns in detil in the remining units. But, before studying these plns, you should be wre bout the relted terminology. This is wht we disuss in Se TERMS USED IN ACCEPTANCE SAMPLING PLANS In this setion, we define some terms used in eptne smpling pln. 1. Lot A lot is the olletion of units or items from whih smple is tken nd inspeted to determine its eptbility. It is found tht lot formtion influenes the effetiveness of the eptne smpling pln. Therefore, formtion of the lot is importnt for the suess of n eptne smpling pln. Some guidelines for the lot formtion re s follows: i) Units in the lot should be homogeneous. It mens tht the lot should onsist of units produed by the sme mhine, sme opertors, using the sme rw mterils nd pproximtely during the sme time period. ii) Units in the lots should be pked so tht the shipping osts nd hndling risks re minimum. This lso mkes the seletion of units in the smple esy.

8 Produt Control 8 The deision bout the eptne or rejetion of ll units of the lot is tken on the bsis of the results of inspetion of the smple. Therefore, the smple should be suh tht it is true representtive of the lot. 2. Sentening The t of epting or rejeting the entire lot is lled sentening the lot. 3. Lot Size The number of units in lot is lled lot size. It is denoted by N. 4. Smple Size The number of units inspeted to sentene lot is lled smple size. It is denoted by n. 5. Lot Qulity The proportion of defetive units in lot is lled lot qulity or proportion defetive. It is denoted by p nd defined s Number of defetive units in lot p... (1) lot size From the definition of p given by eqution (1), you should note tht for the sme lot size, s p inreses, the lot qulity dereses. 6. Aeptne Number When del is finlised between seller nd buyer, they deide on the mximum number of llowble defetive units in smple. This number is lled eptne number nd is denoted by. The rule for eptne or rejetion of lot is tht if the number of defetive units observed in the smple is less thn or equl to the eptne number, the lot will be epted. Otherwise, it will be rejeted. Let us explin the bove terminology with the help of n exmple. Exmple 1: Suppose riket bll mnufturing ompny supplies lots of 500 blls. To hek the qulity of the lots, buyer drws rndom smple of size 20 blls from eh lot nd epts the lot if the inspeted smple ontins t most one defetive bll. Otherwise, he/she rejets the lot. If the lot onsists of 10 defetive blls, find N, n, p nd. Solution: Sine the lot ontins 500 blls, the lot size N = 500. Sine the buyer drws smple of size 20 blls from eh lot to tke the deision bout the lot, the smple size n = 20. The buyer epts the lot if the inspeted smple ontins t most one defetive bll. So the eptne number = 1. The lot onsists of 10 defetive blls. So from eqution (1), the lot qulity is given s Number of defetive units in lot 10 p 0.02 lot size 500 Now tht you hve lernt bout terms suh s lot size, lot qulity, smple size, et., you my like to know: How n we be onfident tht lot of good qulity will be epted nd lot of bd qulity will be rejeted? We n nswer suh questions by lulting the probbility of epting lot of speified qulity. This probbility will be 1 if ll units in the lot re epted

9 nd 0, if ll units in the lot re rejeted. The probbility of epting lot is denoted by P (p) or in short s P. 7. Probbility of epting lot (P ) Suppose tht the lots of the sme size, sy, N, re reeived from the supplier or the finl ssembly line nd submitted for inspetion one t time. The lot is epted if the number of defetive units observed in the smple is less thn eptne number (). Otherwise, it is rejeted. Suppose rndom smple of size n is drwn from eh lot for inspetion. If X represents the number of defetive units in the smple, the lot is epted if X. It mens tht we ept the lot if X = 0 or 1 or 2,... or. Therefore, the probbility of epting the lot is given by P p P P X P X 1 or 2 or 3,..., or... (2) Sine X = 0, 1,..., re mutully exlusive events, from the ddition theorem of probbility, we hve P p P X P X 0 P X 1... P X or P p PX x... (3) x0 We n esily lulte this probbility if we know the probbility distribution of X. Generlly, in qulity ontrol, rndom smple is drwn without replement. So the number of defetive units (X) in the smple follows hypergeometri distribution. However, we know tht when the lot size (N) is lrge ompred to the smple size, i.e, N 10n, the hypergeometri distribution pproximtes binomil distribution with prmeters n nd p where p is the lot qulity. It is esier to lulte the probbilities with the help of the binomil distribution rther thn with the hypergeometri distribution. Therefore, the probbility of getting extly x defetive units in smple of size n using the binomil distribution is given by n x n x P X x C p 1 p... (4) Hene, from eqution (3), we hve x Aeptne Smpling Plns You hve studied in Unit 3 of MST-003 tht if A nd B re mutully exlusive events then P A or B P A P B The nottion n n lso be repersened s n. x C x n x n x x... (5) P p P X P X x C p 1 p x0 x0 However, for rpid lultion, we n use Tble I entitled Cumultive Binomil Probbility Distribution given t the end of this blok. Let us now ompute P (p) for Exmple 1. In this exmple, we hve N 500, n 20, 1, p 0.02 Suppose X represents the number of defetive blls in the smple. The buyer epts the lot if the number of defetive blls (X) in the smple is t most one, i.e., = 1. Therefore, the probbility of epting the lot of qulity p = 0.02 is given by P p P X P X 1 P X 0 P X 1 9

10 Produt Control 1 x0 P X x Sine the lot size (N) is lrge ompred to the smple size (n), i.e., N 10n, X pproximtely follows the binomil distribution with prmeters n nd p where p is the lot qulity. Therefore, the probbility of epting the lot of qulity p is given by 1 1 n x x P p P X x C p 1 p x0 x0 For rpid lultion, we n use Tble I for obtining this probbility. From Tble I, for n = 20, x = = 1 nd p = 0.02, we hve 1 n x nx Cx p 1 p x0 Therefore, the probbility of epting the lot of qulity p = 0.02 is given by n x n x n x P X x C p 1 p x 1 x n x.. (6) x0 n x P p C p 1 p It mens tht if there re severl lots of the sme qulity p = 0.02, bout 94.01% of these will be epted nd bout 5.99% will be rejeted. We n lso lulte this probbility mnully using the sientifi lultor for n = 20, p = 0.02 s follows: P p P X 1 P X 0 P X C C Now, we n see the effet of the lot qulity on the probbility of epting the lot. Suppose the lot in Exmple 1 ontins 20 defetive blls insted of 10. Then the lot qulity is: Number of defetive units in the lot 20 p 0.04 lot size 500 We n lulte the probbility of epting the lot of qulity p = 0.04 with the help of Tble I s disussed bove. From Tble I, for n = 20, x = = 1 nd p = 0.04, we hve P p P X In this se, bout 81.03% lots will be epted nd bout 18.97% lots will be rejeted. If we ompre this probbility with the probbility obtined for p = 0.02, we observe tht s the lot qulity dereses (from p = 0.02 to p = 0.04), the 10

11 probbility of epting the lot lso dereses (from P = to P = ). Aeptne Smpling Plns We n lso see the effet of the eptne number on the probbility of epting the lot. Suppose the buyer epts the lot if the inspeted smple ontins t most two defetive blls, tht is, = 2. Then we n obtin the probbility of epting the lot using Tble I. From Tble I, for n = 20, x = = 2 nd p = 0.02, we hve P p P X P X It mens tht if there re severl lots of the sme qulity p = 0.02, then for the eptne number = 2, bout 99.29% lots will be epted nd bout 0.71% lots will be rejeted. If we ompre this probbility with the probbility obtined for p = 0.02 nd = 1, we observe tht when the eptne number inreses (from = 1 to = 2), the probbility of epting the lot lso inreses (form P = to P = ). It is time for you to puse nd solve some exerises for prtie. E1) A shopkeeper purhses pens from pen ompny in rtons (lots) tht usully ontin one thousnd pens. To hek the qulity of the pens, the shopkeeper selets 25 pens t rndom from eh rton nd visully inspets eh seleted pen for ertin defets. The shopkeeper epts the lot if the inspeted smple ontins t most two defetive pens. Otherwise, he/she rejets the lot. If there re 40 defetive pens in eh rton, find N, n, p, nd P. E2) For smpling pln with n = 5 nd = 0, find the probbility of epting lot tht hs 2% defetive units by ssuming tht the number of defetive units in smple follows binomil distribution. 8. Aeptne Qulity Level (AQL) The seller nd the buyer generlly know tht the supply of ompletely defet-free lots is not possible. So they usully negotite nd rrive t n greement tht the buyer will ept ll lots whih hve t most definite qulity level or definite perentge of defetive units. This definite qulity level is known s eptne qulity level (AQL). It is denoted by p 1. Hene, AQL n be defined s follows: Aeptne qulity level is the qulity level deided in negotition of the seller nd the buyer: If the proportion of defetive units in lot is less thn or equl to AQL, the buyer will hve to definitely ept the lot. Otherwise, the buyer my ept or rejet the lot. Suppose 100% inspetion of ll lots is rried out nd the lot qulity (p) for eh lot is omputed. Then ll lots with lot qulity p AQL will be epted nd ll lots with p > AQL my either be epted or rejeted. In Exmple 1, if the mnufturing ompny nd the buyer deide tht the buyer will ept ll lots whih hve t most 2% defetive units, then eptne qulity level (AQL) of this del is 2%. 11

12 Produt Control Even though the seller nd the buyer deide tht the buyer would ept ll lots of AQL qulity, this my not hppen in prtil situtions. In Exmple 1, the probbility of epting the lot of the lot qulity p = AQL = 0.02 ws P p P X It mens tht if there re severl lots of the sme qulity p = 0.02 = AQL, bout 94.01% of these would be epted nd bout 5.99% would be rejeted. This is obviously risk of the mnufturer beuse it ws greed upon by both tht ll lots of qulity 0.02 will be epted wheres the buyer is rejeting 5.99% of them. This risk is known s produer s risk whih is explined in detil in Se Lot Tolerne Perent Defetive (LTPD) In order to redue the produer s risk, the buyer grees to tolerte lot qulity worse thn eptne qulity level (AQL) up to ertin limit but not beyond it. This limiting vlue is known s the lot tolerne perent or proportion defetive (LTPD). It is lso known s rejetble qulity level (RQL), unstisftory qulity level or limiting qulity level (LQL). LTPD is lso deided t the time when the eptne qulity level is deided in the negotition between the produer nd the buyer. They mke n greement tht the buyer will definitely rejet the lot of qulity equl to or greter thn LTPD. Therefore, LTPD n be defined s follows: Lot tolerne perent or proportion defetive is the qulity level deided in negotition between the produer nd the buyer: If the proportion of defetive units in lot is equl to or greter thn this level, the buyer will definitely rejet the lot. Otherwise, the buyer my ept or rejet the lot. The lot tolerne perent defetive (LTPD) of smpling pln is the level of qulity t whih the lot is routinely rejeted by the smpling pln. LTPD is greter thn AQL. In Exmple 1, if the mnufturing ompny nd the buyer deide tht the buyer will ept ll lots whih hve t most 2% defetive units nd rejet ll lots whih hve 5% or more defetive units, then for this pln the AQL is 2% nd the LTPD is 5%. This level represents the dividing line between good nd bd lots. Therefore, lot tolerne perent defetive is the qulity level of lot tht the buyer onsiders bd nd he/she would like to rejet ll lots tht hve this level of qulity. LTPD is denoted by p 2. In eptne smpling, the eptne or rejetion of the entire lot depends on the onlusions drwn from the smple. Thus, there is lwys hne of mking wrong deision. It mens tht lot of good qulity my be rejeted nd lot of poor qulity my be epted. This leds to two kinds of risks: i) Produer s risk, nd ii) Consumer s risk. We now disuss these risks in some detils. 5.6 PRODUCER S RISK AND CONSUMER S RISK In qulity ontrol, produer my be defined s follows: 12

13 A person or firm or n orgnistion tht produes/mnuftures goods or provides servies for use or onsumption of nother person or firm or orgnistion is known s produer. Aeptne Smpling Plns In Exmple 1, the mnufturing ompny tht produes the riket blls, is the produer. Let us explin wht is ment by produer s risk. Produer s Risk It my hppen in prtie tht smpling inspetion pln leds to the rejetion of lot of stisftory or good qulity. This mens tht there is possibility of rejeting lot hving qulity level less thn or equl to the eptne qulity level (AQL) due to smpling inspetion. If lot of good qulity is rejeted, the produer suffers loss. Therefore, the produer lwys fes the risk of good lot being rejeted. Suh risk is known s produer s risk nd is defined s follows: The probbility of rejeting lot of eptne qulity level (AQL) is known s the produer s risk. It is denoted by P p (p) or in short s P p nd given by p P p P P rejeting lot of eptne qulity level This n be written s p p P p 1 P epting lot of eptne qulity level 1 P (p p 1)... (7) The produer s risk is equivlent to the type I error in hypothesis testing disussed in Unit 8 of MST-004 entitled Sttistil Inferene. Therefore, it is lso denoted by α. For omputing the produer s risk, we hve to ompute the probbility of epting lot of qulity p = AQL. Then we use eqution (7) to ompute the produer s risk. Let us explin how to ompute the produer s risk with the help of n exmple. Exmple 2: Suppose in Exmple 1, the mnufturing ompny nd the buyer gree tht AQL = Find the produer s risk for this pln. Solution: We know tht the produer s risk is p P p 1 P epting lot of eptne qulity level 1 P p For omputing the produer s risk, we first ompute the probbility of epting lot of qulity (p) = AQL = We hve lredy lulted this probbility in Exmple 1. So here we use the result: P p Therefore, the produer s risk for this pln is P p 1 P p p It mens tht if there re severl lots of the sme qulity p = 0.02 s AQL, out of these lots, bout 5.99% will be rejeted. This is obviously risk of the mnufturer (produer) beuse it ws greed upon by both tht ll lots of qulity p =AQL = 0.02 will be epted wheres the buyer is rejeting 5.99% of the lots. Let us now explin the onsumer s risk. 13

14 Produt Control Consumer s Risk A person or firm or n orgnistion tht purhses goods for its own need or onsumption or for use in the prodution of other goods (not for resle to nother person or firm or orgniztion diretly or indiretly) is known s onsumer. In Exmple 1, sine the buyer purhses the riket blls from the mnufturer, he/she is the onsumer. Just s the produer hs risk of lot of good qulity being rejeted, onsumer lso hs risk of buying lot of unstisftory qulity. Suh risk is known s the onsumer s risk. If p 2 is the mximum proportion of defetive (LTPD) in the lot, whih the onsumer is redy to tolerte, the onsumer s risk my be defined s follows: The probbility of epting lot of unstisftory qulity, i.e., LTPD is known s onsumer s risk. It is denoted by P (p) or in short s P. Thus, P p P epting lot of qulity = LTPD... (8) Note tht P (p) is the sme s P (p) for p = LTPD. The onsumer s risk is denoted by β beuse it is equivlent to the type II error desribed in Unit 8 of MST-004. Let us explin how to ompute the onsumer s risk with the help of n exmple. Exmple 3: Suppose in Exmple 1, the mnufturing ompny nd the buyer gree tht LTPD = Find the onsumer s risk for this pln. Solution: We know tht the onsumer s risk is given by It is given tht P p P epting lot of qulity = LTPD P (p 0.05) N 500, n 20, 1 nd LTPD 0.05 We first lulte the probbility of epting the lot of qulity p = 0.05 s follows: P p P X P X 1 We n lulte this probbility s we hve disussed in Exmple 1. From Tble I, for n = 20, x = = 1 nd p = LTPD = 0.05, we hve P p P X Therefore the onsumer s risk is given by P p P p It mens tht if there re severl lots of the sme qulity p = 0.05, out of these bout 73.58% of them will be epted by the buyer even though this qulity is unstisftory. This is obviously the buyer s risk. We tke up n exmple to further explin the terms AQL, LTPD, produer s risk nd onsumer s risk. 14

15 Exmple 4: A mobile mnufturing ompny hs deided to purhse the mobile btteries from bttery mnufturing ompny. Both mnufturing ompnies hve deided tht the btteries re to be supplied in lots of 1000 btteries eh. The lot will be epted up to qulity level p = 0.05 nd rejeted t more thn qulity level p = Aeptne smpling pln is bsed on smple of size 25 drwn from eh lot nd the lot is epted if inspeted smple ontins t most one defetive bttery. Otherwise, the lot is rejeted. Identify whih ompny is the produer nd whih one is the onsumer in the pln. Clulte the produer s risk nd the onsumer s risk. Solution: Here the mobile mnufturing ompny purhses the mobile btteries from bttery mnufturing ompny. So it is onsumer. The bttery mnufturing ompny supplies btteries to the mobile mnufturing ompny. So it is produer. It is given tht N 1000, AQL 0.05, LTPD 0.20, n 25 nd 1 We know tht the produer s risk is defined s Pp P rejeting lot of eptne qulity level 1 P epting lot of qulity = AQL = P p (i) To lulte the produer s risk, we hve to lulte the probbility of epting lot of qulity p = Suppose X denotes the number of defetive mobiles in the smple. The mobile ompny epts the lot if the number of defetive mobiles (X) in the smple is t most one. Therefore, the probbility of epting the lot of qulity p is given by P p P X P X 1 Sine the lot size (N) is lrge ompred to the smple size (n), i.e., N 10n, we n use the binomil distribution nd n use Tble I for obtining the P X 1. probbility Aeptne Smpling Plns From Tble I, for n = 25, x = = 1 nd p = 0.05, we hve P X Therefore, the probbility of epting the lot of qulity p = 0.05 is given by P (p 0.05) From (i), we lulte the produer s risk s follows: p P 1 P p Similrly, we know tht the onsumer s risk is defined s P P epting lot of qulity p LTPD 0.20 P (p 0.20) From Tble I, for n = 25, x = = 1nd p = 0.2, we hve P P p You my like to solve some exerises for prtie. 15

16 Produt Control E3) Identify the onsumer nd produer in the exerise E1. If the shopkeeper nd the pen ompny hve deided tht AQL = 0.03 nd LTPD = 0.10, lulte the produer s risk nd the onsumer s risk. E4) If in the exerise E2, the eptne qulity level (AQL) nd the Lot tolerne perent defetive (LTPD) re 1% nd 5%, respetively, lulte the produer s risk nd the onsumer s risk for this pln. We end this unit by giving summry of wht we hve overed in it. 5.7 SUMMARY 1. The tehnique of ontrolling the qulity of produts in suh wy tht these re free from defets nd onform to their speifitions, is lled produt ontrol. 2. The method of heking, mesuring, testing one or more qulity hrteristis of produt (unit) to determine whether it stisfies the required speifitions or not is lled inspetion. 3. If qulity hrteristi of unit is mesured on ontinuous sle, the inspetion is lled inspetion by vribles. 4. If the tul mesurements of the qulity hrteristi of unit re not tken, but the unit is tegorised s defetive or non-defetive on the bsis of Go-No-Go guges, the inspetion is lled inspetion by ttributes. 5. A lot is the olletion of units or items from whih smple is tken nd inspeted to determine its eptbility. 6. If eh nd every unit of lot is inspeted, the inspetion is known s 100% inspetion. 7. If some items or units re rndomly seleted from lot in suh wy tht seleted smple is true representtive of the entire lot nd eh nd every unit of the seleted smple is inspeted, the inspetion is known s smple inspetion. 8. Aeptne smpling is tehnique in whih smll prt or frtion of units is seleted rndomly from lot nd the seleted units re inspeted to deide whether the lot should be epted or rejeted on the bsis of the informtion supplied by the smple inspetion. 9. The proportion of defetive units in lot is lled lot qulity or proportion defetive, whih is denoted by p nd defined s follows: Number of defetive units in lot p lot size 10. The eptne qulity level (AQL) is the qulity level deided mutully by the mnufturer nd the buyer. If the proportion of defetive units in lot is less thn or equl to AQL, the buyer hs to ept the lot. Otherwise, the buyer my ept or rejet the lot. 11. The lot tolerne perent or proportion defetive (LTPD) is the qulity level deided mutully by the mnufturer nd the buyer. If the proportion of defets in lot is greter thn or equl to this level, the buyer will definitely rejet the lot. Otherwise, the buyer my ept or rejet the lot. 16

17 12. The probbility of rejeting lot of eptne qulity level (AQL) is known s produer s risk P p (p). 13. The probbility of epting lot of unstisftory qulity (LTPD) is known s onsumer s risk P (p). 5.8 SOLUTIONS/ANSWERS E1) The rton ontins 1000 pens nd so the lot size N = Sine the shopkeeper drws smple of size 25 pens, the smple size n = 25. The shopkeeper epts the lot if the inspeted smple ontins t most two defetive pens. Otherwise, he/she rejets the lot. The eptne number = 2 The lot ontins 40 defetive pens Number of defetive pens in the lot 40 p 0.04 lot size 1000 If X represents the number of defetive pens in the smple, then the shopkeeper epts the lot if X = 2. Therefore, the probbility of epting the lot is given by P P X 2 P X 0 P X 1 P X 2 Sine the lot size is lrge s ompred to the smple size ( N 10n ), X pproximtely follows the binomil distribution with prmeters n nd p. Therefore, we n obtin this probbility by using Tble I. From Tble I, for n = 25, x = = 2, nd p = 0.04, we hve E2) It is given tht P P X n = 5, = 0 nd p = 2% = 0.02 Suppose X represents the number of defetive units in the smple, the probbility of epting the lot is given by P P X P X 0 Sine X follows the binomil distribution, we n use Tble I to obtin this probbility. From Tble I, for n = 5, x = = 0, nd p = 0.02, we hve P X Hene, the probbility of epting the lot is given by P P X E3) The shopkeeper purhses pens from pen ompny nd, therefore, he/she is onsumer. The pen ompny is produer sine it sells pens. We hve n = 25, = 2, AQL = 0.03 nd LTPD = 0.10 Aeptne Smpling Plns 17

18 Produt Control The produer s risk is given by Pp P rejeting lot of eptne qulity level 1 P epting lot of AQL = P p 0.03 ` From Tble I, for n = 25, x = = 2 nd p = AQL = 0.03, we hve P p p P 1 P p The onsumer s risk is given by P P epting lot of LTPD qulity P p 0.10 From Tble I, for n = 25, x = = 2 nd p = LTPD = 0.10, we hve P p P P p E4) It is given tht n = 5, = 0, AQL = 1% = 0.01 nd LTPD = 5% = 0.05 The produer s risk is Pp P rejeting lot of eptne qulity level 1 P p 0.01 `From Tble I, for n = 5, x = = 0 nd p = AQL = 0.01, we hve P p p P 1 P p The onsumer s risk is P P epting lot of LTPD qulity P p 0.05 From Tble I, for n = 5, x = = 0 nd p = LTPD = 0.05, we hve P p P P p