CONTROL CHARTS FOR ATTRIBUTES

Transcription

1 UNIT 3 CONTROL CHARTS FOR ATTRIBUTES Structure 3.1 Itroucto Objectves 3.2 Cotrol Charts for Attrbutes 3.3 Cotrol Chart for Fracto Defectve (-Chart) 3.4 -Chart for Varable Samle Sze 3.5 Cotrol Chart for Defectves (-Chart) 3.6 Summary 3.7 Solutos/Aswers 3.1 INTRODUCTION I Ut 2, you have leart about cotrol charts for varables, whch are use to cotrol the measurable qualty characterstcs. You have also leart how to costruct cotrol charts for mea (X-chart) a cotrol chart for varablty (R-chart a S-chart). However, there are may stuatos whch measuremet s ot ossble, e.g., the umber of falures a roucto ru, umber of efects a bolt of cloth, surface roughess of a crcet ball, etc. I such cases, we caot use the cotrol chart for varables. Moreover, cotrol charts for varables ca be use oly for oe measurable characterstc at a tme. Ths may be roblematc f a maufacturg comay or a factory maes a rouct or a art havg more tha oe measurable qualty characterstc, say, 10 qualty characterstcs a the qualty cotroller woul le to cotrol all the characterstcs. The for each qualty characterstc, he/she woul requre a searate cotrol chart for varables. However, t wll be mractcable a uecoomcal to have 10 such charts. I such stuatos, we use cotrol charts for attrbutes, whch rove overall formato about qualty. You wll lear about varous asects of cotrol charts for attrbutes ths ut. I Sec. 3.2, we exla the meag of the term attrbutes ustry a trouce fferet tyes of cotrol charts for attrbutes. I Secs. 3.3 to 3.5, we scuss how to costruct the cotrol charts for attrbutes a terret the result obtae from these charts for costat a varable samle sze, resectvely. I the ext ut, we scuss the cotrol charts for efects. Objectves After stuyg ths ut, you shoul be able to: stgush betwee the cotrol charts for varables a attrbutes; exla the ee for cotrol charts for attrbutes; escrbe fferet tyes of cotrol charts for attrbutes; ece o whch cotrol charts for attrbutes to use a gve stuato; costruct a terret the cotrol chart for fracto efectve (-chart); costruct a terret the -chart for varable samle sze; a costruct a terret the cotrol chart for umber of efectves (-chart). 81

2 Process Cotrol A Go-No-Go gauge s a secto tool whch s use to chec a tem or a ut or a ece agast ts allowe toleraces. The ame Go-No-Go erves from ts use. It meas that we chec the tem a f the tem s accetable (fulfl the secfcatos), we say Go a f uaccetable, we say No-Go. 3.2 CONTROL CHARTS FOR ATTRIBUTES We frst efe the term attrbutes as use qualty cotrol termology. The term attrbutes qualty cotrol refers to those qualty characterstcs, whch classfy the tems/uts to oe of the two classes: coformg or o-coformg, efectve or o-efectve, goo or ba. There are two tyes of attrbutes: ) Where umercal measuremets of the qualty characterstcs are ot ossble, for examle, colour, scratches, amages, mssg arts, etc. ) Where umercal measuremets of the qualty characterstcs are ossble a tems are classfe as efectve or o-efectve o the bass of the secto. For examle, the ameter of a crcet ball ca be measure by the mcrometer but sometmes t may be more coveet to classfy the balls as efectve a o-efectve usg a Go-No-Go gauge (rea the marg remar). I secto of attrbutes, actual measuremets are ot oe, but the umber of efectve tems (efectves) or umber of efects the tem s coute. The sze of efect a ts locato s ot mortat. Items are secte a ether accete or rejecte. There are fferet tyes of cotrol charts for attrbutes for fferet stuatos. We classfy the cotrol charts for attrbutes to two grous as follows: Cotrol charts for efectves, a Cotrol charts for efects. Cotrol charts for efectves are maly of two tyes as gve below: 1. Cotrol chart for fracto efectve (-chart), a 2. Cotrol chart for umber of efectves (-chart). Both cotrol charts for efectves are base o the bomal strbuto. Cotrol charts for efects are also of two tyes as gve below: 1. Cotrol chart for umber of efects (c-chart), a 2. Cotrol chart for umber of efects er ut (u-chart). Cotrol charts for efects are base o Posso strbuto a these charts wll be escrbe the ext ut. I ths ut, we focus o cotrol charts for efectves a scuss the -chart frst. 3.3 CONTROL CHARTS FOR FRACTION DEFECTIVE (-CHART) The most wely use cotrol chart for attrbutes s the fracto (roorto) efectve chart, that s, the -chart. The -chart may be ale to qualty characterstcs, whch caot be measure or mractcable a uecoomcal to measure t. These tems/uts are classfe as efectve or o-efectve o the bass of certa crtera (efects). You have leart that the cotrol charts for varables are use to cotrol oly oe qualty characterstc at a tme. If the qualty cotroller woul le to cotrol two characterstcs, he/she must use two 82

3 cotrol charts for varables. But a sgle -chart may be ale to more tha oe qualty characterstc, fact, to as may qualty characterstcs as we wat. Before escrbg the cotrol chart for fracto efectve (-chart), we exla the meag of fracto efectve. Fracto efectve () s efe as the rato of the umber of efectve terms/uts/artcles fou ay secto to the total umber of tems/uts/artcles secte. Symbolcally, we wrte Cotrol Charts for Attrbutes efectve tems Fracto efectve = Total umber of tems secte (1) For examle, f 500 crcet balls are secte a 20 balls are fou efectve, the fracto efectve of the balls s gve as: Fracto efectve efectve balls Total umber of balls secte Fracto efectve s always less tha or equal to 1 a exresse as a ecmal or fracto. We ow exla how to costruct the -chart. The roceure of costructg a -chart s smlar to that of the cotrol charts for varables. The ma stes of costructg a -chart are as follows: Stes Ivolve Costructo of a -chart Ste 1: We efe o-coformty (efectveess) roerly because t ees o the rouct, ts use, cosumer ees, etc. For examle, a scratch mar a mache mght ot be cosere o-coformty whereas the same scratch o a moble hoe, otcal les, lato, etc. woul be cosere o-coformty. Ste 2: We eterme the subgrou (samle) sze a the umber of subgrous (samles). The selecto of samle sze for cotrol charts for attrbutes s very mortat. The samle sze for cotrol charts for attrbutes shoul be large eough to allow o-coformtes (o-coformg/efectve tems) to be observe the samles. The samle sze for cotrol charts for attrbutes s a fucto of the roorto of o-coformg (efectve) tems. For examle, f a rocess has 2% efectve tems, a samle of 20 tems s ot suffcet because the average umber of efectve tems er samle s It meas that there s oly 40% chace that efectve tems coul be observe the samle. Therefore, msleag ferece coul be mae. So for efectve tems, we woul requre may more samles. I such a stuato, samle sze of 100 shoul be suffcet. Thus, f a rocess s of very goo qualty, the -chart requres large samle sze to eterme lac of cotrol. If the rocess s of oor qualty, small samle sze s suffcet. Geerally, we ece the samle sze as er the table gve below: Percetage of Samle Sze 83

4 Process Cotrol Defectve 0 to 10% 20 0 to 6% to 2.94% to 2.19% Note that etermg the samle sze for a -chart, we requre some relmary observato to obta a rough ea about the roorto of efectve tems a ther average umber. The umber of samles ees uo the roucto rate a cost of samlg ato to other factors. Geerally, 25 samles are gathere to collect suffcet ata. Ste 3: After ecg the sze of the samle a the umber of samles, we select samle uts from the tem rouce raomly, so that each tem has a equal chace of beg selecte. Ste 4: We collect the ata for a -chart the same way as for the cotrol chart for varables. However, stea of recorg measuremets of the tems, we recor the umber of tems secte a the umber of efectve tems. Ste 5: We calculate the fracto efectve for each samle from equato (1). If there are samles of the same sze a 1, 2,..., are the umbers of efectves the 1 st, 2,, th samle, resectvely, the for each samle, we calculate the fracto efectve as follows: 1 2 1, 2,..., (2) Ste 6: We set the 3σ cotrol lmts to f out whether the rocess s uer cotrol or out-of-cotrol wth resect to the fracto efectve. The 3σ cotrol lmts for the -chart are gve by: Cetre le = E Uer cotrol lmt (UCL) = E 3SE Lower cotrol lmt (LCL) = E 3SE (3a) (3b) (3c) For calculatg the cotrol lmts of the -chart, we ee the samlg strbuto of the fracto efectves. You have leart Ut 2 of MST-004 that the mea a varace of the samlg strbuto of the fracto (roorto) efectve s gve by PQ E() P a Var() (4a) Here P s the robablty or fracto or roorto of efectve the rocess a Q = 1 P. We also ow the staar error of a raom varable X s 84 SE(X) var(x)

5 Therefore, the staar error of samle fracto (roorto) efectve s gve by SE Var() PQ (4b) You have leart that the samlg strbuto of the fracto efectves s ot a ormal strbuto. However, f the samle sze s suffcetly large, such that P > 5 a Q > 5, you ow from the cetre lmt theorem, that the samlg strbuto of the fracto efectves s aroxmately ormally strbute wth mea P a varace PQ/. Therefore, the cetre le a cotrol lmts for the -chart are gve as follows: Cotrol Charts for Attrbutes You have stue the Cetral Lmt Theorem Ut1 of MST-004. I CL E() P (5a) r a c t UCL E() 3SE() LCL E() 3SE() I ractce, fracto efectve (P) of the rocess s ot ow. So equatos (5a to 5c) are ot use to costruct the cotrol chart. It s ecessary to estmate the fracto efectve of the rocess. We estmate t usg the samle fractos. The best estmator of the rocess fracto efectve s the average fracto efectve of the samles. Suose, we raw samles of the same sze a for each samle, we calculate the fracto efectve usg equato (2). If,,, are the fracto efectves of the 1 st, 2,, th samle, 1 2 P(1 P) P 3 (5b) P(1 P) P 3 (5c) resectvely, we calculate the average fracto efectve as follows: 1 1,, (6) Geerally, ths formula s use whe we have samle fracto efectves. If the umber of efectve tems for each samle s gve, t s coveet to calculate the average fracto efectve as follows: Sum of efectve tems all samles 1 Total umber of tems secte (7) 1 I ths case, the cetre le a cotrol lmts of the -chart are gve as: CL E() Pˆ (8a) UCL E() 3SE() LCL E() 3SE() ˆP 3 P(1 ˆ P) ˆ (1 ) 3 (8b) ˆP 3 P(1 ˆ P) ˆ (1 ) 3 (8c) 85

6 Process Cotrol Ste 7: Havg set the cetre le a cotrol lmts, we costruct the -chart by tag the samle umber o the X-axs a the samle fracto efectve () o the Y-axs. We raw the cotrol le as a sol le, a the UCL a LCL as otte les. The we lot the value of the samle fracto efectve agast the samle umber. The cosecutve samle ots are joe by le segmets. Ste 8: We terret the result. If all samle ots le o or betwee the uer a lower cotrol lmts, the cotrol chart cates that the rocess s uer cotrol. That s, oly chace causes are reset the rocess. If oe or more ots le outse the uer or lower cotrol lmts, the cotrol chart alarms (cates) that the rocess s ot uer statstcal cotrol a some assgable causes are reset the rocess. To brg the rocess uer statstcal cotrol, t s ecessary to vestgate the assgable causes a tae correctve acto to elmate them. Oce the assgable causes are elmate, we elete the out-of-cotrol ots (samles) a calculate the revse cetre le a cotrol lmts for the -chart by usg the remag samles. These lmts are ow as the revse cotrol lmts. To calculate the revse lmts of the -chart, we frst calculate ew as follows: ew 1 j1 j or ew where the umber of scare samles, 1 j1 j (9) j the sum of fracto efectves the scare samles, a j1 j1 j the sum of umber of efects the scare samles. After fg ew, we recostruct the cetre le a cotrol lmts of the -chart by relacg by ew equatos (8a to 8c) as follows: CL ew (10a) (1 ) 3 (10b) ew ew UCL ew ew (1 ew ) UCL ew 3 (10c) Note 1: If the value of lower cotrol lmt s egatve, we chage the lower cotrol lmt to zero because a egatve fracto efectve s mossble. Note 2: If the -chart cotuously cates a crease the value of the average fracto (roorto) efectve, the maagemet shoul vestgate the reasos beh ths crease rather tha costatly revsg the cetre le a 86

7 cotrol lmts. The ossble reasos beh the crease the efectve tems may be oor comg qualty from veors, tghteg of secfcato lmts, etc. So far, we have scusse varous stes volve the costructo of the -chart. Let us ow tae u some examles to llustrate ths metho. Examle 1: I a comay maufacturg crcet ball, the qualty cotroller sects the balls a classfes them as efectve or o-efectve o the bass of certa efects. The comay maager wats to mata the rocess so that a average of ot more tha 5 ercet of the outut s efectve. Suggest a sutable cotrol chart for ths urose. If the comay ca wor wth a samle of sze 500, calculate the cetre le a cotrol lmts for ths chart. Soluto: Here the balls are classfe as efectve or o-efectve. So we use the cotrol chart for attrbutes. The qualty cotroller has to mata the rocess so that a average of ot more tha 5% of the outut s efectve. It meas that the rocess roorto efectve s gve. So we use the -chart. We are gve that P = 5% = 0.05 a = 500. Therefore, we ca calculate the cetre le a cotrol lmts from equatos (5a to 5c) as follows: CL P 0.05 P(1 P) UCL P Cotrol Charts for Attrbutes P(1 P) LCL P Examle 2: A factory maufacturg small bolts. To chec the qualty of the bolts, the maufacturer selecte 20 samles of same sze 100 from the maufacturg rocess tme to tme. He/she vsually secte each selecte bolt for certa efects. After the secto, he/she obtae the followg ata: Samle Number Proorto Defectve Samle Number Proorto Defectve Estmate the roorto efectve of the rocess. Does the rocess aear to be uer cotrol wth resect to the roorto of efectve bolts? Soluto: We ow that the roorto efectve of the rocess ca be estmate by average roorto efectve of the samles. We ca calculate t usg equato (6) as follows: ˆP

8 Process Cotrol Here the maufacturer vsually secte each selecte bolt a classfe t as efectve or o-efectve. So we use the cotrol chart for attrbutes. Sce the roorto efectve for each samle s gve, we ca use the -chart. Here the rocess roorto efectve s ot ow. So we use equatos (8a to 8c) to obta the cetre le a cotrol lmts for the -chart as follows: CL (1 ) UCL (1 ) LCL We ow costruct the -chart. We raw the cetre le as a sol le a cotrol lmts as otte les o the grah a lot the ots by tag the samle umber o the X-axs a the roorto efectve () o the Y-axs as show Fg Y Proorto Defectve LCL = 0 0 X Samle Number Fg. 3.1: The -chart for roorto of efectve bolts. Iterretato of the result From the -chart show Fg. 3.1, we observe that the ots corresog to samle umbers 4 a 16 le outse the cotrol lmts. So we ca say that the rocess s ot uer statstcal cotrol wth resect to the roorto of efectve bolts. Examle 3: The followg ata are fou urg the secto of the frst 15 samles of sze 100 each from a lot of two-wheelers maufacture by a automoble comay: Samle Number Number of Defectves UCL = CL = Draw the chart for fracto efectve () a commet o the state of cotrol. If the rocess s out-of-cotrol, calculate the revse cetre le a cotrol lmts by assumg assgable causes for ay out-of-cotrol ot.

9 Soluto: To raw the -chart, we ee to calculate the cetre le a cotrol lmts. Here, the rocess fracto efectve s ot ow. So ths case we use equatos (8a to 8c). To calculate the cotrol lmts, we frst calculate the fracto efectve for each samle a the. Samle Number Samle Sze () Defectves () Proorto Defectve ( = /) Total 78 Cotrol Charts for Attrbutes Proorto efectve Numberof efectves Samle sze From the above table, we have Therefore, we ca calculate the cetre le a cotrol lmts as follows: CL (1 ) UCL (1 ) LCL We ow raw the -chart by tag the samle umber o the X-axs a the roorto efectve () o the Y-axs as show Fg

10 Process Cotrol Y Proorto Defectve UCL = CL = Fg. 3.2: The -chart for fracto efectve of two-wheelers. Iterretato of the result From the -chart show Fg. 3.2., we observe that the ots corresog to samle umbers 5 a 12 le outse the uer cotrol lmts. Therefore, the rocess s out-of-cotrol. It meas that some assgable causes are reset the rocess. For calculatg the revse lmts for the -chart, we frst elete the out-of-cotrol ots (5 a12) a calculate the ew usg remag samles. I our case = 100, = 15, = 2, 2 j j 1 j ew ( ) After fg the ew, we recostruct the cetre le a cotrol lmts of the chart usg equatos (10a to 10c) as follows: CL ew ew ( ew ) UCL ew LCL ew ( ew ) ew Samle Number You may le to ause here a chec your uerstag of the -chart by aswerg the followg exercses. E1) Choose the correct oto from the followg: ) The cotrol chart use for attrbutes s the a) -chart b) -chart c) c-chart ) all a), b) a c) j LCL = 0 X

11 ) The cotrol chart for roorto efectve s the a) X- chart b) -chart c) -chart ) c-chart ) The uer cotrol lmt for the -chart s a) 1 3 b) 3 1 c) 3 1 ) 3 1 v) If the lower cotrol lmt has a egatve value a -chart, t s treate as a) zero b) uty c) egatve oly ) ostve oly E2) Lst the fferet tyes of cotrol charts for attrbutes. E3) Wrte the cotrol lmts of the -chart whe the fracto efectve of the rocess ot ow. E4) A aly samle of 30 shrts was tae over a ero of 15 ays orer to motor the maufacturg rocess of the shrts. Each shrt was secte a classfe as efectve or o-efectve. If a total of 22 efectve shrts were fou 15 ays, what shoul be the uer a lower cotrol lmts of the roorto of efectve shrts? E5) A moble maufacturer sects 30 mobles at the e of the ay of roucto a otes the umber of efectve mobles. Ths roceure s cotue u to 12 ays a 2, 1, 3, 0, 2, 1, 0, 5, 2, 0, 3, 1 efectve moble are fou. Is the roucto rocess uer cotrol wth resect to the roorto efectve? Cotrol Charts for Attrbutes 3.4 -CHART FOR VARIABLE SAMPLE SIZE I Sec. 3.3, we have scusse the -chart for costat samle sze, that s, we have secte the same umber of tems each samle a the umber of efectve tems has bee coute. Sometmes, stuatos arse where we caot tae the same umber of tems each samle. Geerally, ths haes whe the -chart s use for 100% secto of outut that vares from ay to ay. The varato the outut er ay ca be ue to breaow of maches, fferet roucto requremets, etc. Whe the samle sze s ot uform (same or costat), we use the -chart for varable samle sze. The roceure of costructg the -chart for varable samle sze s smlar to the -chart wth uform (costat) samle sze excet for the cotrol lmts. Sce the cotrol lmts are fucto of samle sze (), these wll vary wth the samle sze. So for the -chart for varable samle sze, the cotrol lmts are calculate searately for each samle. The cotrol lmts such a stuato are ow as varable cotrol lmts. Geerally, two aroaches are use for costructg varable cotrol lmts for the -chart. Frst Aroach 91

12 Process Cotrol Accorg to the frst aroach, we calculate the cotrol lmts for each samle. Let be the sze of the th samle. The cetre le a cotrol lmts of the th samle for the -chart whe the staar value of fracto efectve P s ow are gve as follows: CL = P (11a) P(1 P) UCL P 3 (11b) P(1 P) LCL P 3 (11c) Whe P s uow, t s estmate by the samle average fracto (roorto) efectve,.e., as we have scusse Sec That meas, we frst calculate the fracto efectve for each samle usg equato (1). If 1, 2,..., rereset the umber of efectves 1 st, 2,, th samles of sze 1, 2,...,, resectvely, the,,..., (12a) a we calculate as follows: or The cetre le remas costat for each samle a s gve by CL (12b) (13a) The cotrol lmts chage ue to varable samle sze. The cotrol lmts for the th samle are gve by ˆ ˆ ˆ P(1 P) (1 ) UCL P 3 3 (13b) ˆ ˆ ˆ P(1 P) 1 LCL P 3 3 (13c) Seco Aroach Accorg to the seco aroach, the cotrol lmts are calculate usg the average samle sze. Ths aroach s use oly whe there s o large varato the samle szes a t s execte that the future samle szes wll ot ffer sgfcatly from the average samle sze. Usg ths aroach, we get costat cotrol lmts just as we get the case of costat samle sze. We calculate the average samle sze as follows: 92

13 (14) 1 The cetre le a cotrol lmts whe P s ow are gve as follows: Cotrol Charts for Attrbutes CL P (15a) UCL P 3 P(1 P) (15b) LCL P 3 P(1 P) (15c) However, whe P s ot ow, the cetre le a cotrol lmts are gve as CL (16a) (1 ) UCL 3 (16b) (1 ) LCL 3 (16c) follows: Note: The -chart for varable samle sze shoul oly be use whe samles of costat sze are ot ossble, uavalable or urealstc. Let us exla how to costruct ths chart. Examle 4: I the roucto of tyres of a automoble, the outut of a gve sze was secte every ay ror to the tyres beg gve to fshe goos stores. The umber of efectve tyres fou every ay secto s summarse the followg table: Number of Samle Tyres Isecte Defectve Tyres Number of Samle Tyres Isecte Defectve Tyres Draw the arorate cotrol chart a commet o the state of the rocess. Soluto: Sce the umber of efectves s gve a samle sze vares, we use the -chart for varable samle sze. We shall solve ths roblem usg both aroaches escrbe ths secto. Frst Aroach 93

14 Process Cotrol Accorg to the frst aroach, the cotrol lmts of the -chart are calculate for each samle. Sce the average fracto efectve of the rocess s ot gve, we use equatos (13a to 13c) to calculate the cetre le a cotrol lmts of the th samle. For ths, we frst calculate the roorto of efectve tyres () for each samle a usg equatos (12a to12b). Number of Samle Number of Tyres Isecte () Defectve Tyres () Proorto of Defectve Tyres ( = /) CL UCL LCL Total From the above table, we have The cetre le a cotrol lmts (show the above table) are calculate usg equatos (13a to 13c) as follows: CL (1 ) UCL , a so o. (1 ) LCL , a so o.

15 We raw the -chart by tag the samle umber o the X-axs a the roorto efectve () of tyres o the Y-axs as show Fg Cotrol Charts for Attrbutes Y Samle Proorto UCL = CL = LCL= Samle Number Fg. 3.3: The -chart for roorto of efectve tyres. Iterretato of the result From the -chart show Fg. 3.3, we observe that the ots corresog to samle umbers 6, 8 a 18 le outse the cotrol lmts. Therefore, we ca say that the rocess s out-of-cotrol wth resect to the roorto of efectve tyres a some assgable causes are reset t. Let us ow solve ths examle by usg the seco aroach. Seco Aroach Accorg to ths aroach, the cotrol lmts are calculate usg the average samle sze a we get costat cotrol lmts just as we get the case of costat samle sze. We ca calculate the average samle sze usg equato (14) as follows: Usg equatos (16a to 16c), we calculate the cetre le a cotrol lmts as follows: CL (1 ) UCL LCL (1 ) The, we lot the -chart (see Fg. 3.4). X

16 Process Cotrol Y Proorto Defectve Iterretato of the result Fg. 3.4: The -chart for roorto of efectve tyres. From Fg. 3.4, we ote that the ots corresog to samle umbers 6, 8 a 18 le outse the cotrol lmts. Therefore, the rocess s out-of-cotrol. Hece, both aroaches gve the same cocluso about the rocess. Now, you ca try the followg exercse a raw the -chart for varable samle sze. E6) To motor the maufacturg rocess of latos, a qualty cotrol egeer raomly selects a umber of latos from the roucto le, each ay over a ero of 20 ays. The latos are secte for certa efects a the umber of efectve latos fou each ay s recore the followg table: Day Latos Isecte Samle Number Defectve Latos Day Latos Isecte Defectve Latos UCL = CL = LCL = Costruct the -chart for varable samle sze wth the hel of both aroaches a commet o the rocess. X 3.5 CONTROL CHART FOR NUMBER OF DEFECTIVES (-CHART)

17 I Sec. 3.4, you have leart that for the -chart, the fracto (roorto) efectve for each samle s calculate a lotte agast the samle umber. Alteratvely, stea of calculatg the roorto efectve, we ca cout the umber of efectve tems the samles a lot them agast the samle umber. The chart thus obtae s ow as the -chart. The -chart s a aatato of the basc -chart. The urose of ths chart s to rove a better uerstag to the oerator who may uersta actual umber of efectve tems easly rather tha the roorto efectve. However, there s oe rawbac of the -chart: It s use oly for equal samle sze because the umber of efectves caot be comare for varable samle sze. For examle, suose we get oe efectve tem a samle of sze 10 a oe efectve tem a samle of sze 100. I both samles, we get oe efectve tem, but the frst samle shows a much hgher rate of roblems, that s 10%, comarso to the seco. The ame s ue to the reaso that the roorto efectve was obtae by vg the actual umber of efectves () by the samle sze (),.e., = /. So =. Therefore, the actual umber of efectves may be reresete by. The uerlyg logc a basc form of the -chart are smlar to the -chart. The rmary fferece betwee the -chart a -chart s that stea of lottg the samle fracto efectves a motorg ther varato, we lot the umber of efectves er samle a motor them. Therefore, ths secto, we escrbe the cetre le a cotrol lmts of the -chart. For obtag the cotrol lmts of the -chart, we ee the samlg strbuto of the umber of efectves (). Recall the samlg strbuto of the umber of efectves escrbe Ut 2 of MST-004. Suose the tems or uts of a rocess (oulato) are classfe to two mutually exclusve grous as efectve (success) a o-efectve (falure) a a raom samle of sze s raw from t. The the umber of efectves follows a bomal strbuto wth mea P a varace PQ where P s the fracto or roorto of efectves the rocess (oulato) a Q = 1 P: E() P a Var() P 1 P (17) We also ow that the staar error of a raom varable (X) s Cotrol Charts for Attrbutes -chart s also ow as -chart. SE(X) var(x) Therefore, the staar error of the umber of efectves s gve by SE Var() P1 P (18) The samlg strbuto of umber of efectves s ot a ormal strbuto. However, f the samle sze s suffcetly large, you ow from the cetre lmt theorem that the samlg strbuto of umber of efectves s aroxmately ormally strbute wth mea P a varace PQ. Therefore, the cetre le a cotrol lmts for the -chart are gve as follows: Cetre le (CL) E() P (19a) Uer cotrol lmt (UCL) E() 3SE() P 3 P(1 P) (19b) Lower cotrol lmt (LCL) E() 3SE() P 3 P(1 P) (19c) 97

18 Process Cotrol I ractce, the fracto (roorto) efectve (P) of the rocess s ot ow. Therefore, t s ecessary to estmate t by the average samle fracto efectve as we have scusse Sce.3.3. If we raw samles of same sze a 1, 2,..., are the umbers of efectve tems 1 st, 2,, th samle, resectvely, we ca calculate the average fracto efectve of the samles as follows: Sum of efectve tems 1 Total umber of tems secte (20) 1 Hece, ths case, the cetre le a cotrol lmts of the -chart are obtae by relacg P by equatos (19a to 19c) as follows: CL E() Pˆ (21a) UCL E() 3SE() Pˆ 3 P(1 ˆ P) ˆ 3 (1 ) (21b) LCL E() 3SE() Pˆ 3 P(1 ˆ P) ˆ 3 (1 ) (21c) Havg set the cetre le a cotrol lmts, we costruct the -chart by tag the samle umber o the X-axs a the umber of efectve tems () the samle o the Y-axs. We raw the cotrol le as a sol le a UCL a LCL as otte les. The we lot the umber of efectve tems for each samle agast the samle umber. The cosecutve samle ots are joe by le segmets. Iterretato of the result If all samle ots le o or betwee uer a lower cotrol lmts, the cotrol chart cates that the rocess s uer statstcal cotrol, that s, oly chace causes are reset the rocess. If oe or more ots le outse the uer or lower cotrol lmts, the cotrol chart alarms (cates) that the rocess s ot uer statstcal cotrol a some assgable causes are reset the rocess. To brg the rocess uer statstcal cotrol t s ecessary to vestgate the assgable causes a tae correctve acto to elmate them. Oce the assgable causes are elmate, we elete the out-of-cotrol ots a calculate the revse cetre le a cotrol lmts for the -chart by usg the remag samles. These lmts are ow as the revse cotrol lmts. To obta the revse lmts for the -chart, we frst calculate the ew as follows: ew 1 j1 ( ) where the umber of scare samles, a j1 j j the sum of umber of efectves the scare samles. (22) 98

19 We the recostruct the cetre le a cotrol lmts of the chart by relacg by ew equatos (21 to 21c) as follows: Cotrol Charts for Attrbutes CL ew (23a) UCL ew 3 ew (1 ew ) (23b) UCL ew 3 ew (1 ew ) (23c) Let us exla how to raw the -chart wth the hel of examles. Examle 5: To motor the maufacturg rocess of latos, a qualty cotrol egeer raomly selects 50 latos from the roucto le, each ay over a ero of 20 ays. The latos are secte for certa efects a the umber of efectve latos fou each ay s recore the followg table: Day Latos Isecte Defectve Latos Day Latos Isecte Defectve Latos Costruct the arorate cotrol chart a state whether the rocess s cotrol. Soluto: I ths case, the qualty cotrol egeer vsually sects each selecte lato to eterme whether t s efectve or ot. So we use the cotrol chart for attrbutes. Sce the umber of efectves s gve, we ca use or -chart. But t s coveet to use the -chart. We o ot ow the rocess fracto efectve. Therefore, we use equatos (21a to 21c) to calculate the cetre a cotrol lmts for the -chart. We are gve that the umber of latos secte each samle () = 50 a the umber of samles () = 20. The total umber of efectves s 1 We calculate from equato (20) as follows: