Chapter 12 - Quality Cotrol Example: The process of llig 12 ouce cas of Dr. Pepper is beig moitored. The compay does ot wat to uderll the cas. Hece, a target llig rate of 12.1-12.5 ouces was established. A total of 30 cas were sampled. Example: A isurace aget wats to determie the satiscatio of his cliets i hadlig claims. Both of the above examples deal with quality ad quality cotrol. I geeral, quality improvemet is cocered with 1. The customers' desires or expectatios 2. Traslatig the desires or expectatios ito some service chage or modicatio 3. Producig product that satises the customer The above would be very easy to do however, o matter how hard we try, perfect product will ot be produced all of the time. Our goal is to try to reduce the amout of variatio i the ed product. We will use statistical methods to idetify ad quatify possible sources of variatio. Garvi established the 8 dimesios of quality (pg 677). These say, basically, the product must satisfy the customers' eeds, give a good performace ad be attractive. These quality techiques are applied to a process - a series of actios that trasforms iputs to outputs over time. iputs outputs * people * products * materials * services * machies * eergy * methods, etc. To improve a process, oe must have a detailed uderstadig of the process ad how it works. A system is a collectio of processes that has a ogoig purpose or missio. Systems uses feedback from its eviromet i order to improve. Example: Possible processes i a credit card compay -Marketig - Billig - Accouts eceivable - Customer Service 38 39 As a meas of idetifyig the eect of various compoets i a system (or a process) oe may try to - model the system: quatitatively describe the system. I ay case, variatio is always preset i the output of all processes. No two items produced by a process are the same. There are 6 major sources of variatio i output: People Machies Methods Measuremet Materials Eviromet I 1930's, Bell Labs advaced the process kow as acceptace samplig. At about the same time, Walter Shewhart recogized that variatio i maufactured products is ievitable but could be accouted for. Shewhart created cotrol charts - which give a picture of how the process is behavig, relative to some predetermied guidelies. The idea behid cotrol charts is to formulate guidelies so that oe is able to idetify variatio i a process brought about by iueces other tha the iputs. Statistical methodology is eeded for us to be the able to ot oly recogize this variatio i processes but also formulate strategies for moitorig this variatio. ecall that variatio is preset everywhere. two possible types of variatio i a process: There are 1. adom Variatio or Variatio due to Commo Causes: Variatio iheret i the process from the methods, materials, people ad the eviromet that are part of the process desig. (See Deitio 12.6) 2. Special Causes Variatio or Variatio due to Assigable Causes: Variatio that is itroduced i to the process by outside iueces, ot part of the process desig. (See Deitio 12.7) No matter how hard we try, radom variatio caot be elimiated. However, through the use of cotrol charts, we will try to determie whether or ot special causes variatio is abudat i a process - subsequetly, oe would try to idetify the source of such variatio. Whe variatio due to special causes is elimiated, we will say that the process is i cotrol. See Deitio 12.4. Follow Example discussed i Sectio 12.2 40 41

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Logic of Cotrol Charts We use cotrol charts to help us dieretiate betwee process variatio due to commo causes ad special causes. Weusecotrol charts to determie whether a process is uder statistical cotrol. A cotrol chart is a time series plot of the idividual measuremets of the output variable to which a horizotal lie called the ceterlie ad two other horizotal lies called the upper cotrol limit ad the lower cotrol limit, respectvely, have bee added. The ceterlie represets the mea of the process. The cotrol limits are placed so that whe the process is i statistical cotrol, the probability ofa measuremet fallig outside the lies is very small (i.e, it is a rare evet). I practice, these limits are positioed a distace of 3 from the ceterlie. If the process is i cotrol ad follows a ormal distributio the probability of a measuremet fallig outside these limits is :0027. These lies are called 3-sigma limits. 46 47 x-chart The previous chart where we used idividual measuremets to moitor the process is called a idividuals chart. The x-chart is used for moitorig the mea of a process. It uses the meas of samples draw from the process. I practice x-chart is ot used aloe sice the use of x-chart requires that the process variatio remais stable. [The -Chart, discussed later, is used for moitorig process variatio]. If a process follows a N( 2 ) distributio the we kow that X 0 1 N @ 2 p A. I order to costruct a x-chart you should at least have 20 samples of measuremets each from the process. Let k = # of samples of size draw from the process, ad x 1 x 2 ::: x k be the meas of each sample. First, we ca compute the followig: Upper Cotrol Limit(UCL): x + p 3 Lower Cotrol Limit(LCL): x ; p 3 The two most importat actios i the costructio of a x-chart are decidig: 1. The sample size. 2. The frequecy with which samples are draw from ogoig process. Samples whose size ad frequecy are such that process chages are most likely to occur betwee those periods are called ratioal subgroups. Example: If it is believed that a process chage is most likelytooccurbetwee days (say, because of settig up or re-loadig of equipmet or raw material eachday) { ratioal subgroup is take as a day dieret workers o cosecutive shifts causes the process to vary from shift to shift { ratioal subgroup is take as a shift. Ceterlie(CL): x = x 1 +x 2 ++x k k 48 49

Estimatio of, the process stadard deviatio. Approach 1: calculate s from each of the k samples ad take theaverage. Approach 2: calculate s from a large sample that was obtaied whe the process was believed to be i cotrol. Approach 3: use the scaled average rage. This is the approach favored by idustry. This approach uses the rages of each of the k samples to estimate : ^ = where = 1 + 2 + + k k 1 2 ::: k are the rages of eachofthek samples. is a costat that makes ^ a ubiased estimator of. Values of ca be obtaied from Table XVII (Appedix B). To read this table you eed to kow the sample size. From the formulas for UCL ad LCL give before we have UCL : x+ 3( = ) p = x+a 2 LCL : x+ 3( = ) p = x+a 2 where the values for A 2 ca be directly read from Table XVII for a specied value of. For the purpose of iterpretatio, a cotrol chart is divided ito zoes: C zoes: the regios 1 of the ceterlie. B zoes: the regios betwee 1 a from the ceterlie (o either side of it). A zoes: the regios betwee 2 ad 3 from the ceterlie (o either side of it). 50 51 Example: 5 samples of size 3 were obtaied of a process, Costruct LCL, CL, UCL for a x-chart. Sample Obs x 1 52, 51, 52 51.667 1 2 56, 52, 53 53.667 4 3 52, 52, 52 52 0 4 51, 53, 51 51.667 2 5 52, 51, 51 51.333 1 260.333 8 Thus x =52:0666 ad =1:6 LCL = x ; A 2 = 52.0666 - (1.023)(1.6) = 50.4298 CL = x = 52.0666 UCL = x ; A 2 = 52.0666 + (1.023)(1.6) = 53.7045 ead Example 12.1, 12.2. Patter-aalysis rules Figure 12.21 summarizes 6 rules used by practitioers to determie whe a process is out of cotrol usig x-charts. 52 53

All these 6 patters are rare evets uder the assumptio that the process is uder cotrol. For example, assume that the process is uder cotrol ad follows a ormal distributio. The we ca calculate the followig probabilities: Pr(poit fallig beyod zoe A o oe side) =.00135 Pr(poit fallig i zoe A o oe side) =.02135 Pr(poit fallig i zoe B o oe side) =.1360 Pr(poit fallig i zoe C o oe side) =.3413 Examples 12.1, 12.2: Cosider the pait-llig process agai. Fillig head becomes clogged several times a moth causig it to dispese less pait later that day. Samplig Strategy: Sample 5 cosecutive cas every hour for the ext 25 hours. atioal subgroupig of 5 cosecutive cas is justi- ed because the idividual measuremets are close eough i time to assume process stays stable durig that period. takig the 5 measuremets every hour is far eough apart to detect ay chage i the mea. From these we ca calculate probabilities of the various patters i Figure 12.21. Example:For ule 1. Pr(poit fallig outside of zoe A)= :00135 2 =:0027: 54 55 Calculate the cotrol limits: 9:99995 + 9:99704 + :::+10:00015 x = =9:9999 25 :0078 + :0092 + :::+ :0077 = = :01028 25 CL: x =9:9999 UCL: x + A 2 =9:9999 + :577(:01028) = 10:0058 LCL: x ; A 2 =9:9999 ; :577(:01028) = 9:9940 where for =5 A 2 = :577 from Table XVII Calculate A, B, C zoes: A-B boudaries: x (2=3)A 2 9:9999 (2=3)(:577)(:01028) = 10:0039 9:9959 B-C boudaries: x (1=3)A 2 9:9999 (1=3)(:577)(:01028) = 10:0019 9:9979 Usig the x-chart: Te ew samples were draw from the pait llig process. Is the process i cotrol durig this time? We simply exted the cotrol limits ad ceterlie to obtai: givig Figure 12.22: 56 57

No poits fallig outside the cotrol limits but six poits i a row steadily decrease. So usig ule 3, this is a idicatio of presece of special causes of variatio. Notice that ules 5 ad 6 are also violated. These lead to the coclusio that the process has goe out of cotrol. Apparetly, the llig head has clogged up about the time sample 26 or 27 was take. -Chart -chart is a cotrol chart that detect chages i process variatio. x-chart plots sample meas: x 1 x 2 ::: x k -chart plot the sample rages: 1 2 ::: k. Chages i the behaviour of idicate chage i process variatio. Costructio of the -chart is similar to that of the xchart. It is based o the samplig distributio of. Assume has mea k ad stadard deviatio. The 3-sigma limits are: 58 59 atioal subgroupig is used for determiig sample size ad frequecies of samplig as i the costructio of the x;chart Cosider the rages from the k samples: 1 2 ::: k. Now we plot CL, LCL ad UCL as horizotal lies to get the cotrol chart. Plot the k sample rages 1 2 ::: k o the cotrol chart. Compute = 1 + 2 ++ k k. is a estimate of k. A estimate of is 0 1 ^ = d B 3 @ C A where d 3 are costats obtaied from Table XVII for agive sample size. Upper Cotrol Limit (UCL): +3^ = +3d 3 = 1+ 3d 3 = D4 Lower Cotrol Limit (LCL): ; 3^ = ; 3d 3 = 1 ; 3d 3 = D 3 where D 3, D 4 are costats obtaied from Table XVII for a give sample size. Costructig A, B, C zoes: Upper A-B boudary: +2d3 Lower A-B boudary: ; 2d3 Upper B-C boudary: + d3 Lower B-C boudary: ; d3 Iterpretig a -Chart: The process is out of cotrol of oe or more of the sample rages fall beyod LCL or UCL or ay ofthe patters i ules 2, 3 or 4 are observed. The process is i cotrol if oe of the out-of-cotrol patters are observed. 60 61

Example 12.3 Costruct a -Chart for the pait fillig process: = :0078+:0092++:0077 25 = :01028 CL : 01028 UCL : (.01028)(2.115) LCL : (.01028)(0) From Table XVII, for =5 D 3 =0 D 4 =2:115. A, B, C zoes: Upper A-B boudary: +2d3 = :01792 Lower A-B boudary: ;2d3 = :00264 Upper B-C boudary: + d3 = :01410 Lower B-C boudary: ; d3 = :00646 From Table XVII for =5, =2:326, d 3 = :864. 62