Control Charts for Means (Simulation)

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1 Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng smulaton. Ths procedure can also be used to study charts wth a sngle observaton at each sample. The n-control mean and standard devaton can be nput drectly or a specfed number of n-control prelmnary samples can be smulated based on a user-determned n-control dstrbuton. The out-ofcontrol dstrbuton s fleble n terms of dstrbuton type and dstrbuton parameters. The Shewhart, Cusum, and EWMA parameters can also be flebly nput. Ths procedure can also be used to determne the necessary sample sze to obtan a gven run length. Smulaton Detals If the n-control mean and n-control standard devaton are assumed to be known, the steps to the smulaton process are as follows (assume a sample conssts of n observatons). 1. An out-of-control sample of sze n s generated accordng to the specfed dstrbuton parameters of the outof-control dstrbuton. 2. The average of the sample s produced and, f necessary for the partcular type of control chart, the standard devaton. 3. Based on the control chart crtera, t s determned whether ths sample results n an out-of-control sgnal. 4. If the sample results n an out-of-control sgnal, the sample number s recorded as the run length for that smulaton. If the sample does not result n an out-of-control sgnal, return to Step Steps 1 through 4 are repeated untl the number of smulatons (Nsm) s reached. The result s Nsm run lengths. 6. The average or medan or specfed percentle of the run length dstrbuton s reported. If the n-control mean and n-control standard devaton are to be smulated based on n-control prelmnary samples (NPrelm), the steps to the smulaton process are as follows (assume a sample conssts of n observatons). 1. NPrelm n-control samples of sze n are generated accordng to the specfed dstrbuton parameters of the n-control dstrbuton. 2. The n-control average and standard devaton are calculated based on the NPrelm smulated n-control samples. 3. An out-of-control sample of sze n s generated accordng to the specfed dstrbuton parameters of the outof-control dstrbuton

2 4. The average of the sample s produced and, f necessary for the partcular type of control chart, the standard devaton. 5. Based on the control chart crtera, t s determned whether ths sample results n an out-of-control sgnal. 6. If the sample results n an out-of-control sgnal, the sample number s recorded as the run length for that smulaton. If the sample does not result n an out-of-control sgnal, return to Step Steps 1 through 6 are repeated untl the number of smulatons (Nsm) s reached. The result s Nsm run lengths. 8. The average or medan or specfed percentle of the run length dstrbuton s reported. Data Dstrbutons A wde varety of dstrbutons may be studed. These dstrbutons can vary n skewness, elongaton, or other features such as bmodalty. A detaled dscusson of the dstrbutons that may be used n the smulaton s provded n the chapter Data Smulator. Formulas for Constructng Control Charts Suppose we have k subgroups, each of sze n. Let j represent the measurement n the j th sample of the th subgroup. Three statstcs that are routnely computed (dependng on the type of control chart) for each subgroup are: The subgroup mean the subgroup range and/or the subgroup standard devaton n j = = 1 n j R = ( n ) ( 1 ) s = n ( j ) j = 1 n 1 2 Estmatng Sgma Control lmts vary accordng to the type of control chart used. These requre an estmate of the process mean, µ (mu), and the process varablty, σ (sgma). A known estmate of µ may be suppled by the user, or t may be estmated by the average of the averages of a number of n-control prelmnary samples, double bar (also known as the grand mean): j = = 1 k When sgma s not nput drectly, there are two methods avalable for estmatng σ. k 290-2

3 Method 1: Estmatng Sgma from the Ranges where R σ R = d k 2 R = = 1 k ( ) E R R d 2 = σ = µ σ Makng the assumpton that the j s follow the normal dstrbuton wth constant mean and varance, we can derve values for d 2 through the use of numercal ntegraton. Method 2: Estmatng Sgma from the Standard Devatons where s σ s = c k 4 s = = 1 k ( ) E s s c 4 = σ = µ σ. Makng the assumpton that the j s follow the normal dstrbuton wth constant mean and varance, we can derve values for c 4 from the followng formula. c 4 = n Γ 2 2 n 1 n 1 Γ 2 Estmatng Sgma when n = 1 When n s one, we cannot calculate R or s snce these requre at least two measurements. In ths case, we use the standard devaton of all k measurements. Unfortunately, ths method does not appromate the wthn-subgroup varaton. Rather, t combnes the wthn and the between subgroup varaton. Xbar Chart Lmts The lower and upper control lmts for the Xbar chart are calculated usng the formula σˆ LCL = z n σˆ UCL = + z n 290-3

4 where z s a multpler (often set to three) chosen to reduce the possblty of false alarms (sgnalng an out-ofcontrol stuaton when the process s n control). Cusum and FIR Cusum Charts The Cusum chart has been shown to detect small shfts n the process average much qucker than the Xbar chart. In PASS we use the Cusum procedure presented by Ryan (1989). Ths procedure may be summarzed as follows: 1. Calculate all statstcs as f you were gong to generate an Xbar chart. 2. Calculate the z usng the formula z = σ 3. Calculate the lower and upper cumulatve sums as follows [ 0 ( ) 1 ] [ 0 ( ) 1 ] S L = ma, z K + S L S H = ma, z K + S H 4. The control lmts are chosen as plus or mnus h. Often, K s set to 0.5 (for detectng one-sgma shfts n the mean) and h s set to Usually, the startng value for S L and S H s zero. Occasonally, however, a fast ntal response (FIR) value of h/2 s used. EWMA Chart Lmts The lower and upper control lmts for the eponentally weghted movng-average (EWMA) chart are calculated usng the formula σˆ LCL = L n 2 R R 2 [ 1 ( 1 R) ] 2 [ 1 ( 1 R) ] σˆ R UCL = + L n 2 R where L s a multpler (usually set to three) and R s smoothng constant. Procedure Optons Ths secton descrbes the optons that are specfc to ths procedure. These are located on the Desgn tab. For more nformaton about the optons of other tabs, go to the Procedure Wndow chapter. Desgn Tab Solve For Solve For Solve for ether the run length dstrbuton or the sample sze. Ths s the parameter dsplayed on the vertcal as of the plot

5 Note: The search for the sample sze may take several mnutes to complete. You may fnd t qucker and more nformatve to solve for run length dstrbuton for a range of sample szes. Smulatons Smulatons Specfy the number of Monte Carlo teratons. Each smulaton conssts of the full process requred to obtan a run length. That s, the number of smulatons here s the number of run lengths that are used to form the run length dstrbuton. A 2007 artcle n Communcaton n Statstcs-Smulaton and Computaton suggests that 5000 to smulatons s usually suffcent. For estmatng percentles near 0 or 1, more smulatons may be needed. For searches for N, you may wsh to use fewer smulatons to determne the ballpark for N and then larger numbers of smulatons for fne-tunng. Mamum Run Length For each smulaton, f the run length reaches the mamum run length, the mamum run length s used for that smulaton. Ths keeps smulatons from gong on ndefntely when run lengths are etremely long. For eample, f the average run length s close to the mamum run length, the mamum run length should be ncreased. Run Length Run Length Summary 1 and 2 Select the frst and second run length dstrbuton summary value to be reported. If a value between 0 and 1 (not nclusve) s entered, the correspondng dstrbuton percentle s reported. For eample, f 0.40 s entered, the run length such that 40% of smulated run lengths are shorter s reported. Percentle or Average When solvng for N, the value entered for "Percentle or Average" ndcates whether the search s based on the average run length or some percentle, such as the medan. If a value between 0 and 1 (not nclusve) s entered, the correspondng dstrbuton percentle s used n the search for N. For eample, f ARL (Average Run Length) s entered, and "Run Length" s 10, the program wll search for the smallest N for whch the average run length s 10.f 0.40 s entered, and "Run Length" s 10, the program wll search for the smallest N for whch 40% of smulated run lengths are shorter than 10. Run Length When solvng for N, ths s run length that s searched for based on the specfed "Percentle or Average". For eample, f ARL (Average Run Length) s entered, and "Run Length" s 10, the program wll search for the smallest N for whch the average run length s 10.f 0.40 s entered, and "Run Length" s 10, the program wll search for the smallest N for whch 40% of smulated run lengths are shorter than

6 Sample Sze N (Sample Sze) Enter a value for the sample sze, N. Ths s the number of observatons n each sample. Ths sample sze s also used when prelmnary samples are used. When a sample sze of one s used, the prelmnary sample standard devaton s estmated wth all the prelmnary data jontly. You may enter a range such as 2 to 10 by 2 or a lst of values separated by commas or blanks. You mght try enterng the same number two or three tmes to get an dea of the varablty n your results. For eample, you could enter " ". Dstrbutons Tab In-Control and Out-of-Control Dstrbutons In-Control Dstrbutons Specfed By Specfy whether a known mean and standard devaton wll be used for the centerlne and standard devaton for boundares, or f the centerlne and standard devaton for boundares wll be based on smulated prelmnary samples. In-Control and Out-of-Control Dstrbutons - Known In-Control Mean (Center Lne) Ths s the centerlne mean from whch boundares and the n-control standard devaton are based. In-Control Standard Devaton Ths s an assumed known standard devaton from whch the boundares are calculated. In-Control and Out-of-Control Dstrbutons Prelmnary Samples In-Control Dstrbuton Ths s the dstrbuton from whch n-control prelmnary samples are smulated. The mean and standard devaton of the n-control samples are used to defne the centerlne and the control boundares for that smulaton. The parameters of the dstrbuton can be specfed usng numbers or letters. If letters are used, ther values are specfed n the boes below. The value "M0" s reserved for the value of the mean under the null hypothess. M0 (In-Control Mean) Ths s the mean of the n-control dstrbuton. Prelmnary samples are smulated based on a dstrbuton wth ths mean. These values are substtuted for the "M0" n the dstrbuton specfcaton gven above. You can enter a lst of values usng the synta or 0 to 3 by 1. Note that whether "M0" s the mean of the smulated dstrbuton depends on the formula you have entered. For eample, "N(M0 S)" does have M0 as ts mean, but "N(M0 S)-N(M0 S)" has a mean of zero

7 Number of Prelmnary Samples For each run length smulaton, ths s the number of samples that are smulated to determne the centerlne and standard devaton for the control chart. For eample, f a control chart uses 50 samples wth N = 5 to establsh the n control centerlne and standard devaton, 50 should be entered here. The program wll generate 50 samples of sze 5 each, obtan means and standard devatons (or ranges) for each sample and use those to produce a centerlne and standard devaton estmate as a bass for the smulated control chart. Commonly recommended numbers of n-control prelmnary samples are 30, 50, and 60. Varyng ths number allows one to see the effect on the dstrbuton of run lengths. Standard Devaton Estmaton Method Specfy whether the overall control chart standard devaton for each smulaton s to be estmated from the ranges of the prelmnary samples or the standard devatons of the prelmnary samples. If N s one, then ths selecton s gnored and the standard devaton s calculated based on all the prelmnary samples jontly. In-Control and Out-of-Control Dstrbutons Outof-Control Out-of-Control Dstrbuton Ths s the dstrbuton from whch out-of-control samples are smulated. The parameters of the dstrbuton can be specfed usng numbers or letters. If letters are used, ther values are specfed n the boes below. The value "M1" s reserved for the value of the mean under the alternatve hypothess. M1 (Out-of-Control Mean) Ths s the mean of the out-of-control dstrbuton. Out-of-control samples are smulated based on a dstrbuton wth ths mean. These values are substtuted for the "M1" n the dstrbuton specfcaton gven above. You can enter a lst of values usng the synta or 0 to 3 by 1. Note that whether "M1" s the mean of the smulated dstrbuton depends on the formula you have entered. For eample, "N(M1 S)" does have M1 as ts mean, but "N(M1 S)-N(M1 S)" has a mean of zero. In-Control and Out-of-Control Dstrbutons S and Other Parameters Parameter Values (S, A, B, C) Enter the numerc value(s) of parameter lsted above. These values are substtuted for the correspondng letter n the dstrbuton specfcatons. You can enter a lst of values usng synta such as or 0 to 3 by 1. You can also change the letter that s used as the name of ths parameter

8 Tests Tab Include Tests Specfy whether to nclude ths test n the Reports and Plots. The specfc optons for each test are specfed below. Shewhart (Xbar) Optons Use Z-Multpler or Probablty Specfy whether the Z-multpler wll be nput drectly or calculated based on the two-sded probablty. The Z-multpler s the value of Z n the Shewhart Xbar chart formula for the lmts: meanlne +/- Z * SDmean Z-Multpler The Z-multpler s the value of Z n the Shewhart Xbar chart formula for the lmts: meanlne +/- Z * SDmean Two-Sded Probablty Ths probablty s used to calculate the Z-multpler to create the Shewhart lmts. The Z-multpler s the value of Z n the Shewhart Xbar chart formula for the lmts: meanlne +/- Z * SDmean Cusum and FIR Cusum Optons K K s the value that s subtracted from the z-score n the CUSUM procedure formula. H H s the out-of-control threshold for the CUSUM procedure. Typcal choces are 4 and 5. FIR FIR or Fast Intal Response s a value that may be used to shorten the run lengths of the CUSUM procedure. FIR s often set to H/2. EWMA Optons R R s the weght assgned to the terms of the movng average chart. It s sometmes known as lambda or w. Values n the range of 0.10 to 0.30 are common. L L defnes the dstance to the boundary. It s essentally a Z-multpler used to create L-sgma lmts. It s sometmes known as q. A typcal value s

9 Eample 1 Run Length Dstrbuton A researcher wshes to eamne the run length dstrbuton for a process montored by a Shewhart (Xbar) chart. S observatons are to make up the sample eamned at each hour. The n-control mean and standard devaton are known to be 5.2 and 3.1, respectvely. The researcher would lke see the run length dstrbuton f the out-ofcontrol mean and standard devaton are 6.2 and 3.1, respectvely. A Z-Multpler of 3.0 s to be used n the control chart for the boundares. Setup Ths secton presents the values of each of the parameters needed to run ths eample. Frst, from the PASS Home wndow, load the procedure wndow by clckng on Qualty Control, and then clckng on. You may then make the approprate entres as lsted below, or open Eample 1 by gong to the Fle menu and choosng Open Eample Template. Opton Value Desgn Tab Solve For... Run Length Dstrbuton Smulatons Mamum Run Length Run Length Summary 1... ARL (Average Run Length) Run Length Summary 2... MRL (Medan Run Length) N (Sample Sze)... 6 Dstrbutons Tab In-Control Dstrbutons Specfed By... Mean, Standard Devaton Drectly In-Control Mean In-Control Standard Devaton Out-of-Control Dstrbuton... N(M1 S) M1 (Out-of-Control Mean) S Tests Tab Shewhart (Xbar)... Checked All Other Tests... Unchecked Use Z-Multpler or Probablty... Z-Multpler Z-Multpler Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Control Charts for a Process Mean Shewhart control lmts were determned by specfyng a centerlne mean and standard devaton drectly. Out-of-Control Dstrbuton (determnes sze of shft): Normal(M1 S) Average Medan In- In- Out-of- Chart Run Run Cntrl Cntrl Cntrl Type Length Length N Mean Z SD LCL UCL Mean M1 S Shewhart Notes Smulatons: Run Tme: 4.0 seconds

10 References Ryan, T.P Statstcal Methods for Qualty Improvement. Wley. New York. Montgomery, D.C Introducton to Statstcal Qualty Control. Wley. New York. Report Defntons Average Run Length s the mean of the run lengths across all smulatons. Medan Run Length s the medan of the run lengths across all smulatons. N s the number of unts measured n each sample. In-control Mean s the assumed known value of the center lne of the control chart. Z s the Z-multpler whch corresponds to the two-sded probablty of a sngle sample mean outsde the control lmts. In-control SD s the assumed known standard devaton that s used n the calculaton of lmts. LCL and UCL are the lower and upper control chart lmts, respectvely. Out-of-Control Mean (M1) s the mean of the dstrbuton from whch out-of-control samples are drawn. It's dfference from the Mean Lne s the shft to detect. Other parameters, often S (standard devaton), defne the dstrbuton from whch out-of-control samples are drawn. Summary Statements For a Xbar control chart wth a mean lne at 5.20 and lower and upper Shewhart control lmts of 1.40 and 9.00, respectvely, samples of sze 6 from the dstrbuton Normal(M1 S) wth mean 6.20 have an average run length of 72.3 and a medan run length of These results are based on 5000 Monte Carlo samples. Indvdual Summares Out-of-Control Dstrbuton (determnes sze of shft): Normal(M1 S) Average Medan In- In- Out-of- Chart Run Run Cntrl Cntrl Cntrl Type Length Length N Mean Z SD LCL UCL Mean M1 S Shewhart Average Run Length 95% CI: (70.3, 74.3) Medan Run Length 95% CI: (48.0, 52.0) Average Run Length and Percentles Avg 1% 5% 10% 25% 50% 75% 90% 95% 99% The results wll vary slghtly because they are based on smulaton. Plots Secton Ths plot shows the dstrbuton of run lengths of 5000 smulated runs

11 Eample 2 Comparng Tests Contnung wth the Eample 1 parameters, the researchers would lke to compare the varous control chart tests avalable. Setup Ths secton presents the values of each of the parameters needed to run ths eample. Frst, from the PASS Home wndow, load the procedure wndow by clckng on Qualty Control, and then clckng on. You may then make the approprate entres as lsted below, or open Eample 2 by gong to the Fle menu and choosng Open Eample Template. Opton Value Desgn Tab Solve For... Run Length Dstrbuton Smulatons Mamum Run Length Run Length Summary 1... ARL (Average Run Length) Run Length Summary 2... MRL (Medan Run Length) N (Sample Sze)... 6 Dstrbutons Tab In-Control Dstrbutons Specfed By... Mean, Standard Devaton Drectly In-Control Mean In-Control Standard Devaton Out-of-Control Dstrbuton... N(M1 S) M1 (Out-of-Control Mean) S Tests Tab All Tests... Checked Use Z-Multpler or Probablty... Z-Multpler Z-Multpler K H... 5 FIR R L

12 Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Control Charts for a Process Mean Shewhart control lmts were determned by specfyng a centerlne mean and standard devaton drectly. Out-of-Control Dstrbuton (determnes sze of shft): Normal(M1 S) Cusum Parameters: K: 0.5, H: 5, FIR: 2.5 EWMA parameters: R: 0.25, L: 3 Average Medan In- In- Out-of- Chart Run Run Cntrl Cntrl Cntrl Type Length Length N Mean Z SD LCL UCL Mean M1 S Shewhart Cusum Cus+Shew FIR Cusum FIR+Shew EWMA EWMA+Shew The FIR tests show the process s out-of-control much sooner than the other tests. Eample 3 Valdaton Usng Montgomery A table n Montgomery (1991), page 298, gves the average run lengths for a mean shft of one standartd devaton to be Cusum (10.4), Cusum wth Shewhart (10.20), Cusum wth FIR (6.35), Cusum wth FIR and Shewhart (6.32). Setup Ths secton presents the values of each of the parameters needed to run ths eample. Frst, from the PASS Home wndow, load the procedure wndow by clckng on Qualty Control, and then clckng on. You may then make the approprate entres as lsted below, or open Eample 3 by gong to the Fle menu and choosng Open Eample Template. Opton Value Desgn Tab Solve For... Run Length Dstrbuton Smulatons Mamum Run Length Run Length Summary 1... ARL (Average Run Length) Run Length Summary 2... MRL (Medan Run Length) N (Sample Sze)... 1 Dstrbutons Tab In-Control Dstrbutons Specfed By... Mean, Standard Devaton Drectly In-Control Mean... 0 In-Control Standard Devaton... 1 Out-of-Control Dstrbuton... N(M1 S) M1 (Out-of-Control Mean)... 1 S

13 Tests Tab Cusum... Checked Cusum + Shewhart... Checked FIR Cusum... Checked FIR Cusum + Shewhart... Checked All Other Tests... Unchecked Use Z-Multpler or Probablty... Z-Multpler Z-Multpler K H... 5 FIR Output Clck the Calculate button to perform the calculatons and generate the followng output. Numerc Results Numerc Results for Control Charts for a Process Mean Shewhart control lmts were determned by specfyng a centerlne mean and standard devaton drectly. Out-of-Control Dstrbuton (determnes sze of shft): Normal(M1 S) Cusum Parameters: K: 0.5, H: 5, FIR: 5 Cusum Parameters: K: 0.5, H: 5, FIR: 2.5 Average Medan In- In- Out-of- Chart Run Run Cntrl Cntrl Cntrl Type Length Length N Mean Z SD LCL UCL Mean M1 S Cusum Cus+Shew FIR Cusum FIR+Shew The average run lengths are very close to those presented n Montgomery (1991)

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