Modified exponentially weighted moving average (EWMA) control chart for an analytical process data

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1 Joural of Chemical Egieerig ad Materials Sciece Vol. (), pp. -, Jauary Available olie at ISSN Academic Jourals Full Legth Research Paper Modified expoetially weighted movig average (EWMA) cotrol chart for a aalytical process data Alpabe K. Patel * ad Jyoti Divecha Directorate of Ecoomics ad Statistics, Sector-8, Gadhiagar-389, Idia. Departmet of Statistics, Sardar Patel Uiversity, Vallabh Vidyaagar, 388, Idia. Accepted 6 December, This article itroduces modified expoetially weighted movig average (modified EWMA) cotrol chart. The modified EWMA cotrol chart is very effective i detectig small ad abrupt shifts i moitorig process mea. This chart is based o modified EWMA cotrol chart statistic, which is a correctio of EWMA chart statistic ad it is free from iertia problem. The advatage of usig modified EWMA chart is its good performace for observatios that are autocorrelated or idepedetly ormal distributed. The performaces of Modified EWMA chart are illustrated alog with EWMA chart for the two types of aalytical process data. The average ru legth (ARL) properties of modified EWMA scheme are derived usig Markov Chai approach. Key words: Abrupt chage, autocorrelated, average ru legth, expoetially weighted movig average (EWMA), modified. INTRODUCTION The cotrol charts amely, Shewart chart (Shewhart, 94), EWMA chart (Roberts, 959) ad CUSUM chart (Page, 954) are ofte used for detectig shifts i a sequece of idepedet ormal observatios with commo variace comig from a particular process. The usefuless of a chart is determied from the timely detectio of shifts as they occur. The literature o evaluatio of properties of these three charts idicates that Shewhart chart detect all sigle large shifts, CUSUM chart detects shifts by accumulatig chages i a directio, while EWMA chart detects shifts through chages accumulated uder expoetial smoothig. I geeral, these three cotrol charts will fail to detect shift i oe case, either the large shift or the small shift i the process. Moreover, these charts caot be used directly i chemical ad pharmaceutical idustries because the observatios from processes i these idustries are frequetly autocorrelated. This autocorrelatio has a large impact o the cotrol charts developed uder the idepedece assumptio. A typical effect of autocorrelatio is to decrease the i-cotrol average ru legth (ARL) leadig to a higher false alarm rate tha for a idepedet process. Moreover, the autocorrelated *Correspodig author. alpakip@yahoo.co.i. processes may also have abrupt chages. Both the situatios are bothersome for these idustries. I particular, i most of the idustrial processes, temperature is a essetial process variable ad the process is sesitive to the cumulative small chages as well as some abrupt chages i temperature values. A abrupt chage is the upredictable large shift occurrig i the process beig moitored for small shifts. Recetly, research efforts have bee put for the process subject to abrupt chages. EWMA chart has received more attetio as it also provides a estimate of forecast of process mea. Yaschi (995) have discussed estimatio of curret process mea i a process subject to small chages, as well as abrupt chages o the basis of EWMA. Capizzi ad Masarotto (3) have provided a class of ew cotrol charts, AEWMA charts o the basis of EWMA to detect shifts of all types for idepedet data. Reyolds ad Stoumbos (6) suggested usig two EWMA cotrols simultaeously, oe for process mea ad aother for process variace to tackle detectio of small shifts with abrupt chages. Woodall ad Mahmoud (5) have proposed a measure of iertia, the sigal resistace, ad show that EWMA suffers from iertia problem more tha the CUSUM. O the other had, several authors have discussed process cotrol methods for autocorrelated observatios. Motgomery ad Mastragelo (99) preseted method

2 Patel ad Divecha 3 that cosists of modelig the autocorrelative structure i the origial data usig EWMA ad applyig Shewhart cotrol charts to the idepedet ormal residuals. Their paper led to discussio, ad oe of the reviewers: Rya (99) stated that, residual chart for AR() will perform poorly uless autoregressive (AR) parameter is egative or extremely close to ad that the derivatio of ARL properties was uclear because most of the time, the true state of autocorrelatio i the process is ot kow. Falti ad Woodall (99) appreciated their method for the virtue of simplicity i the iterest of operators ad egieers. Cox (96) have show that the EWMA is ot a useful predictor for a AR() process havig a autoregressive parameter less tha/3. It is foud that for ARMA (,) processes, the EWMA is useful as a predictor provided that, the AR term sufficietly domiates the movig average (MA) compoet. Harris ad Ross (99) studied the impact of traditioal cotrol charts based o residuals uder several correlatio structures of process ad cocluded that the traditioal EWMA, CUSUM chartig techiques perform poorly i the presece of serial correlatio ad the use of residuals i Shewhart chart is ot effective for detectig small shifts whe the observatios are highly positively correlated. Lu ad Reyolds (999) cosidered the problem of moitorig the mea of a process i which the observatios ca be modeled as a AR() process plus a radom error. They have suggested EWMA cotrol chart based o the residuals from the forecast values of the model usig a itegral equatio method. They showed that whe the level of autocorrelatio is low or moderate, the EWMA chart for residuals ad the EWMA chart for origial observatios require about the same amout of time to detect various shifts; but for high levels of autocorrelatio ad large shifts, the EWMA chart of the residuals is faster a little. I all the efforts, a easy solutio to cotrol chartig is missig. Cotrol charts are effective tools for improvig process quality ad productivity ad simplicity is always demaded by the users. I this article, we itroduce a Modified EWMA (Modified EWMA) chart that combies the features of a Shewhart chart ad a EWMA chart i a simple way ad has the ability to detect small, as well as large shift as soo as possible as required by some idustrial processes with high level of first order autocorrelatio. Modified EWMA cotrol statistic gives weight to the past observatios i slightly differet way tha EWMA ad each curret chage is cosidered with full weight. This corrects EWMA statistic for sufferig from iertia problem. The uderlyig idea is to adapt the weight of the past observatios, past chages, the curret observatio ad the curret chage. Ulike AEWMA cotrol chart which corrects EWMA statistic by cosiderig the differece betwee process value ad cotrol statistic value as the error =Y -X - to detect shifts of differet sizes, Modified EWMA cotrol chart does it by measurig the curret fluctuatio cotaied i EWMA cotrol statistics, as Y - Y -, the differece betwee two cosecutive process values to detect abrupt chages. This is a ew approach ad does ot require cosideratio of score fuctio (error fuctio) ad separate detectio limits; hece it is simpler to operate i compariso to AEWMA chart. We feel that just like EWMA, AEWMA should be applicable to autocorrelated processes, wheever EWMA is apt. The Modified EWMA cotrol statistic is defied as: X = (-)X - Y (Y - Y - ), <. Note that X is the geometric sum of past observatios, past chages, curret observatio ad the curret chage i the process. This make Modified EWMA scheme perform like a EWMA scheme for small shifts ad at the same time detect abrupt chages alike Shewhart scheme. EXPONENTIAL WEIGHT BASED QUALITY CONTROL CHARTS Let Y, =,, be a sequece of first order autocorrelated or idepedet ormal observatios with the process target mea as ad a commo variace, such as sigle measuremets of the process. EWMA cotrol chart A EWMA cotrol chart is based o the statistic: X = (-)X - Y, () where is a suitable costat, such that <, X is th statistic ad Y is th observatio. The quatity X represets the startig value, ofte the target value. This schemes sigals whe X value exceeds specified cotrol limit. I geeral upper ad lower cotrol limits of EWMA Chart are: UCL = L CL = LCL= -L λ ( λ. () ( L is suitable i cotrol width limit, ad is the process stadard deviatio. The detailed discussio o EWMA cotrol schemes ca be foud i Motgomery (996). Adaptive EWMA cotrol chart EWMA cotrol chart caot detect abrupt chage/s, ad thus fail i worst-case situatio. To overcome this

3 4 J. Chem. Eg. Mater. Sci. problem, Capizzi ad Masarotto (3) desiged Adaptive EWMA (AEWMA) cotrol charts. Their class of cotrol charts cosiders the followig statistic: X = X - (e ), X = (3) Where e = Y - X - ad (e ) is a score fuctio. A alarm is raised whe X -µ > h, where deotes the target value of the process mea ad h is a suitable threshold. The h value is maily determied by esurig that the desired mea time betwee false alarms is large. They have desiged AEWMA schemes for three score fuctios, each based o expoetial weight parameter ad specified values of oe or two costats. The AEWMA scheme is effective i detectig shifts of differet sizes, but the moitorig scheme is ot simple. Modified EWMA (Modified EWMA) cotrol chart The desirable properties of a cotrol chart are that it is easy to implemet ad is effective for detectig shifts of all sizes as per techical specificatios. The Modified EWMA chart that we itroduce cosiders past observatios similar to EWMA scheme ad additioally cosiders past chages, as well as latest chage i the process. A Modified EWMA cotrol chart is based o the statistic: X = (-)X - Y (Y - Y - ), (4) where, =,, < is a costat ad the startig value is, X = µ = Y. The mea ad variace of cotrol statistic X are λ respectively, the process mea ad ( λ ( λ ) for autocorrelated process (Appedix ( λ ) ), based o ρ, the first order autocorrelatio coefficiet, ad correlatio structure of the process observatios ad process fluctuatios. Therefore, the upper ad lower cotrol limits of Modified EWMA chart are: UCL = L CL = LCL = -L λ λ ( λ ) λ λ λ λ( λ λ Where is the target mea, is the process variace, (5) ad L are suitable Modified EWMA scheme costats represetig expoetial weight ad i cotrol width limit. This chart reduces to Shewhart chart for highly autocorrelated process for =, L=3. Modified EWMA chart is capable for capturig sigals of small shift i the process, as well as abrupt chages i a autocorrelated process. The process is i cotrol, as log as the values are plotted withi the cotrol limits. A poit plots outside the cotrol limits is iterpreted as evidece that the process is out of cotrol, ad actio are required to fid ad elimiate the assigable causes resposible for this behavior. We kow that EWMA chart suffer from iertia problem because of error i EWMA statistic. Modified EWMA statistic is a corrected EWMA statistic ad EWMA is the best predictor i the class of liear predictors (Yashchi, 993). Hece Modified EWMA is the best predictor of process mea; its mea square error (MSE) is il for autocorrelated process with early oe autocorrelatio (Appedix ). It forecasts the process mea state accurately. This makes Modified EWMA chart free from iertia problem. Now we apply Modified EWMA cotrol scheme moitorig alog with EWMA scheme o temperature data ad capsule weight data from a aalytical process. Applicatio: Modified EWMA chart for highly autocorrelated ormal process Table displays the part of measuremets o temperature colum take every miute from a chemical process that is workig i cotrol ad out of cotrol situatios, abrupt chages ad small shifts occur. The target mea is.468 C, ad the process stadard deviatio is =.3 C ( =.79) (Table ). Here, EWMA (=., L=.84, UCL=.98, LCL=9.738, CL=.468) cotrol scheme could ot detect two abrupt chages at st ad 8 th observatios, ad the small shifts were detected from 7 st observatio. Modified EWMA (=., L=.683, UCL =.99, LCL=9.737) detects those two abrupt chages ad desired small shifts early, timely i 6 st ru. I Modified EWMA statistic, the smaller the value of, the larger is the effect of past history of the process. Therefore, if we select =., the L=.683 is automatically fixed through ARL. Applicatio : Modified EWMA chart for idepedet ormal process The data i Table are part of the measuremets (i grams) take every 3 s from a maufacturig process that is workig i cotrol ad a out of cotrol situatio,

4 Patel ad Divecha 5 Table. Moitorig performace of EWMA ad modified EWMA for the chemical process temperature data. *Shift. No. Y AR() EWMA Modified EWMA =. L =.84 L = * * * * * *.47 * * 3.68* Table. Moitorig performace of EWMA ad Modified EWMA for the pharmaceutical process capsule weight data. *Shift. No. Y EWMA Modified EWMA =.4 L =.477 L = * * * * * * * * abrupt chage occurs. The target mea is 5 g, ad the process stadard deviatio is =.3 g. Data were adapted from Capazzi ad Masarotto (3). Here, EWMA (=.4, L=.477, UCL= 5.6, LCL=4.894, CL=5) cotrol scheme could ot detect ay chages i observatios. Modified EWMA ( =.4, L=.43, UCL = 5.4, LCL=4.896) detects upper shifts at st, 3 rd to 5 th, 7 th to 9 th, ad lower shift at th observatios. From Figure, we ca see that EWMA chart could ot detect ay Shift. O the other had Modified EWMA chart detects all type of shifts as show i Figure.

5 6 J. Chem. Eg. Mater. Sci. EWMA statistic Figure. Plot of EWMA cotrol chart. Modified EWMA statistic Figure. Plot of modified EWMA cotrol chart. PROPERTIES OF MODIFIED EWMA SCHEME AND COMPARISONS WITH EWMA SCHEME ARL properties All the ARL computatios were carried out usig Markovchai approach described i Appedix. ARL properties of Modified EWMA scheme have bee derived for differet choices of {, L} ad i cotrol ARL to be 5 for detectio of small shift oly, because the abrupt chages are detected as they occur. The Modified EWMA scheme itroduced i this article is a modificatio of EWMA scheme for detectig small shifts alog with abrupt shifts. Here, we have compared the ARL values for Modified EWMA scheme with ARL values of EWMA scheme over a wide rage of parameter

6 Patel ad Divecha 7 Table 3. Average ru legths of modified EWMA schemes. Modified EWMA ARL = 5 L Abrupt Shift > > > > > > >.5 > Table 4. Average ru legths of EWMA scheme (Lucas ad Saccucci, 99). EWMA ARL = 5 Shift =.75, L = =.5, L = =., L = =.4, L = values. I Table 3, the ARL values of Modified EWMA for all choices of are smaller tha those for EWMA scheme. ARL values for EWMA schemes as per Markov Chai approach are show i Table 4 for the sake of easy comparisos. Thus, detectio of small, as well as large shifts by a sigle Modified EWMA chart is possible with proper choice of ad kowledge of AR() parameter ad/or correlatio structure of the process observatios ad the process fluctuatios. It should be clear that, the process fluctuatio variace would be smaller for process beig cotrolled for small shifts ad few large abrupt chages. Iertia (Sigal resistace) properties Woodall ad Mahmoud (5) proposed a measure of iertia, the sigal resistace, to be the largest stadard deviatio from target ot leadig to a immediate out-ofcotrol sigal. They defie a simple measure of iertia that they called "sigal resistace". I physics, "iertia" refers to the state of resistace a object has to a chage i its state of motio. Similarly, i statistical process cotrol, "iertia" ca refer to a measure of the resistace of a chart to sigalig a particular process shift. For the EWMA chart, the sigal resistace is: SR(EWMA) = [ h (-) w ] / (6) Where w is the value of the EWMA statistic, h = L λ (. The asymptotic cotrol limits for the EWMA chart are ±h. The sigal resistace for a EWMA cotrol chart i cojuctio with Shewhart cotrol limits is: SR (EWMA Shewhart) = L if h w < ( h λ L) / ( λ ) (7) [ h ( λ ) w]/ λ if ( h λ L) / ( λ ) w h Where L is the value of the multiplier used to obtai the Shewhart limit ad w is the value of the EWMA statistic. Obviously, i this case, the sigal resistace caot exceed the value of the multiplier used to obtai the Shewhart limit, that is, L. The sigal resistace for Modified EWMA cotrol chart is:

7 8 J. Chem. Eg. Mater. Sci. SR(Modified EWMA) = [ h ( λ ) w ]/ λ if h w h (8) Where w is the value of the Modified EWMA statistic, h = L limit. Coclusio λ λ ( λ) is the Modified EWMA decisio λ λ A simple cotrol chart for moitorig small, as well as large shifts i highly autocorrelated or idepedet ormally distributed observatios, such as aalytical processes are give. I additio, the Modified EWMA chart ca also be used to forecast the observatio i the ext period, which ca help aalysts take prevetive actios before process departures to the out-of-cotrol state. REFERENCES Brook D, Evas DA (97). A Approach to the Probability Distributio of CUSUM Ru Legths, Biometrika, 59: Capizzi G, Masarotto G (3). A Adaptive Expoetially Weighted Movig Average Cotrol Chart. Techometrics, 45: Cox DR (96). Predictio by Expoetially Weighted Movig Averages ad Related Methods. J. Roy. Stat. Soc., 3: Crosier RB (986). A New Two-Sided Cumulative Sum Quality Cotrol Scheme. Techometrics, 8: Harris TJ, Ross WH (99). Statistical Process Cotrol Procedures for Correlated Observatios. The Ca. J. Chem. Eg., 69: Lu CW, Reyolds MR Jr. (999). EWMA Cotrol charts for Moitorig the Mea of Autocorrelated Processes. J. Qual. Tech., 3(): Lucas JM, Saccucci MS (99). Expoetially Weighted Movig Average Cotrol Schemes: Properties ad Ehacemets. Techometrics, 3: -. Motgomery DC, Mastraglo CM (99). Some Statistical Process Cotrol Methods for Autocorrelated Data. J. Qual. Tech., 3(3): Motgomery DC (996). Itroductio to Statistical Quality Cotrol., 3 rd Ed, Wiley, p. 9 Reyolds MR Jr, Stoumbos ZG (6). Comparisos of Some Expoetially weighted Movig Average Cotrol Charts for Moitorig the Process Mea ad Variace. Techometrics, 48(4): Page ES (954). Cotiuous Ispectio Schemes. Biometrika, (4): -5. Roberts SW (959). Cotrol chart tests Based o Geometric Movig Averages. Techometrics, : R Laguage: Copyright 4, The R Foudatio for Statistical Computig Versio.. (4--5), ISBN Wardell DG, Moskowitz H, Plate RD (994). Ru Legth Distributios of Special-Cause Cotrol Charts for Correlated Processes. Techometrics, (36): 3-7. Woodall WH, Mahmoud (5). The Iertial Properties of Quality Cotrol Charts. Techometrics, 47: 4. Yashchi E (995). Estimatig the Curret Mea of a Process Subject to Abrupt Chages. Techometrics, 37: 3-33.

8 Patel ad Divecha 9 APPENDIX Appedix : Mea ad variace of modified EWMA cotrol statistic Let {Y, } are first order auto correlated ( ρ ) process observatios followig ormal distributio with mea ad variace. For mea ad variace, cosider expadig Modified EWMA statistic: X = Y (-) X - (Y - Y - ) = Y (-) [Y - (-) X - (Y - - Y - )] (Y - Y - ) = Y (-)Y - (-) X - (-) (Y - - Y - ) (Y - Y ) = Y (-)Y - (-) [Y - (-) X -3 (Y - - Y - 3)] (-) (Y - - Y - ) (Y - Y - ) = Y (-)Y - (-) Y - (-) 3 X -3 (-) (Y - - Y -3 ) (-) (Y - - Y - ) (-) (Y - Y - ) Ad cotiuig like this recursively for X -j, j =, 3,...,, we obtai: X = Here, (-) j Y -j (-) X (-) j (Y -j -Y -j- ). (-) j (Y -j -Y -j- ) accouts for sum of the past ad latest chage/fluctuatios i the process; the uaccouted curret fluctuatios accumulated to time i EWMA statistic: Let X = (-) j Y -j (-) Y Takig expectatio o both side: E(X ) = Y -j- ) (-) j E(Y -j ) (-) E(Y ) (-) j (Y -j -Y -j- ) (-) j E(Y -j - j λ [ ( λ ) ] But λ ( λ ) = = [ ( λ ) ] [ ( λ )] j = E(X ) = [ ( ] µ ( µ = The startig value of process is, X = µ = Y ad < is a costat. The mea is: E(X ) = E((-)X - Y (Y - Y - )) = µ. The variace of Modified EWMA cotrol statistic X is: (i) V (X ) = V( (-) j Y -j ) V((-) Y ) V( Y -j- )) V (X ) = (-) V(Y ) j = (-) j (Y -j - j j λ ( λ ) V ( Y j ) λ ( λ ) Cov ( Y j, Y j ) j j j ( λ ) V( Y j Y ) ( ) [( ),( )] j λ CovY j Y j Y j Y j λ( λ) Cov j ( Y j,( Y j Y j ) ) λ( λ) CovY ( j,( Y j Y j ) ) Sice Y s are autocorrelated ormal with variace, the variace of (Y -Y - ) () is: σ = σ ρσ = ( ρ) σ (small whe ). The weights (-) j decrease geometrically with the age of sample mea. Suppose Y s are correlated to the forward fluctuatio (Y -Y - ) () with commo correlatio ad correlated to the backward fluctuatio (Y -Y ), () with commo correlatio, ad forward fluctuatio (Y -Y - ) are correlated to the backward fluctuatio (Y -Y ), () with commo correlatio 3, the asymptotic variace for large is give as: V( X ) = λ λ( σ ( 4ρ 3 ( ρ )( λ ) σ λ ( λ ) ρσ ( ρ ) σ ( λ ( λ ) ρ ( ρ ) σ ( λ ) ρ ρ σ (ii) ( λ ) ( λ ) I ormal autocorrelated process (a) with ρ3 early egative half ad ρ, ρ early equal ad opposite i sig ad beig moitored for small shifts, (b) with autocorrelatio ρ early oe, the aforemetioed expressio (ii), reduces to: V ( X ) = λ λ( σ ρσ ( ( (iii) λ( ) Let λ ρσ ( λ) be a small value for high value of ρ ad small λ, sometimes eve egligibly small such that

9 J. Chem. Eg. Mater. Sci. Modified EWMA limits equal EWMA limits. Therefore: V(X ) = λ ( λ( (. (iv) The upper-lower cotrol limits for Modified EWMA chart are: ± L λ λ( λ). (v) ( λ) ( λ) Appedix : Cyclical steady state average ru legths of modified EWMA scheme usig Markovchai approach The use of Markov chais to approximate ARL of CUSUM schemes was discussed by Brook ad Evas (97). Crosier (986) modified their Markov chai approach to compute coditioal ad cyclical steady state ARLs for CUSUM scheme. Further, Lucas ad Saccucci (99) customized their approach for ARL properties of EWMA scheme. They evaluated the properties of the cotiuous state Markov chai by discretizig the ifiite state trasitio probability matrix. This procedure ivolves dividig the iterval betwee the upper ad lower cotrol limits ito t = m subitervals of width. The cotrol statistic, X, is said to be i trasiet state i at time, if S i - < X S i for i = -m,- m,..., m, where S i represet the midpoit of the i th iterval. For example, whe the LCL = to UCL = distace is divided ito 5 sub-itervals each of width ( =.9), , ,-.95,.95, , ; the S i s are -.565,-.58,.,.58,.565. approximated by assumig that the cotrol statistic is equal to S i wheever it is i state (i), i = -m, -m,, m. Let Y, =, deote the sequece of process observatios, which are first order autocorrelated ormal with mea ad variace. The the trasiet chage occurrig i state of Modified EWMA cotrol statistic is X -(-) X -, which is equivalet to Y (Y -Y - ). The trasitio i Modified EWMA cotrol chart statistic has the followig mea ad variace: E(S S - ) = E [Y (Y -Y - )] = µ, ad V(S S - ) = V[ Y (Y -Y - )] = V(Y ) V(Y -Y - ) cov{ Y, (Y -Y - )} = (- ρ ) (- ρ ) ρ = where variace, is a small value for early values ρ. The the i cotrol trasitio probabilities are approximated as: P i j = Pr(goig to S j i S i ) Pr [( λ) - {(S j - )-(- )S i } < Y ( λ ) - {(S j )-(- )S i }], i, j = -m, -m,, m. (m = imply t = 5), where represets the stadard ormal distributio fuctio. Appedix 3: Algorithm ad R program for ARL computatio of modified EWMA Step-: Choose the parameter costat, target mea, stadard deviatio, Limit L. Step -: Compute UCL= L λ λ(, ( ( The trasitio probability matrix (t.p.m.), represeted i partitioed matrix form, is give by: LCL= -L λ λ(. ( ( ( ) P=, T where the sub matrix R cotais the probabilities of goig from oe trasiet state to aother, is the idetity matrix, ad is a colum vector of oes. Hece P ij represets the probability that the cotrol statistic goes from state (i) to state (j) i oe step. We have derived algorithm for calculatig cyclical steady state ARL of Modified EWMA scheme usig the Markov chai approach followig Crosier (986) ad Lucas ad Saccuscci (99). The trasitio probabilities i R are Step-3: Choose umber of sub-itervals (states), t = m. Step-4: Compute width w = (UCL - LCL)/t, shift = w/. Step-5: Compute the values of S i i = -m, -m,, m for each sub-iterval. Step-6: Compute the trasitio probability matrix (t. p. m.). Compute R, R = [( λ) - { (S j ) ( - ) S i µ }] - [( λ) - { (S j - ) ( - ) S i µ }] Step-7: Adjust the t. p.m. such that row sums are uity. Step-8: Compute u = [I - R] - Step-9: Compute q = R * I Step-: ARL = q * u, OR ARL = q'[i - R] -