Statistical Process Cotrol Usig Two Measuremet Systems Stefa H. Steier Dept. of Statistics ad Actuarial Scieces Uiversity of Waterloo Waterloo, NL G Caada Ofte i idustry critical quality characteristics ca be measured by more tha oe measuremet system. Typically, i such a situatio, there is a fast but relatively iaccurate measuremet system that may be used to provide some iitial iformatio, ad a more accurate ad expesive, ad possibly slower, alterative measuremet device. I such circumstaces, it is desirable to determie the miimum cost cotrol chart for moitorig the productio process usig some combiatio of the measuremet systems. This article develops such a procedure. A example of its use i the automotive idustry is provided. Key Words: Cotrol Chart; Measuremet Costs. Itroductio Metrology is a importat aspect of maufacturig sice measuremets are ecessary for moitorig ad cotrollig productio processes. However, i may situatios there is more tha oe way to measure a importat quality dimesio. Frequetly the choice betwee the differet measuremet systems is ot clear due to tradeoffs with respect to measuremet cost, time, ad accuracy. Oe particular situatio, that is explored i this article, occurs whe there is a quick ad dirty measuremet device that is iexpesive ad relatively fast, but is ot the most accurate way to measure, ad a slower more accurate ad expesive measuremet device or method. Good examples of this situatio occur i may maufacturig plats. For example, i
foudries the chemistry of molte iro may be checked usig a quick method, called a quick lab, or may be set to a laboratory. I the foudry applicatio, the quick measuremet is used to moitor ad cotrol the process, sice adjustmets to compositio are required immediately ad the lab measuremet takes a umber of hours. The slower lab measuremets are used oly for after the fact cofirmatio. Aother example is the use of i-lie fixture gauges to moitor the productio of egie covers. The fixture gauges provide approximate measuremets for some critical dimesios. A Coordiate Measuremet Machie (CMM) ca be used to determie more precise values. This egie covers example is discussed i more detail later. Whe two measuremet devices are available the curret process moitorig approach is to use results from each measuremet device separately ad ofte for differet purposes. However, from cost ad efficiecy cosideratios it is ot optimal i most cases to use oly oe of the measuremet devices to moitor the process output. I this article a method for usig both measuremet systems i cojuctio to moitor the process mea ad process variability is proposed. The basic idea is straightforward. The first measuremet device is iexpesive ad quick, so we try iitially to make a decisio regardig the state of cotrol of the process based o results from the first measuremet device. If the results are ot decisive, we measure the same sample of uits agai usig the more accurate measuremet device. We assume the testig is ot destructive or itrusive. Notice that this procedure does ot require additioal samplig sice the same sample is measured agai if the iitial results were ot coclusive. Not requirig a additioal idepedet sample is a advatage sice obtaiig aother idepedet sample may be difficult ad/or time cosumig. This idea of usig the secod measuremet device oly i cases where the first measuremet does ot yield clear cut results is motivated by earlier work by Croasdale (974) ad Daudi (99). Croasdale ad Daudi develop double samplig cotrol charts as a alterative to traditioal X cotrol charts. Double samplig charts add warig limits to the traditioal cotrol charts i additio to cotrol limits. The warig limits are used to decide whe a secod idepedet sample is eeded to reach a coclusio regardig the process
stability. Double samplig charts, however, are ot applicable i the two measuremet devices problem sice they assume that the same measuremet device measures all samples ad that measuremet error is egligible. The article is orgaized i the followig maer. I Sectio, cotrol charts for detectig chages i the process mea or variability usig two measuremet devices i combiatio are defied. A example of their use is give i Sectio. I Sectio 4 two measuremet cotrol charts are desiged to miimize measuremet costs subject to a statistical costrait i terms of the false alarm rate ad power of the resultig charts. Fially, i Sectio 5 ad 6 some implemetatio issues are discussed ad a summary of the results is give.. Cotrol Charts for Two Measuremet Systems The results from the two measuremet systems are modeled as follows. Let Y i j = X i + e ij, i =,...,, j =,, () where X i is the true dimesio of the ith uit, Y i ad Y i are the results whe measurig the i th uit with the first ad secod measuremet devices respectively, ad e ij is the measuremet error. We assume the e ij s are idepedet ad ormally distributed with mea zero ad variace σ j, ad that X i ad e ij are idepedet of each other. Assumig that the mea of e ij equals zero, implies that we have compesated for ay log term bias of the measuremet device. The variability of the two measuremet devices (, σ ) are assumed to be well kow. This is a reasoable assumptio sice regular gauge R&R studies for all measuremet devices are ofte required i idustry ad i ay case may be easily performed. Sice each sample may be measured twice we assume the measuremet is o destructive. We also assume that the actual dimesios of the quality characteristic of iterest are ormally distributed with mea ad stadard deviatio equal to µ ad σ respectively. Thus, X ~ N(µ, σ ), ad X ~ N( µσ, ). Also, without loss of geerality, we assume that the i-cotrol process has zero mea ad stadard deviatio equal to oe. I other words, for the i-cotrol process the X variable
4 represets a stadardized variable. For o-ormal quality characteristics a trasformatio to ear ormality would allow the use of the results preseted here. We begi by defiig some terms. Measurig the uits i the sample with the first measuremet device we may calculate Y = Y. If the same sample is measured with the i= secod measuremet device we obtai Y = Y. Based o the distributioal assumptios, i= it ca be show that Y ad Y are bivariate ormal with ( ) = E( Y ) = µ, Var( Y ) = σ σ ( + ), Var( Y ) = ( σ + σ ), ad ( ) = E Cov Y,Y X E Y Cov Y,Y i i ( ( )) + Cov E Y X ( ( ), E( Y X )) = + σ = σ. Note Y ad Y are ot idepedet sice they represet the sample averages obtaied by the first ad secod measuremet device respectively o the same sample of size. Assumig σ <, Y provides more precise iformatio about the true process mea tha Y. However, a weighted average of Y ad Y provides eve more iformatio. Defie w as the average of the i weighted sums give by (). w i = ky + k Y ( ) () i i Based o the momets of Y ad Y we get: E( w) = µ, Var( w) = σ ( ) + σ ( ) σ Cov Y,w ( ( )+ ( )+ ), k k k k + k ( ) = ( σ σ ) We obtai the most iformatio about the true process mea whe the weightig costat k is chose so as to miimize Var( w). Deotig this best value for k as k opt ad solvig gives k opt = σ ( σ + σ ). () Usig k opt, the variace of w ad the correlatio coefficiet relatig Y ad w, deoted ρ w, are give by (4) ad (5) respectively.
5 ( ) = σ Var w k opt σσ + (4) σ + σ σ σ σ σ ( [ + ]+ σ ) σ σ σ σ. (5) + ρ w = ρ( Y,w k opt ) = ([ ][ + ]) The value of k opt will be close to zero if the secod measuremet system is much more precise tha the first device. I that case, w almost equals Y. I geeral, the bigger the discrepacy betwee ad σ the less there is to gai from usig w over Y. The proposed two measuremet X chart operates as follows. I every samplig iterval, take a ratioal sample of size from the process. Measure all uits with the first measuremet device to obtai Y, Y,..., Y. Calculate Y, ad if Y falls outside the iterval [ c,c ], where c is the cotrol limit for the first measuremet device, we coclude the process is out-of-cotrol. If, o the other had, Y falls withi the iterval [ r,r ], where r is the extra measuremet limit ( r c ), we coclude the process is i-cotrol. Otherwise, the results from the first measuremet device are icoclusive, ad we must measure the sample uits agai usig the secod measuremet device. Combiig the iformatio from the two measuremets o each uit i the [ ] sample together, we base our decisios o w. If w falls outside the iterval c, c, where c is the cotrol limit for the combied sample, we coclude the process is out-of-cotrol, otherwise we coclude the process i i-cotrol. This decisio process is summarized as a flowchart i Figure.
6 Take a ratioal sample of size Coclude process is out-of-cotrol Y c [, c ] o Measure with first device Y? Y [ r, c] Measure with secod device w c [, c ] Y r [, r ] yes Coclude process is i-cotrol Figure : Decisio Process for Cotrol Charts for the Process Mea Usig Two Measuremet Systems I may situatios it is reasoable to simplify this procedure by settig c equal to ifiity. As a result of this restrictio, based oly o the results from the first measuremet device, we ca coclude that the process is i-cotrol or that we eed more iformatio, but ot that the process is out-of-cotrol. I applicatios this restrictio is reasoable so log as the time delay for the secod measuremets is ot overly large. A two measuremet cotrol chart desiged to detect chages i process variability, similar to a traditioal S-chart, is also possible. However, if the measuremet variability is substatial it is very difficult to detect decreases i the process variability. Thus, we cosider a chart desiged to detect oly icreases i variability. Also, to simplify the calculatios somewhat we do ot allow sigals based o oly the first measuremet device. This simplificatio is aalogous to the versio of the chart for the process mea where we set c =. The chart is based o two sample stadard deviatios, defied as s = y i y ( ), i = ( ) ad s w = w w i ( ) i= ( ), where w i is give by (). The two measuremet system cotrol chart for detectig icreases i stadard deviatio operates as follows. If s < d, coclude the process is i-cotrol with respect to variability. Otherwise, measure the sample
7 agai with the secod measuremet system. If s w < d w we coclude the process is i-cotrol, otherwise coclude the process variability has icreased. I ay applicatio ivolvig two measuremet devices the first questio that eeds to be aswered is whether just oe of the measuremet devices should be used or if usig them i combiatio will result i substatially lower costs. It is difficult to provide simple geeral rules sice there are may potetially importat factors. However, if the cheaper measuremet device is quite accurate, say <.4 (relative to a process stadard deviatio of uity), the there is little to be gaied by cosiderig the secod measuremet device, ad it is probably best to use oly the first measuremet device. Whe the measuremet variability is larger, a fairly simple rule for decidig whether a cotrol chart based o two measuremet systems is preferable ca be obtaied by cosiderig oly the variable measuremet cost associated with each measuremet device. With measuremet device i, to match the performace of a traditioal Shewhart X cotrol chart with subgroups of size five we eed samples of size 5+ ( σ i ). If the variable measuremet costs associated with the secod measuremet device is ν times the amout for the first measuremet device, the the ratio of the variable measuremet costs for the charts based o measuremet systems oe ad two is R = ν ( + σ ) ( + ). Based o experiece, the greatest gais from usig the two measuremet device cotrol chart results whe R is close to. Geerally for a substatial reductio i costs, say greater tha aroud %, the value of R should lie betwee.6 ad 8. Otherwise, usig oly the secod measuremet device is preferred if R <.6, ad usig oly the first measuremet device would be better if R > 8. More specific cost comparisos are cosidered at the ed of the Desig Sectio.. Example The maufacture of egie frot covers ivolves may critical dimesios. Oe such critical dimesio is the distace betwee two bolt holes i the egie cover used to attach the cover to the egie block. This distace may be measured accurately usig a coordiate
8 measuremet machie (CMM) which is expesive ad time cosumig. A easier, but less accurate, measuremet method uses a fixture gauge that clamps the egie cover i a fixed positio while measurig hole diameters ad relative distaces. I this example, the fixture gauge is the first measuremet device ad the CMM is the secod measuremet device. Previous measuremet system studies determied that for stadardized measuremets =.5 ad σ =.5 approximately; i.e. the CMM has less measuremet variability tha the fixture gauge. We also kow that o a relative cost basis usig the CMM is six times as expesive as the fixture gauge i terms of persoel time. We shall assume that the fixed costs associated with the two measuremet methods is zero. Thus, i terms of the otatio from the sample cost model preseted i the ext sectio we have: f = f =, ν =, ad ν = 6. The mai goal i this example was to cotrol the process mea. As such, i this example we use a two measuremet system cotrol chart oly to detect chages i the process mea. Process variability is moitored usig a traditioal S-chart with the results oly from the first measuremet system. Solvig expressio (9), give i the Desig Sectio of this article, with the additioal simplificatio that c = gives: r =.8, c =.9, with = 5.6 for a relative cost of 5.65. These values are give approximately o Figure. I this optimal solutio the values for r ad c are almost equal. From a implemetatio perspective settig r ad c equal is desirable sice it simplifies the resultig cotrol chart as will be show. With the additioal costrait that r = c, the optimal solutio to (9) is: r = c =.89, = 5.6 with a correspodig cost of 5.67. For implemetatio the sample size is rouded off to five. Thus, the cotrol limits r ad c are set at ±.. The measuremet costs associated with this pla are aroud % less tha the measuremet costs associated with the curret pla that uses oly the first measuremet device, ad aroud 8% less tha the cost associated with usig oly the CMM machie. Figure gives a example of the resultig two measuremet X cotrol chart. O the chart the sample averages based o the first measuremet device are show with a o, while
9 the sample average of the combied first ad secod measuremets (if the secod measuremet is deemed ecessary) are show with a x s. The extra measuremet limit (± r ) for the results from the first measuremet device ad cotrol limit (±c ) for the combied sample are give by the solid horizotal lies o the chart. If the sample average based o the first measuremet lies betwee the solid horizotal lies o the chart we coclude that the process is i-cotrol. Otherwise, if the iitial poit lies outside the extra measuremet limits a secod measuremet of the sample is required. Usig the secod measuremet we calculate the combied sample weighted average w =.Y +.99Y (based o this weightig we could use just Y rather tha w without much loss of power i this example). If w falls outside the solid horizotal lies we coclude the process shows evidece of a assigable cause; otherwise the process appears to be i-cotrol. The dashed/dotted lie deotes the ceter lie of the cotrol chart. I this example, for illustratio, the value. was added to all the measuremets after the 9th observatio to simulate a oe sigma shift i the process mea. Figure shows that i the 5 measuremets a secod sample was required six times, at sample umbers 7,,,, 4 ad 5. However, oly samples,, 4 ad 5 yield a out-of-cotrol sigal. I the other cases, the secod measuremet of the sample suggests the process is still i-cotrol. Of course the umber of times the secod measuremet was eeded after observatio 9 is also a idicatio that the process has shifted. I this applicatio, usig two measuremet cotrol charts results i a reductio i the measuremet costs without affectig the ability of the moitorig procedure to detect process chages.
sample mea.5 - - -.5 Two Measuremet Cotrol Chart 5 5 5 sample umber Figure : Two Measuremet Cotrol Chart for the Process Mea 4. Desig of Cotrol Charts usig Two Measuremet Systems Determiig the optimal desig for two measuremet cotrol charts ivolves determiig the best values for the cotrol limits ad sample size. However, as poited out by Woodall (986 ad 987) purely ecoomic models of cotrol charts may yield desigs that are uacceptable i terms of operatig characteristics. For example, the optimal desig from a purely cost perspective may have such a large false alarm rate that the chart is routiely igored. For this reaso, i this article, the optimal desigs for two measuremet cotrol charts are costraied to satisfy certai miimum operatig characteristics. We first cosider the desig of two measuremet X charts, ad the look at two measuremet S-charts. The MATLAB computer code that determies the optimal desig i both cases is available from the author. 4. Desig of Two Measuremet X Charts Usig the assumptio of ormality, it is possible to determie the probabilities of makig z π ad Q() z = φ( xdx ) be the various decisios illustrated i Figure. Let φ() z = e z the probability desity fuctio ad cumulative desity fuctio of the stadard ormal respectively. Also, deote the probability desity fuctio of the stadardized bivariate ormal as φ z,z,ρ πσ σ ρ exp z ρ z z z ρ. The, (6), (7) ad (8) ( ) + ( ( ) ( )) ( ) = give expressios for the probabilities that the followig evets occur: the procedure cocludes the process is out-of-cotrol (i.e. the procedure sigals) based o results from the first
measuremet; measurig the sample with the secod measuremet is ecessary; ad the combied results from the first ad secod measuremet devices leads to a sigal. ( ) ([ ] ) ( ) [ ] ([ ] ) ([ ] ) p ( µ ) = Pr(sigal o first measuremet) = Pr Y > c OR Y < c = * * + Q( [ c µ ] σ ) Q c µ σ (6) q ( µ ) = Pr(secod measuremet eeded) = Pr r < Y < c OR r > Y > c = Q( [ c µ ] σ ) Q( r µ σ )+ Q r µ σ Q c µ σ * * * * (7) p ( µ ) = Pr(sigal o combied measuremets) ( ( )) = Pr ( w > c OR w < c )& r < Y < c OR r > Y > c = φ( z, z, ρw) dzdz + φ ( z ρ, z, w) dzdz * * * * z [ ( r µ ), ( c µ ) σ] z [ ( r µ ), ( c µ ) σ] * * z [,( c µ ) σw ] z [ ( c µ ) σw, ] φ( z, z, ρw) dzdz + φ ( z, z, ρ w) dz dz * * * * z [ ( c µ ), ( r µ ) σ] z [ ( c µ ), ( r µ ) σ] * * z [,( c µ ) σw ] z [ ( c µ ) σw, ] (8) where σ * = σ + σ ( ), ad σ * w = ( σσ+ σσ+ σσ ) ( σ + σ ). Note that p, p ad q deped o the true process mea ad stadard deviatio. Settig c equal to ifiity results p ( µ ) = for all µ I this article a cost model based o measuremet costs is developed. This measuremet cost model is easy to use sice it requires oly estimates of the fixed ad variable measuremet costs for the two measuremet devices. A more complex cost model that cosiders all the productio costs could be developed based o the geeral framework of Loreze ad Vace (986). However, the productio cost model is ofte difficult to apply, sice costs due to false alarms, searchig for assigable causes, etc. are difficult to estimate i may applicatios. The goal is to miimize the measuremet costs while maitaiig the desired miimum error rates of the procedure. Let f i ad v i deote the fixed ad variable measuremet costs for the ith measuremet system respectively (i =, ). I our aalysis, without loss of geerality, we may set v =, sice the results deped oly o the relative values of the measuremet costs. I additio, to restrict the possibilities somewhat, the fixed cost associated with the first
measuremet device is set to zero, i.e. f =. This restrictio is justified because typically the first measuremet device is very easy ad quick to use, ad would ot require much setup time ( ) ( ) or expese. The, the measuremet cost per sample is + f + v q µ. The best choice for the samplig iterval must be determied through some other criterio, such as the productio schedule. There are a umber of ways to defie a objective fuctio usig the measuremet costs. Sice the process will (hopefully) sped most of its time i-cotrol we miimize the icotrol measuremet costs. Usig this formulatio, the optimal desig of the cotrol chart usig two measuremet devices is determied by fidig the desig parameters that ( ) ( ) miimize + f + v q (9) subject to α = p ( )+ p ( ).7 ad β = p ( )+ p ( ).75 where α is the false alarm rate, i.e. the probability the chart sigals whe the process mea is icotrol, ad β is the power the probability the chart sigals whe the process mea shifts to µ = ±. These particular choices for maximum false alarm rate ad miimum power to detect two sigma shifts i the mea are based o at least matchig the operatig characteristics of a Shewhart X chart with samples of size five. Optimal values for the desig parameters c, c, r ad, that satisfy (9) ca be determied usig a costraied miimizatio approach such as applyig the Kuh-Tucker coditios. This solutio approach was implemeted usig the routie costr i the optimizatio toolbox of MATLAB. Figures ad 4 show the optimal desig parameters for two measuremet charts that satisfy (9) for differet measuremet cost parameters whe settig c equal to ifiity. Figure gives results whe the secod measuremet device also has o fixed costs, while Figure 4 cosiders the situatio where the fixed cost associated with the secod measuremet device is relatively large. Figures ad 4 may be used to determie the desig parameter values that are approximately optimal for two measuremet X charts i terms of i-cotrol measuremet costs. For measuremet costs i betwee those give, iterpolatio ca be used to determie reasoable
cotrol limit values. I practice, the sample size,, must be rouded off to the earest iteger value. Roudig off the sample size effects the power of the cotrol chart, but has o effect o the false alarm rate of the procedure. Of course, roudig dow the sample size decreases the procedure s power, while roudig up icreases the power. Figures ad 4 each cosist of four subplots that show cotour plots of the optimal desig parameters: r, c, ad as a fuctio of ad σ, the variability iheret i the two measuremet devices. Each subplot represets four differet values of ν, the variable measuremet cost associated with the secod measuremet device. Optimal values for r, c, ad i the geeral case where c is allowed to vary are very similar to those give i Figures ad 4. I geeral, the optimal value of c is large ad cosequetly does ot affect the procedure much uless there is a large shift i the process mea.
4 v = σ r *sqrt().6.9.4.8.. σ. c *sqrt()...5 σ. 5.5 6.95 v = r *sqrt().8.7 c *sqrt(). 6 7 σ..5 σ..5 σ. 5.5.4.6.95.9 v = 4 σ. r *sqrt().9.8 σ. c *sqrt()..5.95.9 σ. 6 5.5 7 8 σ. r *sqrt().8.9. σ. v = 6 c *sqrt()..5.9.8 σ. 5.5 6 7 8 9 Figure : Cotour Plots of the Desig Parameters for the No Fixed Cost Case f =, v =, f =
5 σ. r *sqrt().9.8.7. v = c *sqrt().5. σ σ..95.9 5.5 7 6 v = σ. r *sqrt().9.8 σ. c *sqrt()..5.95.9 σ. 6 7 8 5.5 v = 4 σ. r *sqrt().9.8. σ.. c *sqrt()..5.95.9.8 σ. 9 6 8 7 5.5 v = 6 σ. r *sqrt().4 σ. c. *sqrt().5.95.8 σ. 5.5 7 9 8.9..9.7 6 Figure 4: Cotour Plots of the Desig Parameters for the Large Fixed Cost Case f =, v =, f =
6 Figures ad 4 suggest that the parameters r ad c are the most sesitive to chages i the variability of the measuremet devices. I geeral, whe the measuremet costs of the two measuremet devices are comparable, as the first measuremet device becomes more variable ( icreases), icreases, while r decreases. This result makes sese sice it meas we rely more o the secod measuremet device whe the first device is less precise. Coversely as the secod measuremet device becomes more variable (σ icreases), c ad icrease while r icreases margially, sice we rely more o the first measuremet device. v = v = 4 Pr(d measuremet eeded i-cotrol) Pr(d measuremet eeded i-cotrol).5. 5...5 Figure 5: Cotour Plots of the Probability the Secod Measuremet is Required Process i-cotrol, f =, v =, f =.5.75 Now cosider the case where the secod measuremet device is expesive ( f or ν large). As the secod measuremet device becomes less reliable ( σ icreases), agai we observe that c icreases while ad r icrease margially which makes sese. However, the patter appears to be couterituitive whe the first measuremet device becomes less reliable ( icreases) sice ad c decrease margially, but r icreases! Does this mea that we rely more heavily o the iaccurate first measuremet device? Lookig more closely, this apparet cotradictio disappears. As icreases the optimal r also icreases, but this does ot mea that the decisios are more likely to be based o oly the first measuremet device. Whe the accuracy of a measuremet device is poor we expect to observe large deviatios from the actual value. Thus, the observed icrease i r is oly takig this ito accout. Cosider Figure 5 which shows cotours of the probability the secod
7 measuremet is eeded i the two cases: f = ad ν = or 4. The plots i Figure 5 show clearly that as the first measuremet device becomes less accurate we rely o it less eve though, as show i Figure, r icreases. We may also compare the performace of usig two measuremet charts with traditioal X usig oly oe of the measuremet systems. Figure 6 shows the percet reductio i measuremet costs attaiable through the use of the both measuremet systems as compared with the best of the two idividual measuremet systems. I the case where ν equals, the dotted lie shows the boudary betwee where usig each idividual measuremet system is preferred. To the right of the dotted lie (where the measuremet variability of the first measuremet system is large) the secod measuremet system is preferred. Whe ν equals 4 ad 6, the first measuremet devices o its ow is preferred over the secod measuremet device over the whole rage of the plot. ν = ν = 4 ν = 6 σ 5 5 5 σ 5 5 5 σ 5 5 5. 5. 5. 5.5.5.5 σ Figure 6: Cotours plots showig the percet reductio i i-cotrol measuremet costs possible usig the two measuremet X cotrol chart 4. Desig of Two Measuremet S-Charts Now cosider derivig the optimal two measuremet cotrol chart to detect icreases i the process variability. Mathematically, the optimal two measuremet S-chart that miimizes icotrol measuremet costs is determied by fidig the cotrol limits d ad d w that miimize + vp s () () subject to p s (). ad p s ( ).
8 ( ) ( ( )) ( )= Pr ( sw > dw s > d, ) where p s () = Pr s d σ = = χ + d ( ) σ ad p s σ the two measuremet S-chart sigals, i.e. p s σ ( ) equals the probability σ. χ ( x) is the cumulative desity fuctio of a cetral chi-squared distributio with - degrees of freedom. Usig results preseted i the appedix we may accurately approximate p s ( σ ) for ay give actual process stadard deviatio. The choice of. is based o the power possible usig a traditioal S-chart with o measuremet error ad samples of size five that has a false alarm rate of.. Figure 7 shows the expected percet decrease i measuremet costs that result whe usig the optimal two measuremet S-chart rather tha the lowest cost traditioal S-chart based o oly oe of the measuremet systems. Whe ν =, i.e. both measuremet systems are equally expesive, usig just the more accurate measuremet device is always preferred, ad it is ot beeficial to use the two measuremet system approach. Figure 7 suggests that large potetial savigs i measuremet costs are possible usig the two measuremet approach to detect icreases i process variability. ν = ν = 4 ν = 6 σ 5 5 5 5 4 4 σ 5 5 5 4 5 σ 5 5 5 4 5... 5 5 5 5 5 6.5.5.5 Figure 7: Percetage decrease i i-cotrol measuremet costs possible with two measuremet S-chart, assume f = I practice, a process is typically moitored usig both a X ad S-charts. Thus, from a implemetatio perspective usig the same sample size for both charts is highly desirable. For two measuremet charts, sice typically detectig chages i the process mea is a higher priority we use the sample size suggested by the optimal two measuremet X chart. Solvig
9 () shows that the optimal sample size for the two measuremet S-chart is usually smaller tha the sample size suggested for the two measuremet X chart. As a result, by usig the larger sample size the resultig two measuremet S-chart will have better tha the miimum defied operatig characteristics. Derivig the best values for, d ad d w from () we could prepare plots similar to those i Figures ad 4. However, to simplify the desig we cosider a approximatio. Based o the rage of typical values for measuremet costs ad the measuremet variability, ad assumig f =, we obtai usig regressio aalysis the followig approximatios for the optimal cotrol limits: ˆd =. 94 8. σ + 8. σ +. ν, ad () ˆd w = 7.. σ +. σ. ν 7. d ˆ. These approximately optimal limits give good results over the rage of typical measuremet variability. 5. Implemetatio Issues A alterative approach to process moitorig i this cotext is to use a secod sample that is differet tha the first sample; i.e. take a completely ew sample rather tha measurig the first sample agai. This approach is of course a ecessary if the testig is destructive, but it leads to icreased samplig costs as well as difficulties i obtaiig a ew idepedet sample i a timely maer due to autocorrelatio i the process. However, if these samplig cocers ca be overcome, the advatage of usig a additioal sample is that more iformatio about the true ature of the process is available i two idepedet samples tha i measurig the same sample twice. If feasible, takig a ew idepedet sample would be preferred, however, i may cases it is ot possible i a timely maer. I a similar vei, we may cosider situatios where repeated measuremets with a sigle measuremet system are feasible. If repeated idepedet measuremets are possible the, by
averagig the results, we would be able to reduce the measuremet variability by a factor of. If the measuremets are very iexpesive the repeated idepedet measuremet with oe device will evetually yield (usig eough measuremets) a measuremet variability so small that it may be igored. Alterately, we could apply the methodology developed i this article where we cosider the secod measuremet to be simply the results of repeated measuremets o the uits with the first measuremet device. If repeated iexpesive idepedet measuremets usig the first measuremet device are possible usig those measuremets would be the preferred approach. However, this approach will oly work if we ca obtai repeated idepedet measuremets of the uits which is ofte ot the case. 6. Summary This article develops a measuremet cost model that ca be used to determie a optimal process moitorig cotrol chart that utilizes two measuremet devices. It is assumed that the first measuremet device is fast ad cheap, but relatively iaccurate, while the secod measuremet device is more accurate, but also more costly. The proposed moitorig procedure may be thought of as a adaptive moitorig method that provides a reasoable way to compromise betwee measuremet cost ad accuracy. Appedix Usig the otatio of the article, A = ( y y ), ( y y )( wi w) i= i i= i ( y i y wi w w w i ) ( ), ( i i ) = = has a cetral Wishart distributio with - degrees of freedom ad covariace matrix give by Σ = σ σ σ σ +, + k σ + kσ, k ( σ + σ )+( k ) ( σ + σ )+ k( k) σ (Arold, 988). Deotig the
( ) elemets of the matrix A as a ij it ca be show that Pr a c, a c = ( ) j ( ρ ) ρ Γ Γ j Γ j (( ) ) j + ( ( ) ) ( + ) c c I j + ( ), I j + ( ), ρ ρ, ( ) ( ) where ij is a elemet of the covariace matrix, ρ = is correlatio coefficiet, ( ) is the gamma fuctio, ad Idg, Γ x g d t ( )= t e dt, is the icomplete Gamma fuctio. This ifiite sum coverges quickly uless ρ is very close to oe (or mius oe). Ackowledgemets The author would like to thak Jock Mackay for useful discussios, ad Greg Beett for the derivatio preseted i the Appedix. I additio, suggestios from a umber of referees, a associate editor, ad the editor, substatially improved the article. Refereces Arold, S.F. (988), Wishart Distributio, i the Ecyclopedia of Statistical Scieces, Kotz S. ad Johso, N. editors, Joh Wiley ad Sos, New York. Croasdale, R. (974), Cotrol charts for a double-samplig scheme based o average productio ru legths, Iteratioal Joural of Productio Research,, 585-59. Daudi, J.J. (99), Double Samplig X Charts, Joural of Quality Techology, 4, 78-87. Loreze, T.J. ad Vace, L.C. (986), The Ecoomic Desig of Cotrol Charts: A Uified Approach, Techometrics, 8, -. Woodall, W.H. (986), Weakesses of the Ecoomic Desig of Cotrol Charts, Techometrics, 8, 48-49. Woodall, W.H. (987), Coflicts Betwee Demig s Philosophy ad the Ecoomic Desig of Cotrol Charts, i Frotiers i Statistical Quality Cotrol edited by H.J. Lez, G.B. Wetherill, ad P.Th. Wilrich, Physica-Verlag, Heidelberg.