ITB J. Eg. Sci., Vol. 4 No.,, 65-78 65 A Two Cotrol Limits Double Samplig Cotrol Chart by Optimizig Producer ad Customer Risks Dradjad Iriato & Ai Juliai Maufacturig Systems Research Group FTI, Istitute of Techology Badug Email: dradjad@mail.ti.itb.ac.id Abstract. Stadard Shewhart process cotrol chart has bee widely used, but it is ot sesitive i detectig small shift. A umber of alteratives have bee proposed to improve the capability of cotrol chart. The double samplig (DS cotrol chart is aimed at improvig the capability to detect ay small shift out-of cotrol coditio by observig the secod sample without iterruptio. The capabilities of DS cotrol chart were measured as the expected sample size (as a measure of ispectio cost or the cotrol chart power (as a measure for customer risk. Optimizatio of these criteria is used to determie the cotrol limits. I this paper, we optimize both producer ad customer risks uder a certai expected umber of sample sizes as the costrait. Comparig the result to the previous procedure that oly optimize customer risk, the proposed optimizatio procedure gives the same first cotrol limit but smaller secod cotrol limits with higher value of cotrol chart power. Keywords: cotrol chart; customer risk; double samplig; optimizatio; producer risk. Itroductio Regardless of how well desiged or maitaied, ay maufacturig process produces iheret or atural variability as a cumulative effect of uavoidable causes. Cotrol chart is oe amog recogized statistical process cotrol tools that, i geeral, is proactive ad maily aimed at moitorig the process []. A cotrol chart is desiged to accurately idetify atural variability i a maufacturig process as a result of uassigable (or chace causes, or a result of assigable (or special causes, which is cosidered as out-of-cotrol process. I this respect, the stadard Shewhart X cotrol chart has bee widely used [,]. Much of its recogitio is approved because the Shewhart X cotrol chart is simple ad effective. The desig of Shewhart cotrol chart is simply determied by producer risk (the risk to decide that the process is out-ofcotrol whe it is uder-cotrol. At a level of =.7, so that the distaces of cotrol limits from process mea are three times of sample stadard deviatio (see e.g. []. This stadard Shewhart cotrol chart is, however, ot sesitive i detectig small shift that is commo i today precise maufacturig processes. This Received September d, 9, Revised Jue 8 th,, Accepted for publicatio October 8 th,.
66 Dradjad Iriato & Ai Juliai weakess causes the customer risk or the risk of passig o-coformig output whe decidig that the process is uder-cotrol whe it is out-of-cotrol. A umber of alteratives have bee proposed to improve the power or sesitivity of cotrol chart i detectig small shift, such as movig average cotrol chart, cumulative sum cotrol chart ad cotrol chart with warig limits []. We should ote that a quick actio to false idicatio of out-of-cotrol process ca be misleadig. Because of radomess of oise i process data, the presece of a assigable cause is ofte ot immediately observable. It is a mistake to react quickly to a outcome as if it came from a special cause, whe actually it is ot [3]. This coditio is kow as false alarm. Quick detectio without icreasig false alarm rate is desirable. Accordigly, a cotrol chart with warig limit is explored further to get better cotrol o the maufacturig process. A cotrol chart with this objective is useful for cotiuous flow maufacturig system for high volume productio. Reyolds, et al. [4] proposed a cotrol chart with warig limit kow as a variable samplig iterval (VSI cotrol chart. If a out-of-cotrol warig or sigal occurs, ext sample will be take i a shorter samplig iterval; otherwise it is reasoable to take a loger samplig iterval. They also proposed multiple samplig itervals, but oly provided umerical example for two samplig iterval. Istead of varyig samplig iterval, Costa [5] proposed a variable sample size (VSS cotrol chart i dealig with a out-of-cotrol warig or sigal. Costa [5] also metioed a simple model usig oly two sample sizes. VSI ad VSS have the same idea of usig a cofirmatio i the occurrece of out-of-cotrol sigal, ad both are better (higher cotrol chart power i detectig out-of-cotrol coditio with small shift tha stadard Shewhart cotrol chart. The double samplig procedure (DS uses the both ideas of VSI ad VSS simultaeously. I case a out-of-cotrol warig or sigal occurs, i additio to the first sample, the secod sample is added (similar to the idea of VSS. This secod sample with larger sample size is observed with zero or the shortest time iterval (similar to the idea of VSI. The DS cotrol chart was firstly proposed by Croasdale [6]. I this first DS cotrol chart, iformatio from the first ad secod samples is evaluated separately, which thus cofirmatio is doe oly with the secod sample. Daudi, et al. [7] ad Daudi [8] improved Croasdale s DS cotrol chart, ad proposed DS cotrol chart that utilizes the iformatio from both samples at the secod stage. This larger sample size improved the precisio of cotrol chart sice it uses smaller sample stadard deviatio. I estimatig the cotrol chart limits, Daudi's DS cotrol chart optimized of the expected sample size. Istead of miimizig the expected sample size, Iriato ad Shiozaki [9] maximized the power of cotrol chart to determie the
A Two Cotrol Limits Double Samplig Cotrol Chart 67 cotrol chart limits. He, et al. [], ad Costa ad Claro [] have made further developmet of DS cotrol chart. I the ecoomic desig of cotrol charts, there are three categories of costs to cosider [], i.e. costs for samplig ad sample ispectio, costs for ivestigatig out-of-cotrol sigal ad correctig the deviatio, ad costs of producig o-coformig products. Accordigly, there are three motivatios i desigig a cotrol chart ad i estimatig of cotrol chart parameter, i.e. (i miimizig the expected umber of samplig ad ispectio, (ii maximizig capability or probability to detect out-of-cotrol sigal, ad (iii miimizig false alarm (out-of cotrol alarm whe process is uder-cotrol. The secod motivatio is kow as improvig the power of cotrol chart so that miimizig customer risk, while the third motivatio is for miimizig producer risk. The third motivatio is miimizig the time of ioperative system sice the process is cosidered as out-of-cotrol i the presece of commo cause (udercotrol. I this paper, we proposed a method to estimate the cotrol chart limits by optimizig the risks of producer ad customer. By cosiderig both risks, we ca optimize the determiatio of cotrol chart limits uder both coditios, i.e. false alarm coditio whe process is uder-cotrol ad deliverig ocoformig output uder out-of-cotrol process, simultaeously. After itroductio chapter, the outlie of this paper icludes a brief review o the developmet of DS cotrol chart, how to estimate the cotrol chart limits by optimizig the power of the test, ad how to estimate the cotrol limits by optimizig both producer ad costumer risks. At the ed we ca coclude this paper. The DS Cotrol Chart Procedures Croasdale s DS cotrol chart procedure [6] is described as follows (its scheme is also exhibited i Figure :. Take cocurretly the first sample of size ( X i, i,,. ad the secod sample of size ( X i, i,,. from a populatio with mea value ad a kow stadard deviatio.. Calculate the first sample mea X X i /. i 3. Let [ M, M] are limits for the first stage. If X / is i M, M ], the [ process is cosidered to be uder-cotrol, otherwise calculate the sample
68 Dradjad Iriato & Ai Juliai mea X X i / i ad observe the secod stage. X 4. Let [ M, M] are limits for the secod stage. If M, or if / X M, the the process is cosidered to be out-of-cotrol, / otherwise the process is cosidered to be uder-cotrol. For a shift from the mea value ( /, assume the characteristic of output of process follows a ormal distributio fuctio N (,, the probability that the process is moitored as uder-cotrol is give as follows: P [ M ] [ M ] { [ M ]} [ M ] [ M ] { [ M ]} ( where ( is the cumulative distributio fuctio of stadard ormal distributio. The average ru legth is ARL = /( P, ad the expected total sample size is, where P [ M ] [ M ]. ( P X / X / M Take secod sample M Out of cotrol Uder cotrol Uder cotrol - M Take secod sample - M Out of cotrol (First sample (Secod sample Figure Croasdale s DS cotrol chart procedure.
A Two Cotrol Limits Double Samplig Cotrol Chart 69 Differet with Croasdale s procedure, i Daudi s DS cotrol chart [8], the secod sample will be observed oly if the first sample is sigalig a warig, a grey area betwee uder-cotrol ad out-of-cotrol decisios as i Figure. The procedure is described as follows:. Take cocurretly the first sample of size, X i, i,,. ad the secod sample of size, X i, i,,. from a populatio with mea value ad a kow stadard deviatio.. Calculate the sample mea X X i /. i 3. If ( X /( / is i I (see Figure, the process is cosidered to be uder-cotrol. 4. If ( X /( / is i I 3 (see Figure, the process is cosidered to be out-of-cotrol. 5. If X /( / is i I (see Figure, calculate the secod sample mea ( X X i / i ad observe the secod stage. 6. Calculate the total sample mea X ( X X /( ad sample stadard deviatio. X X X 7. If L L or L L, ad if L / / / X or L, the the process is cosidered to be out-of-cotrol, / otherwise the process is cosidered uder-cotrol. Let Z ( X /( / ad Z ( X /( /, the the probabilities that the process is cosidered to be uder cotrol by the first sample ad after observig the secod sample ca be formulated as P a Pr[ Z I] ad P a Pr[ Z I ad Z I 4] respectively, ad the probability that process uder cotrol is Pa Pa Pa.
7 Dradjad Iriato & Ai Juliai X / / X L L Out of cotrol ( I 3 Take secod sample ( I L Out of cotrol Uder cotrol ( I Uder cotrol ( I 4 - L - L Take secod sample ( I Out of cotrol ( I 3 - L Out of cotrol (First sample (Secod sample Figure The Daudi s DS cotrol chart procedure. For a shift from the mea value process is cosidered to be uder-cotrol becomes: P a [ L ] [ L ] * zi ( /, the probability that the { [ cl rc z ] [ cl rc z ]} ( z dz where ( ad ( are the desity ad cumulative distributio fuctios of stadard ormal distributio respectively, * I [ L, L ( L, L ], r, ad c r /. The average ru legth is ARL = /( Pa, ad the expected total sample size is Pr[ Z I ]. Iriato ad Shiozaki [9] aalyzed both DS cotrol charts, ad proved the advatage of Daudi's DS cotrol chart compared to Croasdale s DS cotrol chart i detectig out-of-cotrol sigals. (
A Two Cotrol Limits Double Samplig Cotrol Chart 7 3 Estimatig the Limits of DS Cotrol Chart There are five parameters required to specify the Daudi s DS cotrol charts, i.e. L, L, L, ad. Daudi, et al. [7] suggested a optimizatio procedure for miimizig the expected sample umber to be ispected, which thus the motivatio is to reduce the ispectio cost for moitorig the maufacturig process (producer risk. To fid the solutio, they proposed a heuristic algorithm as follows: (i Determie ad. (ii For a give value of L, both costraits are used to determie the values of L ad L. (iii Fid the optimal compositio of L, L ad L umerically that miimize the objective fuctio for all possible pairs of (,. Differetly, Iriato ad Shiozaki [9] cosidered the power or capability of cotrol chart i detectig the process mea shift. Therefore, the motivatio is to miimize risk of ot kowig that the process mea has deviated (customer risk while settig sample sizes ad so that the expected total sample size is fixed (e.g. =4 or 5 as suggested by Shewhart []. The optimizatio is formulated as follows: Max { [ L ] [ ]} L, L Subject to: L, L zi { [ cl rc z ] [ cl rc z ]} ( z dz *. (3 (i E[total sample size ] =, that is Pr[ Z I ] L [ L ]. (ii Pr[Out of Cotrol ] =, that is { [ L ] [ L ]} { [ cl * zi z] [ cl z]} ( z dz From the first costrait, L ca be expressed i terms of L, which the it reduces the umber of decisio variables. Sice the left had side of the secod
7 Dradjad Iriato & Ai Juliai costrait is a icreasig fuctio of L, the L ca be uiquely determied for fixed L ad L. Usually, stadard Shewhart chart is used as the basis for compariso. The stadard Shewhart X cotrol chart use =5 ad L=3, which thus the producer risk is set at.7. For a shift =.5 ad., the power of the test are.64 ad.8, respectively. Table shows some cotrol limits of DS cotrol charts for some pairs of ad but still give a expected samplig umber =5. It is clear that the DS cotrol chart gives better power tha the stadard Shewhart X cotrol chart. Accordigly, the out-of-cotrol sigal will occur i a shorter iterval tha the stadard Shewhart X cotrol chart. However, it should be oted that the expected sample size icreases as the shift of process mea gets larger. Table Power of DS cotrol chart for some pairs of sample sizes. Sample size L L L =4; =; =5 =4; =3; =5 =4; =5; =5 =4; =6; =5 Power =.5 =..673 3.357 3.7.357.766.674 3.657 3.49.375.88.6744.9999.379.9.966 3.3854 3.557.44.3459.967 3.758 3.87.46.3577.9674.996.467.366.8 3.4575 3.35.6.466.8 3.77.9754.637.476.85.9593.647.48.38 3.46.9966.683.569.38 3.6.959.7.558.389.99.733.55 Table also shows that maximizig power leads to higher value of L (with lower value of L. Usig the fuctio i the first costrait, the higher value of L implies to higher value of L, which is limited to L. If L is very large, the it is o loger ecessary as a out-of-cotrol limit at the first stage. Iriato [] proposed a revised the DS cotrol chart by elimiatig L, ad its scheme is show i Figure 3.
A Two Cotrol Limits Double Samplig Cotrol Chart 73 X / / X L Take secod sample ( I Uder cotrol ( I L Out of cotrol Uder cotrol ( I 4 - L Take secod sample ( I - L Out of cotrol (First sample Figure 3 Revised DS cotrol chart. (Secod sample Table Pa of revised DS cotrol chart for some pairs of sample sizes. Sample size L L =4; =; =5 =4; =3; =5 =4; =5; =5 =4; =6; =5 Power =.5 =..67.783.47.336669.673.787.48.33677.674.783.48.33688.6744.78.486.336879.965.649.4978.4444.966.6486.49783.44494.967.648.49785.44545.9674.6478.49775.44579.79.478.66476.53653.8.47.6647.536484.8.476.6646.5364.85.473.6646.536399.38.463.7384.567977.38.457.73788.56786.38.45.73777.567753.389.443.73766.567647 Accordigly the optimizatio of power i (3 ca be reformulated as follows:
74 Dradjad Iriato & Ai Juliai Max L L L,L [[ ] [ ] { [ L ] [ L ]}.{ [ L ] [ L ]} (4 Subject to: (i E[total sample size ] =, ad the Pr[ Z I ].{ [ L ] [ L ]} (i Pr[Out of Cotrol ] =, ad the [ L ] [ L ]}.{ [ L ] [ ]}. { L. Table shows some cotrol limits of DS cotrol charts for some pairs of ad but still give a expected samplig umber =5. Compared to Table, the revised DS cotrol chart provides higher power, which meas higher capability i detectig shift of process mea. 4 Cosiderig Producer ad Customer Risk As discussed i the first chapter, there are three motivatios i desigig a cotrol chart ad i estimatig of cotrol chart parameter, i.e. (i miimizig the expected umber of samplig ad ispectio, (ii maximizig the power of cotrol chart or miimizig customer risk, ad (iii miimizig false alarm rate or miimizig producer risk. Cosiderig first ad third motivatios, Stadard Shewhart cotrol chart [] is set by determiig a fixed sample size (suggested 4 or 5 ad a false alarm rate. Miimizig cost for ispectio i the first motivatio will lead to miimize the sample size (i sigle samplig or to miimize the expected umber of sample (i double samplig. This first motivatio was used by Daudi [8] i order to estimate cotrol chart limits. Istead of usig the first motivatio, Iriato ad Shiozaki [9] ad Iriato [] secod motivatio. I maufacturig cycle, the process usually starts uder-cotrol ad is moitored usig cotrol chart. If a assigable cause occurs, corrective actio must be doe to elimiate the cause [3]. However, a out-of-cotrol sigal ca occur without a assigable cause, kow as false alarm sigal. I this case a iexperiece operator may shut the system dow ad tries to fid the assigable cause. Sice it is a false alarm, process iterruptio for othig costs the producer. Accordigly, the false alarm rate or producer risk should be miimized (the third motivatio, ad it ca be doe cocurretly with the secod motivatio (i.e. miimizig customer risk.
A Two Cotrol Limits Double Samplig Cotrol Chart 75 Based o the revised DS cotrol chart the false alarm will occur at the secod stage. The false alarm rate is affected by cotrol limits L ad L. Similarly, the rate of customer risk β is also affected by cotrol limits L ad L. Accordigly, we ca develop a optimizatio model as a fuctio of cotrol limits L ad L as follows Mi C C (5 Subject to: E [total sample umber ] = O Pr[ Z I ] { [ L ] [ L ]} O, where C ad C are costs related to for producer ad customer risks respectively. The expected values of both risks are as follows: Pr [Out of cotrol ] = α or O { [ L ] [ L ]} { [ L ] [ L ]} Pr [Uder- cotrol ] = β or [ L ] [ L ] [ L ] [ L ] [ L ] [ L ]. Optimizatio for equatio (5 is ot straightforward, ad thus a heuristic approach is used as i Figure 4. Table 3 shows the optimizatio results for C C ad C C. Sample size =4; =; =5 =4; =3; =5 =4; =5; =5 =4; =6 =5 Table 3 Cotrol limits L ad L. Δ C C C C L L Power L L Power.5.6745 -.6745.478.7..6745.9763.8474.6745.5455.7445.5.9674 -.9674.358.73..9674.9665.8.9674.496.7448.5.86 -.86.93.499..86.55.7449.86.55.77.5.383 -.383.656.387..383.3.77.383.557.695
76 Dradjad Iriato & Ai Juliai Set the values of,,, δ Use,, i the costrait L ca be obtaied? No Icrease Yes Fid L Put these obtaied values,,, L, L, δ i the objective fuctio Miimum cost? No Yes Fiish Figure 4 Algorithm for fidig the optimum solutio. Higher values of C meas that the maufacturig system is expesive, ad thus process iterruptio should be miimized. O the other had, higher value of C meas the risk at the customer is high, ad thus β risk should be miimize. A maufacturer is cosidered more iteral orieted will have C C. The reaso of this orietatio usually is for maximizig the utilizatio of facilities, ad at the ed will result i iteral efficiecy. Otherwise a maufacturer is cosidered as exteral orieted if C C. This is a orietatio towards customer satisfactio. I case where C C, as i Table
A Two Cotrol Limits Double Samplig Cotrol Chart 77 3, there is o feasible solutio for. 5, while i case of C C feasible solutio ca be obtaied. As metioed previously, there are three motivatios i desigig a cotrol chart ad i estimatig of cotrol chart parameter. This proposed method clearly cosidered the secod ad third motivatios, i.e. maximizig capability or probability to detect out-of-cotrol sigal (, ad miimizig producer risk (. Both risks are expressed i cotrol chart limits ad. Compared to Table (oly cosiderig customer risk, with the same arragemets of sample size,, ad, the result (i Table 3 gives the same first cotrol limits ( ad smaller secod cotrol limits (. Sice the expected umber of ispected items is determied by the value of the first cotrol limits (, the proposed method has the same expected cost for ispectio (as the first motivatio. The results also give higher cotrol chart power (, which thus reduces the customer risk. However, smaller secod cotrol limits ( surely icreases the false alarm rate. For example if =4, =, ad =5, the optimum value of =.6745 ad =.9763, the the power is =.8474 (compared to.3369 from Table ad the producer risk =.645 (compared to.7 costrait of Table. If cost at producer is higher (e.g. C C the icreases to.5455, the the power is decreases to.7445 ad the producer risk decreases to.6. 5 Coclusio Despite of the advatage of givig higher capability i detectig out-of-cotrol sigal of mea shift, the DS procedure eeds a complicated calculatio i estimatig its cotrol limits. Efficiecy of the calculatio is improved by chagig the optimizatio problem that implies o the resulted cotrol chart limits ad cotrol chart power. This paper developed a optimizatio procedure for DS cotrol chart uder a motivatio toward both iteral efficiecy ad exteral customer satisfactio, simultaeously. This paper is focused o detectig the process mea shift as the mai objectives i o-lie quality cotrol. Further research will be doe that will give focus o variace cotrol. Ackowledgemet The authors highly ackowledge commets of reviewers that cotribute to a sigificat improvemet of this paper.
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