ECONOMIC STATISTICAL DESIGN OF VARIABLE SAMPLING INTERVAL X CONTROL CHART BASED ON SURROGATE VARIABLE USING GENETIC ALGORITHMS

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1 Voume7 Number4 December206 pp DOI: 0.55/mper ECONOMIC STATISTICAL DESIGN OF VARIABLE SAMPLING INTERVAL X CONTROL CHART BASED ON SURROGATE VARIABLE USING GENETIC ALGORITHMS Tae-HoonLee,Sung-HoonHong 2,Hyuck-MooKwon 3,MinkooLee 4 KoreaAtomicEnergyResearchInstitute,VHTRTechnoogyDeveopmentDivision,Korea 2 ChonbukNationaUniversity,DepartmentofIndustria&InformationSystemsEngineering,Korea 3 PukyongNationaUniversity,DepartmentofSystemsandManagementEngineering,Korea 4 ChungnamNationaUniversity,DepartmentofInformationandStatistics,Korea Corresponding author: Minkoo Lee Chungnam Nationa University Department of Information and Statistics Daehak-ro 99, Yuseong-gu, Daeeon, 3434, Korea phone: e-mai: sixsigma@cnu.ac.kr Received: Juy 206 Accepted:28Juy206 Abstract Inmanycases,aX controchartbasedonaperformancevariabeisusedinindustria fieds. Typicay, the contro chart monitors the measurements of a performance variabe itsef. However, if the performance variabe is too costy or impossibe to measure, and a ess expensive surrogate variabe is avaiabe, the process may be more efficienty controed using surrogate variabes. In this paper, we present a mode for the economic statistica design of a VSIVariabe Samping Interva X contro chart using a surrogate variabe that is ineary correated with the performance variabe. We derive the tota average profit mode from an economic viewpoint and appy the mode to a Very High Temperature ReactorVHTR nucear fue measurement system and derive the optima resut using genetic agorithms. Compared with the contro chart based on a performance variabe, the proposed mode gives aargerexpectednetincomeperunitoftimeintheong-runifthecorreationbetweenthe performance variabe and the surrogate variabe is reativey high. The proposed mode was confined to the sampe mean contro chart under the assumption that a singe assignabe cause occurs according to the Poisson process. However, the mode may aso be extended to other types of contro charts using a singe or mutipe assignabe cause assumptions such asvssvariabesampesize Xcontrochart,EWMA,CUSUMchartsandsoon. Keywords economic design, surrogate variabe, variabe samping interva, TRISO Fue, genetic agorithms. Notations X surrogate variabe, Y performance variabe, n x sampesizeforsurrogatevariabe, n y sampesizeforperformancevariabe, µ x meanofsurrogatevariabe, µ x,0 meanofsurrogatevariabewhentheprocess is in contro, µ x, meanofsurrogatevariabewhentheprocess isoutofcontro, 54 µ y meanofperformancevariabe, µ y,0 mean ofperformance variabewhen the process is in contro, µ y, mean ofperformance variabewhen the process is out of contro, x y c standarddeviationofsurrogatevariabe, standard deviation of performance variabe, magnitudeoftheshiftintheprocessmean measuredin y unit,

2 s q s q λ number of the occurrence of an assignabe cause, ρ correation coefficient between a surrogate variabe and a performance variabe, m number of different samping interva ength, m 2, h -th smaest samping interva engths, =, 2,..., m, k -ththreshodimit, 0 k m k... k, I -th samping interva region using h, =, 2,..., m, IC in contro time, OC outofcontrotime, FA timetoowingtofaseaarm, AC time to owing to assignabe cause, SN 0 expectednumberofsampesinin-contro period, SN expected number of sampes in out-ofcontro period, probabiitythat Xfaoutsidethestcontroimitswhen µ y = µ y,0, probabiitythat Xfaoutsidethestcontroimitswhen µ y = µ y,0 + c y, probabiitythat X beongsto I when µ y = µ y,0, probabiitythat X beongsto I when µ y = µ y,0 + c y, q 2 probabiitythat X faoutsidethe2nd controimitswhen µ y = µ y,0 + c y, R 0 numberoffaseaarmswhenthesignais in action region, numberoffaseaarmswhenthesignais in I region, cost for finding and eiminating an assignabe cause, costforidentifyingafaseaarm, 2 cost incurred from the ost production due toafaseaarm, fixedsampingcost, variabesampingcost, timerequiredtofindanassignabecause, time required to eiminate an assignabe cause, timerequiredtoidentifyafaseaarm, timerequiredtotakeandinterpretasampe, net income per unit time in in-contro state, netincomeperunittimeinout-of-contro state, I netincomepercyce, totaincomepercyce, sampingcostpercyce, R a a 2 a a 3 a 4 b b b 2 b 3 i i 2 I C C 2 τ T ARL 0 ARL ARL L ARL U per-cycecostforsearchingandeiminating an assignabe cause, expected time of occurrence of an assignabe cause between the two adacent sampes, engthofacyce, averagerunengthwhentheprocessis in contro, averagerunengthwhentheprocessis out of contro, ower bound on average run ength when the process is in contro, upper bound on average run ength whentheprocessisoutofcontro. Introduction Contro charts are widey used to monitor and detect different process variations. Controing the process variations prevents the manufacturing of poor products, the need to rework products, and waste. Shewhart contro charts with fixed parameters are typicay sow to detect signas of an assignabe cause. Consequenty, new aternatives to the Shewhartchartshavebeenproposedtomakeupfor this weakness. However, a traditiona approach to samping for contro charts is taken from a process withafixedsizeandfixedtimeintervabetween sampes, and the resuting contro chart is caed a fixed samping intervafsi contro chart. Recent studies have shown that adaptive charts have superior statistica and economic performance compared to FSI contro charts. Recent studies have shown that adaptive charts have superior statistica and economic performance compared to FSI contro charts. Reynodseta.[]introducedtheideaofavariabe samping interva during the production process based on recent data obtained from this process. VSI contro charts have received much attention, for exampe, Reynods and Arnod, Runger and Montgomery, Reynods, Costa and Rahim, Lin et a., Cheweta.andZhangMineta.[2 8]. In a of these studies, an inspection is performed on the quaity characteristic of interestperformance variabe. In some situations, it is impossibe or not economica to directy inspect the performance variabe.insuchcases,theuseofasurrogatevariabe that is highy correated with a performance variabe is an attractive aternative, especiay when inspecting the surrogate variabe is reativey ess expensive than inspecting the performance variabe. In a measurement system for VHTR tristructura-isotropic TRISO fue, for exampe, a direct method for mea- Voume7 Number4 December206 55

3 suring the TRISO fue diameter, which is essentia for the fabrication of the reactor fue, requires very sophisticated equipment and high costs. When a direct measurement of the performance variabe is too expensive, a surrogate variabe that is ineary correated with the performance variabe but ess expensive to measure may be considered instead of the performance variabe. In the measurement system exampe, a Partice Size and Shape AnayzerPSA measurement method is generay ess accurate but much ess expensive than the X-ray measurement method and does not produce radiation because PSA uses aser diffractiontoanayzethesizegradeandshapeofthe partices. Thus, the PSA measurement vaue might beusedasasurrogatevariabe.leeeta.[9]considered an economic design for the X-contro chart using a surrogate variabe under the assumption that a performance variabe might not be used. Additionay,Leeeta.[0]proposedaneconomicdesignidea forthevsi Xcontrochartusingasurrogatevariabe. In this paper, we deveoped an economic statisticadesignforavsi Xcontrochartusingasurrogate variabe with a genetic agorithm under the assumption that the performance variabe might not be used and compared its resuts with a fixed X contro chart design.inthenextsection,anoutineoftheeconomicstatisticamodeforthe Xcontrochartandits underying assumptions are described. In the deveopment of profit function section, the expected cyce timeandexpectednetincomepercycefortheproposed mode are derived. The cost function is then obtainedastheratioofthesetwoquantities.inthe numerica comparisons using genetic agorithms section,theproposedmodeisusedtoobtaintheoptimum designs using the test input vaues in Panagos eta.[,phmhereafter],andisthenappiedtothe measurement system for the nucear fue. In the sensitivity anaysis section, a sensitivity anaysis of the proposed mode is presented for changes in the input parameters. The fina section briefy summarizes the resuts obtained in this paper. Assumptions and mode specifications Assumptions Assume that each product possesses a continuous performance variabe Y that measures the degree to which a product satisfies the stated or impied expectations of customers. If we consider a situation in which measuring the performance variabe is expensive, time-consuming, or even destructive, it is attractivetouseasurrogatevariabe Xthatisineary correated with the performance variabe but ess expensive to measure in this situation. In such cases, we can monitor the production process using a contro chart for ony the surrogate variabe, based on the correation between the performance and surrogate variabes obtained from the experiment performed prior to production. Withthissituationinmind,weproposean X contro chart based on ony the surrogate variabe. Foramorespecificpresentationofthemode,wedescribe the nature of the process conditions and the underying statistica assumptions as foows: The process begins in the in-contro state, with the mean and variance of the performance variabebeing µ y and y,respectivey.anassignabe cause occurs according to a Poisson process with anintensityof λoccurrencesperunitoftime.ifan assignabe cause of the magnitude c occurs, then theprocessmeanshiftsfrom µ x to µ y ± c x. Thesurrogatevariabe Xgiven Y = yisnormaydistributedwiththemean λ + λ 2 yandvariance 2,where λ and λ 2 areknownconstants. λ 2 isassumedtobepositivesothat X and Y have a positive inear reationship. It can be easiyshownthatx, Yfoowsabivariatenorma distributionwiththemeansλ +λ 2 µ y, µ y,variancesλ 2 2 y + 2, 2 y,andcorreationcoefficient ρ = λ 2 y /λ 2 2 y + 2 /2 seetangandlo[2]. Thetimetakentofindanassignabecauseis b, andthetimerequiredtoeiminateitis b.the timetakentoidentifyafaseaarmis b 2,andthe timerequiredtotakeandinterpretasampeis b 3. The cost for finding and eiminating an assignabe causeis a,whiethecostforidentifyingafase aarmis a 2 andthecostincurredfromtheost productionduetoafaseaarmis a 2.Inaddition, thecostofsampingandtestingfor Xvariabesis a 3 + a 4 n x,where a 3 and a 4 arefixedandvariabe sampingcosts,respectivey,and n x isthesampe size.thenetincomesperunittimeofoperation inthein-controandout-of-controstatesare i and i 2,respectivey. Mode specification Based on the previous assumptions, we deveoped amodeforan Xcontrochartusingasurrogate variabe.let h denotethetimeintervabetween X sampes, where the samping interva is varied based onthevaueoftheprecedingsampemean.inthis artice,weassumethatthevsi Xcontrochartsuse afinitenumberofintervaengths h,..., h m,where h... h m and m 2.Thechoiceofasamping interva ength can be represented by a sampinginterva,andet k denotethe -ththreshod imitfactorsforan Xcontrochart.Lettheregion 56 Voume7 Number4 December206

4 between the two contro imits be portioned into m sub-regions as foows: µ 0 k X, µ 0 k X + n n µ 0 + k X +, µ 0 + k X n n for =, 2,...,m, µ 0 k X, µ 0 + k X for = m, n n where 0 k m k m... k, X = x / n x. Ifwedefine Z X = n x X µ x / x,thentheproposed mode can be summarized as foows: Step.Takeasampeofsize n x afteraninterva of h timeunits >= 2. Step2.If Z X < k 2 ifsampingfasinthesecond outermost contro ines, go to step. Otherwise, gotostep3. Step3.If Z X < k ifsampingfasbetween the first outermost contro ines and second outermostcontroines,gotostep4.otherwise,stopthe processandgotostep4. Step4.Iftheaarmisfase,gotostep.Otherwise,gotostep5. Step 5. Identify and eiminate the assignabe cause.gotostep. Figure shows the monitoring procedures of the proposed mode. An economic mode can be formuated by introducing the tota cost function, which refects the reationships between the design parameters of the contro charts and the severa types of costs previousy discussed. Because the underying process for the X contro chart using a surrogate variabe is a renewa reward process, the ong-run expected net income perunitoftimeisgivenby EA = im t E [TIt] /t = EI/ET, 2 where TItisthetotanetincomeuntitime t; Iis thenetincomepercyce,and Tistheengthofthe cyce.thus,aneconomicdesignofthe X contro chart using a surrogate variabe is to determine the vauesof h, n x,and k =,..., msuchthatthey maximize the ong-run expected net income per unit of time. Fig.. The monitoring procedures. Athough the economic design is most effective fromapureyeconomicpointofview,itmighthave undesirabe statistica properties such as high type I and/or type II error probabiities. To make the resuting design satisfy both the cost effectiveness and certain statistica requirements, we may add the desired statistica constraints to the optimization procedure for the design parameters of the proposed mode. We now define the economic statistica designofthecontrochartsasthedesigninwhichthe ong-run expected net income per unit time is maximized subect to a ower bound on the in-contro ARLARL L andanupperboundontheout-ofcontroarlarl U,asinMontgomeryeta.[3]. A mode for the economic statistica design can be formuated as foows: Maximize EA Subectto ARL 0 >ARL L ARL <ARL U, 3 wherearl 0 andarl aretheaveragerunengths whie in contro and out of contro, respectivey. These constraints on ARL can add sensitivity to the shifts in the process mean to the economic mode. NotethatARL 0 andarl arethemeansofthegeometricdistributionswithparameters q s 0 and q s, respectivey. The constraints in3 can be equivaenty expressed as foows: k > Φ, 4 /2ARL L Voume7 Number4 December206 57

5 Φk λ 2 c y nx x + Φ k λ 2 c y nx x 5 <ARL U, where Φ and Φ denotethestandardnorma distribution function and the inverse standard norma distribution function, respectivey. Therefore, the economic statistica design is one that maximizes EA subect to the constraints given in4and5.notethat k shoudbedeterminedusing computationa iterations because5 cannot be expicitysovedwithrespectto k becauseoftwo standard norma distribution functions. Deveopment of the profit function In this section, we derive the expected cyce time and expected net income per cyce for the proposed mode. The cost function, which is the ong-run expectednetincomeperunitoftime,isthenobtained as the ratio of these two quantities. Expected cyce time Intheproposedmode,theprocessbeginsinthe in-contro state and shifts from the in-contro state to the out-of-contro state by an assignabe cause. When a signa is detected, the process is stopped immediatey,andasearchforanassignabecauseisundertaken to see whether it reay exists. If an assignabe cause exists, it is eiminated and the process restarts. Figure2showstheprocesscyceassumedintheproposed mode. Fig. 2. The process cyce. The cyce time for the discontinuous process consists of four periods: an in-contro period, an out-ofcontro period, a period during which the process is stoppedduetoafaseaarmandaperiodforfinding and eiminating an assignabe cause. From the assumptions in the previous section, the expected engthofthein-controperiodis /λ.wenextderive the engths of the three other periods. Expected ength of the out-of-contro period Notethat Dtakesoneof h,..., h m andthatthe distribution of D is determined when the process is in-contro because the samping interva is determined by a previous samping statistics vaue. Reynods et a.[] assumed that PD = h = h, =, 2,..., m. 6 h = Based on the conditiona probabiity, the expectedvaueof D,giventhat D = h,is ED = h PD = h. 7 Because we investigate the process state ony if Xfasinthe I region,theexpectedvaueofthe out-of-contro period, EOC, can be expressed as q EOC = h +ED τ. =2 r 2 r 2 8 Under the assumption that an assignabe cause occurs according to a Poisson process, the expectedtimeof τ,whichisthetimeagbetweentheast preceding samping point and the time at which an assignabe cause occurs, can be expressed as as τ = PD = h Eτ D = h. 9 According to Duncan[4], τ is we approximated Eτ D = h = + λh e λh λ e λh, =, 2,..., m. 2 Expected duration eapsed due to fase aarms Let R 0 bethenumberoffaseaarmsignasinan action region before the process goes out of contro. Wethenobtaintheexpectednumberofsampesin the in-contro period as foows ER 0 = s q s 0 m η = e λh k= e λh. 2 k e λh k 0 AdetaiedderivationisgiveninBaiandLee[5]. The expected duration eapsed due to fase aarmsisthen b 2 timestheexpectednumberof X sampes taken before the shift. That is, 58 Voume7 Number4 December206

6 EFA = b 2 s = e λh q s 0 m k e h k k= e λh. 2 In this mode, we investigate the process state by performingtheprocessif Xfasinthe I region,and the expected duration is zero. 3 Expected duration for finding and eiminating an assignabe cause Whentheprocessisoutofcontro, b +b istakentofindandeiminateanassignabecause.the expected duration for finding and eiminating an assignabe cause is therefore EAC = b 3 n x + b + b. 2 ET = EIC + EOC + EFA + EAC. 3 Expected net income Addingupthesefourperiods,weobtaintheexpectedcycetimefortheprocessas Theexpectednetincomepercyceforthediscontinuous process can be written as EI = EI EC EC 2, 4 where I isthetotaincomepercyce; C isthesampingcostpercyce,and C 2 istheper-cycecostassociated with finding, investigating, and if necessary, eiminating an assignabe cause when an X sampe mean fas outside the action or warning imits. We next derive the expressions for the expected vaues of these costs. Expected tota income per cyce The expected tota income for an in-contro periodis i /λ,whiethatforanout-of-controperiod is i 2 timestheexpectedengthoftheout-of-contro period. The expected tota income per cyce is thus EI = i EIC + i 2 EOC = i λ + i 2 h r q =2 r ED τ. 5 2 Expected samping cost per cyce LetSNbetheexpectednumberofsampesinthe in-contro period. We then get e λh SN 0 = q s 0 m 2 k e λh k 6 η = k= e λh. However, when the process is out-of-contro, the number of sampes required to produce a signa is a geometric random variabe, and the corresponding expected number of samping is given by SN = r 2. 7 Therefore, the expected samping cost per cyce isgivenby EC = a 3 + a 4 n x SN + SN e λh = a 3 + a 4 n x q s 0 m k e λh k k= η e λh + r 2. = 2 8 Expected cost associated with an assignabe cause and fase aarms per cyce The expected cost incurred from fase aarms is given by a 2 + a 2ER 0 + a 2ER, 9 where R isthenumberoffaseaarmswhenthe signaisinthe I region.becausetheexpectedcost incurredfromanassignabecauseis a,thecorresponding expected cost per cyce is given by EC 2 = a 2 + a 2ER 0 +a 2ER + q s + q r where r 2 a a = a 2 + a 2ER 0 + a 2ER + a, ER = q s 0 m η = e λh k= e λh. 2 k e λh k 20 Voume7 Number4 December206 59

7 Theexpectednetincomepercycecanbeobtained by subtracting8 and20 from5. Numerica comparisons using genetic agorithms Theproposedmodeisappiedtothetestexampe considered in PHM for comparison with the economic mode based on the performance variabe. Itisassumedthat ρ = 0.6or0.9,andthevariabe sampingcosta 4 isone-tenthoftheoriginavaues inphmbecausetheprocessismonitoredbyasurrogate variabe, which is usuay much cheaper to measure. The vaues of the corresponding input parametersaregivenintabe.weperformasensitivityanaysisfor m = 2and m = 3butdescribe onyfor m = 2becauseweobservedsimiarresuts for m = 3.Theoptimavauesofthedesignparameters h, n x, k w,and k a for m = 2thatmaximize EAsubecttotheconstraintsin4and5with ARL L = 4andARL U = 500weredeterminedwith Evover, a genetic agorithm optimization too. Genetic agorithms have been used in many engineering areas such as industria engineering, mechanica engineering, aerospace engineering, etc., incuding statisticaprocesscontroinlineta.[6].thecomputationaresutsareshownintabe2.basedonthe resutspresentedintabe2,wemayobservethefoowing: Whenthecorreationishighρ = 0.9,theproposed economic mode yieds a higher income than thatofthephmmodeforacases.however, whenthecorreationisowρ = 0.6,theproposed economic mode yieds a higher income than that of the PHM mode. Thus, the proposed economic modeseemstobeeffectivewhen ρisreativey arge. Whenthecorreationishigh,thesampesize n x issmaerthanorequato n y foracases. Inaofthecaseswhenthecorreationishigh, k ishigherthan4.5,whichiscosetothebounds set for the agorithm. This effectivey means that aminimasampeneedsbetakenusteveryonce in a whie without stopping the process. The reasonforthisresutmightbeduetohighersamping costsandaowercostforidentifyingafaseaarm, combinedwithasmaerdifferenceinthenetincome per unit time between the in-contro state and out-of-contro state. The higher the correation is, the narrower the warningregionk k 2 is.thisimpiesthatthe processneedstonotstoptoinvestigateasignabecause the income earned from the process is arger than the cost for identifying a fase aarm. Inahighcorreationcase,theyiedoftheproposed economic mode improves that of the PHM modebyabout4.9%,andtheyiedoftheeconomic statistica mode is ower than that of the proposedeconomicmodebyabout5%.inaow correation case, however, the proposed economic modeimprovedthatofthephmmodebyabout 4%. This suggests that the statistica constraints onarltendtohaveagreaterinfuenceonthe economic mode when the correation is high. As a resut, the higher the correation is, the more efficient the proposed economic mode is. Tabe Cost and process parameters for the test exampes. Exampe Λ c i i 2 a a 2 a 2 a 3 a 4 b b b 2 b Voume7 Number4 December206

8 Exampe Tabe 2 Optimum designs for the test exampes. Management and Production Engineering Review The PHM Mode Proposed Economic Mode Economic Statistica Mode n y h k EA ρ n x h h 2 k 2 k EA n x h h 2 k 2 k EA Nooptimasoutionstosatisfy the statistica constraints Nooptimasoutionstosatisfy the statistica constraints Nooptimasoutionstosatisfy the statistica constraints Nooptimasoutionstosatisfy the statistica constraints Nooptimasoutionstosatisfy the statistica constraints Nooptimasoutionstosatisfy the statistica constraints Appication to a nucear fue measurement system In this section, the proposed mode is appied to the VHTR TRISO fue measurement system, previousydescribedbykimeta.[6],foracomparison with the economic mode based on a performance variabe.trisofueisatypeofmicrofuepartice. It consists of a kerne, ow-density pyrocarbon, inner high-density pyrocarbon, siicon carbide, and outer high-density pyrocarbon. It is important to obtain theexactfigureofthefuesizeforuniformproduction.whenwemeasuretheouterdiameterofthe VHTRfue,wecanusemethodssuchasaPSA,micrometer, and X-ray. Of the three methods, the use of X-rayY,performancevariabeisthemostaccurate; however, the measurement equipment is too expensive to purchase. On the other hand, the PSA X, surrogate variabe method is ess precise due to ascatteringofight;however,itiseasiertoobtain data without emitting dangerous radiation. Fromtheactuadataanaysis,itisknownthat themeanandvarianceof Y are µ y = and 2 y = 32.8 µm2,respectivey,andthevarianceof Xis x 2 = µm2.itisasoknownthat Xfor thegiven Y = yisnormaydistributedwithamean of yandavarianceof2.57 µm 2 and that the correation coefficient between X and Y is ρ = Tabe3showstheTRISOfuesizesmeasured using the PSA and X-ray methods. The optimum vaues of the design parameters are obtainedintabe4forboththeeconomicmode and the economic statistica mode based on the surrogate variabe, aong with those for the economic mode based on the performance variabe. The foowing vaues for the cost and process parameters areassumedtobe λ = 0.05, c = 2, i = 00, i 2 = 0, a = 200, a 2 = 200, a 2 = 200, a 3 = 3, a 4 =.5, b = 0, b = 0, b 2 = 25,and b 3 = 0.3. Tabe4showsthattheeconomicmodeusingthe surrogate variabe yieds a 2% higher expected net income per hour than the economic mode using the performance variabe. It aso shows that the economic statistica mode yieds the optimum design, which is ony sighty different from that for the economic mode with a minima decrease in the expected perhour net income. We note that the parameter vaues of the economic statistica mode are very robust to changesinbotharl U andarl L. Voume7 Number4 December206 6

9 Tabe 3 TRISO fue sizes measured by the PSA method and X-ray method. Obs PSA X-rays Obs PSA X-rays Obs PSA X-rays Tabe 4 Resuts of the VSI X contro chart using surrogate variabes. Parameters Economic DesignPerformance var. Economic DesignSurrogate var. Statistica Economic Design n xn y h h k k ET EI /ET EC /ET EC 2 /ET EA Sensitivity anaysis The sensitivity of the design is important to find anoptimapanbecauseitisdifficuttoexactyspecifythevauesoftheprocessparameters.inthissection,wefindhowsensitivetheoptimumvauesofthe designparametersh, h 2, k 2 and k aretochanges in certain input parameter vaues using exampe 6 in Sec.5.Thevauesoftheinputparametersarevaried ±30%fromthebasevaue.Basedontheresutsof the sensitivity anaysis, we observed the foowing: Effectsof λand c:fig.3showsthattheoptimum vauesof h and h 2 decreaseasthefaiurerate λ increases. This conforms to our intuition that the process needs to be controed more tighty becauseittendstofaimorefrequentyas λincreases.anincreaseintheshiftsizeoftheprocess meancresutsinanincreaseinthewarningregionk k 2 intheoptimumvaueof hshownin Fig. 4. Fig.3.Sensitivityof h, h 2, k 2,and k to λ. 62 Voume7 Number4 December206

10 Effectsof a, a 2and a 2:Anincreasein a causesadecreaseintheoptimumvaueof h and h 2 showninfig.7.figure8showsbothanincrease in h and h 2 andadecreaseinthewarningregionasthecostforidentifyingafaseaarma 2 increases. Fig.4.Sensitivityof h, h 2, k 2,and k to c. Effectsof i and i 2 :Fig.5showsthatanincreasein theper-hournetincomeinthein-controstatei eadstoadecreasein h and h 2,whereastheoptimumvauesof k 2 increaseasthenetincomeper unittimeintheout-of-controstatei 2 increasesshowninfig.6.thereasonfortheincreasing trendinthewarningregioninfig.6isthatitis more efficient and more economica to check the process state instead of a process stop. Fig.7.Sensitivityofh,h 2,k 2,andk toa. Fig.8.Sensitivityofh,h 2,k 2,andk to a 2. Fig.5.Sensitivityof h, h 2, k 2,and k toi. Therestoftheparametersincuding a 2, b and b showasimiartrendtothoseoftheaboveparameters in the sensitivity anaysis. Concusions Fig.6.Sensitivityof h, h 2, k 2,and k toi 2. WeproposedaneconomicdesignofaVSI Xcontrochartbasedonasurrogatevariabeforacasein which using the performance variabe is impossibe or inappropriate. Compared with the contro chart based on a performance variabe, the proposed modegivesaargerexpectednetincomeperunitoftime in the ong-run if the correation between the performance variabe and the surrogate variabe is reativey high. WhentheproposedVSI Xcontrochartwasappied to a nucear fue measurement system, the numerica comparison resuts show that the VSI mode using a surrogate variabe is more efficient than the Voume7 Number4 December206 63

11 VSI mode using a performance variabe or FSI modefromanetincomepointofview.additionay,if the correation coefficient between the surrogate variabe and performance variabe is higher, it was found that the ong-run expected net income per unit of time is aso increased. Conversey, for a ow eve of the correation coefficient,itcanbemoreusefutouseavsimode using a performance variabe. The proposed mode wasconfinedtothesampemeancontrochartunder the assumption that a singe assignabe cause occurs according to the Poisson process. However, the modemayasobeextendedtoothertypesofcontro charts using a singe or mutipe assignabe cause assumptions such as VSS X contro chart, EWMA, and CUSUM charts. References [] Reynods Jr. M.R., Amin R.W., Arnod J.C., Nachas J.A., X charts with variabe samping intervas, Technometrics, 30, 8 92, 988. [2] Reynods Jr. M.R., Arnod J.C., Optima one-sided Shewhart contro charts with variabe samping intervas, Sequentia Anaysis, 8, 5 77, 989. [3] Runger G.C., Montgomery D.C., Adaptive samping enhancements for Shewhart contro charts, IIE Transactions, 25, 4 5, 993. [4] Reynods Jr. M.R., Shewhart and EWMA variabe samping interva contro chats with samping at fixed times, Journa of Quaity Technoogy, 28, 99 22, 996. [5] Costa A.F.B., Rahim M.A., Economic design of X charts with variabe parameters: the Markov chain approach, Journa of Appied Statistics, 28, , 200. [6] Lin Sung-Nung, Chou Chao-Yu, Wang Shu-Ling, Liu Hui-Rong, Economic design of autoregressive moving average contro chart using genetic agorithms, Experts System with Appication, 39, , 202. [7] Chew X.Y., Khoo Michae B.C., Teh S.Y., CastagioaP.,Thevariabesampingintervarunsum X contro chart, Computer& Industria Engineering, 90, 25 38, 205. [8] Zhang Min, Nie Guohua, He Zhen, Performance of cumuative count of conforming chart of variabe samping intervas with estimated contro imit, Internationa Journa of Production Economics, 50, 4 24, 204. [9] LeeT.H.,LeeJ.H.,LeeM.K.,LeeJ.H.,Economic design of X contro chart using a surrogate variabe, Journa of the Korean Society for Quaity Management, 37, 46 57, [0] LeeT.H.,LeeJ.H.,LeeM.K.,Economicdesignof variabe samping interva contro chart using a surrogate variabe, Journa of Korean Institute of Industria Engineers, 39, , 203. [] Panagos M.R., Heikes R.G., Montgomery D.C., Economic design of X contro charts for two manufacturing process modes, Nava Research Logistics, 32, , 985. [2] Tang K., Lo J., Determination for the optima process mean when inspection is based on a correated variabe, IIE Transactions, 25, 66 7, 993. [3] Montgomery D.C., James C.C., Torng J.K., Frederick P.L., Statisticay Constrained Economic DesignoftheEWMAControChart,JournaofQuaity Technoogy, 27, , 995. [4] DuncanA.J.,Theeconomicdesignof Xchartsused to maintain current contro of a process, Journa of American Statistica Association, 5, , 956. [5] Bai D.S., Lee K.T., An economic design of variabe samping interva X contro charts, Internationa Journa of Production Economics, 54, 57 64, 998. [6] KimW.K,LeeY.W.,ParkJ.Y.,ParkJ.B.,RaS.W., Nondestructive measurement of the coating thickness in the simuated TRISO-Coated fue partice using micro-focus X-Ray radiography, Journa of the Korean Society for Nondestructive Testing, 26, 69 76, Voume7 Number4 December206