The Economic Maintenance Scheduling for a Batch Production System Considering Quality of the Processed Products

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1 Maematical Applicatios i Sciece ad Mechaics The Ecoomic Maiteace Schedulig for a Batch Productio System Cosiderig Quality of e Processed Products JOG HU PARK, SAG CHEO LEE ad HYUG RAE CHO 3 Departmet of Busiess Admiistratio Caolic Uiversity of Daegu Geumak-ri, Hayag-eup, Gyeogsa-si, Gyeogsagbuk-do KOREA icelatte@cu.ac.kr Departmet of Idustrial Systems Egieerig, Egieerig Research Istitute Gyeogsag atioal Uiversity 50 Jiudaero, Jiu-si, Gyeogsagam-do KOREA sclee@gu.ac.kr, hrcho@gu.ac.kr 3 Abstract: This paper itroduces aalytic model to decide e ecoomic prevetive maiteace schedulig for a batch productio system. A schedulig is defied to be optimal if it miimizes e expected cost rate comprised of failure cost, maiteace cost ad uality cost which is assessed from e uality of e processed products usig e cocept of e loss fuctio. The model is developed assumig at a system ca have ot oly failure but also degradatio durig a batch productio ad e degradatio come wi uality-dow of e products. I additio, e maiteace is assumed to restore e system coditio before degradatio. Key-Words: Prevetive maiteace, maiteace schedulig, loss fuctio, batch productio, uality of e processed products Itroductio I today s competitive global marketplace, may compaies believe at good uality is essetial to wi competitive advatage, hece cotiuously make efforts to build high uality productio system. I particular, maufacturers have focus o reducig e variace ad cotiuously moitorig e chage of e process[]. This situatio has led to a eed for takig ito accout uality of e processed products i performace measuremet of a maufacturig process such as OEE(Overall Euipmet Effectiveess), which is ree-part aalysis tool for euipmet performace based o its availability, performace, ad uality rate of e processed products[, 3]. The fact at uality rate of e processed products is a primary factor i OEE implies at we should cosider e uality of e processed products whe to evaluate ot oly e euipmet performace but also ecoomic operatio of euipmet. I additio, Hahm[4] has asserted at ew cocept of maiteace for a productio system should be cosidered whe high uality is eeded. Therefore, e uality of e processed products should be cosidered whe we decide maiteace schedulig for ecoomic operatio of e system, is paper itroduces aalytic model to decide e ecoomic prevetive maiteace schedulig cosiderig uality of e processed products for a batch productio system. The model is developed assumig at a system ca have ot oly failure but also degradatio durig a batch productio ad e degradatio come wi ualitydow of e processed products. The uality-dow of e processed products by degradatio is assessed ito uality cost via loss fuctio ad e maiteace is assumed to restore e system coditio before degradatio. Backgroud Productio is e act of creatig output via process, e process is procedures ivolvig chemical or mechaical steps to trasform iput ito more valuable output[5]. While a productio system ISB:

2 Maematical Applicatios i Sciece ad Mechaics creates products via chemical or mechaical steps, e system, especially machiery, ca be disrupted by chemical or mechaical ifluece. For examples, a machie tool at rotates at high speed ca be axisdeflected by heat strai[6]. The cuttig tool i broachig process ca be chipped, broke, fractured or wor out[7, 8]. Amog ese disruptios, some are able to be detected immediately whe ey happe such as chippig ad breakig, however, oers such as axis-deflectio, fracture ad abrasio are detectable ot immediately but util ispectio period. Therefore, e former ca to be recovered wiout delay but e latter eeds a certai amout of time till recovery, at is to say, e system are uder iaccurate state ad maufactures amout of defective products for a while. I is paper, we are goig to distiguish e former from e latter, as failures from degradatios respectively. I terms of cost, failure icurs a repair cost for recovery but degradatio icurs ot oly a repair cost but also uality cost which is caused by uality dow of e processed products util ispectio. Hece, it is importat to prevetive system degradatio as well as system failure. From e viewpoit of operatig e productio system, it is so axious to maufacture defective products uder iaccurate system state wiout recogitio, erefore, e productio maager employ periodic ispectio to check e system coditio. I a batch productio system, e ispectio is usually performed after a batch productio, check e system coditio ad recover e system if it is degraded. The actio whe e system is degraded is for restorig e system coditio at e startig time, i.e., as good as ew. For example, e cuttig tool is ispected after a batch productio ad replaced if it is much damaged or grided to restoratio if ot[8]. These actios are a kid of prevetive maiteace(pm) for ecoomic operatio of a productio system. The degradatio process durig productio is a kid of deterioratio process of a system, may studies for deterioratio process ad PM have cotiuously researched i ecoomic cocer. Ohisi et al.[9] isists at e system should be replaced as ew whe a system is deterioratig i some degree because e replacemet cost is more ecoomical a maiteace cost ad uality cost, Zhou & Wysk[0] ivestigates ecoomic optimal replacemet period uder small batch productio, Chick & Medel[] itroduces age ad block replacemet models cosiderig wear amout from usage time via reliability fuctio of tool. I additio, Lee et al.[7] ivestigates ecoomic tool life of broachig machie wi tool life process model dividig ito wear process ad succeedig failure process, Oh et al.[8] aalyzes ecoomic life cuttig tool which allows re-gridig based o proportioal age reductio uder age replace policy. The most studies for ecoomic optimal PM have assumed at PM prolog e life of a system, i.e., PM prevet uexpected failure ad erefore prolog e life of a system, ultimately save cost i spite of e maiteace cost. I e productio system, however, PM have also performace-restorig effect as well as lifeprologig effect. A re-grided cuttig tool has same performace to ew oe, a overhauled system would be performed as good as ew. Hece, PM ca save uality cost by reducig e chace to maufacture defective productio, erefore, PM studies should cosider e performace-restorig effect as cost-savig ad take ito accout whe to decide ecoomic optimal PM schedulig. The cotributio of is paper is to assert at e uality-dow of processed products by system degradatio should be cosidered for ecoomic operatio of e productio system ad also itroduce e aalytic model to decide e ecoomic prevetive maiteace schedulig cosiderig e uality cost which assesses e uality-dow of e processed products as cost. The previous PM models have a limit to apply a productio system i at ey do ot take ito accout e uality of e processed products, however, e proposed model try to overcome e limit by e loss fuctio at are used to reflect e ecoomic loss associate wi e uality level of e processed products []. 3 The Quality Cost Model via Loss Fuctio The loss fuctio is a approach to assess e chage of uality level by system degradatio as e uality cost[, 3]. The loss fuctio is used to reflect e ecoomic loss associate wi variatio about, ad deviatio from, e process target or e target value of a product characteristic[], itroduced i e study for ecoomic desig of cotrol chart by Elsayed & Che[4] ad Alexader et al.[5]. Alough Spirig & Yeug[] ad Lofouse[3] itroduce various type of loss fuctio ad eir idustrial applicatios, e uadratic loss fuctio, which is defied as e.() where e process target is m ad e process measuremet is x, is e mostly itroduced[, 3, 4, 6]. ISB:

3 Maematical Applicatios i Sciece ad Mechaics Lx ( ) k ( x m) =. () The coefficiet k i e.() is a costat at depeds o e magitude of characteristic ad e moetary uit ivolved[3]. As show i e.() ad Fig., e total loss icreases parabolically as e deviatio from e target value icreases i e uadratic loss fuctio, which is suitable for assessig uality-dow caused by system degradatio as e uality cost. Fig. The uadratic loss fuctio If a productio system is maufacturig e items as followig ( µσ, ), e expected loss of a product is Lxf xµσ k { σ ( µ m) } ( ) ( ;, ) = k x m f x µσ ( ) ( ;, ) = +. () Appedix A provides detailed developmet. As show e.(), e expected loss ca be represeted by e mea ad stadard deviatio of product characteristics, which implies at e uality cost ca be assessed if e system degradatio causes e chage of product characteristics. We develop aalytic model to decide e ecoomic prevetive maiteace schedulig for a batch productio system uder followig assumptios roughout is paper.. <Assumptios> () The products are maufactured uder a batch productio wi a batch size. () The failure ad degradatio of system are mutually idepedet ad Beroulli trials for each maufacturig wi p ad p respectively. (3) The system failure ca be detected immediately ad if detected, e system is repaired as soo as possible ad begis ew batch productio wi a batch size. (4) The system degradatio is ot able to be detected durig a batch productio. (5) The system is maitaied after every sigle batch productio ad e maiteace is restored e system as good as ew. Above metioed assumptios are commoly acceptable i a batch productio. For example, e broach, which is cuttig tool of broachig machie, ca be failed such as chippig ad breakig, or degraded such as fracturig ad abrasio while e machie maufactures a product. The chippig or breakig ca be detected immediately but e fracturig or abrasio ca be detected oly by ispectio. Therefore, after a batch productio, e machie is ispected to check e coditio of e broach ad maitaied such as replacemet or gridig. After repair or maiteace, e machie begis ew batch productio. However, little attetio is paid to maematical modelig for degradatio process eve ough a few researchers have attempted to represet e degradatio to e chage of product characteristics. Hece, is paper assumes i additio for maematical modelig usig loss fuctio for degradatio process as follows. (6) The system maufactures a product wi characteristics followig ( µσ, ) i ormal coditio. (7) If e system degradatio occurs, e mea of product characteristics moves e value a per product maufacturig after. Assumptio (6) is so atural for products wi omial-e-best type (-type) uality ad assumptio (7) implies e system degradatio icreases chage of product characteristics i proportio as umbers of maufacturig after degradatio occurrece. It ca be geerally acceptable because e degradatio would be goig gradually i more maufacturig. By assumptio (6) ad (7), e product characteristics of a product maufactured i after e degradatio is followig ( µ + ai, σ ), ad e expected loss of e product is k { σ + ( a i) } by e.(). If e batch size is ad system has o failure durig a batch productio, e system degradatio ISB:

4 Maematical Applicatios i Sciece ad Mechaics may happe followig geometric distributio wi parameter p, erefore, e expected uality cost is = + + ( p { ) k σ }. (3) ( p) p k { σ + ( a i) The former term is for e case at e system degradatio occurs durig a batch productio, i.e., betwee e first ad maufacturig, ad e latter term is for o system degradatio occurrece. 4 The Modelig for e Ecoomic Maiteace Schedulig I is sectio, we develop aalytic model for e ecoomic maiteace schedulig usig e uality cost model of sectio 3. A optimal schedulig is * defied to be a batch size( ) which miimizes e expected cost rate comprised of failure cost( C ), f maiteace cost( C ) ad uality cost. m The expected cost rate ca be represeted by e rate of e expected total cost to e expected batch size for a batch process. ote a batch process ot a batch size because a batch productio ca be stopped before maufacturig by assumptio (3). If e system has o failure durig a batch productio, e system maufactures products ad is reewed by assumptio as good as ew by assumptio () ad (5). However, if e system failed before maufacturig, e system would be repaired immediately ad reewed by assumptio (3). Therefore, e expected batch size for a batch process ca be obtaied by i ( i ( p) p) + ( p). (4) The expected total cost for a batch process cosists i failure cost, maiteace cost ad uality cost as above metioed. Firstly, maiteace cost icurs ust oe time, ad secodly, failure cost icurs oly whe e system is failed durig a batch productio, us, it ca be obtaied by Cf ( ( p ) ). (5) failure durig a batch productio whe a batch size is, at is to say, e.(3) computes e uality cost whe system is ot failed util maufacturig. If a system fails at maufacturig, which meas at e system is ot failed util maufacturig, erefore, e uality cost whe system is failed at maufacturig is. (6) ( p) p k { σ + ( a i) = + + ( p ) k { σ } ote at e failure at maufacturig causes a failure cost ad e uality cost o coditio at system is ot failed util maufacturig as show i e.(6). For cociseess of expressio, let us redefie e.(6) as EC [ ] deotig e uality cost o coditio at e system is failed at maufacturig as below. EC [ ] = + ( p ) k { σ }. ( p) p k { σ + ( a i) = + (7) c Similarly, let EC [ ] deote e uality cost o coditio at e system is ot failed util maufacturig as e.(8). c EC [ ] = + ( p { ) k σ }. ( p) p k { σ + ( a i) = + (8) The failure of e system follows geometric distributio wi parameter p, erefore, e expected total cost for a batch process is Fially, uality cost ca be developed as follows. E.(3) is e expected uality cost for e case of o ISB:

5 Maematical Applicatios i Sciece ad Mechaics ( ) c ( p) p EC [ ] + ( p) EC [ ]. (9) = Let CR( ) deote e cost rate whe a batch size is, e optimal maiteace schedulig is a * batch size which miimize e.(0). CR( ) = ( ) ( ) c Cm + Cf ( p) + ( p) p EC [ ] + ( p) EC [ ] =. i ( i ( p) p) + ( p) (0) 4 Coclusio This paper asserts at e uality-dow of processed products by system degradatio should be regarded as uality cost ad also itroduces aalytic model to decide e ecoomic prevetive maiteace schedulig for a batch productio system. O e ecoomic stadpoit, e uality dow of processed products is evaluated via loss fuctio. For uality improvemet of is paper, we are goig to perform umerical aalysis to test e practical usage of e proposed model, ad also ivestigate sesitive aalysis for various choices of variables to provide isights for a effect of e variables ad iter-relatioship of em. Furer studies are also cosidered. Firstly, study for modellig to reflect clearly e chage of e mea ad stadard deviatio of product characteristics by system degradatio. This paper ust assumes at e mea of product characteristics moves e value a per product maufacturig after because ere are few previous studies for at eve ough may researchers agree at system degradatio causes e chage of e mea ad stadard deviatio of product characteristics. I additio, assumptio () i is paper would be able to be exteded to geeral failure distributios. Refereces: [] Su-Keu Seo ad Dae-Kil Seok, Tolerace Aalysis ad Desig for a Mea Shift of Maufacturig Process, Joural of e Korea Istitute of Plat Egieerig, Vol.0, o., 006, pp [] P. Muchiri ad L. Pitelo, Performace measuremet usig overall euipmet effectiveess (OEE): literature review ad practical applicatio discussio, Iteratioal Joural of Productio Research, Vol.46, o.3, 008, pp [3] Jae-Ho Bae, Measuremet of Overall Euipmet Effectiveess Cosiderig Processig Materials ad Meods, Joural of e Korea Istitute of Plat Egieerig, Vol.6, o.3, 0, pp5-33. [4] Hyo Joo Hahm, O e Establishmet of e ew Cocepts of Productive Maiteace ad Euipmet Maagemet, Joural of e Korea Istitute of Plat Egieerig, Vol.4, o.4, 999, pp.5-6. [5] K. S. Kag ad W. S. Jug, Productio ad Operatio Maagemet, Parkyougsa, 00. [6] Dog Hoo Kim, Ju Yeob Sog, Suk Keu Cha ad Ji Suk Choi, Compesatio Apparatus of Heat Distortio ad Real-time Correctio for Kowledge-Evolutio based Machie Tools, Joural of e Korea Society for Precisio Egieerig, Vol.6, o., 009, pp [7] Sag-Cheo Lee, Shi-Ick Kag ad Jug-Wa Hog, A life-process Aalysis of Broachig Tool, IE Iterface, Vol.5, o., 00, pp [8] Sug-Youl Oh, Jug-Wa Hog, Sag-Cheo Lee, ad Chag-Hoo Lie, A Aalysis of Gridig Effects ad Ecoomic Life of Cuttig Tool wi Proportioal Age Reductio Model, IE Iterface, Vol.9, o.4, 006, pp [9] M. Ohishi, H. Kawai, ad H. Mie, A optimal ispectio ad replacemet policy for a deterioratig system, Joural of Applied Probability, Vol.3, o.4, 986, pp [0] C. Zhou, ad M. B. Wysk, Tool Status Recordig ad Its Use i Probability Optimizatio, Joural of Egieerig for Idustry, Vol.4, o.4, 99, pp [] S. Chick ad M. B. Medel, Usig Wear Curves to Predict e Cost of Chages i Cuttig Coditios, Joural of Maufacturig Sciece ad Egieerig, Vol.0, p., 998, pp [] F. A. Spirig ad A. S. Yeug, A Geeral Class of Loss Fuctios wi Idustrial Applicatios, Joural of Quality Techology, Vol.30, o., 998, pp.5-6. [3] T. Lofouse, The Taguchi loss fuctio, Iteratioal Joural of Productivity ad ISB:

6 Maematical Applicatios i Sciece ad Mechaics Performace Maagemet, Vol.48, o.6, 999, pp.8-3. [4] E. A. Elsayed ad A. Che, A ecoomic desig of x cotrol chart usig uadratic loss fuctio, Iteratioal Joural of Productio Research, Vol.3, o.4, 994, pp [5] S. M. Alexader, M. A. Dillma, J. S. Usher ad B. Damodara, Ecoomic desig of cotrol charts usig e Taguchi loss fuctio, Computers ad Idustrial Egieerig, Vol.8, o.3, 995, pp [6] S. H. Park, desig of experimet. Miyougsa, 003. <Appedix A> L x f x ( ) ( ; µσ, ) ( ) ( ; µσ, ) ( ( ; µσ, ) ( ; µσ, ) ( ; µσ, ) ) { ( ) ( ) } { ( ) ( ) ( ) ( ) } { ( ) ( ( ) ) } = + dx = k ( x m) f ( x; µσ, ) dx = k x mx + m f x dx = k x f x dx m xf x dx + m f x dx = k EX mex + m = k EX EX + EX mex + m = k Var X + E X m k { σ ( µ m) }. ISB:

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