Integrating Machine Reliability and Preventive Maintenance Planning in Manufacturing Cell Design

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1 IEMS Vol. 7, No., pp. 4-5, September 008. Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig Kacha Das East Carolia Uiversity, Greeville, NC 7858, USA , dask@ecu.edu R.S. Lashkari Uiversity of Widsor, Widsor, Otario N9B 3P4 CANADA X609, lash@uwidsor.ca S. Segupta Oaklad Uiversity, Rochester, Michiga U.S.A , segupta@oaklad.edu Selected paper from APIEM 006 Abstract. This paper presets a model for desigig cellular maufacturig systems (CMS) by itegratig system cost, machie reliability, ad prevetive maiteace (PM) plaig. I a CMS, a part is processed usig alterative process routes, each cosistig of a sequece of visits to machies. Thus, a level of system reliability is associated with the machies alog the process route assiged to a part type. Assumig machie reliabilities to follow the Weibull distributio, the model assigs the machies to cells, ad selects, for each part type, a process route which maximizes the overall system reliability ad miimizes the total costs of maufacturig operatios, machie uderutilizatio, ad iter-cell material hadlig. The model also icorporates a reliability based PM pla ad a algorithm to implemet the pla. The algorithm determies effective PM itervals for the CMS machies based o a group maiteace policy ad thus miimizes the maiteace costs subect to acceptable machie reliability thresholds. The model is a large mixed iteger liear program, ad is solved usig LINGO. The results poit out that itegratig PM i the CMS desig improves the overall system reliability markedly, ad reduces the total costs sigificatly. Keywords: CMS Desig, Machie Reliability, Maiteace Plaig, Iteger Programmig.. INTRODUCTION The cell desig problem may be described as the process of groupig the machies ito a umber of cells, each capable of processig idepedetly a family of part types (with some parts types processed i more tha oe cell), based o the machiig requiremets of the parts. Iterested readers are referred to Wemmerlov ad Hyer (986), Joies et al. (996), Selim et al. (998), ad Masouri et al. (000) for reviews of the extesive literature o cell formatio techiques. Moder productio equipmets are capable of performig more tha oe operatio, ad as such each part type may be processed usig differet process plas. The obectives of the cell formatio have traditioally bee to reduce throughput times, material hadlig costs, set up times, ad to simplify productio flow ad cotrol (Aski ad Estrada 999). However, most cell formatio work has bee based o the assumptio that the machies are 00% reliable. Machie failures, however, have a sigificat impact o the system performace (e.g., due date compliace, utilizatio, etc.) eve if there are optios to reroute the parts to alterative machies. Ofte, it is ot possible to hadle a machie failure as quickly as the productio requiremets demad. Delays due to machie breakdows ot oly impact the productio rate, they also lead to schedulig problems which decrease the productivity of the etire maufacturig operatio. This issue poits out the importace of machie reliability cosideratio i cell formatio decisios ad durig the operatio allocatio process. Geerally, i a cellular maufacturig system (CMS) the reliability cofiguratio of the machies alog a process pla is a series structure, whereas i a ob shop the reliability structure is parallel, makig it easier to reroute : Correspodig Author

2 4 Kacha Das R.S. Lashkari S. Segupta a part to aother idetical machie i case of ay machie failure. I a CMS, however, itercellular trasportatio arragemets are eeded to hadle the same situatio (Seifoddii ad Dassemi 00). Thus, i a CMS a higher level of machie reliability is eeded to maitai a high CMS performace level. Furthermore, the CMS desig process should iclude machie reliability cosideratio to effectively aticipate ad pla for the adverse effects of machie breakdows. Machies are the maor compoets which accout for a sigificat share of the capital ivestmet i CMS. Although machie reliability plays a importat role i the performace of a CMS, it deteriorates over time as the machie ages. To stem the deterioratio ad to improve the reliability of machies, prevetive maiteace (PM) plas are devised ad put ito effect to miimize the cumulative failure probability of a machie. I the multimachie eviromet of a CMS this requiremet is critical as uplaed breakdows will halt the etire system ad adversely impact the overall CMS performace. Thus we submit that the icorporatio of appropriate PM systems i the plaig of CMS is oe of the most importat requiremets i the moder maufacturig sector. It is oted, however, that PM is ustified oly whe it is cost effective, reduces radom breakdows, ad exteds the useful life of the equipmet. Further, for PM to be effective, the failure rate of the equipmet must be icreasig with time (Jardie ad Tsag 006; Ebelig 997), which is the case for maufacturig machiery (e.g., CNC machies). The obective of this paper is, therefore, to develop a multi-obective CMS desig model which miimizes the system costs ad maximizes the overall system reliability alog the selected process route, assumig the machie reliability to follow a Weibull distributio. The model also icorporates a PM pla i the CMS desig process. The paper is orgaized as follows. Relevat literature is reviewed i the ext sectio. Sectio 3 presets a discussio of the CMS desig model, alog with the machie reliability aalysis, the reliability-based group PM pla, ad a algorithm to implemet the PM pla. A umerical example is provided i Sectio 4 to demostrate the applicability of the model, ad some cocludig remarks are give i Sectio 5.. RELEVANT LITERATURE The umber of research works dealig with the reliability aspects of CMS desig is fairly small. Jeo et al. (998) ad Diallo et al. (00) cosidered machie reliability i their aalysis ad developmet of the CMS. Jeo et al. (998) cosidered alterative routes to develop cell cofiguratios to hadle the problem of a predefied umber of machie breakdows. Their model aimed to miimize waitig costs, late ad early fiish costs ad machie ivestmet costs to solve the machie breakdow problem. Diallo et al. (00) cosidered the machies to be ureliable ad cosequetly attempted to develop a cell cofiguratio with alterative process plas to hadle the machie failures. Most of the machie reliability-based studies i CMS are directed towards performace evaluatio. A umber of studies (Seifoddii ad Dassemi 00; Logedra ad Talkigto 997) emphasized the importace of machie reliability i relatio to the desired output of the CMS. I a CMS, the PM plaig has to focus o the multi-machie eviromet of the cell to address the iterdepedet structure of the CMS. A umber of studies have reviewed the various PM policies i maufacturig systems (e.g., Wag, 00; Dekker et al., 997; Cho ad Parlar 99; Valdez-Flores ad Feldma, 989). Amog the policies that may be applicable to CMS are the fixed group plaed maiteace policy outlied by Dekker et al. (997), or the group maiteace policy suggested by Wag (00). Both plas are based o the cocept of replacig a select group of compoets after a fixed iterval of time, ad addressig the uplaed failures of the compoets durig the iterval through repairs or miimal repairs. Aother group maiteace policy studied by Wilderma et al. (997) cocered the maiteace activities carried out o a group of equipmet ad ivolved a system-depedet set up cost that was the same for all the activities. The groupig of machies saved costs, sice the executio of a group of activities required oly oe set up. Talukder ad Kapp (00) developed a heuristic method for groupig equipmet that would allow the applicatio of PM i a series system with the goal of miimizig the total maiteace-related costs. The Weibull distributio was applied to represet the icreasig failure rates of the equipmet. The study derived a total cost model, ad evaluated the PM itervals by miimizig the total cost for idividual equipmet groups. Kardo ad Frededall (00) developed a maiteace approach for multi-machie situatios. Usig the Weibull distributio, the approach determies the PM itervals such that the cumulative failure probability of a machie stays below a specified limit set by the user orgaizatio. For multi-machie/multi-compoet systems the study cosidered a umber of maiteace policies, two of which may be applicable to PM decisio processes i a CMS. Oe policy is a block replacemet approach i which the compoets are classified ito categories or blocks, based o the similarity of their PM itervals, so as to maitai a maximum tolerable cumulative failure probability. The other policy is to replace all the compoets by determiig the shortest maiteace iterval that maitais a tolerable overall cumulative failure probability. A compariso of the policies leads to the suggestio that a trial ad error approach is eeded to adust the iterval to achieve a miimum possible total cost i a specific situatio. Das et al. (005) developed a PM plaig approach cetered o a effective maiteace iterval for idividual machies i a CMS. The approach has four basic steps: i the first step the model implemets the

3 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig 5 reliability based PM approach ad determies a commo PM iterval for the machies based o a cumulative failure probability upper boud as set by the orgaizatio. I the ext step the model computes, for each machie, the maximum iterval possible by allowig a cumulative failure probability upper boud for each machie. I the third step the model determies the effective iterval for each machie as a iteger multiple of the commo iterval so that the effective iterval is less tha or equal to the maximum possible iterval. I the fial step a maiteace schedule is developed depedig o the effective iterval for each machie. Recetly, Das et al. (006) proposed a CMS desig model icorporatig maufacturig system cost, machie system reliability ad effective PM iterval based o the PM plaig approach i Das et al. (005). A umerical example was solved to ivestigate the applicability of the model i the maufacturig cell desig. The results idicated that PM has the potetial to reduce the maufacturig system cost ad icrease the machie system reliability performace whe compared with the results of the cell desig without ay PM cosideratio. Extedig the work of Das et al. (006), the preset paper itegrates the PM pla with machie system reliability ad maufacturig system cost i the form of a mathematical model of the cell desig process. A umerical example problem is solved to ivestigate the CMS desig performace i terms of machie system reliability, ad maiteace cost of the CMS. 3. MODEL DEVELOPMENT 3. Machie Reliability Aalysis i a Process Pla To examie the cocept of machie reliability i the cotext of a CMS, we cosider a small cell cosistig of four part types to be processed o five machies. Table presets a typical routig table for the set of part types with iformatio about the operatios of the part types, ad the machies capable of performig these operatios. I geeral, each part type may be processed uder various process plas, ad uder a give process pla each operatio of a part type may be performed o oe or more machies, givig rise to a umber of process routes. For istace, part type may be processed usig ay of the eight process routes listed i Table. Cosiderig process route #6 as a example, the system reliability correspodig to the machies alog this route is: R 3 4( t) = R( t) R3( t) R4 ( t) () where R (t) is the reliability of machie at time t. Assumig that machie failures follow a Weibull distributio with the characteristic life θ ad the shape parameter, the reliability fuctio for a machie is: R t t) = exp[ ( ) ] () θ ( ad equatio () is ow writte as: Or, R R ( t) = = {, 3, 4} exp[ t θ ( ) ] (3) t ( t) = exp[ ( ) ] (4) θ = {, 3, 4} After coversio to logarithmic scale may be writte as: t LIR t R t θ 3 4( ) = l = (5) 3 4 ( ) {, 3, 4} where LIR -3-4 (t), hereafter referred to as the reliability idex, is the atural log iverse of the reliability of the machie sequece M-M3-M4 correspodig to process route #6. For the Weibull distributio we have: MTBF θ = Γ + / ) (6) ( which, upo substitutio i equatio (5), results i: LIR t Γ ( + / ) t ( ) = (7) 3 4 MTBF {, 3, 4} where MTBF is the mea time betwee failures for machie. I a similar fashio, the reliability idices for each process route correspodig to each ( combiatio may be evaluated. Sice for each part oly oe process route may be chose, the obective would be to select process routes such that the sum of their reliability idices would result i a optimum level of overall reliability for the Table. A typical routig table for a set of part types. Part types 3 4 Process plas Operatios 3 M3, M4, M M5 M, M, M3 M4 M4 M M4, M5 M3 M, M3 M M5 M, M3, M M4 M4, M5 M, M3 M4, M5 M M, M4 M, M4 M M, M3 M5 M4

4 6 Kacha Das R.S. Lashkari S. Segupta CMS. This is i fact oe of the obectives of the CMS desig model to follow. Table. Process routes for part type i Table. Process Routes Process Pla Machie Sequece i Process Route M3-M4 M3-M5 M-M4 M-M5 M-M3-M M-M3-M4 M4-M3-M M4-M3-M4 3. Reliability-Based Group PM Pla A reliability-based PM plaig for CMS is developed with the aim of determiig the largest possible PM iterval to miimize the total maiteace cost by reducig the umber of maiteace actios while keepig the idividual machie failure probabilities below a predefied Upper Boud as may be specified by a orgaizatio. To set the limit o the cumulative failure probability of machies we have followed the approaches of Johso (959) ad Kardo ad Frededall (00). Assumig that tp is the iterval at which PM is carried out, the cumulative failure probability of a machie at time tp may be expressed as: F tp) = exp[ ( tp / θ ) ] (8) ( Usig equatio (8), we may determie the PM iterval tp whe a Upper Boud o the cumulative failure probability of machie at time tp is set by the orgaizatio; that is, determie: tp θ {l } F ( tp) /, =,,, (9) such that, F (tp) Upper Boud, =,,, m (0) Based o these equatios, the followig optimizatio model (to be idetified as OptimIterval model heceforth) may be proposed to determie the optimal PM iterval: OptimIterval Model Maximize tp subect to: tp F ( tp) θ {l } =,,, m () / F (tp) Upper Boud, =,,, m () The solutio to the OptimIterval model is illustrated through umerical example. Numerical Example. We cosider a example ivolvig 4 machies. The Table 3. Machie data for the example problem. Machie b cp MTBF MTTR θ cfr cpr Co M M M M M M M M M M M M M M Note: b is machie capacity i hours; cp is the machie o-utilizatio pealty cost; cfr is the failure maiteace cost; cpr is the PM cost; C o is the fixed cost of PM, to be explaied later. $50

5 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig 7 reliability data icludig MTBF, MTTR (mea time to repair), as well as the cost data are displayed i Table 3. The MTBF, MTTR,, ad θ values are geerated radomly. The values are assumed to be greater tha oe to cosider icreasig failure rate of machies, ad a Upper Boud cumulative failure probability of 0.5 is assumed. The solutio to the OptimIterval model is preseted i Table 4. The optimal PM iterval tp is 40.3 hours. To get a isight ito the solutio, the cumulative failure probability of each machie at tp = 40.3 hours is computed ad displayed i colum 3 of Table 4. Table 4. Solutio of OptimIterval Model. Machie Commo Iterval tp F(tp) Tmax M M M M M M M hours M M M M M M M It may be observed that oly the cumulative failure probability of machie M6 has reached the Upper Boud of 0.5, whereas for other machies the cumulative failure probability is less tha the Upper Boud. This implies that, by implemetig PM actios after every 40.3 hours, the failure probabilities of the machies are maitaied at or below the reliability threshold set by the Upper Boud level. I colum 4 of Table 4 the maximum possible PM iterval, Tmax, for each machie is displayed. The Tmax value for a machie is obtaied by solvig equatio () usig the Upper Boud as the value of F(tp). It is evidet that for all the machies, other tha M6, Tmax 40.3 hours. For istace, for machie M the cumulative failure probability at tp = 40.3 hours is 0.03; however, from equatio (), at a value of F (tp) = 0.5, we obtai Tmax = 56.3, implyig that machie M may be maitaied at itervals of 56.3 hours without violatig the cumulative failure probability Upper Boud of 0.5.Maitaiig machies such as M at itervals of 40.3 hours results i too This example will be used i the sectios that follow; therefore, Table 3 presets the complete set of data. may maiteace actios uecessarily. By defiig a effective maiteace iterval for a machie, we ca avoid the uecessary PM actios ad still maitai a threshold o the machie failure probabilities. This idea uderlies the developmet of the followig algorithm which addresses the above limitatio. 3.3 Algorithm for Effective Maiteace Plaig Step. Specify the values of Co (the fixed cost of carryig out each PM actio), cpr (estimated average PM cost for machie to take it back to as-good as ew coditio), cfr (estimated failure repair cost for machie ),Upper Boud, θ, ad Step. Compute the optimum PM iterval tp usig the OptimIterval model as described above Step 3. Compute the maximum PM iterval, Tmax, for each machie by settig F (tp) = Upper Boud i equatio (): Tmax ( Upper Boud) = θ {l } (3) / Step 4. Compute the total cost TC(T) over the plaig period T usig the above iputs i the followig sequece. Y Tmax = (4) tp efftp N = tp Y, (5) T = (6) efftp N = max{ N, =,,, m} (7) max CPMcell = N max Co + N cpr (8) CFMcell = m = m = efftp N cfr ( ) θ (9) TC ( T) = CPMcell+ CFMcell (0) Y, N iteger I this model, Y computes the equivalet umber of optimum itervals correspodig to Tmax for machie, efftp represets the effective PM iterval applicable to machie, ad N is the umber of prevetive maiteace actios to be scheduled for machie. CPMcell ad CFMcell represet, respectively, the total PM cost ad the total failure repair cost for the CMS machies. cpr is the PM cost per occasio; cfr is the failure maiteace cost; ad C o is the fixed cost of PM.

6 8 Kacha Das R.S. Lashkari S. Segupta The expressios for CPMcell ad CFMcell are derived based o the total cost model i Jardie ad Tsag (006). For details, the iterested readers may refer to Das et al. (006) It may be oted here that if the plaig period T is ot exactly a iteger multiple of the effective iterval efftp, equatio (6) will result i the last PM iterval beig a partial oe, ad as such the correspodig failure maiteace cost (equatio 9) as well as the PM cost for this period will be uderestimated; however, this cost decrease is, for all practical purposes, egligible ad will ot affect the desig outcome. Step 5. Record CPMcell, CFMcell, TC(T), N, N max Step 6. Develop the PM schedule for the group of machies accordig to N Numerical Example The algorithm is illustrated usig the data i Table 3 for umerical example. At the cumulative failure probability Upper Boud of 0.5, the solutio of the OptimIterval model i Step of the algorithm is the same as that preseted i Table 4. For practical cosideratios, the optimum PM iterval of 40.3 hours is set as tp 40 hours, at which the cumulative failure probability of each machie is computed ad displayed i colum 3 of Table 4. As was poited out earlier, the optimum tp correspods to machie M6 whose cumulative failure probability is at the Upper Boud level of 0.5. The failure probabilities of all the other machies computed at tp 40 are less tha 0.5; equivaletly, the correspodig tp for these machies would be higher tha 40 hours if their failure probabilities are set at the Upper Boud value. This is doe by implemetig Step 3 of the algorithm, which computes the maximum PM itervals, Tmax, for machies other tha M6, as give i the last colum of Table 4. The detailed output from Step 4 of the algorithm is preseted i Table 5. Equatios (4) ad (5) evaluate the equivalet umber of commo PM itervals correspodig to the Tmax values, ad the effective PM iterval for each machie, respectively. For example, for machie M3, the maximum prevetive maiteace iterval of 08. hours ca be writte as 08. (40)(5), implyig that Y 3 =5, ad therefore, the effective PM iterval for machie M3 is 00 hours. Equatio (6) computes the umber of times PM actio is carried out o each machie durig the plaig period T. I our case, T = 000 hours, thus, M3 udergoes a total of 000/00 = 0 PM actios durig the plaig period of 000 hours. Equatio (7) computes N max = 50, the maximum umber of times PM is carried out i the cell. Based o the Y values, a PM schedule may ow be defied. I this case there are 50 PM actios, therefore, whe Y =, the PM schedule for machie is i periods,,, 50. Whe Y =, the PM schedule for machie is i periods, 3, 5,, 49, ad so o. Equatio (0) computes the total maiteace cost, TC (T) = $08,6, for the cell over the plaig period T. The compoets of TC(T) are the PM costs, CPMcell = $73,370 (equatio 8), ad the total failure repair costs, CFMcell = $ 34,756 (equatio 9). 3.4 PM ad Machie Reliability Aalysis We cosider the CMS discussed i Sectio 3., where there is a PM schedule defied by the orgaizatio based o the algorithm itroduced i Sectio 3.3. Havig determied tp, the commo maiteace iterval, a machie will udergo a PM actio after every Y. tp time uits, for a total of N times durig the plaig period T, Table 5. Prevetive maiteace pla determied by the algorithm. Machies Effective PM iterval effit # of PM Actios (N) # of PM itervals (Y) Schedule of maiteace CPMcell ($) CFMcell ($) M 0 7 3, 4, 7,, 49 M 80 5, 3, 5,, 49 M , 6,,, 49 M ,, 3,, 50 M , 7, 3,, 49 M ,, 3,, 50 M , 7, 3,, 49 M , 4, 7,, 49 M , 6,,, 49 M0 80 5, 3, 5,, 49 M 40 50,, 3,, 50 M , 5, 9,, 49 M ,, 3,, 50 M , 4, 7,, 49 73,370 34,756

7 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig 9 after which its reliability may be writte as: N R ( T ) = [ R ( Y tp)] R ( T N Y tp), () assumig that the machie is restored to its origial coditio after a PM actio is admiistered (Ebelig 997). For the Weibull distributio equatio () becomes: Y tp R = ( T ) exp[ N ] θ T N Y tp exp[ ] θ () Agai, usig process route #6 i Table as a example, we ow substitute equatio () i equatio () to obtai: 3 4( T) = = ], 3, 4 R Y tp exp[ N θ =, 3, 4 which may be simplified as: l R = ( T ) 3 4 T N Y exp[ θ {, 3, 4} tp ) ] Y tp [ N θ + (3) Table 6. Typical part type iformatio for the umerical example. Part (Demad) (36) (976) 3(88) 4(0) 5(946) 6(935) 7(388) 8(766) 9(5) 0(986) Parameters Operatios accordig to process pla Operatios accordig to process pla Machie M M4 M3 M7 M8 M3 M3 M6 M5 M3 M9 M8 M4 M Time (mi) Cost($) Machie M3 M0 M8 M M6 M3 M M0 M9 M6 M8 M7 M8 M9 M8 M9 Time (mi) Cost($) Machie M0 M6 M M M3 M M9 M8 M3 M4 M7 M4 Time (mi) Cost($) Machie M M M9 M3 M8 M M M3 M4 M0 M3 M7 M8 M9 Time (mi) Cost($) Machie M3 M3 M9 M6 M M4 M8 M4 M6 M Time (mi) Cost($) Machie M M5 M7 M9 M3 M M4 M7 Time (mi) Cost($) Machie M9 M3 M M9 M6 M5 M4 M M3 M7 M4 M0 M6 M Time (mi) Cost($) Machie M4 M7 M4 M M M4 M0 M M M M3 M4 M4 M Time (mi) Cost($) Machie M M0 M M5 M3 M6 M3 M9 M7 M M8 M0 M M Time (mi) Cost($) Machie M M M5 M3 M3 M M4 M3 M3 M5 M7 M Time (mi) Cost($)

8 0 Kacha Das R.S. Lashkari S. Segupta T N Y tp θ ] (4) Recallig that, for each machie, the plaig period T is divided ito a umber of effective itervals (equatio (6)), ad that we igore the last (partial) PM iterval i case the plaig period T is ot a iteger multiple of the effective iterval, equatio (4) ow reduces to: Y tp N R T θ l = (5) 3 4( ) {, 3, 4} Usig equatio (6), equatio (5) is ow writte as: LIR 3 4 ( T) = {, 3, 4} Y tp Γ( + / ) N MTBF = LIR ( tp) (6) = {, 3, 4} where we have defied: Y tp Γ ( + / ) LIR ( tp) = N (7) MTBF 3.5 CMS Desig Model I this sectio, we describe the itegratio of the machie reliability ad maiteace plaig cocepts ito the multi-obective desig model for a cellular maufacturig system. It is assumed that there is a set of machies =,,, m to process a set of part types I =,,, with uiform demads d i durig the plaig period T. The reliability data for the machies are available i terms of MTBF, MTTR,, ad θ. A part type i may be processed uder ay of the process plas p =,,,. For the sake of brevity, a part type-process pla combiatio will be represeted by ( from hereo. The operatios performed o a ( combiatio are o =,,,, ad the machies that ca perform operatio o of ( are represeted by the set J ipo. The correspodig refixturig cost ad the operatio cost are represeted by CR o ( ad CO o (,, respectively. The 0- decisio variable X oc ( equals if operatio o of ( is performed o machie i cell c, ad zero otherwise. The obective is to group the machies ito a umber of cells, ad to assig each part type to oe or more cell for processig so as to miimize the total costs ad maximize the overall system reliability idex as defied i sectio Obective Fuctios The first obective fuctio F computes the total system costs cosistig of the variable cost of maufacturig operatios (VCM), the iter-cell material hadlig cost (MHC) ad the pealty cost associated with machie outilizatio (MNC): Miimize F = VCM + MHC + MNC (8) The variable cost of maufacturig operatios, VCM, may be expressed as: VCM = di { CO i= C c= X oc p= ( o= J ipo o ( + CR o ( } (8a) The iter-cell material hadlig cost MHC computes the cost of movig the parts from cell c to cell ĉ: MHC = d i i= p= o= J ˆ ipo J ip ( o+ ) c, cˆ C H ic ˆ cˆ X oc ( X ˆ ˆ ( ( o+ ) c where, H icĵĉ is the cost of movig a uit of part type i from machie, after performig operatio o i cell c, to machie ĵ i cell ĉ for the ext operatio (o+). It is oted that MHC is a o-liear fuctio, which may be liearized by replacig the product term X oc ( X ˆˆ( ) ( o+ ) c ip by a biary liearizatio variable, Y ˆ ˆ (, which satisfies costraits (36) ad (37) below. It is evidet that Y ( oc c oc ˆ cˆ takes the value of if ad oly if a uit of part type i is moved from machie i cell c, after performig operatio o, to machie ĵ i cell ĉ for operatio (o+). Thus, the resultig expressio for MHC is: MHC = d i i= p= o= J ˆ ipo J ip( o+ ) c,ˆ c C H ˆ ˆY ˆ ˆ ( (8b) ic c oc c Fially, the term MNC computes the pealty cost for the proportio of the time machie is ot utilized: MNC = m = TO o ( [ i cp d ( + TR A ( T ) b i= o p= C o= ( ] ) X ( c= oc (8c)

9 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig where b is the capacity of machie, A (T) is the iheret availability of machie, ad A (T)b represets the effective capacity of machie. I additio, TO o ( ad TR o ( are, respectively, the operatio ad refixturig times correspodig to operatio o of ( o machie, ad cp is the pealty cost of o-utilizig the capacity of machie. The secod obective fuctio F computes a measure of the iverse of the system reliability, i atural logarithmic scale, over the set of all the ( combiatios: Miimize F = i= p= o= J ipo C c= LIR ( tp) X oc ( (9) where LIR (tp) was defied i equatio (7). Equatio (9) geerates a composite expressio by addig up the reliability idices alog all the feasible process routes for each ( combiatio. Durig the optimizatio process, the operatio allocatio variable X oc ( is compelled to assig oly oe machie to each operatio of the ( i order to comply with costraits (30) ad (3), which will follow. Cosequetly, for each (, the solutio will iclude the reliability idices of the machies for oly oe selected process route Costraits. Each part type is assiged to a sigle process pla. The biary variable Z( equals oe if ad oly if part type i is processed uder process pla p. p= Z( =, i (30). For a give ( combiatio, each operatio of the process pla is assiged to oe of the available machies i oe of the cells. J ipo C c= X oc ( = Z(, i, p, o (3) 3. A machie is assiged to at most oe cell. The variable M c equals if machie is assiged to cell c, ad 0 otherwise. C c= M c (3) 4. There is a user-defied upper limit o the umber of machies allowed i a cell. m = M c UM c (33) 5. A machie has to be assiged to a cell c before ay operatio could be allocated to that machie. i= p= o= X oc ( M c,, c (34) 6. The allocated operatios to a machie will ot exceed its effective capacity. i= d i p= o= [ TO ( + TR ( ] X ( o b M c A ( T ),, c (35) where A (T), the iheret availability of machie, is approximated as (Ebelig 997): MTBF A ( T ) MTBF + MTTR 7. Costrait for liearizig o-liear fuctio for MHC i equatio (8b) as described above: X oc( + X ˆˆ( Y ˆˆ( ( o+ ) c occ i p, o {,,, }, J ˆ J, c, cˆ (36), ipo, ip( o+ ) Xoc ( + X ˆˆ( Y ˆˆ(, ( o+ ) c occ i p, o {,,, }, J ˆ J, c, cˆ (37), ipo, ip( o+ ) 8. The last costrait set imposes itegrality o relevat variables X oc (, M c, Z (, Y ˆˆ ( {0, }, occ i p, o, J, ˆ J, c, cˆ (38), ipo ip( o+) 4. A NUMERICAL EXAMPLE To illustrate the applicability of the model, a umerical example ivolvig 4 machies ad part types is preseted. The solutio is obtaied usig the commercial solver LINGO 9. The total umber of variables, iteger variables ad the costraits are, respectively, 5537, 55340, ad 86. Table 6 displays a portio of the processig data for the first 0 part types. The relevat machie iformatio for this example was already give i Table 3. As ca be see i Table 6, each part type may be processed usig oe of the two process plas. For example, uder process pla, part type has four operatios; operatio may be assiged to either machie M or machie M4; the operatio time o M is.65 miutes, ad the correspodig cost is $3.09. Also, the demad for part type is 36 uits durig the plaig period. The iformatio i Table 3 icludes, for each machie, the machie capacity o 0, oc

10 Kacha Das R.S. Lashkari S. Segupta (b), ad the pealty cost (cp) associated with the outilizatio of the machie capacity, as well as the reliability parameters MTBF, MTTR,, θ, ad the related maiteace costs. For example, for machie M, the capacity is 000 hours, the pealty cost is $85 per percetage o-utilizatio of machie capacity, MTBF is 99 hours, ad MTTR is 7 hours; the parameters of the Weibull distributio are =.64, ad θ = 334.9; the prevetive Table 7. Compariso of model results with ad without prevetive maiteace cosideratio. Compariso Factors Sceario Sceario Itegrated machie reliability ad PM Machie reliability oly, o PM CASE Miimize F (Obective fuctio I) oly F value $755,84.60 $755,84.60 F compoets VCM ($) $754,9.50 $754,9.50 MHC($) $ $ MNC($) $763.0 $763.0 F value 376.7,906.5 Cell Cofiguratio Cell M, M, M3, M5 M, M9, M0, M3 Cell M4, M7, M, M4, M7, M, Cell 3 M6, M9, M0, M3 M, M3, M6 Cell 4 M8, M, M4, M5, M8, M4, M CASE Miimize F (Obective fuctio II) oly F value 96, ,3.30 F compoets VCM ($) 94, ,78.0 MHC($),450.00, MNC($) F value Cell cofiguratio Cell M, M3, M5 M, M3, M7 Cell M8, M9 M9, M Cell 3 M3, M0, M4 M, M5, M0, M3 Cell 4 M4, M7, M CASE 3 Miimize F s.t. F Є Є = 4.6 Є = F value 95, ,65.0 F compoets VCM ($) 94, , MHC($) MNC($) F value Cell cofiguratio Cell M3, M7, M9, M M Cell M4, M5 M3, M7, M9, M Cell 3 M, M8, M0, M4 M, M5, M0, M3 Maiteace Activity Costs: CPMcell ($) 73, CFMcell ($) 34, ,4.00 TC(T) ($) ,4.00

11 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig 3 ad failure maiteace costs are $49 ad $334, respectively. To evaluate the secod obective fuctio F, we eed to compute LIR (equatio (7)) for each ( combiatio which i tur depeds o the values of tp, N, ad Y. These parameters are already computed ad listed i Table 5. To examie the effects of the PM itegratio o the CMS desig, the umerical example is also solved without the cosideratio of prevetive maiteace i the CMS desig. This is achieved by settig N =, Y = ad tp = T = 000 hours i equatio (7) ad thereby trasformig it ito the followig form: LIR 000 Γ ( + / ) ( tp) = (39) MTBF Accordigly, Table 7 summarizes the solutio results uder two scearios: sceario whe PM is cosidered, ad sceario whe it is ot. Uder each sceario, three cases are cosidered. I the first case, the multi-obective model is solved usig a hierarchical approach to optimize the first obective fuctio, F, oly, subect to costraits (30)-(38), ad igorig the secod obective fuctio, F. The solutio correspods to a F value of $755, (which is a lower boud o this obective fuctio); the value of secod obective fuctio, F, evaluated at this solutio poit is uder sceario, ad uder sceario. Furthermore, uder either sceario, the solutio geerates four cells, although the cell compositios i the two scearios are differet. For example, uder sceario, cell cosists of machies M, M, M3, ad M5, whereas uder sceario cell I cosists of machies M, M9, M0, ad M3. As is evidet, there is a sigificat decrease i the value of the secod obective fuctio uder sceario, implyig a improved reliability performace whe PM is cosidered. I the secod case, the model is solved to optimize the secod obective fuctio, F oly, subect to the same costraits as before, ad igorig the first obective fuctio F. Uder sceario, the solutio results i a F value of 4.6 (which is a lower boud o this obective fuctio), ad the value of first obective fuctio, F, evaluated at this solutio poit is $96,37.0. Uder sceario, the respective values of the two obective fuctios are ad $99,3.30. Oce agai, there is a sigificat decrease i the F value uder sceario compared to sceario, idicatig that the reliability performace may be greatly improved by itegratig PM plaig ito the CMS desig process. The solutio i the secod case ivolves four cells i sceario ad three cells i sceario, with widely differet cell compositios. I the third case, the multi-obective model is solved usig a pre-emptive solutio approach, placig priority o the secod obective fuctio, ad optimizig the first obective fuctio subect to costraits (30)-(38) ad the additioal costrait: Obective fuctio F ε where ε = 4.6 uder sceario, ad ε = uder sceario. As may be observed i Table 7, uder either sceario, the solutio achieves the F target values (i.e., 4.6 uder sceario ad uder sceario ) ad results i a F value of $ uder sceario, ad $885, 65.0 uder sceario. Oce agai, the results idicate the CMS reliability performace improvemet as a cosequece of the PM cosideratio i the cell desig. The solutio i this case ivolves three cells uder each sceario, although the cell compositios are widely differet as expected. Fially, Table 7 compares the total maiteace cost uder the two scearios as well. The maiteace costs are computed by followig the group PM plaig approach ad the correspodig algorithm i Sectio 3.3. Uder sceario, whe PM is cosidered, the total maiteace cost TC(T) for the CMS durig the plaig period is: $08,6.00, which cosists of the PM cost of CPMcell = $73, (equatio (8)) ad the failure repair cost of CFMcell = $34, (equatio (9)). Whe PM is igored, the machies are oly subect to radom failures, ad there are o PM-related costs. Therefore, CPMcell = 0, ad to compute CFMcell i equatio (9), we set tp = T, N =, ad Y =. The total maiteace cost for the CMS i this case is $84,4.00. As is evidet, the itegratio of PM plaig cocepts ito the CMS desig process etails substatial beefits by improvig the system reliability performace ad thus reducig the uplaed machie dowtime cost to a large extet. 5. CONCLUSIONS We have preseted a CMS desig model which cosiders machie reliability ad prevetive maiteace plaig assumig that machie failure times follow a Weibull distributio with icreasig failure rate. The model is i the form of a large scale multi-obective 0- iteger program. This is a ew approach that itegrates machie reliability, system costs, ad prevetive maiteace plaig i the overall desig of the CMS. The model cosiders the alterative process routes to process a part type, evaluates the system reliability correspodig to the machies alog a process route, ad seeks to maximize the overall reliability of the cell while miimizig the overall system costs. A umerical example is provided to demostrate the applicatio of the model. The results idicate that the cosideratio of PM plaig i the CMS desig process leads to a sigificat improvemet i the reliability performace of the system, ad a sizeable reductio i the

12 4 Kacha Das R.S. Lashkari S. Segupta total maiteace cost. Fially, the model is computatioally feasible, ad as is demostrated here, it may be solved usig commercial software (e.g., Ligo 9). ACKNOWLEDGMENT This study was fuded by the Natural Scieces ad Egieerig Research Coucil, Caada, through a research grat to the secod author. REFERENCES Aski, R. G. ad Estrada, S. (999), Ivestigatio of cellular maufacturig practices. I S. A. Irai (ed), Hadbook of Cellular Maufacturig Systems, Wiley, New York, Chapter, Cho, I.D., ad Parlar, M. (99), A survey of maiteace models for multi-uit systems. Europea Joural of Operatioal Research, 5, -3. Das, K., Lashkari, R.S., Segupta, S. (006), Itegratio of machie reliability ad prevetive maiteace plaig i the desig of cellular maufacturig systems, Proceedigs of 7 th Asia Pacific Idustrial Egieerig Maagemet Systems Coferece, Bagkok, Thailad, 03-3 Das, K., Lashkari, R.S., Segupta, S. (005), A study of prevetive maiteace plaig for CMS. Proceedigs of the st Iteratioal Coferece o Operatios ad Supply Chai Maagemet, Bali, Idoesia, -8. Diallo, M., Perreval, H., ad Quillot, A. (00), Maufacturig cell desig with flexible routig capability i presece of ureliable machies. Iteratioal Joural of Productio Ecoomics, 74, Dekker, R., Va Der Schoute, F., ad Wilderma, R. (997), A review of multi-compoet maiteace models with ecoomic depedece, Mathematical Methods of Operatioal Research, 45, Ebelig, C. E. (997), A Itroductio to Reliability ad Maitaiability Egieerig, McGraw-Hill, New York. Jardie, A. K. S. ad Tsag, A. H. C (006), Maiteace, Replacemet ad Reliability, CRC Press, Boca Rato, FL. Jeo, G., Broerig, M., Leep, H. R., Parsaei, H. R., ad Wog, J. P. (998), Part family formatio based o alterative routes durig machie failure. Computers ad Idustrial Egieerig, 35, Johso, L. G. (959), The statistical treatmet of fatigue experimets, Research Laboratories, Geeral Motors Corporatio, IVA, 05. Joies, J. A., Kig, R. E., ad Culbreth, C. T. (996), A comprehesive review of productio orieted maufacturig cell formatio techiques. Iteratioal Joural of Flexible Automatio ad Itelliget Maufacturig, 3, 6-0. Kardo, B., ad Frededall, L. D. (00), Icorporatig overall probability of system failure ito a prevetive maiteace model for a serial system. Joural of Quality i Maiteace Egieerig, 8, Logedra, R., ad Talkigto, D. (997), Aalysis of cellular ad fuctioal maufacturig system i the presece of machie breakdow. Iteratioal Joural of Productio Ecoomics, 53, Masouri, S. A., Husseii, S. M. M., ad Newma, S. T. (000), A review of moder approaches to multicriteria cell desig. Iteratioal Joural of Productio Research, 38, 0-8 Seifoddii, S., ad Dassemi, M. (00), The effect of reliability cosideratio o the applicatio of quality idex. Computers ad Idustrial Egieerig, 40, Selim, H. M, Aski, R. G., ad Vakharia, A. (998), Cell formatio i group techology: review, evaluatio ad directios for future research. Computers ad Idustrial Egieerig, 34, 3-0. Talukder, M. S., ad Kapp, G. M. (00), Equipmet assigmet to multiple overhaul blocks i series system, Joural of Quality i Maiteace Egieerig, 8, Valdez-Flores, C., ad Feldma, R. M. (989), A survey of prevetive maiteace models for stochastically deterioratig sigle uit systems. Naval Research Logistics, 36, Wag, H. (00), A survey of maiteace policies of deterioratig systems: ivited review, Europea Joural of Operatioal Research, 39, Wilderma, R. E., Dekker, R., ad Smit, A. C. J. M. (997), A dyamic policy for groupig maiteace activities, Europea Joural of Operatioal Research, 99, Wemmerlov, U., ad Hyer, N. L. (986), Procedures for the part family machie group idetificatio problem i cellular maufacturig, Joural of Operatios Maagemet, 6, -47. APPENDIX Idices c {,,, C} cells i {,,, } part types {,,, m} machies p {,,, } process pla for part type i Ip a part type-process pla combiatio o {,,, } operatios of ( J ipo {,,, m} set of machies that ca perform operatio o of (

13 Itegratig Machie Reliability ad Prevetive Maiteace Plaig i Maufacturig Cell Desig 5 Parameters A (t) b CO o ( CR o ( cp CPMR d i H icĵĉ MTBF MTTR N tp TO o ( availability of machie at time t available time o machie durig the plaig period cost of performig operatio o of ( o machie cost of refixturig a uit of ( for operatio o o machie pealty cost of o-utilizatio of the capacity of machie average cost of PM per occasio for machie demad for part type i durig plaig period cost of movig part type i from machie i cell c to machie ĵ i cell ĉ to perform the ext operatio mea time betwee failures for machie mea time to repair for machie umber of PM itervals for machie durig the plaig period commo PM iterval for the machies i a cell time to perform operatio o of ( o TR o ( UM Y θ Decisio variables machie time to refixture ( for operatio o o machie maximum umber of machies i a cell equivalet umber of commo itervals tp applicable to machie (a iteger) shape parameter of Weibull distributio for machie characteristic life of Weibull distributio for machie M c = if machie is assiged to cell c; 0 otherwise X oc ( = if operatio o of ( is performed o machie i cell c; 0 otherwise Y oc ˆ c ˆ( = if ( moves to machie ĵ i cell ĉ to perform operatio (o+) after performig operatio o o machie i cell c; 0 otherwise Z( = if part type i is processed uder process pla p; 0 otherwise