Operatin-Dependent Maintenance Scheduling in Flexible Manufacturing Systems M. Celen 1, D. Durdanvic 1 1 Dept. f Mechanical Engineering, University f Texas at Austin, Austin, TX, United States Abstract In highly flexible and highly integrated manufacturing systems such as semicnductr manufacturing, the dynamic interactins between equipment cnditin, peratins executed n the tls and prduct quality necessitate int decisin-making in maintenance scheduling and prductin peratins. T address these prblems, we devise an integrated decisin-making plicy ptimizing a custmizable bective functin with respect t peratin-dependent degradatin mdels and prductin target. Optimizatin was achieved using a metaheuristic methd based n the results f discrete-event simulatins f a generic cluster tl. The results shw that peratin-dependent maintenance decisin-making utperfrms the case where maintenance decisins are made withut cnsideratins f peratin-dependent degradatin dynamics. Keywrds: Scheduling, Maintenance, Dispatching 1 INTRODUCTION Maintenance is an essential part f manufacturing peratins ensuring that adequate prductin resurces are available t achieve desired prductivity and quality in a manufacturing system. There are mainly tw types f maintenance peratins; reactive maintenance (RM) which ccurs when a tl/machine actually fails and preventive maintenance (PM) which is perfrmed n a tl/machine befre actual failure ccurs. Even thugh RM is unavidable, it usually csts much mre and requires mre maintenance time when cmpared t PM [1]. Fr example, in an autmtive assembly plant a minute f unexpected dwntime can cst as much as $0000 []. Hence, prper scheduling f PM is always desired. PM can be rughly characterized as reliability-based maintenance (RBM), where maintenance is perfrmed at certain time r usage intervals, and cnditin-based maintenance (CBM), where maintenance is perfrmed when the cnditin f machine requires a repair. RBM cnsiders the lng-run average f the system dynamics and hence the decisins made are ptimal nly in the steady-state. On the cntrary, since CBM is mre dynamic and the decisins are based n the cnditin f the system at that time pint, the effects f thse decisins can be ptimized and adusted within an arbitrarily chsen time hrizn [3]. The benefits f CBM arise frm the ability t preventively maintain the system nly when necessary, thus, saving resurces and imprving system availability [4]. In Flexible Manufacturing Systems (FMS), PM decisins are cnsiderably harder t make, as the machines have the capability f cnducting different manufacturing peratins and/r prducing at varius speeds. In such systems, degradatin f a machine depends highly n the peratins perfrmed n that machine. Thus, selectin f peratins executed n a machine directly affects PM decisins by changing the degradatin dynamics. On the ther hand, PM actins interrupt prductin and change the system reliability and equipment availability, which in turn directly affects decisins as t which peratins shuld be perfrmed n which piece f equipment [5]. The necessity f int decisin making as a result f such dynamic interactin between PM and recnfiguratin actins has been addressed nly recently. Yang et al. [6] cmbine age-based maintenance (ABM) decisins with peratins decisin making in an envirnment where there are nly tw peratin mdes available: fast and slw. Naturally, fast peratin mde leads t faster degradatin but high prductin, and vice versa. This effect is mdeled by peratin specific reliability functins. In their study, pssible schedules are generated via a Genetic Search Algrithm [7] and the value f each schedule is evaluated by a discrete event simulatin system. It was shwn in their results that intly ptimizing thrughput and maintenance peratins results in decreased maintenance time and increased prfits. A much mre cmplex situatin is cnsidered in Zhu et al. [5]. They develp an integrated recnfiguratin and ABM plicy (IRABM) n a single-prduct parallel-serial system with recnfigurable capability. With recnfiguratin cmes the ability t imprve system thrughput and reduce the likelihd f a system-wide failure. Nevertheless, there is als an assciated cst f peratin transfer frm the degraded machine t the less degraded ne, necessitating a tradeff between the benefits and drawbacks f peratins recnfiguratin. It has been shwn in their study that IRABM utperfrms ABM in terms f lwer expected ttal cst. The previus tw publicatins mdeled peratindependent degradatin thrugh peratin specific reliability functins, which makes them very suitable fr RBM appraches. The wrk by Zhu et al. [8] utilizes Markv mdels t mdel the degradatin f the machine cnditins which makes it mre suitable fr CBM applicatins. They use simulatin based ptimizatin in a lad-sharing system, where different cmpnents f the system share an verall lad. In such systems, lads allcated t each machine vary, thus affecting degradatin dynamics f each machine since the degradatin f a machine depends n the amunt f lad assigned t that machine. It was shwn that Integrated Lad Allcatin and Cnditin Based Maintenance Plicy (ILACBM) results in increased availability f equipment when cmpared t the traditinal CBM.
As machine degrades, the utging prduct quality (yield) usually decreases. The earliest wrk cnsidering the effects f equipment cnditin n prduct quality and incrprating it int maintenance scheduling is reprted in [9]. A decisin making plicy that simultaneusly determines maintenance and prductin schedules fr a multiple-prduct single machine system was develped by cnsidering the fact that machine cnditin can affect the yield f different prduct types. In a later study, Slan [1] extends the wrk in [9] by cnsidering varying prductin targets and multiple maintenance actins. Slan and Shanthikumar [10] cnsider multiple machines and add b dispatching decisins t the decisin-making prcess develped in [9]. They cmpare the perfrmances f sme predetermined dispatching rules (such as FCFS, SPT, selecting the lt with highest yield, etc.) and sme special maintenance plicies (such as fixed state, fixed time, etc.). Lee et al. [3] present a mre elabrate wrk n CBM and dispatching with yield cnsideratins in a semicnductr manufacturing envirnment. In this wrk, as an imprvement t [9], there are n priri predetermined CBM plicies. Instead, CBM plicies fr different wafer types are determined via discrete event simulatin and genetic algrithm based ptimizatin f prduct-specific maintenance triggering states. Dispatching f bs t the machines is als btained using a genetic algrithm. The authrs reprt that using wafer type-dependent CBM plicies results in increased yields. By nt restricting dispatching and maintenance plicies as in [9], Lee et al. wrk n a system which is a clse representatin f an actual cluster tl in a semicnductr fab. Hwever, they verlk the fact that degradatin is an peratindependent prcess and assume that each peratin affects the degradatin f machines in the same way. In ur study, we cnsider a multiple-prduct/multiplemachine system where each prduct requires several peratins fr cmpletin. These peratins are executed n nn-identical machines, each f which can execute a certain subset f peratins. Degradatin prcesses f the machines are mdeled as peratin dependent Markv mdels, with the utput quality f the prducts decreasing as the cnditin f the machine degrades accrding t a knwn prduct-specific stchastic mdel. Our aim is t facilitate maintenance decisin-making in highly flexible manufacturing systems (where ne has the ability t d multiple peratins in multiple statins), based n the afrementined peratin-dependent degradatin mdels. The decisin-making will be dne by maximizing a custmizable reward functin, taking int accunt rewards f prductin and the csts f maintenance peratins. PROBLEM STATEMENT In this study, we fcus n a flexible manufacturing system with m manufacturing statins labeled and we assume that in thse statins we are prducing a set f prduct types. Let be the set f all peratins that can be executed by the statins f that manufacturing system. Each prduct type, is assciated with a sequence f peratins, where is the number f peratins needed t manufacture that prduct type. Since any tw prduct types and may have sme cmmn peratins, the intersectins, are nt necessarily empty sets. It is assumed that each statin can execute peratins, where is the number f peratins that can be executed by that statin. Any tw statins and may be able t execute sme cmmn peratins, which means that,, are nt necessarily empty sets. That means sme peratins can be executed by mre than ne statin, making it necessary t chse which statin t use fr a given peratin. The gal will be t prduce f each prduct type,, within a certain missin time. In this paper, we acknwledge that different peratins have different degradatin effects n the statins and mdel the degradatin prcess via cncatenated peratin-dependent Markv mdels. The degradatin prcess f each statin is characterized by a set f peratin-dependent Markv Chains defined ver a cmmn state space, in which each f the states dentes a degradatin state f the statin (1 denting the "gd as new" state and denting the "failed" state). Fr any statin, the prbability f transitining frm state t state if peratin is executed in it is defined as, tgether these prbabilities frm the peratin-dependent state transitin matrices. All Markv chains are assumed t be unidirectinal, mdeling the well-accepted intuitin that the cnditin f a statin can nly wrsen ver time, unless a maintenance peratin is dne. Let dente the state f statin that triggers preventive maintenance when peratin is executed in it. It means that if the current statin s degradatin state is greater than r equal t, then peratin culd nt be perfrmed in that statin, unless a maintenance peratin is dne n it befrehand. Thus, as time passes and statins degrade, each statin "lses" mre and mre peratins that can be dne in it, unless a maintenance actin is invked n it t restre its cnditin. In additin, let us take int accunt the fact that as the degradatin level f a statin increases, the prbability f success f any peratin executed in that statin decreases. We assume that fr any statin the prbability f success f a given peratin is a knwn functin f the peratin and the statin state, dented as. Our bective is t find a cmbined plicy f maintenance triggering and dispatching f peratins acrss a system that maximizes a reward functin that cnsiders the benefits f prductin, csts f maintenance and penalties fr unmet prductin gals. It is assumed that the sequence in which prducts are manufactured is fixed. Let dente this fixed prduct sequence meaning that first f prducts will be made, fllwed by f prducts, etc. Hence, ur prblem can be expressed as fllws: maximize S 1 1 S1 S N 1 S 1 S S N subectt where S 1 m S m S N m E R n nmr c r nmpc p a ( N n ) p W W [ w,w,,w Reward fr each cmpleted wafer f type, Cst f reactive maintenance Cst f reactive maintenance 1 Unmet prductin gal penalty unit penalty fr wafer type, Number f gd wafers prduced f type missin time T Nw ] (P1) during
Ttal number f reactive maintenance Ttal number f preventive maintenance Maintenance plicy fr chamber 𝑐𝑖 { The set f peratin-dependent preventive maintenance triggering states btained by slving prblem (P1) will be referred t as the Operatin-dependent CBM Plicy. In rder t evaluate the benefits f peratin-dependent maintenance decisin-making, we will cmpare slutins f (P1) with the traditinal CBM plicy. In traditinal Operatin-independent CBM Plicy, it is assumed that the maintenance triggering states fr each statin are independent f the peratin executed in it (i.e. fr each statin i). This prblem can be expressed as fllws: maximize S1 S Sm E R n nm r c r nm p c p W subect t [w 1,w,,w Nw ] a (N n )p W (P) In bth prblems (P1) and (P), fr the peratins that can be executed by mre than ne statin, peratins dispatching is based n the intuitive paradigm f always dispatching it t the least degraded statin. This dispatching plicy is supprted by results frm [3]. Obective functin values btained frm (P1) and (P) will be used t evaluate relative efficacy f the tw plicies and explre cnditins when ne utperfrms the ther. 3 SOLUTION PROCEDURE 3.1 Operatin-independent CBM plicy determinatin Fr a manufacturing system with statins, where each statin has M degradatin states, the slutin space fr prblem (P) cnsists f candidate slutins (since degradatin state 1 wuld never be bserved in a CBM plicy as it dentes a trivial plicy f maintaining the statin after each peratin). The slutin space fr a simple 5-statin manufacturing system with 5 degradatin states has candidate slutins. Since ur fcus in this study was n a small manufacturing system (see Results sectin), prblem (P) can be slved using cmplete enumeratin. The expected prfit f each candidate slutin is determined by evaluating it via discrete-event simulatin f the target system ver multiple replicatins. 3. Operatin-dependent CBM plicy determinatin Slutin Representatin Fr an m-statin manufacturing system with N peratins that can be executed in these statins, a slutin fr peratin-dependent CBM plicies can be represented with a matrix illustrated in Figure 1. In this matrix, th clumn represents the maintenance triggering states f statin fr each peratin and similarly th rw represents the triggering states f each statin when peratin is prduced in them. There are bviusly up t candidate slutins in the slutin space, thugh it can be decreased by acknwledging that nt all peratins can be executed in all statins. Nevertheless, even the afrementined reductin usually results in a slutin space that is t large fr cmplete enumeratin, especially when candidate slutins are evaluated via replicated discrete-event simulatins. Candidate slutin matrix representatin Maintenance plicy fr peratin Figure 1: Slutin representatin fr peratin-dependent CBM plicy. Hence, in rder t find practical, applicable and near ptimal slutin, we used a Tabu Search algrithm [11] based n the results f discrete-event simulatins. As illustrated in Figure, a set f feasible slutins is generated by the tabu search algrithm and fed int the discrete-event simulatr. The bective functin value relevant t each candidate slutin is btained frm multiple replicatins f discrete-event simulatins f the target manufacturing system. This expected value is fed back int the tabu search algrithm as the bective functin f the ptimizatin prcess. Figure : Decisin-making by tabu search algrithm based n simulatins Tabu Search Algrithm Tabu search (TS) is a lcal search technique that enhances the explratin perfrmance by using advanced memry structures f a cmputer. Once a candidate slutin has been determined, it is marked as 'tabu' s that the same slutin is nt visited by the algrithm ver a certain number f iteratins. The search starts frm an initial slutin (randmly seeded r chsen based n sme prblem specific infrmatin), mves iteratively frm a slutin t a nn-tabu slutin in the lcal neighbrhd f and terminates when sme stpping criterin is satisfied. The flwchart f the tabu search algrithm implemented in this study is given in Figure 4. The slutin representatin, initial slutin determinatin, neighbrhd generatin, tabu and aspiratin cnditins, and stpping criteria used in this study are explained belw. Slutin Representatin: The representatin f a slutin in matrix frm was explained thrughly in Slutin Representatin sectin. Initial Slutin Determinatin: The initial slutin fr the TS is based n the slutin f the prblem (P). The result f (P), dented as, is cnverted int the peratin-dependent CBM slutin representatin as shwn in Figure 3. Figure 3: Cnversin f peratin-independent CBM plicy slutin t initial slutin f TS Neighbrhd Generatin: A cell in the matrix representing the current slutin is selected randmly and its value is perturbed. An example f neighbrhd generatin fr a system with 3 statins, 4 peratins and 5 degradatin states is illustrated in Figure 5.
At each iteratin, randm cell selectin is repeated fr times and thus candidate slutins are generated at each iteratin. seen as different prduct types prduced in this system. In rder t assess the perfrmance f the newly prpsed ptimizatin methdlgy, we used the AutMd sftware package [13] t simulate a 5-chamber cluster tl t prduce 3 types f wafers. The parameters used are given in Tables 1, and Figure 6. Figure 4: Flwchart f tabu search algrithm fr peratin-dependent CBM plicy ptimizatin Table 1: Summary f parameters 0.98 0.0 0 0 0 0.90 0.08 0.0 0 0 0.93 0.05 0.0 0 0 0 0.98 0.0 0 0 0 0.90 0.08 0.0 0 0 0.93 0.05 0.0 0 P Op 1 0 0 0.98 0.0 0 P Op 0 0 0.90 0.08 0.0 P Op 3 0 0 0.93 0.05 0.0 0 0 0 0.98 0.0 0 0 0 0.93 0.07 0 0 0 0.97 0.03 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Figure 5: Illustratin f neighburhd generatin in TS algrithm Tabu Cnditin: Returns t previus candidate slutins are prhibited fr the next iteratins. Aspiratin Cnditin: If the best slutin amng candidate slutins is btained by a tabu mve, but yields an expected prfit higher than the best prfit btained s far, then that slutin is selected as the next incumbent slutin regardless f its tabu status. Stpping Criterin: The algrithm terminates whenever maximum number f iteratins is reached r a nnimprving mve is made fr a certain number f cnsecutive iteratins. 4 RESULTS Operatin-dependent maintenance decisin-making was tested n an example f a cluster tl, a highly sphisticated and integrated machine rutinely used in semicnductr manufacturing. A cluster tl is a mini manufacturing system f interacting subsystems (chambers and material handling system) and can be seen as a quintessential FMS since each chamber can be used t execute varius peratins with varius perating parameters. Each chamber f this cluster tl can be seen as a manufacturing statin and different wafer layers prduced in these chambers can be seen as different peratins perfrmed in these chambers, while different wafer types pushed thrugh this system can be 0.94 0.04 0.015 0.005 0 0.98 0.015 0.005 0 0 0 0.94 0.04 0.015 0.005 0 0.98 0.015 0.005 0 P Op 4 0 0 0.94 0.04 0.0 P Op 5 0 0 0.98 0.015 0.005 0 0 0 0.94 0.06 0 0 0 0.98 0.0 0 0 0 0 1 0 0 0 0 1 0.95 0.04 0.01 0 0 0.93 0.04 0.0 0.01 0 0.97 0.03 0 0 0 0 0.95 0.04 0.01 0 0 0.93 0.04 0.0 0.01 0 0.97 0.03 0 0 P Op 6 0 0 0.95 0.04 0.01 P Op 7 0 0 0.93 0.06 0.01 P Op 8 0 0 0.97 0.03 0 0 0 0 0.95 0.05 0 0 0 0.97 0.03 0 0 0 0.97 0.03 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0.90 0.05 0.03 0.0 0 0.99 0.01 0 0 0 0.89 0.07 0.03 0.01 0 0 0.90 0.05 0.03 0.0 0 0.99 0.01 0 0 0 0.89 0.07 0.03 0.01 P Op 9 0 0 0.90 0.07 0.03 P Op 10 0 0 0.99 0.01 0 P Op 11 0 0 0.91 0.07 0.0 0 0 0 0.90 0.10 0 0 0 0.99 0.01 0 0 0 0.91 0.09 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0.9 0.06 0.0 0 0 0.95 0.03 0.0 0 0 0 0.9 0.06 0.0 0 0 0.95 0.03 0.0 0 P Op 1 0 0 0.9 0.06 0.0 P Op 13 0 0 0.95 0.03 0.0 0 0 0 0.95 0.05 0 0 0 0.96 0.04 0 0 0 0 1 0 0 0 0 1 Figure 6: Operatin-dependent transitin prbability matrices Table : Prbability f success infrmatin fr each peratin
As mentined befre, the results fr peratinindependent CBM plicy were btained by cmplete enumeratin. 40 replicatins f each candidate slutin were simulated t btain the expected prfit f that slutin and was determined t be the best peratin-independent CBM plicy. In rder t ptimize the peratin-dependent CBM plicy, tabu search algrithm was run fr 0 iteratins (tests with lnger iteratins shwed that 0 iteratins were enugh t btain the best results). At each iteratin 10 randm cells were selected fr neighbrhd generatin. Hence, 30 candidate slutins were generated at each iteratin. 40 replicatins f each slutin were simulated t btain the expected prfit f that slutin. Admissible mves were marked as tabu fr = 5 iteratins. As illustrated in the first tw rws f Table 3, peratindependent CBM plicy results in higher expected prfit when cmpared t the peratin-independent CBM plicy. (a) LCL PROFIT UCL STDDEV Original Parameters Identical Wafer Reward Identical PM and RM Cst Very Expensive RM Cst Operatinindependent Operatindependent Operatinindependent Operatindependent Operatinindependent Operatindependent Operatinindependent Operatindependent 353.5 4337.4 514. 40.4 373.7 4605.6 5478.6 436.5 9175.5 108.5 11389.5 553.5 9077.6 10443.8 11809.9 683.1 4111. 4756 5400.8 3.4 45.6 5006 5759.4 376.7 3148.8 4049.6 4950.5 450.4 3146.6 4173.1 5199.7 513.3 Table 3: Cmparisn f the results f peratindependent and peratin-independent CBM plicies (b) Fr the best peratin-dependent CBM plicy, it was bserved that the maintenance triggering states and have been changed frm 3 (the state implied by the peratin-independent CBM plicy) t 4 and 5, respectively. Bth peratins degrade the chambers slwly and this result prves the intuitin that less frequent maintenance crrespnding t later triggering f maintenance is ptimal fr slwer degrading peratins. In additin t that, bth peratins are executed in the manufacturing f cheaper wafers. The ptimized maintenance triggering states als cnfirm ur intuitin that as the wafer reward decreases, the effect f cmpleting that wafer successfully n the prfit decreases, thus favring later triggering f maintenance. In rder t see the perfrmance f ur prpsed methd in the presence f different parameters, we have created three case studies by changing the wafer rewards and reactive maintenance csts. 4.1 Case 1: Identical Wafer Rewards fr All Wafers In this case we wanted t see the effects f changes in wafer rewards n peratin-dependent CBM plicies. In the previus sectin, it was seen that a decrease in wafer rewards wuld lead t later triggering f maintenance. T further prve this intuitin, we set and tested the prpsed methd with this setting. Fr this parameter setting, was determined as the best peratin-independent CBM plicy via cmplete enumeratin. (c) (d) Table 4: Cmparisn f maintenance triggering states fr different parameter sets
As shwn in the 3rd and 4th rws f Table 3, with identical wafer reward setting, ur prpsed methd still yields a higher expected prfit than the peratinindependent CBM plicy. Cmparing the maintenance triggering states in Tables 4.a and 4.b, it can be bserved that increasing wafer rewards results in earlier triggering f maintenance, which further cnfirms ur intuitin. 4. Case : Identical PM and RM Cst When the csts f PM and RM are the same and if the prbability f success (peratin yield) was nt affected by chamber degradatin, we wuld let all the chambers run until failure. Hwever, since we assume that success prbabilities are affected by chambers' cnditins, we cannt expect the chambers t run until failure. Nevertheless, intuitin suggests that later triggering f maintenance wuld ccur if the RM cst is decreased t be the same as the PM cst, since the difference between scheduled and unscheduled maintenance becmes smaller. T test this intuitin, we decreased the cst f RM and set. Fr this parameter setting, was chsen as the best peratinindependent CBM plicy via cmplete enumeratin. The effect f the RM cst decrease can immediately be seen when the peratin-independent CBM plicy f this case is cmpared t the plicy btained with the riginal parameters ( was increased by ). Cmparisn f the prfits in the 5th and 6th rws f Table 3 shws that ur methd yields higher expected prfit. In additin, it can be bserved by cmparing Tables 4.a and 4.c that mst f the maintenance triggering states call fr later triggering f maintenance, agreeing with ur intuitin that a decrease in RM cst wuld result in later maintenance triggering. 4.3 Case 3: Very Expensive RM When the cst f reactive maintenance is extremely high, it wuld be intuitive t avid reactive maintenance as much as pssible, even if it means sacrificing the prductin, at the expense f frequent PM actins that avert cstly RM. After increasing the RM cst frm 50 t 1000, we btained the best peratin-independent CBM plicy as. In cmpliance with ur intuitin, cmparisn f Tables 4.a and 4.d shws that a high RM cst results in earlier triggering f maintenance. Even s, cmparisn f the expected prfits in rws 7 and 8 f Table 3 shws that peratin-dependent CBM plicy yields higher expected prfits when cmpared t peratin-independent CBM plicy. 5 SUMMARY This paper presents a slutin methdlgy t determine an intelligent CBM plicy fr a manufacturing system cmpsed f multiple manufacturing statins that can execute multiple peratins. T the best f ur knwledge, this is the first maintenance scheduling study in a manufacturing system cnsisting f multiple manufacturing statins that cnsiders bth peratindependent degradatin mdels and a mdel f the prbability f manufacturing success yield, which is bth peratin and degradatin dependent. A simulatin mdel incrprating the degradatin and success prbability infrmatin is develped t btain the expected prfits fr candidate CBM plicies. As a benchmark, peratin-independent CBM plicies are determined via cmplete enumeratin. These plicies are cmpared with the newly prpsed methdlgy that ptimizes the peratin-dependent maintenance plicy via a tabu search algrithm. Fr all the scenaris cnsidered, the peratin-dependent CBM plicies btained by the newly prpsed slutin methdlgy are shwn t utperfrm the benchmark peratinindependent CBM plicies. One shuld nte that in this paper, we assume that the sequence in which different prduct types are prduced is a priri given. Hwever, there exists a ptential benefit in simultaneusly ptimizing the decisins f maintenance and prduct type sequence. Therefre, develpment f an integrated decisin making plicy fr maintenance scheduling and prduct dispatching, and evaluating the effects f different parameters n this plicy will be the tpics f the future research. 6 REFERENCES [1] Wang, H., 00, A Survey f Maintenance Plicies f Deterirating Systems, Eurpean Jurnal Of Operatinal Research, 139:469-489. 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