Optimal Safety Stocks and Preventive Maintenance Periods in Unreliable Manufacturing Systems.

Transcription

1 Int. J. Production Economics 07 (007) doi:0.06/j.ijpe Optimal Saety Stocks and Preventive Maintenance Periods in Unreliable Manuacturing Systems. A. Gharbi*, J.-P. Kenné** and M. Beit** Université du Québec, École de technologie supérieure, 00, Notre Dame Street West, Montreal (Quebec), Canada, H3C K3 * Automated Production Engineering Department, Production Systems Design and Control Laboratory ** Mechanical Engineering Department, Production echnologies Integration Laboratory Abstract We consider a manuacturing system with preventive maintenance that produces a single part type. An inventory is maintained according to a machine age dependent hedging point policy. We conjecture that, or such a system, the ailure requencies can be reduced through preventive maintenance resulting in possible increase in system perormance. raditional preventive maintenance policies, such as age replacement, periodic replacement, are usually studied without inished goods inventories. In the cases where the inished goods inventories are considered, restrictive assumptions are used, such as not allowing breakdown during the stock build up period and during backlog situations due to the complexity o the mathematical model. In order to solve this problem, we develop a more realistic mathematical model o the system, and derive expressions o the overall incurred cost used as the basis or optimal determination o the jointly production and preventive maintenance policies (i.e. production rates and preventive maintenance requency, depending on inventory levels o the produced parts). Such a cost consists o inventory, backlog, corrective and preventive maintenance costs. he work reported here has a signiicant practical application (no restriction on ailures occurrence and backlog situations) in the context o production planning o manuacturing systems. Numerical examples are included to illustrate the importance and the eectiveness o the proposed methodology. Keywords: Preventive maintenance, Inventory, Production, Reliability, Manuacturing systems.. Introduction A ailure prone part production inventory system is considered in this paper. he system produces a single product type to satisy an exogenous demand process. o hedge against the uncertainties in the both production and the demand processes, provision or inished inventory buer between the system and the demands is kept. Demands that arrive when the inventory buer is empty are back ordered and are, thereore not lost as in available models (Das and Sarkar (999), Rezg et al. (004) and reerences therein). It has been largely shown in the literature that implementing preventive maintenance strategies or several randomly ailing production units can be an eective way to extend their lives and reduce operating costs (Barlow and Proschan (965), Savsar (997), Chelbi and Ait-Kadi 007. his manuscript version is made available under the CC-BY-NC-ND 4.0 license

2 (004) and reerences therein). he reader is reerred to Savsar (006) or details on other maintenance policies and their eects on the productivity and availability o a manuacturing system. A overview o relevant literature reveals that signiicant contributions, in the perormances optimisation o manuacturing systems, have been proposed based on (i) preventive maintenance, (ii) production control and (iii) jointly production and maintenance optimisation models. hose models are considered individually or simultaneously and are restricted to simpliied assumptions that sometimes provide less realistic preventive maintenance or production policies. In the last ew decades, maintenance planning has been an active area o research ocused on the reliability theory as presented recently by Chelbi and Ait-Kadi (004). Hence, a replacement policy which ensures maximum utilisation o the useul lie o a component beore its preventive replacement is an obvious option or large and costly components. Age replacement policy (ARP) is one such option over block replacement policy (BRP) or group replacement policy (GRP). For details on such policies, the reader is reerred to Barlow and Proschan (965), Ajodhya and Damodar (004) and reerences therein. One o the basic and simple replacement policies is the age replacement policy, where the unit is replaced upon a ailure or a preix age, whichever occurs irst (see Hong and Jionghua (003), Ajodhya and Damodar (004)). Given that ARP is based on age dependent preventive maintenance periods instead o ixed periods, as in BRP, it remains more realistic and hence attracts many researchers. We reer the reader to extended versions o the age replacement policies and their implantation presented in Ajodhya and Damodar (004). he related policies are no realistic in the context o manuacturing systems given that requent machine breakdowns inevitably create bottlenecks or the process. Hence, preventive maintenance (to reduce likelihood o machine breakdowns) combined to the control o inished goods inventory is a potential way o reducing the overall incurred cost. he aorementioned models are classiied herein as static models given that the obtained policies are based on the mean values o the involved stochastic processes. In addition, the dynamics o the inished goods inventory is not considered in those models or a large class o manuacturing systems. Conversely, manuacturing systems with unreliable machines have been modelled using the so called stochastic optimal control theory in which ailures

3 and repairs processes were supposed to be described by homogeneous Markovian processes. he related optimal control model ails in the category o problems presented in the pioneering work o Rishel (975). Investigation in the same direction provided the analytical solution o the one-machine, one-product manuacturing system obtained by Akella and Kumar (986). Preventive maintenance planning problems are combined to the production control to increase the availability o the production system and hence to reduce the overall incurred cost (see Boukas and Haurie (990)). A preventive maintenance model or a production inventory system is developed in Das and Sarkar (999) using inormation on the systems conditions (such as inished product demand, inventory position, costs o repair and maintenance, etc.) and a continuous probability distribution characterizing the machine ailure process. An analytical model o BRP and saety stock strategy is ormulated by Ki-Ling and Warren (997), using also restrictive assumptions such as: the time to accomplish build-up and depletion o saety stock is small relative to the mean time to ailures (MF). he model presented in Salameh and Ghattas (00) combines ARP and saety stock to show that one need to built an inventory just beore the preventive maintenance. It is assumed in Salameh and Ghattas (00) that extra capacity is maintained to buer against uncertainties o the production processes and that there is no possible breakdown o the machine beore the preventive maintenance date. Without the assumption made by Salameh and Ghattas (00) on the machine dynamics, the stochastic optimal control theory is used in Boukas et al. (995), Gharbi and Kenne (000, 005), Kenne and Gharbi (999) and in Kenne and Boukas (003) to deine an machine age dependent production and preventive maintenance policies. Such policies are based on non homogeneous Markov models, and hence are restricted to exponential distributions describing operational and down times o the involved machines. he purpose o this paper is to investigate the joint implementation o preventive maintenance and saety stocks in a more realistic manuacturing environment using a stochastic model not restricted to markovian processes as mentioned previously. he main results presented in this paper extends the works o Ki-Ling and Warren (997), Salameh and Ghattas (00), Rezg et al. (005) and reerences therein. Hence, we investigate herein 3

4 joint implementation o preventive maintenance and saety stocks in unreliable production environment with back order (i.e., unmet demands during the repair period are not lost) and with possible ailure at any age o the machine (or example during built up o the saety stock). Available models, based on reliability theory, are not able to remove the aorementioned restrictions due to the complexity o the mathematical model. Removing theses restrictions, as in this paper, involves additions concepts needed or the determination o the incurred cost expression used, here as criteria index. he reminder o this article is organized as ollows. In section, we deine the assumptions used in the model. he probability model and the control policy are outlined in section 3. Using the properties o the probability model, we develop expressions or the measures o the system perormance and present the optimality conditions in section 4. A numerical example is presented in section 5. Sensitivity and results analysis are provided in section 6 and concluding remarks are given in section 7.. Notations and model assumptions hrough the paper, the ollowing notation will be used: C C C C u() inventory holding cost per unit tine penalty cost or each unit o unmet demand cost o corrective maintenance cost o preventive maintenance production rate o the system u mx maximal production rate d b () t Ft () rt () demand rate random variable describing the time to machine breakdown probability density unction o b cumulative distribution unction o b ailure rate o the machine 4

5 pm random variable describing time to perorm preventive maintenance qt () Qt () probability density unction o pm cumulative distribution unction o pm cm random variable describing time to perorm corrective maintenance gt () Gt () S probability density unction o cm cumulative distribution unction o cm stock threshold level or saety stock level scheduled time to preventive maintenance We consider the production o a single item on a production process with a capacity umx as to satisy a constant demand rate d items units per unit o time with d u mx so. ypical examples o such a production process include stamping and press punching in the automotive industry and die casting (see Ki-Ling and Warren (997)). he capacity umx represents the maximum production output rate, such as one where all three shits o operation are utilized. With such a system, we irst identiy the stochastic process that account or all random events, namely production, ailure, repair and preventive maintenance. We obtain a probability structure on those stochastic processes which are then exploited to obtain system perormance measure. Failure and repair events are due to machine breakdown, which occurs in a random manner and constitute a major source o uncertainties in the production process. Whenever a breakdown occurs, corrective maintenance is perormed, during a random amount o time, to restore the machine to its initial condition (i.e., the machine is assumed to be new and its age is set to zero). During the repair periods, one o the ollowing two situations occurs: - demands or items are met only by saety stock; - all unmet demands are backlogged. Hence, the considered machine is capable o catching up with the unmet demand without interrupting the normal production process as soon as production resumes. For this situation, there is a possibility o having another breakdown during catch-up period. In 5

6 order to compute the incurred cost using mean values o the involved stochastic processes, most previous models, in relevant literature, assumed that such a possibility is negligible. he likelihood o machine breakdown is reduced when preventive maintenance is scheduled and combined to production planning. Each time, immediately ater the maintenance operation is perormed, the machine is restored at its initial working condition. In addition, we based the model under consideration on the ollowing assumptions: (A-) he production unit is subjected to stochastic breakdowns and repairs (A-) I a machine ailure occurs during a production phase, corrective repair is started immediately and ater repair, the machine is restored back to the same initial working condition. In addition, a preventive maintenance action (as a corrective one) renews the production system (i.e., the age o the machine is set to zero). (A-3) he mean value o the time requirement or a preventive maintenance operation is short when compared with the mean time to machine breakdown. (A-4) he demand rate o the product is a known constant whereas the production rate (which is greater than the demand rate) depends on the decision variables S and. (A-5) Shortages may occur due to longer repair time. In that case, all the unsatisied demands are backlogged (A-6) A suicient capacity is present to allow accumulation o saety stock in the beginning o each machine lie cycle; but the time to accomplish build-up and depletion o saety stocks is not necessarily small relative to mean time to ailure (MF) as in available models (i.e., a breakdown could arises during this time). (A-7) here is no restriction on any o the operational, repair and preventive maintenance time distribution. (A-8) Breakdowns o the machine don t aect the quality o products. 3 Analytical model and control policy Let the system state be denoted by (, x) where indicates the system status, and x indicates the surplus level. Note that the surplus is positive when it represents inventory and negative when it represents a shortage situation. Let (, F) be a measurable space and 6

7 F : t 0 an increasing class o sigma-ields representing the history o the (, x) process. t A sample value corresponds to an x - trajectory which is continuous, and a sequence o -values without accumulation points. Given also the discontinuity on the machine age trajectory (set to zero ater each corrective or preventive maintenance operation), the control problem in the case o joint determination o saety stock levels and machine age preventive maintenance, as in this paper, is o the type o piecewise deterministic random process. Hence, the production system under study can be considered as a deterministic system as long as no machine breakdowns or stoppage occur. he set o admissible control policies is a amily o Ft -adapted processes with values in mx ( ) u( ) : 0 u( ) u. In addition, a suicient capacity is present to allow accumulation o saety stock in the beginning o each machine lie cycle. he time to accomplish build-up and depletion o saety stocks is not necessarily small relative to MF. Hence, there is a possibility o having another breakdown during catch-up period. he preventive maintenance policy is not implicitly bounded (i.e., not included in ( ) ) given hat there is no ixed maximal preventive maintenance rate in the proposed model. he maintenance epoch is one o the control variable that we are looking or. Let u() denotes the production rate that may vary with time and with the state/capacity o the machine. hereore, u( ) 0 and is subject to the random process ( t), t 0. With the total surplus x() and the demand rate d ]0, ), the continuous part o the system dynamics is described by the ollowing dierential equation: dx() t u( t) d x(0) x () dt where x is the initial surplus level. Recall that all unmet demands are backlogged and a penalty cost is incurred per item on a per unit basis. he discrete part o the system dynamics is described by the system status ( t),,3 with ( t) i the machine is operational, ( t) i the machine is under repair and ( t) 3 i the machine is under preventive maintenance. he machine state 7

8 moves rom the dierent modes o the process () t according to random variables b, cm and pm deined as time to machine breakdown, corrective maintenance time and preventive maintenance time. At mode and or a given age, the production rate is given by an extended version o the so called hedging point policy (HPP) deined by a threshold level S. Such a policy is given by: u i mx x S u( x) d i x S 0 otherwise In such a policy, an optimal inventory level S is maintained during time o excess capacity availability to hedge against uture capacity shortage brought by machine ailures. I the current inventory level exceeds the optimal inventory level S, one should not produce at all; i less, one should produce at the maximum rate u mx exactly produce enough to meet demand d. () ; while i exactly equal, one should he ARP is combined to the hedging point policy, deined by equation (), and the result leads to the proposed control policy. Recall that the ARP consists o a preventive maintenance which is perormed at a scheduled time that depends on the age o the machine. he proposed policy hence depends on two parameters, namely S and, and is completely deined or given values o those parameters. While producing, the machine and surplus dynamic both involve dierent scenarios used in the next section to develop optimality conditions rom which values o the proposed control policy parameters are determined. 4. Optimality conditions he expected cost per unit o time, used herein as optimisation perormance criteria, includes the surplus costs, the preventive and corrective maintenance costs. Note that surplus costs consist o inventory cost or positive surplus and backlog cost or negative surplus. 4.. Failure and surplus costs 8

9 he inventory and backlog costs are determined through the investigation o two dierent scenarios based on the act that the inventory level reaches the optimal inventory level or not. Scenario no. : here is a breakdown during the building o a saety stock S at the rate umx d and the involved repair process ended with inventory or at a positive surplus level. he inished goods inventory in such a situation is illustrated in igure. he mean time to repair the machine without backlog is given by: where process. a a tmw t g t dt r t (3) 0 a umx d t, a is the ailure age and a r is the ending age related to the repair d stock S U mx d A d U mx d a 0 r a age Figure : Surplus sample path or a ailure during the phase o saety building with inventory at the end o the repair process. Using the dashed areas illustrated on igure, the inventory beore and ater the ailure is represented by Sur INV : u d tmw (4) mx Sur A a a u d tmw d ( a ) da INV mx 9

10 where 0 and U S d mx. he repair process ended with inventory i the involved repair time is less than a ( u d) / d. mx Scenario no. : here is a breakdown during the building o a saety stock S at the rate umx d and the involved repair process ended with backlog or at a negative surplus level. he surplus in such a situation is illustrated in igure. stock S U mx d d 0 a a o a r a age A A3 U mx d Figure : Surplus sample path or a ailure during the phase o saety building with backlog at the end o the repair process. In this scenario, the ailure occurs at the age a. he repair time exceeds a u d d ( max ) / and hence involves integration rom a to as in equation (5). he backlog is represented by the surace Sur : Bkg Sur A t a g t dt a da u d d d Bkg ( ) a d d umx d mx umx d (5) he cost or a ailure during the phase o saety building is given by equation (6) which is obtained by grouping equations (4) and (5) rom scenarios and (i.e., 0

11 C t C Sur C Sur ). Note that obtained expression o the cost is multiplied os INV Bkg by ( a ) da which is the probability to have a ailure at age a. Cost umx d a C s tmw umx d Umxd a tmw d d 0 umx d d d u mx d a d d umx d C t a g t dt a da (6) Scenario no. 3: here is a ailure at the saturation phase, in which the stock level is kept at level S and the production rate is reduced to d, and the involved repair process ended with inventory or at a positive surplus level. he inished goods inventory in such a situation is illustrated in igure 3. stock S U mx d A3 d U mx d 0 ts a ar age Figure 3: Surplus sample path or a ailure during the saturation phase with inventory at the end o the repair process he ailure occurs at a machine age located between t S /( u d) and the scheduled preventive maintenance time. Note that t s is the age at which the saety stock levels is reached. he mean operational time o the machine is described by m ( ) given by: 0 s m t t dt R (7) mx

12 he mean time to repair o the machine without backlog is described by tmw and given by: s d 0 tmw ar a t g t dt (8) he inventory surace related scenario 3 (i.e., dash area A 3 illustrated in igure 3), is given by: tmw Sur 3 3 ( ) INV A S m S tmw d (9) Scenario no. 4: here is ailure at the saturation phase and the involved repair process ended with backlog or at a negative surplus level. he surplus in such a situation is illustrated in igure 4. stock S U mx d 0 ts a ao a r A4 a age Figure 4: Surplus sample path or a ailure during the saturation phase with backlog at the end o the repair process. he mean time to repair o the machine with backlog (i.e., repair time greater than S/ d) is described by tmw 3 and given by: S tmw a a t g tdt d (0) 3 r s d he backlog surace related to scenario 4 (i.e., dash area A 4 illustrated in igure 4), is given by: 4 d d Sur Bkg A4 ( tmw3 ) umx d ()

13 Given that the ailure occurs ater t S /( u d) and beore, equation () is rewritten s mx using Fx ( ) which is the cumulative distribution unction o the time to machine breakdown evaluated at a. One obtain the ollowing expression: 4 S S d d Sur Bkg F F s t g tdt umx d d d umx d () he cost or a ailure during the saturation phase is given by equation (3) which is obtained by grouping equations (9) and () rom scenarios 3 and 4 (i.e., C t C Sur C Sur ). 3 4 os 34 INV Bkg Cost C S m S tmw d tmw 34 S S d d C F F s t g tdt umx d d d umx d (3) 4.. Preventive maintenance and surplus costs he proposed model includes preventive actions, described by the distribution unction () p. wo dierent scenarios are considered using the surplus sign (positive or inventory and negative or backlog) at the end o the preventive maintenance action. Scenario no. 5: he preventive maintenance action starts at and ends with inventory or at a positive surplus level as illustrated in igure 5. stock S U mx d A5 d 0 t s a m age Figure 5: Surplus sample path or a preventive maintenance at with inventory at the end o the preventive maintenance process 3

14 he mean time to preventive maintenance without backlog (i.e., maintenance time less than S/ d) is described by tpm and given by: s d am tpm t qtdt 0 (4) where am is the ended age o the preventive maintenance process. he inventory surace related scenario 5 (i.e., dash area A 5 illustrated in igure 5), is given by: 5 ts tpm Sur INV A5 S( ) S tpmd (5) Scenario no. 6: he preventive maintenance action starts at and ends with backlog or at a negative surplus level as illustrated in igure 6. stock S U mx d d 0 a o a m A6 age Figure 6: Surplus sample path or a preventive maintenance at with backlog at the end o the preventive maintenance process he mean time to preventive maintenance with backlog (i.e., maintenance time greater than S/ d) is described by tpm and given by: S tpm am ( t ) qtdt (6) S/ d d he backlog surace related scenario 6 (i.e., dash area A 6 illustrated in igure 6), is given by: d d pm 6 Sur Bkg A6 t u mx d (7) 4

15 he cost due to preventive maintenance at is obtained by grouping equations (5) and (7) rom scenarios 5 and 6 (i.e., C t C Sur C Sur ). he obtained sum is 5 6 os 56 INV Bkg multiplied by R ( ) which is the probability to go up to preventive maintenance. Cost ts tpm C S( ) S tpm d 56 R S d d C s t qtdt d d umx d (8) 4.3. Overall cost and optimality conditions he production cost includes corrective and preventive maintenance costs obtained by multiplying the involved penalty by the occurrence probability. he cost generated by the preventive maintenance action is obtained by multiplying the preventive maintenance cost (i.e., C ) by the probability o its occurrence (i.e., R which is the reliability unction o the machine evaluated at ). he cost generated by the corrective maintenance action is also obtained by multiplying the corrective maintenance cost (i.e., C ) by the probability o its occurrence (i.e., F which is the ailure cumulative unction o the machine evaluated at ). he maintenance cost (corrective and preventive) is then given by: CM R C F C (9) According to assumption (A-) and equation (7), the duration o a production cycle is approximated given by: where Cycle _ ime m( ) R MP F MR (0) MP and MR are mean time to preventive maintenance and mean time to repair. he overall expected cost is obtained by summing scenarios costs (equations (6), (3) and (8)) and maintenance costs given by equation (9). By dividing the obtained sum by the cycle time given by equation (0), one obtains the overall expected cost per unit time L(), given by the ollowing equation: Cost Cost34 Cost56 CM L( S, ) m( ) R MP F MR () 5

16 he optimality conditions and hence the age dependent optimal value o the saety stock (threshold level) and preventive maintenance epoch are given by the ollowing equations: L( S, ) S L( S, ) Due to the complexity o previous expressions (see equation () and dependent equations), proving the convexity o L( S, ) and obtaining the analytical optimality conditions rom equations () and (3) become more complex. Hence, instead o solving () and (3) to obtain optimal values o the involved parameters, a numerical procedure, using a simple enumeration is presented in the next section. he best easible solution * * ( S, ) is given urther rom the application o the proposed numerical procedure. S 0 0 () (3) 5. Numerical procedure and example he ollowing iterative numerical procedure has been developed to ind the optimal control policy, characterized by parameters S * and *, and the optimal overall incurred cost. u, d, C, C, C, C, ( t), ( t), ( t), S, S,,, MR, MP,, S Input: mx r p min max min max inc inc Step : Set S : Smin and : min - Step 3: Compute tmw, ( i,,3), and t pm, tpm using equations (3), (8), (0), (4) and (6) i Step 4: Compute Cost ( ) S using equation (6) Step 5: Compute m( ), Cost34 ( S, ), Cost56 ( S, ) and CM ( ) using equations (7), (3), (8) and (9) Step 6: Compute the total cost L( S, ) using equation () Step 7: I Step 8: I max then set : inc max and go to Step 5; else go to Step 8 S S then set S : S Sinc and go to Step 3; else ind the solution Output: that minimise L( S, ) * * S and = optimal threshold level and time or preventive maintenance 6

17 * * L( S, ) = optimal cost Stop he previous numerical scheme proceeds as ollows: a) read input data b) consider computational grid on and S or given lower and upper bounds min max (, ) and min max (, ) S S respectively (see step ). c) Compute the overall cost (see steps 3 to 6) d) or each easible schedule preventive maintenance time (i.e., ), consider a discrete time interval inc and solve the min max optimality condition at time t to obtain the optimal cost and the associated threshold level (see step 7) e) or each easible schedule threshold level S (i.e., min max S S S ), consider a discrete stock interval Sinc and solve the optimality condition at time t to obtain the optimal cost and the associated preventive maintenance time (see step 7) ) return the lowest cost and the associated S and called hereinater optimal threshold and preventive maintenance time (i.e., * * L( S, ), * * S and ). We consider a airly general example problem as a vehicle or providing urther details on the solution o the optimisation control problem under study. Also presented are numerical results that provide urther insight to the problem. For an illustrative purpose, assume umx item per unit o time and the production process is run to satisy a constant demand rate d 0.65 item per unit o time. In addition, the ollowing parameters are adopted or the basic case (dierent others cases are considered later during a sensitivity analysis): + - C $5000, C 3000, C 5, C 50. he time to breakdown b is Weibull with and =00 (i.e., 88.6 ). he time to repair and the time to preventive maintenance are lognormal with =0, and =5, 0.5 respectively. cm cm pm pm 7

18 Using the proposed iterative numerical procedure and the previous data, we obtain an overall cost unction represented by its contour plot in igure 7. he optimal cost or this example S is.7 and 67. * * L * ( ) $87 and the corresponding control parameters are: Maintenance period Figure 7: Contour plot o the overall cost unction and Optimal control policy parameters or the illustrative example he next section presents the robustness o the developed model through a sensitivity analysis and illustrates the useulness o the approach proposed. 6. Sensitivity analysis Four classes o studies are considered through the variations o corrective maintenance, preventive maintenance, inventory and backlog, costs. For those classes, we illustrate the sensitivity analysis through igures 8 to. It is interesting to note the ollowing rom igure 8, obtained with the variation o the corrective maintenance cost C rom $3000 to $80000 with C $3000, C $5 and - C $50 : 8

19 Stock Preventive maintenance period - he scheduled production time or preventive maintenance * decreases with the increasing o the corrective maintenance cost due to the act that the saety stock decreases; and hence increase the possibility to have a backlog situation (see igure 8(a)). - he optimal threshold level decreases with the increasing o the corrective maintenance cost and converges to an asymptotic value or large values o such a cost as the scheduled preventive maintenance time (see igures 8(a) and 8(b)). It is clear rom igure 8 that the corrective maintenance cost has a signiicant inluence on the optimal threshold level and the scheduled preventive maintenance time and hence to the overall incurred cost Corrective maintenance cost x (a) Corrective maintenance cost x 0 4 (b) Figure 8: Optimal production and preventive maintenance policies or dierent corrective maintenance costs. 9

20 Preventive maintenance period he asymptotic behaviour observed both in igures 8(a) and 8(b) states that or large value o the corrective maintenance cost, the preventive maintenance period attempts a minimal value that results to a small value o the stock level. his is due to the act that a minimal value o preventive maintenance period corresponds to more requent preventive maintenance actions that avoid ailures or which excessive costs are considered. From igure 9, obtained with the variation o the preventive maintenance cost C rom $00 to $50000 with C $5000, C $5 and - C $50, we note the ollowing: - he scheduled production time or preventive maintenance * increases with the increasing o the preventive maintenance cost given that one need to reduce the requency o preventive maintenance due to their excessive cost (i.e., or large values o C ). Such a structure is illustrated in igure 9(a). - he optimal threshold level increases with the increasing o the preventive maintenance cost and converges to an asymptotic value or large values o such a cost as the scheduled preventive maintenance time (see igure 9(b)). he lower requency o preventive maintenance at large values o C increases the breakdown requency o the machine. o hedge against possible disruption o the inventory due to ailures, large values o threshold levels are recommended as shown by the obtained results (see igure 9(b)) Preventive maintenance cost x 0 4 (a) 0

21 Stock Preventive maintenance cost x 0 4 Figure 9: Optimal production and preventive maintenance policies or dierent preventive maintenance costs It is interesting to note an asymptotic behaviour o both preventive maintenance period and stock level rom igure 9. Such a behaviour, is due to the interaction between the preventive maintenance period and the stock level which obviously depends on the MR and MF o the considered distributions. (b) From igure 0, obtained with the variation o the inventory cost C rom $ to $5 with C $5000, C $3000 and - C $50, we note the ollowing: - he scheduled production time or preventive maintenance * decreases with the increasing o the inventory cost due to the act that more requent preventive maintenance is needed to avoid excessive inventory levels (i.e., not recommended or large values o C ). Note that large values o the inventory cost (see igure 0(a)) * converges also to an asymptotic value or - he optimal threshold level decreases with the increasing o the inventory cost and converges to zero or large values o such a cost as shown in igure 0(b).

22 Stock Preventive maintenance period Inventory cost (a) Inventory cost Figure 0: Optimal production and preventive maintenance policies or dierent inventory costs (b) he higher requency o preventive maintenance at large values o C decreases the breakdown requency o the machine. Hence there is no need to keep a signiicant saety stock level or large values o C (i.e., * S 0 as C ). From igure, obtained with the variation o the inventory cost C rom $0 to $400 with C $5000, C $3000 and C $5, we note the ollowing: - he scheduled production time or preventive maintenance * decreases with the increasing o the backlog cost to increase the availability o the production system through requent preventive maintenance at large values o C (see igure (a)). - he optimal threshold level increases with the increasing o the backlog cost and converges to an asymptotic value or large values o such a cost as the scheduled preventive maintenance time (see igure (b)).

23 Stock Preventive maintenance period Backog cost (a) Backog cost Figure : Optimal production and preventive maintenance policies or dierent backlog costs (b) As previously, the higher requency o preventive maintenance at large values o C decreases the breakdown requency o the machine and the relatively small value o the inventory cost ensure to the system a comortable inventory level to hedge against potential ailures that could generate backlog and excessive cost. he results presented in this paper indicate that, as expected, the optimal production policy or the considered manuacturing system is characterized by two parameters namely optimal threshold level * S and scheduled preventive maintenance period *. he control policy () is completely deined by values S and * * 3

24 he trends o the curves shown in igures 8 to conirm the robustness o the proposed approach through a sensitivity analysis. his is perormed by threshold levels and scheduled preventive maintenance periods versus an overall incurred cost including inventory, backlog, corrective and preventive costs. he asymptotic behaviour, well illustrated in igures 8 to, shows clearly that obtained results make sense and that the proposed approach is robust. For the system considered previously and related to a one-machine, one-product manuacturing system, the production and preventive maintenance policies are completely known or given parameters S and. For a more complex manuacturing system consisting o m machines producing n dierent part types, the production and preventive maintenance policies depend on the parameters,, m and S,, S n. As a result, m n parameters or actors could be used to deine the control policy in the context o a multiple parts, multiple products manuacturing system. he experimental design approach, combined to simulation and analytical models could be used to determine the eects o considered actors on the incurred costs and to determine their optimal values. Details on experimental design and simulation modelling could be ind in Gharbi and Kenne (000). 7. Conclusion A production inventory and preventive maintenance system with general characteristics and realistic assumptions has been considered here. he primary objective o the study was to determine when to perorm the preventive maintenance, i any, on the system and the level o the saety stock so as to improve the system perormance (i.e., the overall incurred cost). he mathematical model o the system provided an useul tool or deriving the expressions or the system perormance measure. It was demonstrated, through a numerical example problem, how the cost based measure can be used as a basis o determining optimal level o the saety stock and the scheduled preventive maintenance period. As a result we were able to use a numerical search method to locate the optimum point (i.e., optimum parameters or production and preventive maintenance). he randomness involved in various operational aspects o the system makes it airly diicult to analyse. 4

25 Furthermore, our assumptions o general probability distributions or all o the associated random variables (except the time between demand arrived, assumed constant) make the analysis o the system more involved. In the absence o closed orm or the incurred cost, we used numerical methods to evaluate and hence to determine optimal values o the control parameters. Reerences AJODHYA N. D. and DAMODAR A., 004, Age replacement o components during IFR Delay ime, IEEE transactions on reliability, 53(3), AKELLA, R. and KUMAR, P. R., 986, Optimal control o production rate in a ailureprone manuacturing system. IEEE ransactions on Automatic Control, AC-3, 6-6. BARLOW, R. E. and PROSCHAN, F., 965, Mathematical heory o Reliability, Wiley, New York. BIELECKI,. and KUMAR, P. R., 988, Optimality o Zero-Inventory Policies or Unreliable Manuacturing Systems, Operations Research, 36, BOUKAS, E. K. and HAURIE A., 990, Manuacturing Flow Control and Preventive Maintenance : A Stochastic Control Approach, IEEE rans. on Automatic Control, Vol. 33, No. 9, pp BOUKAS, E. K., KENNE, J.P. and ZHU, Q., 995, Age Dependent Hedging Point Policies in Manuacturing Systems, American Automatic Control Council, Seatle, Washington, june -3. CHELBI A., and AI-KADI D., 004, Analysis o a production/inventory system with randomly ailing production unit submitted to regular preventive maintenance, European Journal o Operational Research, 56, DAS,.. and SARKAR, S., 999, Optimal Preventive Maintenance in a Production Inventory System, IIE transactions on quality and reliability engineering, vol. 3, pp GHARBI, A. and KENNÉ, J. P., 000, Production and Preventive Maintenance Rates Control or a Manuacturing System: An Experimental Design Approach, Intenational Journal o Production Economics, Vol. 65 No. 3,

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