Maintenance Scheduling and Production Control of Multiple-Machine Manufacturing Systems

Transcription

1 Coputers & Industral Engneerng 48 (2005) do:0.06/.ce Mantenance Schedulng and Producton Control of Multple-Machne Manufacturng Systes A. Gharb a and J.-P. Kenné b a Autoated Producton Engneerng Departent, Unversty of Quebec, École de technologe supéreure, Producton Systes Desgn and Control Laboratory 00, Notre Dae Street West, Montreal (Quebec), Canada, H3C K3. b Mechancal Engneerng Departent, Unversty of Quebec, École de technologe supéreure, 00, Notre Dae Street West, Montreal (Quebec), Canada, H3C K3. Abstract Ths paper deals wth the producton and preventve antenance control proble for a ultple-achne anufacturng syste. The obectve of such a proble s to fnd the producton and preventve antenance rates for the achnes so as to nze the total cost of nventory/backlog, repar and preventve antenance. A two-level herarchcal control odel s presented, and the structure of the control polcy for both dentcal and non-dentcal anufacturng systes s descrbed usng paraeters, referred to here as nput factors. By cobnng analytcal forals wth sulaton-based statstcal tools such as experental desgn and response surface ethodology, an approxaton of the optal control polces and values of nput factors are deterned. The results obtaned extend those avalable n exstng lterature to cover non-dentcal achne anufacturng systes. A nuercal exaple and a senstvty analyss are presented n order to llustrate the robustness of the proposed approach. The extenson of the proposed producton and preventve antenance polces to cover large systes (ultple achnes, ultple products) s dscussed. Key words: Preventve Mantenance, Herarchcal Control, Flexble Manufacturng Systes, Sulaton, Experental Desgn, Response Surface Methodology (RSM).. Introducton The proble of controllng anufacturng systes wth unrelable achnes was forulated as a stochastc control proble by Older and Sur (980). Falure and repar processes were supposed to be descrbed usng hoogeneous Markov processes. The related optal control odel falls under the category of probles studed prevously by Rshel (975). Slar nvestgatons have resulted n the analytcal soluton of the one-achne one-product anufacturng syste control proble obtaned by Akella and Kuar (986). In the case of nonhoogeneous Markov processes nvolvng states and control-dependent transton rates, the control proble becoes ore coplex. In ths sphere, Boukas and Haure (990) consdered the Ths anuscrpt verson s ade avalable under the CC-BY-NC-ND 4.0 lcense

2 fact that the falure probabltes of a achne depend on ts age, and they added the possblty of perforng preventve antenance to the exstng odels. The related age-dependent set of dynac prograng equatons were solved nuercally for a gven anufacturng syste. However, wth the nuercal schee presented by Boukas and Haure (990), t reans dffcult to obtan a general structure for the optal control of a large class of anufacturng systes. A potental way of copng wth such a dffculty s to develop heurstcal ethods based on the reducton of the sze of the consdered control proble. Hence, dfferent approaches have been proposed n the exstng lterature wth a vew to dervng sple near-optal control polces for anufacturng systes. The concept of hedgng pont polcy, ntroduced by Kea and Gershwn (983), s one of the sple ways avalable for fndng suboptal control polces n the producton plannng and antenance schedulng of anufacturng systes. For further detals on ths concept, we refer the reader to the age-dependent hedgng pont concept presented by Boukas et al. (995) and by Kenne and Gharb (999). Because of the coputaton of threshold levels, the dervaton of suboptal polces based on ths concept sees to be dffcult for a large class of anufacturng systes. Another approach s to develop herarchcal control ethods based on the partcular structure of the syste. Ths can be done by usng the sngular perturbaton approach. Such an approach anly nvolves reducng the sze of the control proble accordng to the dscrepancy between the te scales of events nvolved. By replacng fast processes wth ther respectve ean values, one can construct a deternstc ltng proble, whch s coputatonally ore tractable. Detals on ths approach can be found n Kokotovc et al. (986), Lehoczky et al. (99), Seth and Zhang (994) and Soner (993). In ths paper, we wll frst defne the structure of the optal control polces, both for dentcal and for non-dentcal achne anufacturng systes. Based on such structures, we wll then extend the producton and antenance rates control odel presented n Kenne and Boukas (2003) n order to deterne the control polcy n a ore general case ncludng non-dentcal achne anufacturng systes. The resultng structure s descrbed through a set of paraeters we call nput factors. We resort to a cobnaton of analytcal and sulaton-based experental approaches to fnd an approxaton of the optal control polces for producton and preventve antenance by deternng the values of nput factors. In the proposed approach, the paraeterzed near-optal control polcy s used as an nput for the sulaton odel. For each entry consstng of a cobnaton of paraeters, the cost ncurred s obtaned. It s fro ths relatonshp that the best control factor values are deterned and a relatonshp between nput factors and such a cost s gven. The applcaton of such an approach s otvated by the works of Kenne and Gharb (999) and Gharb and Kenne (2000). We refer the reader to these works for a lterature revew on the applcatons of sulaton and statstcal ethods such as experental desgn and RSM n the sphere of anufacturng systes control. The reander of the paper s organzed as follows: In secton 2, the optal control proble s descrbed both for dentcal and non-dentcal achne systes. The proposed control approach s descrbed n secton 3. The logc of the sulaton odel s descrbed n secton 4. In secton 5, the experental desgn approach and response surface ethodology are outlned. A nuercal exaple and a senstvty analyss are also presented n secton 5. Concludng rearks are presented n secton 6. 2

3 2. Proble Stateent In ths secton, we present an explct forulaton of the stochastc optal control proble related to the producton control and preventve antenance schedulng of anufacturng systes wth non-dentcal achnes. Based on the large sze of the optalty condtons obtaned, we next present a sngular perturbaton for of the control odel for ultple dentcal achne anufacturng systes. The structure of the control polcy n such a stuaton s extended to defne that polcy n the case of non-dentcal achnes, for whch optalty condtons are dffcult to solve. The syste under study conssts of achnes producng n dfferent part types. The operatonal ode of achne can be descrbed by a stochastc process t. Such a achne s avalable when t s operatonal t and unavalable when t s under repar t 2 or under preventve antenance t 3. We then have t, 2,3. We can descrbe the anufacturng syste ode by the rando vector t t,, t wth values n. Let a a,, a and, antenance rates respectvely. The process t, be the vectors of achne ages and preventve s odelled by a contnuous te Markov chan defned by achne ages and control dependent transton rates atrx wth The transton rates processes t,,, M. Q a, a, a, 0,,,M, wth M card Q defned as follows: are derved fro the cobnaton of those of the dependent Our approach s used when the rate of change n the achne states s uch hgher than the rate at whch the cost s dscounted. In ths paper, we assue a constant deand rate wthout any loss of generalty. Two te scales are then consdered: the dscountng cost event and the achne state process te scales. When the dfference between the two te scales s very large, the te can be splt. Thus, the transton rates for the syste can be expressed as q, where q and the dscount rates are of the sae agntude. The sngular perturbaton paraeter s used here to express the herarchcal structure of the proposed approach. Wth, an equvalent deternstc proble can be derved fro the forulaton of the ntal stochastc proble. Both control probles (stochastc and deternstc) are descrbed by dynac prograng equatons (DPE) presented later n ths secton. In order to ncrease the syste capacty or the avalablty of the achnes, we assue that the transton rate fro the operatonal ode to the preventve antenance ode for each achne s a control varable called t,,,. The syste behavour s descrbed by a hybrd state coprsng both a dscrete and a contnuous coponent. The dscrete coponent conssts of the dscrete event stochastc process 3

4 t, whle the contnuous coponent conssts of contnuous varables x,, x n a,, x and a a correspondng to the nventory/backlog of products and the cuulatve ages of achnes. These state varables are descrbed by the followng dfferental equatons: x t u t d x 0 () x t ut a0 a a f (2), where x, a and d are gven ntal surplus or backlog, ntal achne ages and deand rates vectors respectvely. Let x~ x,a and ~ u u,. Let u,, u n producton rates. The set of feasble control polces, s gven by: u denote the vector of K and 0 n u t, t, u t 0, u t t ax n p p z, u p u p 0 (3), where p s the processng te of the part type p, and ax s the axu preventve antenance rate of each achne. Let G, x~, u~ be the nstantaneous cost defned as follows: x~,u ~ G, c x c x c, (4), B where c + and c - are costs ncurred per unt produced parts for postve nventory and backlog respectvely, x + = (ax (0, x ),, ax (0, x n )), x - = (ax (-x, 0),, ax (-x n, 0)) and c are gven constants used here for preventve antenance and repar actvty costs. Our obectve s to control the producton rate u and the preventve antenance rate or the control polcy u~ so as to nze the expected dscounted cost gven by: x~,u pt J, e G, x~,u dt x0 x, a0 a, 0 0 (5), subect to constrants gven by equatons () to (4). The value functon of such a proble s: x~, nf u~ J, B x~,u ~ (6) The value functon gven by equaton (6) s locally Lpschtz, convex and s the unque vscosty soluton of the followng HJB equatons (see Kenne (997) and Seth and Zhang (994)), x n. x, x G, x, u q, x u (7), 4

5 where, and v x, x~ u d f u s the gradent of v n ~ x. d p T T n up lo n 3 2 (8), h where T up, Tlo and h are the upper value, the lower value and the dscrete step of the state varable T. For each product, =,, n, the producton rate u, =,,, has three possble ~ values 0, d, U ; ths corresponds to 3 +n ponts for the set of u. For each achne, the ax preventve antenance rate s chosen between two values, 0 and ; ths gves 2 ponts for the achnes. The denson d, as n equaton (8), s very large for a ultple-part, ultpleachne anufacturng syste. Gven that there s no way to solve HJB equatons (7) analytcally, nuercal ethods based on d are usually used to characterse the optal control polcy. A sngular perturbaton approach s used to defne a herarchcal control schee based on ltng probabltes and deternstc optalty condtons, as presented here n the case of dentcal achnes. Such an approach s unusable n the case of non-dentcal achne anufacturng systes. 0 For dentcal achnes, let t, t the nuber of operatonal achnes at te t wth values n 0,,, control the producton rate u(t) and the preventve antenance rate or the control polcy u~ u so as to nze the expected dscounted cost gven by: J x~,u pt, e G, x~,u dt x0 x, a0 a, 0 0 The value functon of the control proble consdered s gven by: be a fnte-state stochastc process correspondng to. Our obectve s to (9), x~ nf J u~, x~,u ~ (0) Such a functon s also locally Lpschtz, convex and s the unque vscosty soluton of the followng HJB equatons:, x~ n u d x~,u ~ q, ~ x, x~ f u a, x~ G, x () u~ 0 The optal control polcy, u s the soluton of HJB equatons descrbed by (). Gven the large sze of () for ultple-achne, ultple-part type anufacturng systes, t s necessary here to proceed to the reducton of the syste sze through the ltng control proble. Such a proble s based on the statonary dstrbuton of the stochastc process, whch s coputed here n ters of the ean values of the achne ages and preventve antenance rates denoted by a M and respectvely. If a and ω, =,, are replaced n the odel by a M and, a constant transton rates atrx a, Q M s obtaned. The fnte state Markov chan assocated 5

6 to such a atrx s then hoogeneous, wth statonary or ltng probabltes gven by:, 2,, vq. 0 and v (2) Wth these probabltes, the convergence property of the ntal stochastc control proble to the ltng proble when 0 s establshed n Soner (993). Based on the herarchcal odel presented n Kenne (997) and Kenne and Boukas (2003) statng that ltng probabltes are obtaned for gven, the overall optzaton proble can be descrbed by the followng deternstc HJB equatons: x~ n 0 kax n uk n x~ k d x~ G x~, a u d x k. M (3) The structure of the stochastc control presented n Boukas and Haure (990) and Kenne (997) provde a achne age-dependent control polcy through the soluton of (3). The constructon of the correspondng stochastc control polcy structure provdes the followng producton and preventve antenance polces: Producton Control Polcy: The structure of the optal producton polcy could be gven by: t t t u ax s x u x~ d s x (4), 0 s x where Z s the threshold value of product and u ax u ax wth u ax descrbng the axu producton rate of product on achne for a ultple-achne, ultple-product anufacturng syste. Preventve Mantenance Polcy: The structure of the optal achne age-dependent preventve antenance polcy depends on the ean age value a M of the achne and a paraeter, and s defned as follows: f a t a ax M x~ (5) 0 otherwse Note that f 0, preventve antenance actons are perfored before the achne age reaches a M ; otherwse they are perfored after a M. The values and sgn of depend anly on the nstantaneous cost paraeters defned n equaton (4) for a gven anufacturng syste. For such a syste, a achne ean age value a M s deterned usng sulaton through off-lne experents. The followng achne age-dependent breakdown probablty dstrbuton s used to descrbe the dynacs of any achne. 6

7 a exp k a t (6), where k,,,, a s defned as the nuber of parts produced snce the last nterventon on the achne (repar or preventve antenance). A achne age ncreasng falure rate (IFR) s gven by (6), wth k. A well-known IFR, such as Webull, can be recovered by choosng sutable values for k. Note that the optal control polcy s well defned by paraeters and, whch we call here nput factors, for a gven a M. are gven constants and the achne age t In the next secton, we present a heurstcal control approach used here to estate the optal values of such factors. The proposed approach s based on a cobnaton of analytcal and sulaton odels, experental desgn and response surface ethodology. 3. Control Approach Results obtaned fro tradtonal ethods of producton and preventve antenance schedulng of ultple-achne anufacturng systes are not generally enough to provde a cofortable level of desred perforance. To prove these ethods, the descrptve capactes of conventonal sulaton odels are cobned wth analytcal odels, experental desgn and response surface ethodology. Ths approach has been successfully used n the cases of sngleachne and ultple-dentcal-achne anufacturng systes (see Kenne and Gharb (999) and Gharb and Kenne (2000)). A block dagra of the resultng control approach s depcted n Fg.. FMS Control Proble Dscrete event Sulaton Model Perforances evaluaton (costs of: nventory, backlogs and antenance) 5 2 Analytcal odel (Herarchcal approach & optalty condtons) Experental desgn (ANOVA & factors effects) 6 3 Nuercal ethods structure of the optal control polcy paraeterzed near optal control polces Response Surface Methodology Regresson analyss Optzng the estated cost functon 7 4 Control Factors Z and δ =, ; =, n Near-optal control polcy u(z * ) and ω(δ * ) 8 Fg.: Proposed control approach 7

8 The structure of the proposed control approach presented n Fgure, conssts of the followng sequental steps:. The Control proble stateent of the anufacturng syste, as n secton 2, conssts of the representaton of the producton plannng and antenance schedulng proble through a stochastc optal control odel based on control theory. Hence, the proble of the optal flow control for the anufacturng syste consdered s descrbed n ths frst step, whch contans a specfcaton of the obectve of the study. That obectve s to fnd the control varables (u, ω) called the producton and preventve antenance rates n order to prove the related output (.e., the ncurred cost). 2. The optalty condtons, descrbed by the HJB equatons, are obtaned fro the proble stateent of the prevous step. It s shown n ths step that the value functon, representng the ncurred cost, s the soluton of the HJB equatons, and the correspondng control polcy (producton and preventve antenance rates) s optal. When the rate of change n the achne states s uch hgher than the rate at whch the cost s dscounted, the te can be splt, and the sngular perturbaton approach used to develop optalty condtons for both stochastc and deternstc control probles. The control polcy of the stochastc proble s constructed fro that one of the correspondng deternstc proble. 3. The nuercal ethods are used n ths step to solve the HJB equatons of the ltng proble, gven that there s no way of solvng the equatons analytcally when t coes to real anufacturng systes (ultple-achne, ultple parts). 4. The control factors Z, =, n for producton plannng and δ, =, for preventve antenance schedulng, descrbe the nuercal control polcy obtaned, extended to the stochastc proble. 5. The sulaton odel uses the near optal control polcy defned n the prevous step as nput for conductng experents n order to evaluate the perforances of the anufacturng syste. Hence, for gven values of the control factors, the cost ncurred s obtaned fro the sulaton odel presented n secton The experental desgn approach defnes how the control factors can be vared n order to deterne the effects of the an factors and ther nteractons (.e., analyss of varance or ANOVA) on the cost through a nal set of sulaton experents. 7. The response surface ethodology s then used to obtan the relatonshp between the ncurred cost and sgnfcant an factors and nteractons gven n the prevous step. The obtaned regresson odel s then optzed n order to deterne best values of factors called here Z * for producton, and δ * for preventve antenance schedulng. 8. The near-optal control polcy (u(z * ), ω(δ * )) s then an proved age-dependent hedgng pont polcy to be appled to the anufacturng syste. The applcaton of the proposed control approach gves the producton and preventve antenance rates descrbed by equatons (4) and (5) respectvely for best values of factors Z * and δ *. 4. Sulaton Model A dscrete event sulaton odel that descrbes the dynacs of the syste ()-(2), s developed usng the Vsual SLAM language (Prtsker and O Relly, 999). Ths odel conssts of several networks, each of whch descrbes a specfc task n the syste (.e., deand generaton, control polcy, states of the achnes, nventory control..., etc.). The dagra of the proposed sulaton odel s shown n Fg. 2 wth the followng notaton block descrptons: 8

9 Fg. 2: Dagra of sulaton odel. The INITIALIZATION block ntalzes the varables (current surplus, producton rates, ncurred cost, etc) 2. The Deand Arrval block perfors the arrval of a deand for product at each d - unt of te. A verfcaton s then perfored on the nventory value of product, and the nventory or the backorder s updated. 3. The CONTROL POLICY segent block s defned n the prevous secton (see equaton (4) for the achne producton rates). The control polcy s defned by the output of the FLAG block. Ths block s used to peranently verfy the varaton n the stock level x (t). If x (t) > Z, then the producton rate s set to a zero value; otherwse the producton rate s set to the deand rate (x (t) = Z ) or to the axu producton rate (x (t) < Z ). 4. The PARTS PRODUCTION block perfors the producton of fnshed goods. 5. The update the nventory block perfors the varaton of the nventory level when a fnshed goods producton or a deand arrval occurs (.e. producton of fnshed goods ncreases nventory f there s no backorder or t satsfes the cuulatve deands, and hence decreases backorders). Off-lne runs of the sulaton odel, for a two-dfferent-achne, one-part type anufacturng syste, usng control polcy descrbed by (4) for Z =20 s llustrated n Fg. 3 for a product stock traectory. 9

10 STOCK TIME Fg. 3: A product stock traectory (Z =20). It s nterestng to note that: () the nventory level ncreases to Z and reans at ths value; () the nventory level decreases durng repar or preventve antenance tes; () the decreasng level of the nventory depends on the repar and the preventve antenance tes, whch are dfferent for each achne. 6. The falure-repar block perfors two functons: t defnes the te-to-falure of each achne, and repars broken ones. Usng the breakdown probablty dstrbuton defned by (6), n Fg. 4, we present the achne age traectores for two achnes (k = 0-4, k 2 = ), obtaned fro off-lne runs of the sulaton odel. For exaple, achne, fro ntal te to the frst up te (where the breakdown occurs), the achne age ncreases fro zero to 65. The achne age s then set to zero durng the repar te. When the achne s repared, t produces parts and ts age ncreases agan a M 20 Produced parts Fgure 4: Machne age traectory. a M2 Mach. Mach Te Fg. 4: Machne age traectores 0

11 7. The preventve antenance block defnes the te at whch we should send each achne out for preventve antenance. The ean value of the ages at whch achne breakdowns occur s the ean age of achne denoted here as (a M ). The preventve antenance on that achne should be done around that age (see equaton (5) for the achne preventve antenance rates). We deterne a M through off-lne sulaton runs. For the exaple llustrated n Fg. 4, these values are: a M = 28 and a M2 = 89. The falure-repar, the preventve antenance and the update the nventory blocks update the ncurred cost block. 8. The updates the ncurred cost block calculates n a real te the cost of nventory, backlogs and correctve and preventve antenance. 5. Experental Desgn and Response Surface Methodology To llustrate the approach presented n ths paper, we consder a two-achne, one-product anufacturng syste. The optal flow control for the anufacturng syste consdered s forulated as n secton 2, wth optalty condtons gven by HJB equatons () and (3) for the two levels of the proposed herarchcal approach. The optal control polcy s approxated by a heurstc control polcy defned n ters of desgn factors, as n equatons (4)-(5) for the achne producton and preventve antenance rates. The obectve of the proposed approach s to fnd the best paraeters of the control varables u(.) and (.), (.e., Z, δ and δ 2 ), n order to prove the related output (.e., the ncurred cost). The sulaton odel descrbes the dynacs of the syste usng the control polcy paraeterzed by the factors Z, δ and δ 2. These factors are consdered as the nput of such a odel, and the correspondng ncurred cost s defned as ts output. Fro the values of the nput factors and the correspondng cost values, the experental desgn approach deternes nput factors and/or ther nteractons that have sgnfcant effects on the output. Sgnfcant factors or nteractons are then consdered as nput of a response surface ethodology, n order to ft the relatonshp between the cost and the nput factors. The optal values of the nput factors, called Z *, δ * and δ 2 * are deterned fro ths estated relaton. The related achne age-dependent odfed hedgng pont polcy s then an proved hedgng pont polcy to be appled to the anufacturng syste. Due to the convexty of the value functon (see secton 2), we selected a 3 3 response surface desgn. The experental desgn s used to study and understand the effects that soe paraeters, naely Z, δ and δ 2, for the anufacturng syste, have on the perforance easure (.e., the cost). 5. Nuercal Exaple The followng are the nuercal values of the constants used prevously: d=2; U ax =.5; 2 U ax =.6; c + =; c - =0; c 2 = 60; c 2 2= 50; c 3 = 00; c 3 2=80; q 2 =0.05; q 2 2 = 0.022; q 2 =0.045; q 2 2 = 0.042; q 3 =0.8; q 2 3 =0.67; k = 0-4 ; k 2 = Note that, n the case of non-dentcal achnes, c 2, and c 3 are used as repar and antenance costs of achne, respectvely. In

12 addton, q 2 s the breakdown rate of achne, and q 2 and q 3 are respectvely the correctve and preventve antenance rates for achne. Based on off-lne sulaton runs, where the nu and the axu values of the factors were observed, the ndependent varable levels were chosen as n Table. Table : Level of ndependent varables Factor Low Level Center Hgh Level Descrpton Z Stock level δ Mantenance for M δ Mantenance for M 2 Three replcatons were conducted for each cobnaton of the factors, and therefore, 8 (3 3 x 3) sulaton runs were ade. To reduce the nuber of replcatons, we used a varance reducton technque called coon rando nubers (Law and Kelton (2000)). We conducted soe prelnary sulaton experents usng 3 replcatons, and notced that the varablty allows the effects to be dstngushed. 5.2 Result Analyss The statstcal analyss of the sulaton data conssts of the ultfactor analyss of varance (ANOVA). Ths s done usng a statstcal software applcaton, such as STATGRAPHICS, to provde the effects of the three ndependent varables (Z, δ and δ 2 ) on the dependant varable (Cost). The ANOVA table correspondng to the generated data s llustrated n Table 2. Fro Table 2, as all the p-values are less than 5%, we conclude that the an factors Z, δ and δ 2, ther quadratc effects, as well as ther nteractons are sgnfcant at the 0.05 level. The R-squared value of fro the ANOVA table, states that 94% of the total varablty s explaned by the odel (Montgoery (200)). Table 2: ANOVA Table Source Su of squares Df Mean Square F-Rato P-Value A:Z ,0000 B: d C:d AA AB BB BC CC Blocks Total error Total (corr.) R-squared = percent 2

13 The resdual analyss was used to verfy the adequacy of the odel. A resdual versus predcted value plot and noral probablty plot were used to test the hoogenety of the varances and the resdual noralty, respectvely. We conclude that the odel s satsfactory, and there s no need for the transforaton of response varables or for addtonal replcatons for the sulaton odel. The second order odel s then gven by: Cost = Z δ δ Z Z * δ Z * δ δ δ * δ δ 2 2 (7) d Z d Fg. 5: Estated response surface Z The near-optal control polcy to be appled to the anufacturng syste consdered s defned by the nu of the cost functon (7) located at Z * =22.99, δ * =2.36 and δ 2 * = as shown n Fg. 5. A cost value of 5.76 s obtaned wth such a control polcy. To crosscheck the valdty of the soluton, Z * =22.99, δ * =2.36 and δ 2 * = were used as nput to the sulaton odel. The cost value obtaned was 5.6, whch falls n the 95% confdence nterval ( X n n 2 S t = [50.39; 5.93] ), obtaned usng n=0 replcatons of n, n 2 the sulaton odel. Z *, δ * and δ * 2 defne the best-odfed hedgng pont polcy to be appled to the anufacturng syste consdered. Wth the aforeentoned optal values of the ndependent factors or nput paraeters, the cost s nzed and the correspondng control polcy s the best approxaton of the optal control one. The followng control polcy s to be appled to the anufacturng syste presented n ths exaple: If nventory level of product s greater than 23, then the producton rate s set to zero value; If nventory level of product s equal to 23, then produce at the deand rate; If nventory level of product s less than 23, then produce at the axu producton rate. If the age of achne s greater than 5.64 (a M - δ * = ), then send achne to preventve antenance f the nventory level of product s equal to 23. If the age of achne 2 s greater than (a M2 - δ * 2 = 89-(-5.59)), then send achne 2 to preventve antenance f the nventory level of product s equal to 23. 3

14 5.3 Senstvty Analyss. A set of nuercal exaples are consdered on the senstvty of the obtaned control polcy wth respect to nventory, backlog, correctve and preventve antenance costs (.e., c +, c -, c 2, c 3 ). The followng varatons, llustrated n Table 3, are explored and copared to a basc case. Decreasng c + : ths ust result n a tendency to ncrease the stock level n order to avod further backlog costs. Increasng c + : ths ust result n a tendency to decrease the stock level n order to avod further nventory costs. Decreasng c - : ths ust result n a tendency to decrease the stock level n order to avod further nventory costs. Increasng c - : ths ust result n a tendency to ncrease the stock level n order to avod further backlog costs. Decreasng c 2 (correctve antenance cost): ths ust result n a tendency to delay the preventve antenance perod (fewer preventve antenance actvtes). Increasng c 2 (correctve antenance cost): ths ust result n a tendency to advance the preventve antenance perod (ore preventve antenance actvtes). Decreasng c 3 (preventve antenance cost): ths ust result n a tendency to advance the preventve antenance perod (ore preventve antenance actvtes). Increasng c 3 (preventve antenance cost): ths ust result n a tendency to delay the preventve antenance perod (fewer preventve antenance actvtes). Table 3: Senstvty analyss table c + c - c 2 c 2 2 c 3 c 3 2 * Z * δ * δ 2 Cost * Reark Basc case Z * ncreases Z * decreases Z * decreases Z * ncreases δ * decreases δ * ncreases δ * 2 decreases δ * 2 ncreases δ * ncreases δ * decreases δ * 2 ncreases δ * 2 decreases Through the above analyss, t clearly appears that the results obtaned ake sense, and that the proposed approach s robust. In the second part of ths secton, we wll dscuss how to control ore coplex anufacturng systes. For an achne, n products anufacturng syste as n ths paper, we obtan an +n factors experental desgn (.e., one factor for each achne and product). For large values of and n, a ore approprate experental desgn ust be explored, snce the coplete 3 +n s very dffcult to pleent. In such a stuaton, a two-step desgn approach s recoended: 4

15 . Use of two-level fractonal factoral desgns (.e. 2 f-p ) as flter n order to elnate nonsgnfcant factors and/or nteractons (Montgoery, 200). 2. Use of experental desgn related to sgnfcant factors or nteractons. The Box-Benhken or Box-Wlson central coposte desgns are coonly used at ths level (Montgoery, 200). However, the Box-Wlson desgn s preferred because we reuse all the results of the experents perfored durng the screenng step. The proposed approach sgnfcantly reduces the nuber of sulaton runs, and should gves rse to near-optal control polces for ore coplex anufacturng systes. 6. Concluson In ths paper, we have extended the concept of hedgng pont polcy to the producton and preventve antenance control proble of a ultple, non-dentcal achne anufacturng syste. The proposed approach was based on the cobnaton of the herarchcal control odel, sulaton experents, experental desgn and RSM. Frst, we nvestgated a near-optal control polcy of a achne age dependent Markov process through the constructon of the stochastc control polcy fro one of the dependant deternstc odels. We then assocated to such a polcy paraeters called ndependent varables. A sulaton odel was developed to descrbe the dynac of the producton syste under the proposed odfed hedgng pont polcy. An experental desgn approach was then used to nvestgate the effects of specfc factors on the cost ncurred durng the producton horzon. The proposed approach cobnes the sulaton ethod wth the statstcal ethod to provde the estaton of the cost functon related to the control proble consdered. A response surface ethodology was used to perfor ths functon n ters of sgnfcant an factors and nteractons gven by the experental desgn approach. Fro the estaton of the cost functon, the best values of control paraeters were easly coputed. 7. References Akella R. and Kuar P. R. (986), Optal Control of Producton Rate n a Falure Prone Manufacturng Syste. IEEE Trans. On Autoatc Control, Vol. AC-3, No. 2, pp Boukas, E. K. and Haure A. (990) Manufacturng Flow Control and Preventve Mantenance: A Stochastc Control Approach. IEEE Transactons on Autoatc Control, Vol 33 No 9, Boukas E.K., Kenne J.P. and Zhu Q. (995) Age-dependent Hedgng Pont Polces n Manufacturng Systes. Aercan Autoatc Control Councl, Seattle, Washngton, June Gharb A. and Kenne J. P. (2000) Producton and Preventve Mantenance Rates Control for a Manufacturng Syste: An Experental Desgn Approach. Internatonal Journal of Producton Econocs, Vol 65 No 3, pp Kenne J.P. (997), Planfcaton de la Producton et de la Mantenance des Systèes de Producton: Approche Hérarchsée. Ph.D thess, Ecole Polytechnque de Montréal, Unversté de Montréal. Kenne, J. and P. Boukas, E. K., (2003) Herarchcal Control of Producton and Mantenance Rates n Manufacturng Systes. Journal of Qualty n Mantenance Engneerng, Vol. 9, No.. 5

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