Adaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates

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1 Int J of Mthemtic Sciences nd Appictions, Vo, No 3, Septembe Copyight Mind Rede Pubictions wwwjounshubcom Adptive Conto of Poduction nd Mintennce System with Unknown Deteiotion nd Obsoescence Rtes Fwzy A Bukhi Deptment of Sttistics nd Opetions Resech, Coege of Science, King Sud Univesity, POBox 4, Riydh 4, Sudi Abi fbukhi@ksuedus Abstct This ppe des with continuous-time mode of poduction nd mintennce system whose pefomnce deteiotes ove time The pobem of dptive conto of poduction-mintennce system with unknown deteiotion te is pesented Using feedbck the schedued poduction te, peventive mintennce te, nd updting ue of deteiotion nd obsoescence tes wi be deived Keywods: Adptive conto, Lipunov technique, Obsoescence te, Poduction inventoy system, Peventive mintennce eve INTRODUCTION Appictions of optim conto theoy to mngement science, in gene, nd to poduction pnning, in pticu, e poving to be quite fuitfu; see [] Ntuy, with the optim conto theoy, optim conto techniques cme to be ppied to poduction pnning pobems The poduction inventoy pobem fo mnufctuing systems, which is subject to uncetinties such s demnd fuctutions, mchine fiues, nd othes, hs ttcted the ttention of numeous eseches Duing the pst yes, numbe of methods hve been epoted fo detemining economic quntities fo diffeent poducts on singe o mutipe mchines []-[4] A peventive mintennce mode fo poduction inventoy system is deveoped in [] using infomtion on the systems conditions (such s finished poduct demnd, inventoy position, costs of epi nd mintennce, etc) nd continuous pobbiity distibution chcteizing the mchine fiue pocess A mnufctuing system with peventive mintennce tht poduces singe pt type is consideed in by [6] An inventoy is mintined ccoding to mchine-ge-dependnt hedging point poicy In this ppe they hve conjectue tht, fo such system, the fiue fequencies cn be educed though peventive mintennce esuting in possibe incese in system pefomnce Tdition peventive mintennce poicies, such s ge epcement, peiodic epcement, e usuy studied without finished goods inventoies In the cses whee the finished goods inventoies hve been consideed, estictive ssumptions hve used, such s not owing bekdown duing the stock buid up peiod nd duing bckog situtions due to the compexity of the mthemtic mode This ppe studies poduction-inventoy mode which poduces singe item with cetin poduction te nd seeks n dptive ctu poduction nd inventoy tes This mode hs dynmic ntue nd n dptive conto ppoch seems pticuy we suited to chieve its go eve nd poduction te Aso, the unknown deteiotion te wi be deived fom the conditions of symptotic stbiiztion bout the stedystte Besides djusting its poduction te, we so ssume tht the fim hs set n inventoy go eve nd penties e incued fo the inventoy eve to devite fom its go nd fo the desied poduction te to devite fom the ctu poduction te The est of the ppe is ognized s foows Foowing this intoduction, the mode is intoduced in Section in the cse of n infinite pnning hoizon Section 3 discusses the pobem of dptive conto of poduction nd mintennce system The Lipunov technique is used to pove the symptotic stbiity of the system bout its stedy stte The conditions of the symptotic e used to deive the schedue poduction te nd peventive mintennce te In Section 4 we pesent some iusttive exmpes Section concudes the ppe MATHEMATICAL MODEL 73

2 Fwzy A Bukhi Conside mnufctuing system tht poduces singe poduct, seing some units nd dding othes to inventoy We stt by witing the diffeenti eqution modes These cn be convenienty fomuted in tems of Moeove, the foowing vibes nd pmetes e used: x( t ) : Inventoy eve t time t, p( t ) : Popotion of good units of end item poduced t time t, S( t ) : Demnd te t time t, u( t ) : schedued poduction te t time t, ( t) : Ntu deteiotion te t time t, m( t ) : Peventive mintennce te ppied t time t to educe the popotion of defective units, ( t) : Obsoescence te of pocess pefomnce in the bsence of mintennce We mke the foowing ssumptions: Thee is non-negtive initi inventoy eve: x() x A demnd must be stisfied: x( t) Negtive poduction is not owed The mintennce eve is bounded by owe imit of zeo nd n uppe imit of m( t) M p( t) A functions e ssumed to be nonnegtive, continuous nd diffeentibe Assume tht t the time instnt t, the demnd occus t te S( t ), the schedued poduction t te u( t ), the peventive mintennce te m( t ), nd the obsoescence te ( t) It foows tht the inventoy eve x( t) nd the peventive mintennce te p( t) evove ccoding to the foowing fist ode diffeenti equtions: x ( t) p( t) u( t) S( t) ( t) x( t), () p ( t) m( t) ( ( t) m( t) ) p( t), Whee dot mens the diffeentition with espect to time nd the initi stte is x() x, p() p These bnce equtions wee so used by [6]-[8] without deteiotion te This ppe wi geneize the bove equtions which wee used by Dnny et to contin deteiotion te fo inventoy items The stedy-sttes of this system cn be deive by setting x ( t) nd p ( t) Tht is: m x ( u p S ), p, () m Whee x, p nd S e the vues of inventoy eve, popotion of good units te, nd demnd te t the stedy-stte In wht foows, we wi dispy the numeic soution of the system () fo the foowing set of pmetes vues: P V u S m + s i n + s i n ( t ) ( t + ) 74

3 Adptive Conto of Poduction nd Mintennce System With the foowing vues of initi inventoy eve, popotion of good units: x( t) nd p( t) 6 espectivey The numeic esuts e iustted in Figue to Figue c We concude tht the stem of the whoe system in the sme diection the popotion of good units 7

4 Fwzy A Bukhi Next, in wht foows we wi discuss the dptive conto of the poduction nd mintennce mode with unknown deteiotion te using Lipunov technique 3 ADAPTIVE CONTROL PROBLEM In this section, we wi discuss in detis the dptive conto pobem of the poduction nd mintennce mode with unknown deteiotion te Both the schedued poduction te nd peventive mintennce te wi be deived Aso the updting ue of deteiotion te wi be deived fom the conditions of symptotic stbiity of the efeence mode To study the dptive conto pobem we stt by consideing the equied vues of both inventoy go eve nd, popotion of good units' te s thei vues t the stedy-stte of system () Assume tht the unknown deteiotion te hs the estimto ˆ( t) nd ou efeence mode wi tke the foowing fom: x ( t) p( t) u( t) S( t) ˆ ( t) x( t), (3) p ( t) m( t) ( ( t) m( t) ) p( t), This system wi be stisfied by the speci soution () if the estimto of the unknown deteiotion te tends to the e vues of deteiotion te Tht is: ˆ( t) In ode to study the dptive conto pobem, we stt by obtining the petubed system of the poduction nd peventive mintennce mode bout its stedy sttes ( x, p ) To obtin this petubed system we wi intoduce the foowing new vibes: e ( t) x( t) x, e( t) p( t) p, v( t) u( t) u, v( t) m( t) m, ˆ (4) ( t) ( t), ( t) ( t), d( t) S( t) S, Substituting fom (4) into (3) nd tking into ccount the stedy- stte soution (), one gets: e v ( t)( pe ( t)) ue ( t) ( ( t)) e( t) x ( t) d( t), () e v( t) ( ( t) v( t))( p e( t)) ( m) e ( t), The system soution () wi be used to study the dptive conto pobem using the Lipunov technique Theoem (Refeence [9]) Let e( t), be n equiibium point fo gene nonine system modeed n n by: e f ( e, t) whee t nd e( t) : Let D be domin contining the equiibium point e( t) nd define: : D be continuousy diffeentibe function such tht: (), nd ( e), e D : If ( e), e( t) D, then e( t) is stbe This mens tht given ; >, n (, ] such tht B { e( t) : e( t) } D If ( e, t), e( t) D, then e t is symptoticy stbe, ie, e() D, e() e( t) s t Using this theoem, we wi show tht ou cosed oop system is symptoticy stbe The foowing theoem gives the desied poduction te nd updting ue fo deteiotion te ensue the symptotic stbiity of the efeence evese ogistic inventoy mode with uncetin diffeent types deteiotion nd dispos te Theoem The soution (, ) x p of the system () is symptoticy stbe in Lipunov since if nd ony if: thee exist poduction te nd peventive mintennce te: 76

5 Adptive Conto of Poduction nd Mintennce System u t S( t) S k ( x( t) x) u ( p( t) p) p( t) ( ) u, m t k ( p( t) p) m p( t) ( ), thee exist two continuous functions such tht : t t ( t) [ c f( p, ) d ] e, t t x e ˆ( ) [ (, ) t t c f x d ] e, Whee,,, i i (7) k i e positive e conto gins pmetes tht cn be seect by the fim, whie c nd c e two e constnts, nd f, nd f e: t f( p, t) p( t)( p( t) p) e, t (8) f( x, t) ( x ( t) x x( t) ) e, Poof: The poof of this theoem is bsed on the choosing of the suitbe Lipunov function fo the system petubed system which consists of the two systems () nd (7) Assume this function hs the fom: ( e, e, ) ( ei i ), (9) i This function is positive definite function of the vibes e i (t), (i=,) nd ( t) The tot time deivtive of the Lipunov function (8) ong the tjectoy of the systems () nd (7) gives: ( k ) e ( m k ) e, () [ ] (6) Since, k, m k, i, ( i, ), then the tot time of the Lipunov function is negtive definite function of the vibes ei ( t), ( t), ( i,), so ei ( t), nd vi ( t), whee ( i,), nd ( t) the speci soution is symptoticy stbe in the Lipunov sense This competes the poof Lemm: Thee e wys schedue poduction te u(t) nd peventive mintennce te m(t) fo t> The poof of this Lemm is obvious It foows immeditey fom the second condition of theoem Theefoe, using the nonine feedbck (6) fo schedued poduction te, nd peventive mintennce te nd updting ue fo the unknown deteiotion te, the poduction nd mintennce system with unknown deteiotion is symptoticy stbe bout its stedy stte Now substituting the dptive contoed the schedued poduction te nd peventive mintennce te (6) into () we get the foowing contoed system: x ( t) k ˆ ( x( t) x) ( t) x( t) x p ( t) k ( p( t) p) ( m)( p( t) p) ( ( t) ) p( t) ( t) p( t)( p( t) p) ( ( t) ) () ˆ ( t) x ( t) x x( t) ˆ ( ( t) ) The bove nonine system of diffeenti equtions is used to study the time evoution of inventoy eve, the popotion of good units' te nd dynmic estimtos of deteiotion te It ppes fom () tht the nytic soution of the system is difficut to deive since it is non-ine nd theefoe we sove it numeicy in the next section 4 Numeic Soution 77

6 Fwzy A Bukhi To iustte the soution pocedue descibed in section 3, we cy out the numeic soution of the system () using numeic integtion ppoch The objective of this section is to study nd discuss the numeic soution of the pobem to detemine n dptive conto sttegy fo poduction nd peventive mintennce inventoy mode with unknown deteiotion nd obsoescence tes The numeic soution goithm is bsed on numeic integtion of the system () using the Runge-Kutt method This numeic soution wi dispyed gphicy fo diffeent vues of the pmetes The sensitivey nysis of numeic soution of the system () is discussed fo diffeent cses of the system pmetes nd initi sttes A Constnt demnd te In this exmpe we discuss the numeic soution of the petubtion of constnt demnd te The foowing set of pmete vues is ssumed: P V P V u S S m p x k k 3 with the foowing vues of initi inventoy eve, popotion of good units: x(), p() 7, (), nd () espectivey The numeic esuts e iustted in Figue to Figue f 78

7 Adptive Conto of Poduction nd Mintennce System 79

8 Fwzy A Bukhi We cn see tht the inventoy eve nd the deteiotion te e osciting bout thei go eves nd tes, espectivey Whie the popotion of good units' te, the obsoescence te, the schedue poduction te, nd the peventive mintennce te e exponentiy tend to thei go tes B Exponenti demnd te In this exmpe we discuss the numeic soution of the petubtion of exponenti demnd te The foowing set of pmete vues is ssumed: P V P V u S S m 3 3t e 3 p x k k 3 With the foowing vues of initi inventoy eve, popotion of good units: x(), p() 7, (), nd () espectivey The numeic esuts e iustted in Figue 3 to Figue 3f 8

9 Adptive Conto of Poduction nd Mintennce System 8

10 Fwzy A Bukhi We cn see tht the inventoy eve nd the deteiotion te e osciting bout thei go eves nd tes, espectivey Whie the popotion of good units' te, the obsoescence te, the schedue poduction te, nd the peventive mintennce te e exponentiy tend to thei go tes CONCLUSION In this ppe we discussed the pobem of dptive conto of poduction nd mintennce system with unknown deteiotion nd obsoescence tes The numeic soution showed tht the inventoy eve nd the deteiotion te oscite bout thei go eves nd tes, espectivey Whie the popotion of good units te, the obsoescence te, the schedue poduction te, nd the peventive mintennce te e exponentiy tends to thei go tes The peventive mintennce te nd the obsoescence te e obtined REFERENCES [] SP Sethi nd GL Thompson, Optim Conto Theoy: Appictions to Mngement Science nd Economics, nd ed, Kuwe Acdemic Pubishes, Dodecht [] SK Goy nd BC Gii, Recent tends in modeing of deteioting inventoy, Euopen Joun of Opetion Resech, vo 34,, pp -6 [3] B Pote nd F Tyo, Mod conto of poduction-inventoy systems, Intention Joun of Systems Engineeing, vo 3, no 3, 97, pp 3-33 [4] Y Sm, Optim conto of simpe mnufctuing system with estting costs, Opetions Resech Lettes, vo 6,, pp 9-6 [] T T Ds, nd S Sk, Optim peventive mintennce in poduction inventoy system, IIE Tnsctions on Quity nd Reibiity Engineeing, vo 3, 999, pp 37 [6] J Kenne, A Ghbi, nd M Beit, Age- dependent poduction pnning nd mintennce sttegies in uneibe mnufctuing system with ost se, Euopen J of Opetion Resech, vo 78,7, pp 48-4 [7] I Dnny, P Cho, L Abd, nd M P, Optim poduction nd mintennce decision when system expeience ge dependent deteiotion, Optim conto ppictions & methods, vo 4, 993, pp 3-67 [8] A Ghbi nd J Kenne, Poduction nd peventive mintennce tes conto fo mnufctuing system: An expeiment design ppoch, Int J Poduction Economics, vo 6,, pp 7-87 [9] H K Khi, Nonine Systems, 3 d ed, Pentice-H, New Jesey, 8