An Optimal Control Approach to InventoryProduction Systems with Weibull Distributed Deterioration


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1 Journal of Mahemaics and Saisics 5 (3):64, 9 ISSN Science Publicaions An Opimal Conrol Approach o InvenoryProducion Sysems wih Weibull Disribued Deerioraion Md. Aiul Baen and Anon Abdulbasah Kamil School of Disance Educaion, Universiy Sains Malaysia 8 USM, Penang, Malaysia Absrac: Problem saemen: We sudied he invenoryproducion sysem wih woparameer Weibull disribued deerioraion iems. Approach: he invenory model was developed as linear opimal conrol problem and by he Ponryagin maximum principle, he opimal conrol problem was solved analyically o obain he opimal soluion of he problem. Resuls: I was hen illusraed wih he help of an example. By he principle of opimaliy we also esablished he Riccai based soluion of he HamilonJacobiBellman (HJB) equaion associaed wih his conrol problem. Conclusion: As an applicaion o quadraic conrol heory we showed an opimal conrol policy o exis from he opimaliy condiions in he HJB equaion. Key words: Invenoryproducion sysem, Weibull disribued deerioraion, linear quadraic regulaor, principle of opimaliy, opimal conrol. INRODUCION Many managemen science applicaions involve he conrol of dynamic sysems, i.e., sysems ha evolve over ime called coninuous ime sysems or discreeime sysems depending on wheher ime varies coninuously or discreely which is a rich research area []. We are especially ineresed in he applicaion of opimal conrol heory o he producion planning problem. InvenoryProducion sysem consiss of a manufacuring plan and a finished goods warehouse o sore hose producs which are manufacured bu no immediaely sold. he advanages of having producs in invenory are: Firs hey are immediaely available o mee demand; second, by using he warehouse o sore excess producion during low demand periods o be available for sale during high demand periods. ypically, he firm has o balance he high producion coss and find he quaniy i should produce in order o keep he oal cos a a minimum. Now a days he opimal conrol heory has been applied o differen invenoryproducion conrol problems where researchers are involved o analye he effec of deerioraion and he variaions in he demand rae wih ime in logisics. he model of invenoryproducion sysem considered he invenory depleion no only by demand bu also by iem's deerioraion []. hese ypes of problems have been sudied by several researchers o deermine he opimum order quaniy for differen demand paerns [,,4]. he model was presened for deerioraing iems wih ime proporional demand [4] whereas a heurisic model was developed ha allowed he variaion in boh replenishmencycle lengh and he sie of he order []. A linear quadraic regulaor problem was used wih known disurbance o an invenoryproducion sysem wih iems ha deeriorae a a known consan rae []. hey used a firsorder model o represen he invenory sysem in deriving he opimal producion policy. Imporance of iems of deerioraing in invenory modeling is now widely acknowledged, as shown by [3,3]. A number of sudies have been done wih he assumpion ha he deerioraion rae follows he Weibull disribuion [5,6,,,,5]. he assumpion of he consan deerioraion rae was relaxed [7] using a woparameer Weibull disribuion o represen he disribuion of ime o deerioraion. his model was furher generalied [9] aking a hreeparameer Weibull disribuion. A woparameer Weibull disribuion deerioraion is adoped o develop an invenory model wih a finie rae of replenishmen [8]. In general, in formulaing invenory models, wo facors of he problem have been of growing ineres o he researchers, one being he deerioraion of iems and he oher being he variaion in he demand rae wih ime. his sudy develops an opimal conrol model and uilies he opimal conrol heory o obain opimal producion policy for invenory producion sysems Corresponding Auhor: Md. Aiul Baen, School of Disance Educaion, Universiy Sains Malaysia, 8 USM, Penang, Malaysia. 6
2 J. Mah. & Sa., 5 (3):64, 9 where he novely we ake ino consideraion in our research is ha he ime of deerioraion is a random variable followed by he woparameer Weibull disribuion. his disribuion can be used o model eiher increasing or decreasing rae of deerioraion, according o he choice of he parameers. he probabiliy densiy funcion of a woparameer Weibull disribuion is given by: η g() =η e, > where, η> is he scale parameer, > is he shape parameer. he probabiliy disribuion funcion is: G() = e η, > he insananeous rae of deerioraion of he onhand invenory is given by: Since our objecive is o minimie he seup and he invenory coss, he objecive funcion can be expressed as he quadraic form: { [ ˆ ] [ ˆ ] } minimie J = h x() x() + C u() u() d he inerpreaion of his objecive funcion is ha we wan o keep he invenory x as close as possible o is goal ˆx and also keep he producion rae u as close o is goal level û. he quadraic erms h[x() x()] ˆ ; and C[u() u()] ˆ impose 'penalies' for having eiher x or u no being close o is corresponding goal level. he dynamics of he sae equaion of his dynamic model which says ha he invenory a ime is increased by he producion rae and decreased by he demand rae and he insananeous rae of deerioraion η of Weibull disribuion can be wrien as according o: g() G() A() = =η, > () Applicaions of opimiaion mehods o producion and invenory problems dae back a leas o he classical economic order quaniy model. Some references ha apply conrol heory o producion and invenory problems are [4,8,9,5,6,9,4]. In he presen sudy, we assume ha imedependence of he demand rae. Deerioraion rae is assumed o follow a woparameer Weibull disribuion. he purpose of he sudy is o give an opimal producion policy which minimies he cos for invenory producion sysems where iems are deerioraing wih a Weibull disribuion. o esablish he Riccai based soluion form o he Hamilon Jacobi Bellman (in shor, HJB) equaion associaed wih he opimal conrol problem is also of our ineres. Finally an aemp has been made o give an opimal conrol policy from he opimaliy condiion in he HJB equaion. MAERIALS AND MEHODS o build our opimal conrol model, we consider ha a firm can manufacures a cerain produc, selling some and socking he res in a warehouse. We assume ha he demand rae varies wih ime and he firm has se an invenory goal level and producion goal rae. We also assume ha he firm has no shorage, he insananeous rae of deerioraion of he onhand invenory follows he woparameer Weibull disribuion and he producion is coninuous. 7 dx() [u() y() x()]d x() x, x = η = > () x() = he invenory level in he warehouse a any insan of ime [,] ˆx() = An invenory goal level wha is se by he firm aking ino consideraion he available sorage space, h = he invenory holding cos incurred for he invenory level o deviae from is goal u() = he firm manufacured unis of he producion rae a any insan of ime [,] û() = he producion goal rae C> = he uni cos incurred for he producion rae o deviae from is goal y() = he demand rae > = Represens he fixed lengh of he planning horion he mehodologies applied in his sudy are as follows: Ponryagin maximum principle [] and he principle of opimaliy [3]. Ponryagin maximum principle is used o solve he opimal conrol problem analyically and o obain he opimal soluion of his problem. he principle of opimaliy is also used o esablish he Riccai based soluion of he Hamilon JacobiBellman (HJB) equaion associaed wih his conrol problem. RESULS AND DISCUSSION Developmen of he opimal conrol model: By he virue of () he insananeous sae of he invenory
3 J. Mah. & Sa., 5 (3):64, 9 level x() a any ime is governed by he differenial equaion: dx() +η = d x() = x and x( ) = x() u() y(),, (3) his is a linear ordinary differenial equaion of firs order and is inegraing facor is: exp{ d} exp{ }. = η = η Muliplying boh sides of (3) by exp{ η } and hen inegraing over [,], we have: x() exp{ η } x() = [y() u()]exp{ η }d, (4) Subsiuing his value of x() in (4), we obain he insananeous level of invenory a any ime [,] is given by: x() =, [y() u()]exp{ η }d [y() u()]exp{ η }d. exp{ η } In order o develop he opimal conrol model, we sar by defining he variables (), k() and v() such ha () = x() x(), ˆ k() = u() u() ˆ (5) = ˆ η ˆ (6) v() u() y() x() By adding and subracing he las erm η ˆ x() from he righ hand side of Eq. 6 and rearranging he erms we have: = η ˆ + η d() [ (x() x()) u() y() x()]d Hence: d() = [ η () + u() y() η ˆ x()]d (7) Now subsiuing (5) and (6) in (7) yields: d() [ () k() v()]d. = η + + (8) ˆ he opimal conrol model becomes: minimie J = {h[() ] + C[k() ]}d (9) subjec o an ordinary differenial equaion: d() [ () k() v()]d (), = η + + = > () his form is a sandard Linear Quadraic Regulaor (LQR) problem wih known disurbance v() defined in (6). he general form of his LQR opimal conrol problem for a finie ime horion [,] is he following: () minimie J = { ()Q()() + k ()R()k()}d subjec o an ordinary differenial equaion: d() = [A ()() + B ()k() + v()]d () =, > () Q() and R() = Real symmeric posiive semidefinie marices of appropriae dimension A () and B () = he sysem dynamics marices > = fixed Soluion o he opimal conrol problem: Soluion by ponryagin maximum principle: In order o find exremals for his opimal conrol problem () and (), we apply Ponryagin maximum principle o form he Hamilonian as: { ()Q()() + k ()R()k()} H(x, φ,k,) =. +φ (){A ()() + B ()k() + v()} hen he necessary condiions of opimaliy give he cosaes equaions as: d φ () = Hd = ()Q() A () φ () d; d φ () = Q()() A () () φ d. he exremal conrol vecor is given by: from which we have: H = R()k() + B () φ () = k (3) k() = R ()B () φ () (4) 8
4 J. Mah. & Sa., 5 (3):64, 9 because R is nonsingular. hen from (), he exremal sae vecor saisfies: d() = [A ()() B ()R ()B () φ ()]d (5) Now by combining (3) and (5), he saecosae equaions can be wrien in marix as follows: d() A () () B ()R ()B () = d φ() Q() A () () φ he soluion o his sysem of linear differenial equaion is of he form: () () =ϕ(,) φ() φ() A () B ()R ()B () ϕ (,) : = Q() A () is he sae ransiion marix. Now by he ransversaliy condiion φ () = for LQR problem, we obain: and () = A ()() B ()R ()B () φ () φ () = Q()() A () φ () = (6) If A () is nonsingular for all in [,] hen from (6), we have: φ () = A () Q()() which esablish a linear relaionship beween φ () and (): φ () = M()() (7) M() = A () Q(). (8) ha minimies (). Now differeniaing boh sides of (7) and, hen by () and (9), we have: d φ () = dm()() + M()d() = dm()() + M()[A ()() + B ()k() + v()] () = dm()() + M()A ()() M()B ()R ()B ()M()() + M()v() Subsiuing (7) o (3) we have: d φ () = Q()() A ()M()() () Combining () and (), we obain: [dm() + M()A () + A ()M() M()B ()R ()B ()M() + Q()]() + M()v() = () Since () mus hold for any value of () and for all in [,] we mus have: dm() = [M()B ()R ()B ()M() M(){A () + A ()} Q()]d (3) is called a Riccai equaion and M() is he Riccai marix. he boundary condiions are: () =, φ () = and M() = By comparing Eq. 9, we have: A() = η, B =, Q= h and R= C hen from (8) we obain: M() = h( η ) and he opimal conrol policy () becomes: Example : If we choose: k() = hc ( η ) () A = (byη=, = ), B =, Q= h = and R = C= hen subsiuing (7) in (4), we obain he opimal conrol which is given by a linear feedback law: k() = R ()B ()M()() (9) 9 hen he opimal conrol model () and () becomes over an finie ime horion [,]: minimie J(k()) = { () + k ()}d (4)
5 J. Mah. & Sa., 5 (3):64, 9 subjec o he conrol sysem: d() = [ () + k() + v()]d () =, > (5) So he opimal (sae) feedback conrol is given by: k() = M()() (6) he Riccai Eq. 3 becomes wih he scalar values (chosen above): dm() M () M() + = (7) Now by he mehod of separaion of variables o (6): hen: dm() = [M () + M() ]d dm() C = + M() + M() + + from which we have: M() + ln = + C M() + + where, C is a consan. (8) Seing q() = M() + and subsiuing o (8) hen we have: q() = Ne q() + where, N is a inegraion consan. Now by using he boundary condiion for M () = we have: q() = M() + = = ( + ) Ne Ne and ( ) + ( )e M() = ( ) + + ( )e (9) Subsiuing (9) ino (6) we have he opimal (sae) feedback conrol policy: ( ) + ( )e k() = () ( ) + + ( )e Riccai soluion by dynamic programming principle: n n Suppose w(,):r R R is a value funcion whose value is he minimum value of he objecive funcion obained earlier (4) and (5) (choosing he same values: A = (byη=, = ), B =,Q= h = and R = C= for he invenory sysem given ha we sar i a ime in sae. ha is: w(,) = inf J(k()) where, he value funcion w(,) is finie valued and wice coninuous differeniable on (, ) [,) By he Principle of Opimaliy [3], i is naural ha w(,) solves he following Hamilon JacobiBellman (in shor, HJB) equaion: w (,) + min[k () + k()w(,)] ()w (,) k + v()w (,) + () =, < (3) wih he erminal boundary condiion w(,) = and where w(,) and w(,) are he parial derivaives of w(,) wih respec o and respecively. In order o solve he HJB Eq. 3 we minimie he expression inside he bracke of (3) and aking derivaive wih respec o k() seing i o ero. hus he procedure yields: from which N = e + e q() = + + e + herefore we obain: ( ) ( ) k () = w(,) (3) Subsiuing (3) ino (3) yields he equaion: w (,) w (,) ()w (,) = v()w (,) () (3)
6 J. Mah. & Sa., 5 (3):64, 9 known as he HJB equaion. his is a parial differenial equaion which has a soluion form: w(,) = a() () (33) hen: a() = = (34) w (,) a()(), w (,) () Subsiuing (33) and (34) ino (3) yields: a() [ a () + a()(v() ) + ]() = (35) Since (35) mus hold for any value of, we mus have: a() + + = a()(v() ) a () is called a Riccai equaion where: ( v() ) ( τ )( v() ) a() = a()e + e {a ( τ ) }d τ. herefore, (33) is a soluion form of (3). Since (33) is a soluion of he HJB Eq. 3, hen he opimal conrol k() can be wrien as k() = a()(), where a() is known consan. An applicaion o quadraic conrol heory: We sudy he opimal conrol problem (5) o minimie he producion cos over all M subjec o an ordinary differenial sae equaion: d () = [A () () + k () + v()]d () = (36) where, A() = η and M denoes he class of all progressively measurable F() adapive processes k() such ha: We assume w(,) R is a coninuous, nonnegaive, convex funcion saisfying he polynomial growh condiion such ha: m+ ( ) w() P +, R, m N for some consan P > + (39) Le us choose he parameers of Weibull disribuion in such way so ha his disribuion can be used o model decreasing rae of deerioraion and hen we can assume ha: A() + v() < for sufficienly large demand rae y() (4) Lemma : Under (39), he differenial equaion: d () = [A () () +ρ (w ( ())) + v()]d () = (4) admis a unique soluion (), where: k if k, ρ () = if k <, if < (4) Furher, for any m N +, here exiss P > such ha: m + ( ) E () P (43) Proof: Since ρ(w ()) is bounded, Eq. 4 admis a unique srong soluion () wih: E n () < / Using I ôs formula we have: (44) m+ lim E () = for he response () o k(). (37) By he same line as (3) we consider he Bellman equaion associaed wih he problem (4): w (,) + min[k () + k()w (,)] + A ()()w (,) k v()w (,) (),if Q: (, ) [,) + + = = w(,) =, > (38) n n { } () = + n A () + v() () d { } n n + n ρ(w(()))() sgn(())d n + n A() + v() () d n n + n ρ(w(()))() sgn(())d,
7 J. Mah. & Sa., 5 (3):64, 9 where, A() =η is he insananeous rae of deerioraion of he onhand invenory followed by he woparameer Weibull disribuion. Now by (4) and aking expecaion on boh sides: n n E () n + ne ρ(w(()))() sgn(())d (45) n (n) = + E S ()d (n) n = ρ S () n (w ( ())) () sgn( ()) By (4) i is easily seen ha ρ(w ( ())) k if b for sufficienly large b>. Clearly : Also: < (n) sup E S () ( () < b) E S () E k. (n) ( () b) ( () b) In addiion, by (44) we see ha he righhand side of equaion (45) is bounded from above. his complees he proof. heorem : We assume (39). hen he opimal conrol k() is given by: k () =ρ(w ( ())) (46) Proof: Le us noe ha = min{k + k} and he 4 k / minimum is aained by ρ (). We apply I ôsformula for convex funcions [7] o obain: where, { τ n} is a sequence of localiing sopping imes for he local maringale. By (39) and (43) of Lemma, we have: m+ ( ) E[w( ())] P + E () (m+ ) P + E () P + ( P ( + )) <. (48) Dividing boh sides by and leing we ge: liminf E[w( ())] = (49) Hence () saisfies (37). Leing n o (47) and using (49) we obain: { } E[ () + k () d] w() from which J(k ) w(). Now by (48) we have J(k ) w() < hence, k = (k ()) M. Le k M be arbirary. By he same line as above, we have: { } E[w(())] w() E[ () + k() d], k M By (39) and (37): liminf E[w(())] = hus we can obain he desired resul. CONCLUSION = + { + + } + w ( ())]d w( ()) w() [ A () () k () v() w ( ()) By virue of (38) τn τ n = + { } E[w( ( ))] w() E[ () k () d] (47) his sudy has described he soluion of an invenoryproducion sysem wih Weibull disribuion deerioraing iems using boh Ponryagin maximum principle and Dynamic programming principle. he resuling producion conrol policy has minimied he objecive funcion of he oal cos wih he applicaion o quadraic conrol heory. hese models can be exended in many ways. For example, if oher coss such as he sorage cos are included; or insead of
8 J. Mah. & Sa., 5 (3):64, 9 minimiing he oal cos, one may wan o maximie he oal profi where he uni revenue rae is boh funcion of ime and of he invenory level. However, we need o give numerical illusraive examples for his opimal conrol of a producioninvenory sysem wih Weibull disribuion deerioraing iems. ACKNOWLEDGEMEN he researchers wish o acknowledge he suppor provided by Fundamenal Research Gran Scheme, No. 3/PJJAUH/678, Universii Sains Malaysia, Penang, Malaysia for conducing his research. REFERENCES. Andijani, A. and M. ALDajani, 998. Analysis of deerioraing invenoryproducion sysems using a linear quadraic regulaor. Eur. J. Opera. Res., 6: hp://ca.inis.fr/?amodele=affichen&cpsid= 563. BahariKasani, H., 989. Replenishmen schedule for deerioraing iems wih imeproporional demand. J Opera. Res. Soc., 4: hp:// 3. Bellman, R., 957. Dynamic Programming. Princeon. Princeon Universiy Press, New Jersey. 4. Bensoussan, A., Jr. E.G. Hurs and B. Naslund, 974. Managemen Applicaions of Modern Conrol heory. Elsevier, New York, ISBN: 7433, pp: Chakrabary,., B.C. Giri and K.S. Chaudiri, 998. An EOQ model for iems wih Weibull disribuion deerioraion, shorages and rended demand: An exension of Philip s model. Compu. Opera. Res., 5: hp://ca.inis.fr/?amodele=affichen&cpsid= Chen, J.M. and S.C. Lin. 3. Opimal replenishmen scheduling for invenory iems wih Weibull disribued deerioraion and imevarying demand. Inform. Opimi. Sci., 4: Cover, R.P. and G.C. Philip, 973. An EOQ model for iems wih Weibull disribued deerioraion. AIIE. rans., 5: DOI:.8/ Feichinger, G. and E.J. Dockner, 985. Opimal pricing and producion in an invenory model. Eur. J. Opera. Res., 9: hp://ideas.repec.org/a/eee/ejores/v9y985ip4556.hml 9. Gaimon, C., 988. Simulaneous and dynamic price, producion, invenory and capaciy decisions. Eur. J. Opera. Res., 35: hp://ideas.repec.org/a/eee/ejores/v35y988i3p4644.hml. Ghare, P.N. and G.F. Schrader, 963. A model for exponenially decaying invenories. J. Ind. Eng., 5: Ghosh, S.K. and K.S. Chaudhuri, 4. An orderlevel invenory model for a deerioraing iem wih Weibull disribuion deerioraion, imequadraic demandand shorages. Adv. Model. Opimi., 6: 35. hp:// Goel, V.P. and S.P. Aggarwal, 98. Pricing and ordering policy wih general Weibull rae of deerioraing invenory. Indian Journal of Pure and Applied Mahemaics, : hp:// insa/insa_/5a_68.pdf 3. Goyal, S.K. and B.C. Giri,. Recen rends in modeling of deerioraing invenory. Eur. J. Opera. Res., 34: 6. hp://ideas.repec.org/a/eee/ejores/v34yip 6.hml 4. Happing, U. and H. Wang, 99. An economic ordering policy model for deerioraing iems wih ime proporional demand. Eur. J. Opera. Res., 46: 7. hp://ideas.repec.org/a/eee/ejores/v46y99ip 7.hml 5. Harl, R.F. and S.P. Sehi, 984. Opimal conrol problems wih differenial inclusions: Sufficiency condiions and an applicaion o a producion invenory model. Opimal Conrol Appli. Mehod., 5: hp://prolog.univie.ac.a/cgibin/pubs_absrac.cgi?pubnr= 6. Hwang, C.L., L.. Fan and L.E. Erickson, 967. Opimal producion planning by he maximum principle. Manage. Sci., 3: hp:// 7. Karaas, I. and S.E. Shreve, 99. Brownian Moion and Sochasic Calculus. SpringerVerlag, New York, ISBN: , pp: Misra, R.B., 975. Opimum producion lo sie model for a sysem wih deerioraing invenory. In. J. Prod. Res., 3: DOI:.8/ Pekelman, D., 974. Simulaneous priceproducion decision. Opera. Res., : hp://or.journal.informs.org/cgi/conen/absrac/ /4/788 3
9 J. Mah. & Sa., 5 (3):64, 9. Philip, G.C., 974. A generalied EOQ model for iems wih weibull disribuion. AIIE. rans., 6: DOI:.8/ Ponrygin, L.S., V.G. Bolyanski, R.V. Gamkrelidge and E.F. Mischenko, 96. he Mahemaical heory of Opimal Processes. 6h Edn., John Wiley and Sons, New York, pp: 36. hp://books.google.com.pk/books?id=udnsaaaa MAAJ&q=he+mahemaical+heory+of+opimal +processes&dq=he+mahemaical+heory+of+op imal+processes&pgis=. Sehi, S.P. and G.L. hompson,. Opimal Conrol heory, Applicaions o Managemen Science and Economics. nd Edn., Springer, USA., ISBN: , pp: Shah, N. H. and Y.K. Shah,. Lieraure survey on invenory model for deerioraing iems. Econ. Ann., 44: 37. hp://scindeks.nb.rs/aricle.aspx?arid=33645s&lang=en 4. Spreukouski, A.Y., 967. A problem in opimal sock managemen. J. Opim. heor. Appli., : 34. DOI:.7/BF Wu, J.W. and W.C. Lee, 3. An EOQ invenory model for iems wih Weibull disribued deerioraion, shorages and imevarying demand. Inform. Opimi. Sci., 4: 3. 4
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