Optimal Control Approach to Production Systems. with Inventory-Level-Dependent Demand

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1 Optmal Control Approach to Prodcton Systems wth Inventory-Level-Dependent Demand Egene Khmelntsky and Ygal Gerchak Department of Indstral Engneerng, Tel-Avv Unversty, Tel-Avv 69978, Israel Fax: , E-mal: Aprl 1 Revsed September 1 Abstract Demand occasonally depends postvely on the amont of dsplayed stock, especally n the case of novelty or mplse prchase tems. Sch nventory-level dependence of demand rate has been ncorporated nto some contnos revew nventory control models, bt only ones wth demand rate whch does not vary wth tme, no shortages, nfnte prodcton rate and per-nt-tme cost mnmzaton. We propose an optmal control model whch relaxes all these assmptons. That model generalzes a known optmal control model by addng nventory-level dependence of demand to the state dynamcs. The framework s one of dsconted proft maxmzaton. We apply the maxmm prncple and obtan three possble snglar regmes. The specal case of tme-nvaryng demand s solved explctly. The ncapactated case s analyzed n detal for a wave-mode tme dependence of demand, and a nmercal example s gven. 1. Introdcton The Prodcton Control lteratre typcally assmes that demand, whether determnstc or stochastc, or tme-varyng or not, s exogenos frm s prodcton/nventory polcy cannot affect t. Bt, t has been observed by marketng practtoners and researchers that dsplayed stocks can have a stmlatng effect on demand (e.g., Corstjens and Doyle 1981). At least two types of stml effects of nventory on demand have been dentfed. The frst and more obvos s referred to as the 1

2 selectve effect, where more tems n nventory provde cstomers wth more to choose from and ths ndce them to prchase more. Ths wll happen where nts of an tem are not dentcal, a cstomer may lke the feelng of a wde selecton. For certan baked goods, low stocks may rase cstomers percepton that the nts are left-overs and not fresh. The second type of stmls of dsplay qantty s the advertsng effect. Large dsplays often rase the percepton among cstomers that the tem s poplar n the market, whch may sgnal a good vale and case them to prchase more (or more often). Some stores dsplay hge nventory ples of prodcts lke blank vdeo-tapes, sffcent to meet many years of demand, clearly amng at stmlatng demand. See Gerchak and Wang (1994) and Balakrshnan et al. () for a revew of relevant lteratre. As for ncorporatng nventory-level dependence nto tradtonal nventory models, Johnson (1968), Gerchak and Wang (1994) and Wang and Gerchak () have done so for stochastc perodc revew models, and nmeros stdes startng wth Baker and Urban (1988) throgh Balakrshnan et al. () for determnstc contnos revew models. The latter task s complex de to the dffclty of provng jont concavty n lot-sze and re-order pont, becase the holdng costs are a solton of a dfferental eqaton. Ths work proposes a dfferent approach to modelng and analyzng determnstc prodcton/nventory systems wth nventory-level dependent demand rate optmal control (Bensossan, Crochy and Proth 198, Seth and Thompson, Kogan and Khmelntsky ). The optmal control approach proved ts effcency n many economc, manageral and ndstral applcatons (see, e.g. Khmelntsky and Caramans 1998, Zhang, Yn and Bokas 1). The advantages of ths approach n or context are as follows: ) t permts sng a fnte or (dsconted) nfnte horzon objectve; ntal and/or termnal nventory level(s) can be specfed; ) the prodcton rate can be pper bonded; ) demand rate can be tme-varyng as well as nventorylevel dependent. In case of a shortage, t can declne n the magntde of the shortage; v) model can be solved to characterze the optmal prodcton polcy.

3 The model generalzes those analyzed by Bensossan et al. (198, Ch. III), and Kogan and Khmelntsky (, Ch. ) n addng nventory-level dependence of demand to the state dynamcs. Snce revene s not constant anymore, the framework s one of proft maxmzaton; we se an nfnte horzon dsconted formlaton for concreteness. On the other hand, or model does not have setp costs, whch were a key ngredent n the EOQ-based formlatons. Note that althogh the resltng nventory depleton pattern s remnscent of some pershable nventory models, here nventory s depleted by sales, whch contrbte to revene. We apply the maxmm prncple and obtan three possble snglar regmes. The specal case of tme-nvaryng demand s solved explctly. An ncapactated model wth wave-mode tme dependence of demand s analyzed n detal and a nmercal example s provded.. Problem formlaton Let X( be the nventory level at tme t, and d(x, the demand rate at tme t when nventory level s X. Let X be the postve porton of X [ X = max( X,)] and c the holdng cost rate per nt. Let X be the negatve porton of nventory [ X = mn( X,)] and c the shortage penalty rate per nt. Let p dscont rate. The control s the prodcton rate fncton (, whch mght be pper bonded by U. Let C() be the cost rate correspondng to a prodcton rate. Then, the problem s to maxmze the total proft of the frm (revene mns nventory and prodcton costs): Max J = e ρt [ pd( X, c X ( c X ( C( ( ) ]dt (1) s.t. X = ( d( X,, X () = X, () ( ( U. () Note that prodcton costs can depend on prodcton rate n a non-lnear (plasbly, convex) manner, whch s partclarly mportant f U s nfnte.

4 . Capactated problem wth lnear prodcton cost The necessary condton of optmalty for the problem (1)-() takes the form of the maxmm prncple, whch, for the case of a lnear prodcton cost, C() = c, declares that there exsts a contnos co-state fncton ψ ( satsfyng the co-state eqaton c, f X ( > d( X, ρt ρt ψ ( = ( ψ ( pe ) e c, f X ( < (4) X [ c, c ], f X ( = Another reqrement of optmalty s that the maxmzaton of the Hamltonan fncton H = e ρt [ pd( X, c X ( c X ( c ( ] ψ ( [ ( d( X, ] (5) wth respect to ( reslts n U, ( =, [, U ], Assmptons on the problem parameters are f ψ ( > ce f ψ ( < ce f ψ ( = c e ρt ρt ρt (6) The fncton d(x, s dfferentable w.r.t. t and twce dfferentable w.r.t X except for X=; for X=, the sbdfferental of d(,, denoted by d(, s a known nterval d (, = [ a(, b( ], a(, b(. Here sbdfferental d(, * * s defned as the set { x d( X, d(, Xx, X }. d( X, X s non-negatve for X p>c. The frst two lnes n (6) nqely defne the prodcton rate as a fncton of the co-state varable (fll prodcton and no prodcton regmes respectvely). The thrd lne n (6) presents a snglar prodcton regme, for whch the optmal control vale cannot be obtaned by Hamltonan 4

5 maxmzaton. To determne ( along the snglar regme, we let the snglar condton ψ t c e ρt ( ) = hold n an nterval of tme. Then, over ths nterval t c e ρt ( ) = ρ. (7) ψ By sbstttng (7) n (4), we obtan three types of snglar regmes SR1: X(= over the snglar nterval; SR: X(> over the snglar nterval; SR: X(< over the snglar nterval. Consderng the state and co-state eqatons () and (4) nder the three snglar regmes, we determne the followng condtons whch are necessary for the regmes occrrence. The SR1 regme can potentally occr only at those tme ntervals where [ a(, b( ] ( ρc [ c, ]) ( p c ) c Ø and d(, U. The optmal control s (=d(,. The SR regme can potentally occr only at those tme ntervals where Xˆ ( d( Xˆ (, U and X ˆ ( >. The optmal control s (= X ˆ ( d( Xˆ (,, where X ˆ ( t ) s a solton of eqaton d( X, ρc c = X p c. The SR regme can potentally occr only at those tme ntervals where ~ ~ X ( d( X (, U ~ and X ( <. ~ ~ ~ The optmal control s (= X ( d( X (,, where X ( t ) s a solton of eqaton d( X, ρc c = X p c. 5

6 .1 Example Consder a specal case when the demand fncton does not change n tme, d(x,=d(x). That s the type of demand consdered by prevos nventory-dependent demand lteratre thogh wthn d( X ) dfferent economc settng (emphaszng setp cos. Also assme that < X for X>, and that ρ < c. Then, the SR regme does not exst and the SR regme exsts for only one vale of c the bffer level, ˆX, whch s constant n tme. If both d() U and d(xˆ ) U, then there are two steady state soltons that satsfy the necessary optmalty condtons, X ( and X ( Xˆ. In order to compare the objectve vales of these two soltons, we have to take nto accont not only the steady state tself, bt also how the solton converges to the steady state from the ntal vale X. To be specfc, let the steady state be reached at X ( Xˆ and d d ( X ) = d max αx e ( d, max d ) e βx, f f X X <. Sch a choce of the d(x) fncton reflects the selectve and advertsng effects of nventory dscssed n Secton 1,.e. d(x) ncreases when X ncreases. Ths d(x) ndcates also the lost sales effect for shortages,.e. d(x) decreases when X - ncreases. In both cases d(x) converges, ether to d max when X goes to nfnty, or to zero when X - goes to nfnty. Now, by ntegratng the state eqaton (), we obtan where 1 X ( = ln e β B A Aβ ( e 1), β ( X A t U d A = dmax, max, f f X X < Xˆ > Xˆ and B = d max d. For the parameters U=1, d max =11, d =5, X =, p=5, c =.5, c =., =. and =., we calclated Xˆ =7.9, d (Xˆ ) =9.76, and the tme pont tˆ at whch the trajectory enters the snglar 6

7 regme X(= Xˆ s eqal to The objectve vale s J= Smlarly, for the other steady state X= we fond that the steady state level s reached at t=.47 and the objectve vale s J=86.4. If the convergence nterval s neglgble, then the comparson between the two optons s based on the steady state parameters only. In sch a case, we fnd that the solton X ( s better than ˆ the other one, ( ) ˆ d( X ) d() c X t X (, f <, Xˆ p c.e. f ˆ ( p c )[ d( X ) d()] < c Xˆ, whch s nttve. Otherwse, the second solton s preferable over the frst one. 4. Uncapactated problem wth non-lnear prodcton cost In ths secton we assme that the prodcton cost fncton C() s strctly convex and dfferentable. In sch a case, the prodcton costs are ncreasng sharply, so a hgh prodcton rate s not lkely to occr, thereby allowng s to drop the capacty constrant. The maxmzaton of the Hamltonan reslts n, [ C ] f ψ ( e -ρt ( = 1 ρt -ρt ( e ψ ( ), f ψ ( > e C () C () (8) where ψ ( satsfes the same co-state eqaton (4). The demand fncton s assmed to be dfferentable w.r.t. X, except for X=, and dfferentable w.r.t. t except for a fnte nmber of tme ponts. Unlke n the prevos (lnear, capactated) case, here there s only one type of snglarty, at X=. If the snglar regme occrs n an nterval of tme, then over that nterval ρt ( = d(,, ψ ( = e C ( d(, ). The necessary condton for the regme to occr s 4.1 Solton method ( p C ( d(, ) )[ a(, b( ] ( C ( d(, ) C ( d(, ) d (, [ c, c ]) ρ Ø. (9) Consder a wave-mode demand,.e. for any X, d(x, ncreases over an ntal tme nterval and d(x, drops down thereafter (see examples n Fgre 1), whch s rather plasble n many 7

8 statons. The wave-mode demand can, for example, reflect a typcal prodct lfe cycle, whch conssts of for major segments: start-p, rapd growth, matraton and declne (see Nahmas (1997)). For sch cases, the nmber of tme ntervals over whch condton (9) holds, s at most three. That s becase for large postve and large negatve vales of d (,, condton (9) does not hold (for example, at the jmp ponts n Fgre 1b). Fgre 1. Examples of a "wave-mode" demand. Wthot loss of generalty, we assme that the nmber of tme ntervals over whch (9) holds s exactly three. If there are less, then the solton method smplfes. Denote the tme ntervals over whch (9) holds by τ 1, τ and τ. Snce the tme horzon s not bonded, the fnal regme mst be snglar. Otherwse, the solton s not bonded,.e. X ( goes to nfnty. However, the two ntermedate ntervals τ 1 and τ may or may not have snglarty. Therefore, the problem of fndng the optmal solton becomes a combnatoral one. Denote the startng tme pont by τ,.e. τ ={}, and by τ..., j the solton whch does have snglarty τ 1 j wthn the ntervals τ, k=1,,j, and does not have snglarty wthn other ntervals. k There are for possble soltons τ τ 1, τ 1, τ, τ. (1) 8

9 For these soltons the state-co-state dynamcs over the snglar arcs, denoted by τ, s known to be t X ( = and t c e ρ ψ ( ) =. The state-co-state dynamcs over the reglar arcs, whch lnk ether two snglar arcs or τ wth a snglar arc, s determned from the followng state and co-state eqatons: X ( = d( X,, 1 [ C ] ( e ψ ( ), f ψ ( e f ψ ( > e -ρt ρt -ρt C () C () (11) d ( X, ρt ρt c, f X ( ψ ( = ( ψ ( pe ) e. (1) X c, f X ( < To lnk a par of τ s, the method makes se of a shootng procedre for solvng two-pont bondary-vale problems (Seth and Thompson,.). Specfcally, to lnk τ wth τ, =1,,, the procedre looks for sch vale of ψ () that allows the solton of the system (11)-(1) to enter the snglar arc at a tme pont looks for sch pont ot t τ n t τ. Smlarly, to lnk τ wth n τ, 1 < n, the procedre of extng snglarty at τ that allows the solton of the system (11)- n (1) to enter the snglar arc at a tme pont tn τ n. A solton τ... τ, j τ 1 j for whch t t, k=1,,j s locally optmal, snce t satsfes the necessary condtons of n k ot k optmalty. If, among for possble soltons (1) there are two or more local optma, the solton wth the least objectve (1) s the globally optmal one. 4. Example The parameters of the problem were chosen as: X =, p=5, ρ =., α =.1, β =., c =., c - =1, C ( ) =.. To exemplfy the solton method, we choose the d(x, fncton n sch a way that t has a wave pattern as a fncton of t for each X, as dscssed n Secton 4.1, and t has a monotone ncreasng pattern as a fncton of X for each t, as dscssed n Secton.1. Formally, 9

10 d d( X, = d max ( ( d ( e αx, max ( d ( ) e βx, f f X X <, where d max 7, ( = 11, 4, f t < 4 f 4 t < 7 f 7 t, d ( = 5,, f t < 4 f 4 t < 7 f 7 t For the adopted parameters, the snglar regme condton (9) holds at three ntervals: τ 1 = (,4), τ = (4,7) and τ = (7, ). By shootng, we tred to lnk each par of the ntervals. It was fond that the pars τ and τ 1 cannot be lnked. In spte of the fact that all the pars n the trajectory τ 1 are lnked, the trajectory tself does not exst. Ths s becase 1 >. Ths, the only trajectory whch does exst s τ where both pars of ntervals n ot t t1 are lnked and n ot t t <. Fgres -4 present the optmal solton. 5. Conclsons The optmal control approach allowed s to formlate problems of nventory/prodcton control at consderable level of generalty. In partclar, tme dependence (as well as nventory level dependence) of demand s a featre whch exstng nventory models do not have. We acknowledge, however, that or contnos prodcton formlaton does not accont for setp costs, the key ngredent of EOQ-lke models n lteratre. We provded specfc nsghts, analytcal and nmercal, nto two scenaros whch may be vewed as captrng a smlar realty: capactated prodcton wth lnear prodcton costs and ncapactated prodcton wth convex prodcton costs. References Baker, R.C. and T.L. Urban (1988) A Determnstc Inventory Model wth an Inventory-Level- Dependent Demand, Jornal of the Operatonal Research Socety, 9, Balakrshnan, A., M.S. Pangbrn and E. Stavrlak () Stack Them Hgh, Let em Fly : Lot- Szng Polces when Inventores Stmlate Demand. Workng paper, Department of 1

11 Management Scence and Informaton Systems, Penn State Unversty, Unversty Park, Pennsylvana. Bensossan, A., M. Crochy and J-M Proth (198) Mathematcal Theory of Prodcton Plannng, North-Holland, Amsterdam. Corstjens, M. and P. Doyle (1981) A Model for Optmzng Retal Space Allocatons, Management Scence, 7, 8-8. Gerchak, Y., and Y. Wang (1994) Perodc Revew Inventory Models wth Inventory-Level Dependent Demand, Naval Research Logstcs, 41, Johnson, E.L. (1968) On (s,s) Polces, Management Scence, 15, Khmelntsky, E., and M. Caramans (1998) One-Machne n-part-type Optmal Set-p Schedlng: Analytcal Characterzaton of Swtchng Srfaces, IEEE Transactons on Atomatc Control, 4, Kogan, K. and E. Khmelntsky () Schedlng: Control-Based Theory and Polynomal-Tme Algorthms, Klwer Academc Pblshers, Dordrecht. Nahmas, S. (1997) Prodcton and Operatons Analyss, McGraw Hll, rd ed. Seth, S.P. and G.L.Thompson () Optmal Control Theory: Applcatons to Management Scence, nd ed., Klwer Academc Pblshers, Dordrecht. Wang, Y., and Y. Gerchak (1) Spply Chan Coordnaton when Demand s Shelf-Space Dependent, Manfactrng and Servce Operatons Management, (1), Zhang, Q., G.G. Yn, and E.-K. Bokas (1) Optmal Control of a Marketng-Prodcton System, IEEE Transactons on Atomatc Control, 46,

12 - Fgre. Inventory level X(. Fgre. Demand d(x, (bold lne) and prodcton rate ( (thn lne). Fgre 4. Snglar co-state dynamcs e ρt Cost (d(,) (bold lne) and co-state varable ψ ( (thn lne). 1