We consider the planning of production over the infinite horizon in a system with timevarying

Transcription

1 Ifiite Horizo Productio Plaig i Time-varyig Systems with Covex Productio ad Ivetory Costs Robert L. Smith j Rachel Q. Zhag Departmet of Idustrial ad Operatios Egieerig, Uiversity of Michiga, A Arbor, Michiga 4809 We cosider the plaig of productio over the ifiite horizo i a system with timevaryig covex productio ad ivetory holdig costs. This productio lot size problem is frequetly faced i idustry where a forecast of future demad must be made ad productio is to be scheduled based o the forecast. Because forecasts of the future are costly ad difficult to validate, a firm would like to miimize the umber of periods ito the future it eeds to forecast i order to make a optimal productio decisio today. I this paper, we first prove that uder very geeral coditios fiite horizo versios of the problem exist that lead to a optimal productio level at ay decisio epoch. I particular, we show it suffices for the first period ifiite horizo productio decisio to solve for a horizo that exceeds the logest time iterval over which it ca prove profitable to carry ivetory. We the develop a closed-form expressio for computig such a horizo ad provide a simple fiite algorithm to recursively compute a ifiite horizo optimal productio schedule. (Forecast Horizo; Dyamic Lot Sizig; Time-varyig Costs). Itroductio The productio lot sizig problem is a model for the cotrol of productio over a multiperiod plaig horizo (Deardo 982). It is oe of the most frequetly used sigle-item determiistic ivetory plaig models (Federgrue ad Tzur 99). The objective is to schedule productio over the plaig horizo so that demad is satisfied at miimum cost. Stadard assumptios are that demad is determiistic (i.e., kow i advace) ad backorderig is ot allowed (i.e., demad caot be satisfied by future productio). The fudametal ecoomic tradeoff here is the balace of reductios i cost of productio agaist correspodig icreases i costs of carryig ivetory. I the presece of ecoomies of scale o the cost of productio, it ca prove profitable to produce more tha the curret period s demad ad carry ivetory forward to satisfy future demad, thereby lowerig the average cost of productio (cycle stock motive). Eve i the absece of ecoomies of scale i productio costs, the future cost of productio may exceed the cost of curret productio plus ivetory carryig costs agai leadig to curret productio that exceeds curret demad (speculative motive (Chad ad Morto 986)). The choice of plaig horizo to employ is a difficult issue, because the system beig modeled typically has a log but otherwise idefiite lifespa. A resolutio of this problem is to utilize a ifiite horizo to model the uderlyig log but ukow fiite horizo lifespa of the system. I the geeral case of time-varyig demad ad cost, the resultig model presets a challegig problem to solve (the statioary case reduces to the classic ecoomic lot size (ELS) model (Harris 93). Early efforts to solve ifiite horizo versios of the problem allowed time-varyig demad but restricted costs to be either statioary liear (Thompso ad Sethi 980, Morto 978a), ostatioary liear (Kureuther ad Morto 973), or statioary covex (Kureuther /98/4409/33$05.00 Copyright 998, Istitute for Operatios Research ad the Maagemet Scieces MANAGEMENT SCIENCE/Vol. 44, No. 9, September b2d se5 Mp 33 Thursday Oct 08 0:7 AM Ma Sci (September) se5

2 Ifiite Horizo Productio Plaig ad Morto 974, Morto 978b, Modigliai ad Hoh 955, Lee ad Orr 977). Their approaches to establishig the existece of ad procedures for discovery of solutio ad forecast horizos variously used either margial aalysis to boud optimal productio levels (Morto 978; Kureuther ad Morto 973, 974; Morto 978), or Lagrage multipliers to decouple preset from future decisios by forcig edig ivetory to be zero (Modigliai ad Hoh 955, Lee ad Orr 977). Most of these papers provide forward algorithms together with a stoppig rule that, if met, results i discovery of a solutio horizo to the uderlyig ifiite horizo problem. However, existece of solutio ad forecast horizos were oly established uder additioal sigificatly stroger assumptios o the cost structure of the problem. The so-called dyamic lot size versio of the problem where productio costs are fixed-plus-liear ad ivetory holdig costs are liear has bee extesively studied i the ostatioary case. Although the recet focus has bee o computatioal breakthroughs i solvig fiite horizo versios of the problem (see, e.g., Aggarwal ad Park 990; Federgrue ad Tzur 99; ad Wagelmas, Va Hoesel, ad Kole 989), the properties exploited there have, i some cases, bee used to establish coditios o fiite horizo versios of the ifiite horizo problem that guaratee early decisio agreemet with optimal decisios of the ifiite horizo problem. Such a fiite horizo is called a solutio horizo. Whe the agreemet does ot deped o problem data (i this case demad) beyod this solutio horizo, it is also called a forecast horizo because oly data over this horizo eeds to be forecasted to establish ifiite horizo optimal early decisios (Bes ad Sethi 988). Although solutio ad forecast horizos may fail to exist here, Federgrue ad Tzur (99, 992) provided a stoppig rule that is guarateed to be met wheever they do exist (see also Chad ad Morto 986). A importat property of the dyamic lot size problem is the mootoicity of the last period with productio i the plaig horizo N. This last property has bee extesively exploited to geerate forecast horizo existece ad discovery results for the dyamic lot size problem ad its variatios (see, e.g., Wager ad Whiti 958; Zabel 964; Eppe et al. 969; Thomas 970; Blackbur ad Kureuther 974; Ludi ad Morto 975; Besoussa et al. 983; Chad 982; Chad, Sethi, ad Proth 990; ad Chad, Sethi, ad Sorger 989). See also Heyma ad Sobel (984) for a geeral review of usig policy mootoicity i homogeeous MDP problems. I this paper, we cosider the ifiite horizo versio of the geeral lot sizig problem uder disecoomies of scale i productio ad ivetory holdig costs. This covexity assumptio is equivalet to the coditio that margial productio ad holdig costs be odecreasig. For example, this icludes the case where ivetory costs are liear ad where a firm experieces a higher overtime rate for productio exceedig the stadard capacity followed by a still higher uit cost for exceedig overtime capacity through outsourcig. The optimizatio problem to be solved falls withi the class of doubly ifiite covex programmig problems, because there are both a ifiite umber of variables (productio levels) ad costraits (demad satisfactio i each period). There is a extesive literature o solutio ad forecast horizo approaches to solvig such geeral problems i ifiite horizo optimizatio (see, e.g., Bea ad Smith 984, 993; Bes ad Sethi 988; ad Schochetma ad Smith 989, 992). However, a key assumptio there that guaratees that geeral purpose algorithms will successfully discover a equivalet fiite horizo problem is uiqueess of a ifiite horizo optimal solutio. Although this coditio is believed to be typically met i practice, it is difficult to verify. I this paper, we explore istead a ovel algorithmic approach for fidig solutio ad forecast horizos that systematically exploits mootoicity of optimal early decisios i horizo N whe productio ad ivetory holdig costs are covex. This focus o early decisio mootoicity, as opposed to late decisio mootoicity as i the treatmet of the dyamic lot size problem where costs are cocave, leads to a closed form expressio for a forecast horizo guarateed to yield optimal early productio decisios for the ifiite horizo problem. As we will show, the legth of the forecast horizo is the logest iterval of time over which it ca prove profitable to carry ivetory. The paper is orgaized as follows. I 2, we formulate the ifiite horizo model of the problem. I 3, we prove that uder very geeral coditios, solutio 34 MANAGEMENT SCIENCE/Vol. 44, No. 9, September 998 3b2d se5 Mp 34 Thursday Oct 08 0:7 AM Ma Sci (September) se5

3 Ifiite Horizo Productio Plaig horizos exist leadig to fiite horizo versios of the problem that yield optimal solutios to the ifiite horizo problem. I 4, we give a closed-form expressio for computig a solutio (ideed forecast) horizo ad a simple recursive procedure for computig a optimal ifiite horizo productio schedule. 2. Problem Formulatio Cosider a sigle-product firm where a decisio for productio must be made at the begiig of each period, Å,2,...Wewill adopt the followig otatio wherei Å,2,... Costats ad fuctios: D Å the demad durig period (oegative itegers) a Å the discout factor for the time value of moey (0 õ a õ ) I 0 Å the ivetory o had at the begiig of period (oegative iteger) c (x) Å the cost of producig x uits of the product durig period (oegative) h (x) Å the cost of holdig x uits of ivetory edig period (oegative) Decisio variables: P Å the productio level durig period (oegative itegers) I Å the ivetory o had at the ed of period (oegative itegers) We will use the superscript (*) to deote optimality. With the above otatio, we ca formulate this ifiite horizo problem, labeled Q, as 0 Å (Q) Miimize: a [c (P ) / h (I )] () Subject to: I / P 0 D Å I, Å,2,... (2) 0 P 0, I 0, Å, 2,... (3) P, I : iteger, Å, 2,... (4) where I 0 is give. As we ca see from (2), if we kow the productio levels P i all periods, we ca determie the ivetory levels I. Therefore, it suffices to fid a optimal productio schedule P *, P *, P *,... Note, how- 2 3 ever, that this is a doubly ifiite iteger oliear programmig problem ad is therefore a formidable problem to solve. 3. Existece of Solutio Horizos We ow ivestigate coditios uder which a fiite horizo versio of the problem has a optimal first decisio that is i agreemet with a ifiite horizo optimal first decisio. If we ca fid a optimal ifiite horizo first decisio P* by solvig a fiite horizo versio of the problem, we ca roll forward oe period ad form a ew ifiite horizo problem with ew iitial ivetory I* Å I 0 / * P 0D to obtai a optimal ifiite horizo secod decisio for the origial problem. This rollig horizo procedure ca the recursively recover a optimal ifiite horizo productio schedule. I this sectio, we formulate the N-horizo trucated versio of the problem ad show that, uder covex productio ad ivetory holdig costs, optimal productio levels of the N-horizo problem are icreasig i N. We the idetify coditios uder which a N- horizo optimal th decisio, N, coverges as N r to a ifiite horizo optimal th decisio. Fially, we establish existece of a fiite horizo versio for solvig the ifiite horizo problem. 3.. The N-Horizo Problem We formulate the N-horizo problem, labeled (Q(N)), correspodig to the origial ifiite horizo problem (Q) as: N 0 Å (Q(N)) Miimize: a [c (P ) / h (I )] (5) subject to: I / P 0 D Å I, Å,2,...,N (6) 0 P 0, I 0, Å,2,...,N (7) P, I : iteger, Å,2,...,N. (8) Let S R be the set of all feasible productio schedules to (Q), S(N) R N the set of feasible productio schedules to (Q(N)), P(N) ay feasible productio schedule to (Q(N)), ad I(N) the edig o had ivetories resultig from the productio schedule P(N), N Å,2,...Weowadopt our first assumptio o (Q) ad hece (Q(N)), i.e., that both productio ad ivetory holdig costs are covex: MANAGEMENT SCIENCE/Vol. 44, No. 9, September b2d se5 Mp 35 Thursday Oct 08 0:7 AM Ma Sci (September) se5

4 Ifiite Horizo Productio Plaig A0. Productio ad ivetory holdig costs are covex, i.e., c ( ) ad h ( ) are covex fuctios with c (0) Å h (0) Å 0 for all Å,2,... The followig lemma provides that for the N-horizo problem (Q(N)), icreasig demad leads to a mootoe icrease i productio. LEMMA (VEINOTT [964]). Let P*(N) Å (P* (N), P 2* (N),..., P N* (N)) be ay optimal solutio for a vector (D, D 2,...,D N ) of demads. If oe of these demads is icreased by uit, it is optimal to icrease oe of these productio levels by uit. The proof of this lemma ca be foud i Deardo (982). Cosider ow the demad profile for a N / - horizo problem where D N/ Å 0. Sice, without loss of optimality, we ever leave positive ivetory at the ed of a horizo, we coclude I N * Å 0 at a optimal solutio for D N/ Å 0. The by the priciple of optimality, * * P (N) Å P (N / ) (9) whe D N/ Å 0. Hece applyig the lemma repeatedly as D N/ is icreased oe uit at a time, we have * * P (N) P (N / ), for N Å, 2,... (0) for ay fixed D N/. Followig the same argumet, we also have P*(N) P*(N / ), for all N, N Å,2,... () Hece, we have prove the followig corollary. COROLLARY. P*(N) is mootoically icreasig i N for ay fixed, N Optimal Solutio ad Value Covergece of the N-Horizo Problems Before we discuss covergece of optimal solutios of the N-horizo problems, we eed the followig additioal otatio ad assumptios. Let C(P) be the objective fuctio of (Q) for P S ad C* Å C(P*). Also let C(P(N); N) be the objective fuctio of (Q(N)) for P(N) S(N) ad C*(N) Å C(P*(N); N). Furthermore, we adopt the followig additioal assumptios o (Q): A. There exists a fiite cost feasible productio schedule to (Q), i.e., C(P) õ for some feasible productio schedule P S. A2. The margial costs of productio are uiformly bouded from above ad away from zero, i.e., 0 õ d c (P ) 0 c (P 0 ) g g õ for all itegers P ú 0 ad all Å,2,... Assumptio (A) is eeded for a solutio to (P) to exist while (A2) is a regularity coditio that bouds optimal productio ad ivetory levels. We ow show that P*(N) coverges as horizo N r to a ifiite horizo optimal th decisio P* õ for all (Theorem ). That is, lim Nr P*(N) Å P* uder the above coditios. This compoetwise covergece of P*(N) Å (P* (N), P 2* (N),..., P N* (N), 0, 0,...) to P* Å (P *, P 2*,...) as vectors i R is precisely product covergece i R (Schochetma ad Smith 992), so we may equivaletly write that P*(N) r P* asn r. We establish this covergece by first showig that P*(N) coverges to a ifiite horizo feasible solutio as N r (Lemmas 2 ad 3) ad the that value ad hece solutio covergece holds for all Å, 2,... (Lemma 4 ad Theorem ). LEMMA 2. There exist fiite productio bouds PV, Å,2,...,so that P*(N) PV õ for all N ad Å, 2,... PROOF. Suppose ot, the there exists some ad subsequece N k, k Å,2,...,such that lim P*(N ) Å. (2) kr By assumptio (A2), ad hece k lim c (P*(N k)) Å (3) kr lim C*(N k) Å. (4) kr However, by (A), with P(N k) the first Nk decisios i P, C*(N k) C(P(N k); N k) C(P) õ. (5) This cotradicts equatio (4). 36 MANAGEMENT SCIENCE/Vol. 44, No. 9, September 998 3b2d se5 Mp 36 Thursday Oct 08 0:7 AM Ma Sci (September) se5

5 Ifiite Horizo Productio Plaig By Corollary, at ay decisio epoch,, there exists a mootoically icreasig sequece of optimal decisios P*(N), N Å, 2,... ByLemma 2, this sequece of values is bouded from above. Therefore, P*(N) must coverge as N goes to ifiity, i.e., lim P*(N) Å PP õ (6) Nr exists for all Å,2,... It remais to show PP is ifiite horizo optimal. LEMMA 3. PP S, i.e., PP is ifiite horizo feasible. PROOF. Fix Å,2,...The PP Å lim P *(N) Å lim P *(N) j j j jå jå Nr Nr jå lim D 0 I Å D 0 I. (7) Nr jå j 0 j 0 jå Hece PP is ifiite horizo feasible. LEMMA 4. lim Nr C*(N) Å C*, i.e., optimal value covergece holds. PROOF. Sice, without loss of optimality, productio is bouded i every period by Lemma 2, this follows from the geeral optimal value covergece result of Theorem 3.2 i Schochetma ad Smith (989). We ca ow prove our pricipal result that fiite horizo optima mootoically coverge upwards as horizo legthes to a ifiite horizo optimal solutio. THEOREM. PP is ifiite horizo optimal, ad hece P*(N) coverges mootoically upward to a ifiite horizo optimal productio schedule, i.e., P*(N) F PP as N r for all Å,2,... PROOF. From (3.6), ad oegativity of the costs, for ay positive iteger M, M Å 0 P P a [c (P ) / h (I )] M 0 Å Å lim a [c (P*(N)) / h (I*(N))] Nr N 0 Å lim a [c (P*(N)) / h (I*(N))] Å C* Nr by Lemma 4. Now take the limit as M r o both sides of the above iequality to get M 0 Å C(P) P Å lim a [c (P P ) / h (I P )] C*. (8) Mr From Lemma 3, PP S ad hece PP is ifiite horizo optimal. Theorem allows us to easily exted Veiott s mootoicity lemma to the ifiite horizo case. COROLLARY 2. Suppose Assumptios (A0) through (A2) hold. Let P* be ay optimal solutio for a vector (D, D 2, ) of demads. If oe of these demads is icreased by uit, it is optimal to icrease oe of these productio levels by uit. PROOF. Let HP be the optimal ifiite horizo productio schedule for the demad vector (D,..., D j /,...) ad let HP*(N) be the correspodig optimal productio volume i period uder this demad schedule for a plaig horizo of N periods. Note that for all Å,2,..., P Å lim P*(N) lim P*(N) Å P P (9) Nr Nr by Lemma ad Theorem. Also P Å lim P * (N) lim P *(N) / k k k kå Nr kå Nr kå Å / lim P *(N) Å / P P k k (20) kå Nr for all Å, 2,... sice without loss of optimality edig ivetory i period N is zero for all plaig horizos N Å,2,...(9)together with (20) imply that P Å Pˆ for all periods but oe i which P Å Pˆ /. The coclusio of Theorem is called optimal solutio covergece while that of Lemma 4 is called optimal value covergece. Optimal value covergece supports a method aalogous to successive approximatios as applied to homogeeous MDP problems (Deardo 982). These may be viewed as equivalet to solvig successively loger horizo problems as we iterate (the iitial guess of value fuctio is see here as a termial value at the ed of horizo). Optimal value covergece implies that for N large eough, the correspodig optimal N-horizo pla kå MANAGEMENT SCIENCE/Vol. 44, No. 9, September b2d se5 Mp 37 Thursday Oct 08 0:7 AM Ma Sci (September) se5

6 Ifiite Horizo Productio Plaig P*(N) achieves a cost arbitrarily close to that achieved by a optimal ifiite horizo solutio P*, i.e., P*(N) ad P* are close i value. But P*(N) is ot a ifiite horizo feasible solutio. Optimal value covergece is therefore of limited use, approximatig ifiite horizo optimal cost, but ot solutios, while it is the latter we eed to implemet. Still we may at times be able to exted P*(N) feasibly over the ifiite horizo at small cost to achieve a ifiite horizo feasible solutio with early the same cost as P*. The solutio covergece result of Theorem is however far more powerful, because policies ad ot just costs are arbitrarily well approximated by sufficietly log fiite horizo optimal solutios. I fact, the approximatio to early decisios is without error i this case as we ote i the ext subsectio Solutio Horizos for Solvig the Ifiite Horizo Problem By Theorem, lim P*(N) Å P*, Å, 2,... (2) Nr where P* Å Pˆ is a ifiite horizo optimum. This implies that for ay e ú 0, there exists a horizo, N e () such that Let e Å. The so that ÉP*(N) 0 P*É õ e, for all N N (). (22) e for all N N (). I particular, ÉP*(N) 0 P*É õ (23) P*(N) Å P* (24) * * * P (N) Å P, for all N N (25) where N* Å N () so that N* is a solutio horizo. That is, there exists a fiite horizo N* sufficietly distat that a optimal first period productio lot size for ay horizo that log or loger yields a ifiite horizo optimal first period productio lot size. By forward dyamic programmig, let f (i) be the preset value of the optimal cost from period through period with edig ivetory level i i period, where i (I 0 0 D j ) /. The jå f (i) Å mi {f (i / D 0 P ) 0 0 P D /i 0 / a [c (P ) / h (i)]} where f 0 (i) Å 0 for i Å I 0 ad otherwise. If we kew the value of the solutio horizo N *, we could the solve for f N* (0) to get a ifiite horizo optimal first period productio level * * * P Å P (N ). By (24), we ca similarly compute the th period ifiite horizo optimal productio decisio for all Å, 2,...Wecathe recursively fid (P *, P 2*, ) Å P* with zero error. We tur to the computatio of solutio (ad forecast) horizos i Computig Solutio ad Forecast Horizos We have show i the previous sectio that there exists a solutio horizo N* such that P*(N) Å P*, for all N N* at ay decisio epoch. I this sectio, we seek a method to compute solutio horizos for all Å, 2,... ad a correspodig simple algorithm to compute a optimal ifiite horizo solutio P* for all. Cosider P *(N) as N icreases. By Corollary, * * P (N / ) P (N). (26) Therefore, the optimal first decisio either remais the same or icreases as N icreases. Suppose the latter, that is, Sice, moreover, * * P (N / ) ú P (N). (27) P*(N / ) P*(N), for all N (28) by Corollary, at least oe additioal uit of ivetory is produced i period ad held for N periods to satisfy a uit of demad i period N /. Evidetly, by (27) it is the less costly to satisfy a uit of demad i period N / by productio i period tha by productio i later periods, ad i particular tha by productio i period N /. Let s be a lower boud o the margial 38 MANAGEMENT SCIENCE/Vol. 44, No. 9, September 998 3b2d se5 Mp 38 Thursday Oct 08 0:7 AM Ma Sci (September) se5

7 Ifiite Horizo Productio Plaig cost of carryig a additioal uit of ivetory, i.e., by covexity of h, we may set Table The Forecast Horizo i Days for the First Ifiite Horizo Optimal Productio Level The by (A0), s Å if {h ()} 0. N N 0 N N/ Å s( 0 a )/( 0 a) a h () a g 0 c () so that N N* where N* is give by ( 0 a)c () / s N* Å log a (29) ( 0 a)g / s where X represets the smallest iteger strictly greater tha X. We coclude the * * * P Å P (N ) (30) is a ifiite horizo first decisio depedig oly o D, D 2,..., DN* where N* is give by (29). That is, N* is a forecast horizo for the first productio decisio. Followig the same argumet, we ca compute the forecast horizo for the secod productio decisio, ad so o. A tighter boud o a forecast horizo N ca be obtaied by utilizig specific problem data to compute the greatest umber of periods it is ecoomic to carry ivetory. That is, a forecast horizo is provided by the largest period of time it ca prove profitable to hold a uit of ivetory produced i period. Note that N* is idepedet of all demads. It also oly depeds o the values of bouds o ivetory ad margial productio costs. To get a feelig for the magitude of our forecast horizo, we look at some examples. I the simple case where productio costs are statioary ad liear over time, g Å sup{sup[c (P ) 0 c (P 0 )]} Å c () P ú0 ad N * Å. I other words, as we would expect, we oly eed to kow the demad i the first period to make the optimal first decisio regardless of the ivetory costs sice o ivetory is eeded whe productio cost does ot vary over time. Cosider ow the case where the productio costs are piecewise liear or eve oliear. I this case if we set r u g Å uc (), u ú (i.e., the margial productio cost will ot exceed uc ()) ad s Å vc () where v is the ivetory charge as the sum of a proportio of productio cost, opportuity costs, taxes, isurace costs, the value loss over time (e.g., certai products have to be sold by discout), floor space retal costs, etc., the 0 a / v N* Å log a. ( 0 a)u / v For various ivetory charges v per day, discout factor a Å /( / r/365) per day where r is the iterest rate per year, we computed N * for u Å to 2. The results are show i Table. We chose ivetory costs uusually high here to illustrate how short these forecast horizos ca be. However, eve i the case of moderate ivetory costs, forecast horizos ca be sigificatly reduced by a more detailed aalysis usig more precise cost iformatio to provide better bouds o the miimal forecast horizo.,2 We are idebted to Awi Federgrue ad a aoymous referee for suggestios that sigificatly improved the clarity of this paper. 2 This work was supported i part by the Natioal Sciece Foudatio uder grats DDM , DMI , ad DMI Refereces Aggarwal, A., J. K. Park Improved algorithms for ecoomic lotsize problems. Workig paper, IBM Thomas J. Watso Research Ceter, Yorktow Heights, NY. MANAGEMENT SCIENCE/Vol. 44, No. 9, September b2d se5 Mp 39 Thursday Oct 08 0:7 AM Ma Sci (September) se5

8 Ifiite Horizo Productio Plaig Bea, J., R. L. Smith Coditios for the existece of plaig horizos. Math. Oper. Res , Coditios for the discovery of solutio horizos. Math. Programmig Besoussa, A., J. Crouhy, J. M. Proth Mathematical Theory of Productio Plaig. Advaced Series i Maagemet. North- Hollad, Amsterdam. Bès, C., S. Sethi Cocepts of forecast ad decisio horizos: Applicatios to dyamic stochastic optimizatio problems. Math. Oper. Res Blackbur, J, D., H. Kureuther Plaig horizos for the dyamic lot size model with backloggig. Maagemet Sci Chad, S Lot sizig for products with fiite demad horizo ad periodic review policy. Europea J. Oper. Res , T. E. Morto Miimal forecast horizo procedures for dyamic lot size models. Naval Res. Logist. Quart , S. Sethi, J. M. Proth Existece of forecast horizos i udiscouted discrete-time lot size models. Oper. Res ,, G. Sorger Forecast horizos i the discouted dyamic lot size model. Workig paper, Uiversity of Toroto, Toroto, Caada. Deardo, E Dyamic Programmig: Models ad Applicatios. Pretice-Hall, Eglewood Cliffs, NJ. Eppe, G. D., F. J. Gould, B. P. Pashiga Extesios of the plaig horizo theorem i the dyamic lot size model. Maagemet Sci Federgrue, A., M. Tzur. 99. A simple forward algorithm to solve geeral dyamic lot sizig models with periods. Maagemet Sci , Fast solutio ad detectio of miimal forecast horizos i dyamic programs with a sigle idicator of the future: Applicatio to dyamic lot-sizig models. Workig paper, Graduate School of Busiess, Columbia Uiversity, New York. Harris, F. W. 93. How may parts to make at oce. Factory: The Magazie of Maagemet , 52. Reprit 990, Oper. Res Heyma, D., M. Sobel Stochastic Models i Operatios Research. Vol. II. McGraw-Hill, NY. Kureuther, H. C., T. E. Morto Plaig horizos for productio smoothig with determiistic demads. Maagemet Sci , Geeral plaig horizos for productio smoothig with determiistic demads. Maagemet Sci Lee, D. R., D. Orr Further results o plaig horizos i the productio smoothig problem. Maagemet Sci Ludi, R. A., T. E. Morto Plaig horizos for the dyamic lot size model: Zabel vs. protective procedures ad computatioal results. Oper. Res Modigliai, F., F. E. Hoh Productio plaig over time ad the ature of the expectatio ad plaig horizo. Ecoometrica Morto, T. 978a. The ostatioary ifiite horizo ivetory problem. Maagemet Sci b. Uiversal plaig horizos for geeralized covex productio schedulig. Oper. Res Schochetma, E. E., R. L. Smith Ifiite horizo optimizatio. Math. Oper. Res , Fiite dimesioal approximatio i ifiite dimesioal mathematical programmig. Math. Programmig Thomas, L. J Price-productio decisios with determiistic demad. Maagemet Sci Thompso, G. L., S. P. Sethi Turpike horizos for productio plaig. Maagemet Sci Veiott, A. F., Jr Productio plaig with covex costs: A parametric study. Maagemet Sci Wagelmas, A., S. Va Hoesel, A. Kole Ecoomic lot-sizig: A O( log )-algorithm that rus i liear time i the Wager- Whiti case. CORE discussio paper o Uiversité Catholique de Louvai, Louvai-la-Neuve, Belgium. Wager, H. M., T. M. Whiti Dyamic versio of the ecoomic lot size model. Maagemet Sci Zabel, E Some geeralizatios of the ivetory plaig horizo theorem. Maagemet Sci Accepted by Awi Federgrue; received September 3, 996. This paper has bee with the authors 2 moths for 2 revisios MANAGEMENT SCIENCE/Vol. 44, No. 9, September 998 3b2d se5 Mp 320 Thursday Oct 08 0:7 AM Ma Sci (September) se5