OPERATIOS RESEARCH Vol. 52, o., Jauary February 2004, pp. 05 5 iss 0030-364X eiss 526-5463 04 520 005 iforms doi 0.287/opre.030.0080 2004 IFORMS Ifiite Horizo Productio Schedulig i Time-Varyig Systems Uder Stochastic Demad Torpog Cheevaprawatdomrog,Robert L. Smith Departmet of Idustrial ad Operatios Egieerig, The Uiversity of Michiga, A Arbor, Michiga 4809, {toychee@yahoo.com, rlsmith@umich.edu} We cosider ifiite horizo productio schedulig uder stochastic demad. All problem data are allowed to vary across periods, icludig demad distributios, costs, ad reveues. A forecast horizo, whe it exists, is a fiite problem horizo with the property that the correspodig first-period optimal productio decisio remais optimal regardless of demad ad cost projectios beyod this horizo. Thus, a forecast horizo allows us to reduce the amout of future data we eed to forecast to solve for a optimal first decisio for the ifiite horizo problem. I this paper, we establish the existece of a forecast horizo uder the assumptios that () costs ad reveues are time-varyig liear, ad (2) demad is ever evetually zero. A key result for establishig the existece ad computatio of forecast horizos is the mootoicity, ad hece covergece, of optimal first-period policies as the horizo icreases of fiite horizo versios of the ifiite horizo problem. A closed-form formula is provided for computig a forecast horizo that depeds oly o the discout factor ad uiform upper ad lower bouds o demad ad uit productio ad ivetory holdig costs. I particular, its value is idepedet of, ad determied i advace of, forecastig the demad distributio. We show that the effect of ucertaity i demad is to icrease the forecast horizo associated with a determiistic problem by a costat plus a factor equal to oe plus the ratio of these upper to lower bouds o per-period demad. The associated forecast horizo ca be surprisigly short, eve a few days, whe the ivetory costs are high. Subject classificatios: Dyamic programmig: applicatios. Productio schedulig: applicatios, approximatios. Area of review: Maufacturig, Service, ad Supply Chai Operatios. History: Received May 200; revisio received December 2002; accepted Jauary 2003.. Itroductio Productio schedulig problems with log but idefiite horizos are ofte modeled as ifiite horizo problems. I the time-varyig case, a possible way to solve these problems is through use of the forecast horizo approach. This allows for the determiatio of a first optimal decisio by cosiderig just a fiite horizo of forecasted data. That is, we attempt to fid a fiite horizo with the property that data beyod this horizo do ot affect the optimality of the first-period policy. This ca resolve the followig dilemma. O the oe had, we eed a sufficiet amout of data so that our decisio is ot shortsighted. O the other had, we would like to miimize the amout of forecastig we eed to do because it is usually expesive ad difficult to justify. Smith ad Zhag (998) have recetly established the existece of a forecast horizo for the productio plaig problem i time-varyig systems with covex costs ad determiistic demad. They also provide closed-form formulas for the calculatio of forecast horizos. Our goal is to exted these results to the stochastic demad case. There is a large literature focusig o the productio schedulig problem with time-varyig stochastic demad ad statioary liear costs. Morto ad Petico (995) provide ear-myopic bouds ad heuristics for fiite horizo problems. Lovejoy (992) presets bouds ad stoppig times for usig myopic policies i the case where the demad i each period ca deped upo the previous period s demad. Morto (978) provides optimal cost ad policy bouds that are mootoe i horizo legth, as well as coditios uder which policies for each period of a fiite horizo problem coverge to a uique optimal policy of the ifiite horizo problem. Kleidorfer ad Kureuther (978) prove that optimal policies are mootoe i demad, ad provide bouds o policies as well as a algorithm for fidig a forecast horizo whe it exists. Karli (960a, b) shows, i the case where the purchasig cost is statioary covex, that optimal policies are mootoe i demad, ad optimal policies are myopic if the demad is stochastically icreasig. I the case where costs are ostatioary, Alde ad Smith (992) provide value error bouds for solvig fiite horizo solutios of the uderlyig Markov decisioprocess problem (see also Heradez-Lerma ad Lasserre 990 for a extesio of this work to Markov cotrol policies). Sethi ad Cheg (997) exted S s policy optimality to the Markovia demad ifiite horizo case. For ostatioary liear costs, Federgrue ad Tzur (996) provide a algorithm for detectig a miimal forecast horizo, if it exists, i a problem with restricted orderig policies. Veiott (965) presets coditios uder which myopic policies are optimal. Sobel (98) discusses coditios for a Markov decisio process to have a myopic optimal solutio. Topkis (998) proves that i a fiite horizo prob- 05
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig 06 Operatios Research 52(), pp. 05 5, 2004 IFORMS lem, optimal productios are mootoe i demad as costs remaied fixed. Zhag (998) establishes the existece of a solutio horizo by usig mootoicity of the optimal policy uder the assumptio that the demad is kow i each period before we make the decisio for that period. Garcia ad Smith (2000b) establish existece ad discovery of forecast horizos for a geeral dyamic optimizatio problem by assumig that there exists a forecast idex such that the first-period optimal actios are mootoically icreasig i that idex. Garcia ad Smith (2000a) use a similar approach to that i Garcia ad Smith (2000b) to establish the existece ad discovery of forecast horizos i the stochastic demad productio schedulig problem uder the assumptio that for ay fixed fiite plaig horizo, there exist first-period optimal solutios that are mootoe with respect to stochastic demad. Such policy mootoicity has bee established uder may coditios (see, for example, Karli 960a, Kleidorfer ad Kureuther 978, Morto 978, ad Topkis 998). However, a key claim i Garcia ad Smith (2000b), that the optimal first productio of the -horizo problem is equal to that of the +-horizo problem with zero demad i the last period, is ot true for stochastic demad problems. This fact ivalidates the opportuity to coclude policy mootoicity by embeddig a -horizo problem withi a + -horizo problem, thus rederig their existece proof ivalid for the stochastic demad problem. Also, the optimal first decisio of the -horizo problem is ot ecessarily a lower boud to that of the ifiite horizo problem, so that the algorithm proposed may prematurely stop with a horizo that fails to be a forecast horizo. I this paper, we geeralize Smith ad Zhag (998) to the case where demad is stochastic. We allow costs to be time varyig but liear. Also, we do ot make the assumptio that demad is kow prior to decidig o productio, as i Zhag (998). We preset a forward algorithm that is guarateed to fid a forecast horizo. I particular, we prove that it will fiitely termiate uder the regularity coditio that miimum demad i a period is strictly positive ifiitely ofte. We also provide a simple formula for the ratio of a forecast horizo legth for the stochastic demad problem to that of the determiistic demad problem i terms of lower bouds o the miimum per-period demads ad upper bouds o the maximum per-period demads. This result, together with the closedform upper boud for a forecast horizo for the determiistic demad problem preseted i Smith ad Zhag (998), provides a closed-form forecast horizo for the stochastic demad problem. This paper is orgaized as follows. The ifiite horizo productio schedulig problem is formulated i 2. Sectio 3 itroduces a fiite horizo versio of the origial ifiite horizo problem that provides a lower boud o the first period s optimal productio decisio. The existece of a forecast horizo is established by showig that the iitial policies of the fiite horizo problems mootoically icrease, ad hece evetually agree with a ifiite horizo optimal policy as the horizo legthes. Sectio 4 itroduces aother fiite horizo versio (the atural trucated versio) of the ifiite horizo problem ad establishes that policies mootoically decrease as the horizo legthes. A algorithm is provided for detectig whe these upper ad lower bouds agree, thus resultig i discovery of a ifiite horizo optimal iitial productio level. A computable boud o the value of a horizo for which this stoppig criterio is met is provided. This forecast horizo is provided as a closed-form formula for which illustrative umerical values are provided. The resultig horizos ca be surprisigly short, thus simultaeously resultig i miimal data oe eeds to forecast to compute a optimal first-period productio decisio, as well as resultig i a efficiet forward algorithm for its computatio. 2. The Ifiite Horizo Problem (L) We exted the productio schedulig problem formulatio i Chapter 6 i Deardo (982) to the case where the costs as well as the demad distributios are allowed to vary across periods. We begi each period by observig a iteger-valued begiig ivetory level for each period = 2 3 We the decide the iteger-valued productio level to brig ivetory up to its after-productio level. Productio is assumed to be istataeous. After that, we pay the ivetory holdig cost to carry the after-productio ivetory to the ed of the period, at which time we meet the stochastic demad. We satisfy as much demad as possible, but o backloggig is allowed, so that we are i the sales-lost case. After satisfyig demad, if there is ivetory left, the it becomes the begiig ivetory of the ext period. For each period, i = the begiig ivetory i for period where i 0 iteger ad the begiig ivetory of period, i,isgive ad fixed. j = the after-productio ivetory j for period where j i iteger. c = the uit productio cost c for period where c > 0. h = the uit ivetory holdig cost h for period where h > 0. r = the uit sale price r for period where r > 0. D = the radom iteger demad D for period. d = a lower boud d o demad for period so that D d for all with probability oe. d = a upper boud d o demad for period so that D d for all with probability oe. = the discout factor, 0 <. Assumptio. Productio levels are uiformly bouded; i.e., there exists p< such that 0 j i < p for all.
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig Operatios Research 52(), pp. 05 5, 2004 IFORMS 07 Assumptio 2. Demads are uiformly bouded; i.e., there exists d< such that d < d for all. Assumptio 3. Margial productio ad ivetory holdig costs ad reveues are uiformly bouded; i.e., there exist c <, h<, ad r < such that c < c h < h, ad r < r for all. Assumptio 4. For each period, it is profitable to produce ad hold to satisfy the demad i the same period; i.e., r >c + h. I additio, it is ot profitable to lower the ext period productio cost by ot satisfyig curret demad; i.e., r >c +. Let Z i j deote the preset value at the begiig of period of the expected et profit i period, ifwe begi period observig a begiig ivetory i ad decide to produce to brig the ivetory level up to j, j i.wehave Z i j = c j i h j + r Emij D I additio, if D = d, the i + = j d +, where x + = max0x. A strategy = 2 3 is a sequece of policies, oe for each period where a policy for period is a colum vector of after-productio ivetory decisios, oe for each feasible state, i.e., possible begiig-ivetory level. We deote by i the after-productio ivetory level give by strategy whe we are i state i begiig period. deotes the set of all feasible strategies. By Assumptio, the set of feasible policies is fiite i each period, so that is compact i the product topology of compoetwise covergece. Give a strategy, let P m deote the probability trasitio matrix for ivetory levels begiig i period ad edig at the begiig of period m uder. Let Z deote the colum vector whose ith compoet is give by Z i i, i.e., the coditioal expected preset value begiig period of the et profit of producig to ivetory level i, give that the process eters ivetory level i at the begiig of period. Lettig Z m deote the expected total discouted et profit icurred i periods through m uder strategy, wehave m Z m = P Z = Fially, let Z be the preset worth of expected et profit over the ifiite horizo uder strategy. The, Z = P Z = We would like to fid a feasible strategy that maximizes the expected total discouted et profit over the ifiite horizo. Our ifiite horizo problem L is the max Z Because all per-period et profits are uiformly bouded, the discout factor is less tha oe, ad the set of feasible strategies is compact, the maximum above is attaied (Bea ad Smith 984). 3. Existece of Forecast Horizos 3.. The Fiite Horizo Lower Boud Problem L I this subsectio, we costruct a -horizo lower boud problem L for L, ad formulate it as a dyamic programmig problem. We call this problem a lower boud problem because it ca be show i later subsectios that i each period, the smallest optimal decisio for each state i this -horizo lower boud problem is a lower boud for optimal decisios of a ifiite horizo problem. The -horizo lower boud problem L is a -horizo problem i which the problem data from periods through is the same as that of the ifiite horizo problem. However, the udiscouted et profit i period is replaced by a termial value depedig oly o the state i that we eter period, idepedet of the ifiite horizo problem. We will assume without loss of geerality that excess ivetory over demad is to be sold at the ext-period productio cost, which is the recovered by productio i the ext period. This pair of cost ad reveue flows will exactly cacel each other out. I short, we must pay the productio cost for the begiig ivetory of each period. A strategy = 2 is a sequece of policies, oe for each period up to period. Because the et profit for period is idepedet of the decisio take, we do ot iclude a policy for period. However, it is sometimes coveiet to view as its policies followed by a arbitrary extedig sequece of feasible policies over the ifiite horizo. We take these to be policies resultig i zero productio. (The cotext withi which we ivoke the symbol should make clear which iterpretatio we are placig o it.) A policy for period is a colum vector of decisios, oe for each state. Thus, i is the decisio provided by strategy if we are i state i i period. is the set of all possible strategies, which is fiite by Assumptio. Give a strategy, let P m deote the probability trasitio matrix startig i period ad edig at the begiig of period m uder, m = 2. We ow defie the termial value fuctio for period. We set the udiscouted profit eterig state i i period to be c + h/ i, idepedet of the decisio take i period. ote that this represets the buy-back cost c of ivetory sold at the ed of the previous period plus the cost of carryig that ivetory idefiitely over the ifiite horizo. This cost, together with o prospect of sellig this ivetory, discourages ivetory buildup for period, thus lowerig the optimal productio level for period as iteded. Let Z deote the expected discouted et profit icurred i period through uder. The, Z = P Z + P Z =
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig 08 Operatios Research 52(), pp. 05 5, 2004 IFORMS where Z is the vector whose ith compoet is the termial value c + h/ i. We would like to fid a feasible strategy that maximizes the expected discouted et profit for this problem over its horizo. The -horizo lower boud problem L is the max Z Because expected discouted et profit is fiite ad the set of feasible strategies is fiite, the maximum above is agai attaied. Because L is a fiite horizo problem, we ca solve L by employig the followig dyamic programmig (DP) fuctioal equatios. Let F y deote the maximum expected discouted et profit from period through if we choose to produce the amout of product y i period. I additio, let V x be the maximum expected discouted et profit from period through if we begi period at ivetory level x, or equivaletly (because the previous periods sold edig ivetory is produced back), if we eed to produce at least x; i.e., V x = max F yyx V x = ( c + where for all <, y < ad () h ) x (2) F y = M y + EV + y D + ad (3) M y = c + h y + r Emiy D + c + Ey D + (4) = r c h y r c + Ey D + (5) because miy D = y y D +. We ca iterpret M y as the expected discouted et profit icurred durig period if we decide to produce the amout y at the begiig of period. 3.2. The Optimal Policy Structure for the -Horizo Lower Boud Problem L I this subsectio, we show that optimal policies are i the form of a threshold produce-up-to level. We ow recall Lemma 5 i Supplemet 2 i Deardo (982). Lemma. Let h be a covex fuctio o the iterval S, ad let g be a covex fuctio o a iterval T that cotais hx x S. Ifg is odecreasig o T, the ghx is covex o S. Proof. See Deardo (982). Lemma 2. If hx is covex ad gx is oicreasig cocave, the ghx is cocave. Proof. ote that gx is odecreasig covex ad ivoke Lemma. Lemma 3. If X is a radom variable ad fyx is cocave i y for each fixed value x, the EfyX is cocave. Proof. Follows from the fact that EfyX is a covex combiatio of cocave fuctios fyx. Lemma 4. For all <, M y is cocave. Proof. For all <, to show that M y is cocave, it suffices to show that each of its terms i (5) is cocave. Its first term r c h y is liear, ad thus cocave. By Lemma 3, Ey D + is covex because y x + is covex i y for each x. By Assumptio 4, r c +, ad we get r c + 0. Therefore, its secod term r c + Ey D + is cocave. Hece, M y is cocave i y. Theorem. I the -horizo lower boud problem L, there exist itegers S ad S for all such that the followig equivalece holds: F s = max { V x = F F yy0 y ad (6) s x s F x x > s (7) if ad oly if S s S. Proof. We will prove this theorem by iductively showig that for all, V x is oicreasig cocave. From (2), V x is oicreasig cocave. ow suppose for some <, V+ x is oicreasig cocave. To show that F y = M y + EV+ y D + is cocave, it suffices to show that each compoet is cocave. M y is cocave by Lemma 4. For each fixed x, because y x + is covex ad V+ x is oicreasig cocave by assumptio, Lemma 2 implies that V+ y x+ is cocave. Thus, EV+ y D + is cocave by Lemma 3. Hece, F y is cocave. Because F y is cocave i y ad, by Assumptio, the maximum i (6) is over a fiite set of itegers, we have that there exist itegers S ad S such that s satisfies S s S if ad oly if s satisfies (6). Because F y is cocave, it is odecreasig whe y s ad oicreasig whe y>s. Thus, (7) follows. To complete the iductio, from (7), we have that V x is oicreasig cocave. From Theorem, we coclude that S ad S are threshold values represetig, respectively, the smallest ad largest produce-up-to quatities optimal for eterig period i the -horizo lower boud problem L.
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig Operatios Research 52(), pp. 05 5, 2004 IFORMS 09 3.3. Policy Mootoicity L I this subsectio, we prove that both the miimum ad the maximum optimal produce-up-to levels of the -horizo lower boud problem L are mootoe odecreasig i the horizo legth. Let f x deote the margial optimal expected discouted et profit of producig x uits with o begiig ivetory at the begiig of period ; i.e., f x = F x F x x Also, let v x deote the margial optimal expected discouted et profit of startig period with begiig ivetory x; i.e., v x = V x V x x We will set x = V x+ V x +. We have, for all <, f x = F x F x = M x + EV + x D + M x + EV + x D + = m x + E + x D (8) where m x=m x M x (9) = c +h x+r EmixD +c + Ex D + c +h x r Emix D c + Ex D + = c +h +r w x+c + u x by (4) (0) =r c h r c + u x by (5), () where w x=emixd mix D 0 ad (2) u x=ex D + x D + (3) We ca iterpret w x as the margial expected demad satisfied i period if we decide to produce x. I additio, we ca iterpret u x as the margial expected begiig ivetory of period + if we decide to produce x i period. Because F x is cocave from the proof of Theorem, f x is oicreasig. Also, by Theorem, S s S just i case s attais the maximum i (6). Hece, f x > 0 0 <xs f x = 0 S <x S (4) f x < 0 S <x I additio, from (7), we have for S s S {, v x = 0 x s f x x > s (5) Lemma 5. For all x, v x 0, v x f x, ad v x is oicreasig i x. Furthermore, { v x = 0 whe f x 0 f x whe f x < 0 (6) Proof. From (4) ad (5), we have (6). Thus, v x 0 ad v x f x for all x. Furthermore, we kow from the proof of Theorem that F x is cocave i x. Hece, f x is oicreasig i x. Thus, v x is also oicreasig i x from (5). Lemma 6. v + x v x for all ad x. Proof. Case I: x S +. From (5) ad (2), ( v + x = 0 c + h ) = v x Case II: x>s +.Wehave v + x = f + x from (5) = c + h + r w x + c + u x ( c + + h ) u x from (8), (0), (3), ad (2) c + h hu x by (2) c + h h by (3) c + h h ( = c + h ) = v x Theorem 2. For all ad <, S ad S are mootoe odecreasig i. Proof. Proof by iductio. By Lemma 6, v + x v x for all x. Suppose for some <, v + + x v+ x for all x. From (8), f + y f y =m y+e [ + + y D ] m y E [ + y D ] =E [ + + y D + y D ] (7) By the iductio hypothesis, + + y D + y D 0 with probability oe. Thus, for all y, f + y f y (8) The, from (4), f + S f S >0. We have the, by (4), S + S. I additio, S + S because if S + < S, the by (4), f + S + <0. However, by (8), f S f S 0 by (4) ad we have reached a cotradictio. To complete the iductio, it is left to show v + x x for all x. v
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig 0 Operatios Research 52(), pp. 05 5, 2004 IFORMS Case I: x S S +. By (5), v + x = 0 = v x f Case II: S <x S +. By (4), (5), ad the fact that x is oicreasig from cocavity of F x, v + x = 0 >f S + f x = v x Case III: x>s + S. By (5) ad (8), v + x = f + x f x = v x 3.4. Optimal Policy ad Value Covergece for the -Horizo Lower Boud Problem L I this sectio, we show that optimal policies ad values of the -horizo lower boud problem coverge to those of the ifiite horizo problem. We begi with a lemma. Lemma 7. If as for all whe, = 2, the Z Z as. Proof. See the Appedix. ote that as for all is equivalet to writig as i the product topology of compoetwise covergece. Because feasibly exteded is i for all ad is compact ad i particular closed, we coclude that beig feasible for all implies its limit must be i, i.e., feasible. We tur to showig optimal policy covergece. Because Theorem 2 states that S ad S are mootoe icreasig i ad bouded above by Assumptio, S lim S ad S lim S must exist for all. Because for all = 2 both S ad S, = + + 2 is a covergig sequece of itegers, there exist for each = 2 such that S = S ad S = S The, for all S with S S S, we have S = lim S, where S S + S S for all. ote that S S S for all ad, hece, is a optimal produce-up-to threshold for the -horizo lower boud problem,. We coclude that all thresholds S betwee S ad S are limits of sequece of optimal -horizo lower boud problem thresholds. It remais to show that these thresholds are ifiite horizo optimal, i.e., that optimal policy covergece holds. Theorem 3. Optimal policy covergece holds; i.e., if S S S for all ad S S as, the S is a optimal produce-up-to threshold for the ifiite horizo problem (L). Proof. Let represet the policy of the -horizo lower boud problem correspodig to produce-up-to level S ad let correspod to the policy for the ifiite horizo problem of produce-up-to level S. Hece, as. We have already oted that because, the limit policy must be ifiite horizo feasible; i.e.,. Let ad ote that is feasible for the -horizo problem so that Z Z Hece, lim Z lim Z (9) But by Lemma 7, because, lim Z = Z (20) ad also by Lemma 7, because ˆ where ˆ, we get lim Z = Z (2) Hece, from (9), (20), ad (2) we coclude Z Z for all. Hece, is ifiite horizo optimal. ote from (20) i the proof of the previous theorem we get lim Z = Z where Z Z is the optimal value for the -horizo lower boud problem ad Z Z is the optimal value for the ifiite horizo problem. That is, we have demostrated optimal value covergece as well as optimal policy covergece. Also ote that we have prove that the ifiite horizo problem L has a optimal produce-up-to threshold structure, iherited from the fiite horizo problems. 3.5. Existece of Forecast Horizos for L I this subsectio, we costructively prove existece of a forecast horizo by showig that a horizo sufficietly log to geerate a cumulative demad exceedig the largest optimal iitial productio level must be a forecast horizo. I particular, we show that is a forecast horizo where } = mi { S < d k = 2 3 (22) k= Roughly speakig, if we thik of ivetory beig depleted uder a FIFOpolicy, o demad beyod horizo is optimally satisfied by productio i period. I this sese, is similar to the determiistic case of Smith ad Zhag (998), where a forecast horizo is the logest horizo over which it is optimal to carry a uit of ivetory. The followig assumptio assures that defied above exists ad is fiite.
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig Operatios Research 52(), pp. 05 5, 2004 IFORMS Assumptio 5. Demad is strictly positive ifiitely ofte; i.e., d > 0 ifiitely ofte. Lemma 8. I the -horizo lower boud problem L, for all,if0xd, the f y for all y x depeds oly o the problem data i period. Proof. For all x d, from (8), (), (3), ad the fact that x d,wehavef x = r c h, which depeds oly o the problem data i period. ote the importace of the assumptio of lost sales i cocludig Lemma 8. Lemma 9. I the -horizo lower boud problem L, for all, ifx>d, it is sufficiet to kowthe problem data i period ad to kow f+ y for all y x d to determie the value of f y for all y x. Proof. For all x>d ad all y x, from (8), f y = m y + E [ + y D ] = m y + E [ V + y D + V + y D +] d = m y + k=d [ V + y k+ V +y k+] PD = k Because k y implies V+ y k+ = V+ y k+ = V+ 0, wehave y f y = m [ y + V + y k+ V +y k+] k=d y PD = k = m y + v + y kpd = k k=d Settig k = y k, wehave y d f y = m y + k = v + k PD = y k (23) Lemma 5 implies that v+ k depeds o f + k ad k y d x d, ad thus the result follows. Lemma 0. For all y S + ad, f y is idepedet of the problem data beyod. Proof. For all, by applyig Lemma 9 recursively, startig with = ad edig with =, we have that, i geeral, it is sufficiet to kow the problem data i period 2 2 ad to kow f y for all y S + 2 k= d j to compute the value of f y for all y S +. However, by the defiitio of, S < k= d k, which implies S + 2 k= d k d. Thus, by Lemma 8, f y for all y S + 2 k= d j depeds oly o the problem data i period. Hece, for all y S +, f y is idepedet of the problem data beyod. Theorem 4. We have S = S ad S = S for all ; i.e., is a forecast horizo. Proof. From (4), f S + <0. For all,by Lemma 0, f y for all y S + is idepedet of the problem data beyod. Thus, f y = f y for all y S +. Because from (4) S ad S ca be computed if f y, y S + is kow, S = S ad S = S idepedetly of the iformatio beyod. 4. Computig Forecast Horizos for L 4.. The -Horizo Upper Boud Problem I this subsectio, we defie a -horizo upper boud problem U ad formulate it as a dyamic programmig problem. Aalogous to the lower boud problem L,we call U a upper boud problem because it will be show later that the largest optimal decisio for each state i each period of the -horizo upper boud problem is a upper boud to the correspodig optimal decisios of the ifiite horizo problem. A -horizo upper boud problem U is a -horizo problem i which the problem data from periods through are the same as that of the ifiite horizo problem except that there is a termial value fuctio which gives the udiscouted et profit for period depedig o the state of period, i. We ow defie the termial value fuctio Z for the -horizo upper boud problem U. That is, let Z represet the colum vector i which the ith compoet of the vector Z i, is the et profit i period give that we eter state i at the begiig of period. For all i, weset Z i = 0 ote that without loss of optimality it will ever prove worthwhile for a problem with horizo exceedig to geerate a ivetory level greater tha that yielded by U. The reaso is the c received per uit ivetory sold at the ed of period iu places a margial value o edig ivetory of, at most, c. It is thus o more costly to produce additioal ivetory i period at its cost c per uit. The problem U thus results i a maximal begiig-ivetory period ad i a correspodig maximal productio level begiig period, which upper bouds productio for the ifiite horizo problem. I summary, as we shall show below, the ordiary -horizo trucatio of the ifiite horizo problem upper bouds optimal first-period productio. Let Z deote the expected preset value of the et profit icurred i periods through uder i the -horizo upper boud problem U. The, Z = P Z + P Z = P Z = =
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig 2 Operatios Research 52(), pp. 05 5, 2004 IFORMS We would like to fid a feasible strategy that maximizes the expected total discouted et profit over the horizo. The -horizo upper boud problem U is as follows: { max Z } Because the set of feasible strategies is fiite, the maximum above is attaied. By dyamic programmig, we ca solve U by solvig the optimality equatios, V = max yyx F y, as i (), with (2) replaced by V x = 0 0 which implies ˆv x = 0 0 x< d (24) <x< d (25) ote that hat i fuctios idicates that the fuctios are for the -horizo upper boud problem U. Theorem 5. I the -horizo upper boud problem U, there exist itegers T ad T for all such that the followig equivalece holds: F s = max F y ad (26) yy0 { V x = F s if ad oly if T x s F x x > s (27) s T. Proof. Because V x is oicreasig cocave, the rest of the proof is similar to that of Theorem. Lemma. ˆv + x ˆv x for all x>0 ad = 2 Proof. For all = 2 ad x>0, we have by Lemma 5 that ˆv + x 0 =ˆv x are moo- Theorem 6. For all ad <, T toe oicreasig i. ad T Proof. The result ad proof for the lower boud problem goes through for the upper boud problem if we replace by + ad + by throughout. For example, V x becomes V + x, ad V + x becomes V x. The, by Lemma, v + x =ˆv x ˆv + x = v x The rest of the proof of Theorem 2, the, still holds. I Theorem 2, we have proved that for all, S S + ad S + S which implies that T + T ad T T + T for all, which i tur implies that T mootoe oicreasig i. ad T are We ow itroduce optimality equatios for the ifiite horizo problem. Let F y deote the maximum expected discouted et profit from period o i the ifiite horizo problem if we choose to produce y i period. I additio, let V x be the maximum expected discouted et profit from period o i the ifiite horizo problem if we begi period at ivetory level x. Similarly, f y F y F y, y>0 ad v x V x V x, x>0. Furthermore, let S ad S deote the miimum ad the maximum produce-up-to levels for the ifiite horizo problem, respectively. We have V x = max y yyx (28) F y = M y + EV + y D + (29) for all = 2 3 Lemma 2. S S T 2 3 Proof. Omitted. ad S S T,, = 4.2. Algorithm for Detectig Forecast Horizos We begi with a forward algorithm that solves the -horizo problems L ad U for ever greater horizos util ad if we get agreemet i the largest ad smallest correspodig iitial productio levels for the first period. Solutio Algorithm Step 0. =. Step. Solve L ad U.If S = T ad S = T, stop. Otherwise, let = +, the go to Step. The followig theorem assures us that if the algorithm stops, the we have ideed solved for the ifiite horizo problem s optimal first decisio. Moreover, the optimality of this iitial productio level is uaffected by data beyod the horizo for which the stoppig coditio was met. Theorem 7. If the algorithm stops at =, the is a forecast horizo. Proof. If the algorithm stops at =, by Theorem 2, Theorem 6, ad Lemma 2, we have S = T ad S = T for all idepedetly of the iformatio beyod. Theorem 8. The algorithm fiitely termiates; i.e., the stoppig coditio is evetually met.
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig Operatios Research 52(), pp. 05 5, 2004 IFORMS 3 Proof. It is sufficiet to show that S = T ad S = T where is defied by (22). By Theorem 0, for all y S +, f y is idepedet of the problem data beyod. However, the data from periods through ofl ad U are the same. Thus, f y = fˆ y for all y S +, which implies that S = T ad S = T. Theorem 9. If T s S for some, the s is a ifiite horizo optimal produce-up-to level for period. Proof. Cosider all. By Theorem 6, T is mootoically decreasig i. Therefore, T s By Lemma 2, T S is bouded below by S. Thus, s (30) By Theorem 2, S is mootoically icreasig i. Therefore, s S (3) From (30) ad (3), for all, S s S Thus, s is a optimal produce-up-to level for period for all lower boud problems with horizo. From the discussio i 3.4, we ca costruct a covergig sequece of optimal strategies for -horizo lower boud problems, = 2 3 4 such that s is a optimal produce-up-to level for period of all lower boud problems with horizo. The, s must be a optimal produce-up-to level of the limit of this covergig sequece. If s were ot ifiite horizo optimal, this would cotradict Theorem 3. Thus, s is a ifiite horizo optimal produce-up-to level for period. Theorem 9 is useful because it may allow us to obtai i advace a ifiite horizo optimal policy ot oly for the first period, but also for future periods, before the stoppig rule i the Solutio Algorithm is met. 4.3. Forecast Horizo Bouds I this sectio, we preset a formula for the ratio of the forecast horizo legth of the stochastic demad problem to the determiistic demad problem i terms of lower bouds o the miimum ad upper bouds o the maximum perperiod demads to be ecoutered. This ratio, together with the closed-form formula for a upper boud o the miimal forecast horizo for the determiistic demad problem preseted i Smith ad Zhag (998), results i a closed-form formula for a forecast horizo for our stochastic demad problem. Let be a forecast horizo whe the demad is determiistic, as give i Smith ad Zhag (998). We preset their formula here i our otatio, { } c + h = log c + h where h = if h >0 ad x deotes the smallest iteger strictly greater tha x. Oe may iterpret as a horizo log eough so that the discouted cost of producig oe uit ad holdig it through period is greater tha the discouted cost of producig this uit i period +; i.e., c + k h> c k= Suppose ow there exist d< ad d< such that d = sup d ad d = if d. Let = d/d. Defie = 2 +. We will show that is a forecast horizo for our stochastic demad problem. Lemma 3. For all ad x> d, f x c h + c + + v+ x d. Proof. For all x> d, x D with probability oe. From (3), we have u x = Ex D + x D + = From (8) ad (), because u x =, we have f x = c h + c + + E + x D However, v+ x is oicreasig i x by Lemma 5, ad x> d by assumptio. Thus, with probability oe, + x D v + x d Therefore, f x c h + c + + v + x d Lemma 4. For all >, S k= d. Proof. Proof by cotradictio. Suppose for some we have S > k= d k. Because S > k= d k > d,by Lemma 3 we get f S c h + c 2 + v 2 S d From Lemma 5, v 2 S d f 2 S d. Therefore, f S c h + c 2 + f 2 S d Applyig the same procedure with f S replaced by f2 S d,wehave f S c h + c 2 c 2 h 2 + 2 c 3 + 2 f 3 S d d 2
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig 4 Operatios Research 52(), pp. 05 5, 2004 IFORMS Applyig the same procedure recursively, we have S c k h k + c + f k= ( + v S d k ) k= From Lemma 5, v x 0 for all x. Thus, S c k h k + c + f k= By defiitio of, f S <0 This cotradicts (4). Therefore, for all, S k= d k. Theorem 0. is a forecast horizo for stochastic demad problems. Proof. Because is a forecast horizo for stochastic demad problems, it is sufficiet to show S < d k k= which implies by (22) that. By defiitio of d, k= d k k= By defiitio of, d = d (32) d d (33) By defiitio of d, d k / d so that k= d k= d k (34) Because we defie = 2 +, / >, we get k= d k > d k (35) k= However, by Lemma 4, d k S (36) k= From (32) to (36), we have k= d k > S which implies that. Therefore, is a forecast horizo. Table. The forecast horizo i days for the first ifiite horizo optimal productio level for a ratio of maximum to miimum daily demads of at most two. r v.2.4.6.8 2 0.2 02 4 6 8 0 2 0.2 0 6 0 4 8 22 0.2 005 0 8 26 34 42 0. 02 4 6 8 0 2 0. 0 6 0 4 8 22 0. 005 0 8 26 34 42 0.05 02 4 6 8 0 2 0.05 0 6 0 4 8 22 0.05 005 0 8 26 34 42 Corollary. If = +, the 2 +. Proof. = 2 + 2 + + From Corollary, we coclude that the effect of ucertaity i demad is to icrease the horizo we eed to forecast demad over by a costat plus a factor of at most oe plus the ratio of maximum to miimum per-period demads. Also ote that the stoppig rule of the Solutio Algorithm of 4.2 is always met at ay forecast horizo. We coclude that the Solutio Horizo Algorithm will termiate after at most iteratios. Table, derived from Table i Smith ad Zhag (998), gives umerical values for for a variety of parameters. I the table, we have assumed the ratio of maximum to miimum daily demads is at most two. The maximum uit productio cost is the factor u times the uit productio cost i period ; i.e., u = c/c while miimum ivetory cost is v times the first-period productio cost, i.e., v = h/c. The table provides the forecast horizo for various ivetory charges v per day ad iterest rates r per year for u, respectively, oe ad two. The results are show i Table. Because the Solutio Horizo Algorithm termiates withi a horizo of at most, we coclude from Table that it termiates very rapidly for these parameter values with a ifiite horizo optimal productio decisio after oly a few icremets of the horizo. Moreover, forecast horizos are evidetly remarkably short for problems with sigificat ivetory costs, sometimes just a few days i legth. Appedix Lemma 7. If as for all whe, = 2 the Z Z as. Proof. Let k k be large eough that = for all k whe k. Such a k exists because is u
Cheevaprawatdomrog ad Smith: Ifiite Horizo Productio Schedulig Operatios Research 52(), pp. 05 5, 2004 IFORMS 5 iteger ad uiformly bouded over for all. Moreover, k as k. By the defiitio of Z ad Z, we have for all, Z Z = P Z P Z = = P Z ow, for all k, because = for all k, we have P = P ad Z = Z for all k Thus, for all k ad k, Z Z = P Z P Z = = P Z k =k + k k P Z =k + P Z Hece, for all k, lim Z Z k =k k P Z k P Z + k k P Z =k + lim P Z = k k P Z =k + k k P Z =k because lim P Z =0. Assumptios, 2, ad 3 imply that expected costs ad reveues per period are uiformly bouded. Therefore, Z ad Z are uiformly bouded; i.e., there exist M< such that each compoet of Z ad Z is bouded by M for all. Hece, we have k P Z + k P Z =k =k =k k 2M 2M Thus, takig the limit as k,weget lim Z Z lim k 2M k = 0 Hece, lim Z Z =0or lim Z = Z Ackowledgmets This work was partially supported by the atioal Sciece Foudatio uder Grats DMI-973723, 982074, ad 9900267. Refereces Alde, J., R. L. Smith. 992. Rollig horizo procedures i ohomogeeous Markov decisio processes. Oper. Res. 40(Suppl. 2) S83 S94. Bea, J., R. L. Smith. 984. Coditios for the existece of plaig horizos. Math. Oper. Res. 9(3) 39 40. Deardo, E. V. 982. Dyamic Programmig, Models ad Applicatios. Pretice-Hall, Eglewood Cliffs, J. Federgrue, A., M. Tzur. 996. Detectio of miimal forecast horizos i dyamic programs with multiple idicators of the future. aval Res. Logist. 43 69 89. Garcia, S., R. L. Smith. 2000a. Solvig ostatioary ifiite horizo stochastic productio plaig problems. Oper. Res. Lett. 27 35 4. Garcia, S., R. L. Smith. 2000b. Solvig ostatioary ifiite horizo dyamic optimizatio problems. J. Math. Aal. Appl. 244(2) 304 37. Heradez-Lerma, O., J. Lasserre. 990. Error bouds for rollig horizo policies i discrete-time Markov cotrol processes. IEEE Tras. Automatic Cotrol 35(0). Karli, S. 960a. Dyamic ivetory policy with varyig stochastic demad. Maagemet Sci. 6 23 258. Karli, S. 960b. Optimal policy for dyamic ivetory process with stochastic demad. SIAM 8 6 629. Kleidorfer, P., H. Kureuther. 978. Stochastic horizos for the aggregate plaig problem. Maagemet Sci. 25 020 03. Lovejoy, W. S. 992. Stopped myopic policies i some ivetory models with geeralized demad processes. Maagemet Sci. 38 688 707. Morto, T. E. 978. The ostatioary ifiite horizo ivetory problem. Maagemet Sci. 24 474 482. Morto, T. E., D. W. Petico. 995. The fiite horizo ostatioary stochastic ivetory problem: ear-myopic bouds, heuristics, testig. Maagemet Sci. 4 334 343. Sethi, S., F. Cheg. 997. Optimality of (ss) policies i ivetory models with Markovia demad. Oper. Res. 45 93 939. Smith, R. L., R. Q. Zhag. 998. Ifiite horizo productio plaig i time varyig systems with covex productio ad ivetory costs. Maagemet Sci. 44 33 320. Sobel, M. J. 98. Myopic solutio of Markov decisio processes ad stochastic games. Oper. Res. 29 995 009. Topkis, D. M. 998. Supermodularity ad Complemetarity. Priceto Uiversity Press, Priceto, J. Veiott, A. F. 965. Optimal policy for a multi-product, dyamic, ostatioary ivetory problem. Maagemet Sci. 2 206 222. Zhag, R. Q. 998. ew results i ifiite horizo productio plaig. Techical report, Departmet of Idustrial ad Operatios Egieerig, Uiversity of Michiga, A Arbor, MI.