OPERATIONS RESEARCH Vol. 55, No. 2, March April 2007, pp. 367 377 iss 0030-364X eiss 526-5463 07 5502 0367 iforms doi 0.287/opre.060.0359 2007 INFORMS Capacity Maagemet for Cotract Maufacturig Diwakar Gupta Graduate Program i Idustrial ad Systems Egieerig, Departmet of Mechaical Egieerig, Uiversity of Miesota, Mieapolis, Miesota 55455, guptad@me.um.edu Lei Wag SmartOps Corporatio, Oe North Shore Ceter, 2 Federal Street, Pittsburgh, Pesylvaia 522, lwag@smartops.com Cotract maufacturers sell capacity uder differet terms to differet buyers. I a commo practice, the supplier offers a market stadard price ad lead-time combiatio for products listed i its catalog. At the same time, it strikes cotracts with some high-volume customers who require recurrig delivery of a custom product at a short otice, usually timed to meet the buyer s productio schedule. Whereas the maufacturer is obligated to satisfy demad from these customers, it ca dyamically choose which trasactioal orders to accept. I this paper, we aalyze two scearios. I the first sceario, the maufacturer produces cotractual orders o a make-to-order basis, ad i the secod, it maitais fiished goods ivetory for such orders. At each decisio epoch, the maufacturer decides which trasactioal orders to accept to maximize its log-ru expected profit. Whe cotractual items are made to stock, the maufacturer also chooses how much extra capacity to allocate to cotractual items, over ad above what is eeded to meet realized demad. We establish the structure of the optimal policies ad propose scalable heuristics whe optimal policies are hard to compute/implemet. Our models are the used to study the effect of demad variability o the optimal profit. We also show how the amout of capacity available, relative to demad, chages the desirability of servig either trasactioal-oly or both markets. Subject classificatios: maufacturig: cotract maufacturig, strategy; ivetory/productio: applicatios, policies; dyamic programmig/optimal cotrol: Markov. Area of review: Maufacturig, Service, ad Supply Chai Operatios. History: Received October 2004; revisios received September 2005, March 2006; accepted March 2006.. Itroductio May cotract (custom) maufacturers of compoets ad maufactured materials such as steel, plastics, ad glass face demad from two types of customers. A large umber of oe-off orders are from trasactioal (type-2) customers who are offered a market-stadard price ad lead-time combiatio for product families listed i the maufacturer s catalog. At the same time, aual cotracts are struck with some high-volume cotractual (type-) customers who require recurrig delivery of a custom product at a short otice, usually timed to meet the buyer s productio schedule. The key differece is that the supplier is obligated to satisfy orders from the latter (or icur shortage pealties), whereas it ca select which trasactioal orders to satisfy without pealty. However, oce a trasactioal order is accepted, it must be satisfied withi the promised delivery time, or else tardiess pealties accrue. We model two scearios. I the first model, called the MTO productio mode, all orders are produced i a maketo-order (MTO) fashio, whereas i the secod model, referred to as the MTS-MTO mode, the maufacturer may produce cotractual items i the make-to-stock (MTS) fashio. The MTO model is appropriate whe the cotract is for total busiess volume ad the actual item demaded by the cotractual customer might chage from oe order to aother, or ivetory is perishable/too expesive to hold. It is also appropriate for certai service providers, such as cargo carriers, who offer differet lead times at differet prices (stadard ad premium services). I both models, the maufacturer observes type- ad type-2 demads at discrete decisio epochs ad decides how may type-2 orders to accept to maximize the expected total discouted profit. Whe type- items are made to stock, the maufacturer also chooses how much extra capacity to allocate to type- items, over ad above what is eeded to meet the realized demad. We formulate the maufacturer s problem as a Markov decisio process (MDP) ad prove certai properties of the optimal value fuctio, which are the used to obtai optimal/heuristic policies. I the MTO sceario, the relevat decisio is how may type-2 orders to accept. We show that this decisio depeds oly o the total backlog of type-2 items ad ot o their due dates. We the prove that the optimal acceptace policy is a easy-to-implemet threshold policy that specifies a accept-up-to level (bookig limit). Whe maufacturers ca produce to stock, the correspodig MDP has multidimesioal state ad actio spaces, ad the optimal value fuctio does ot appear to have ay useful structure, which makes it difficult to compute a optimal policy. Therefore, we preset two implemetable heuristics. Our approach ca be used to address several maagerially importat issues. For example, it is possible to quatify the beefit to a maufacturer of egotiatig cotracts that limit the variability of type- demad by offerig limited quatity flexibility to the buyer. Our aalysis also shows that servig oly trasactioal orders may be more profitable whe capacity is 367
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig 368 Operatios Research 55(2), pp. 367 377, 2007 INFORMS tight. Fially, our models ca help evaluate the icremetal beefit of each extra uit of capacity ad of havig loger market-stadard lead time. The remaider of this paper is orgaized as follows. I 2, we provide a particular example that is partially resposible for our iterest i this class of problems. This sectio also cotais a brief literature review. The maufacturer s problem is formalized i two mathematical models. Notatio ad model formulatios are preseted i 3, followed by aalyses i 4 ad 5. We preset tests of goodess of heuristics, as well as maagerial isights, i 6 ad cocludig remarks i 7. The olie appedix is available at http://or.pubs.iforms.org/pages/collect.html. 2. Motivatig Example ad Literature Review We are motivated, i part, by our iteractios with a itegrated steel mill (ISM) that makes may differet types of fiished steel coils; for example, galvaized, ti- or chrome-plated, or hydro formed ito tubes. Each product also has a uique dimesioal ad steel-grade specificatio, with the result that the ISM produces literally thousads of ed products. However, the basic productio steps ad the process sequece are, by ad large, the same for all ed products with the exceptio of the fial step that may ivolve processig through a differet fiishig lie (see Deto et al. 2003 for details). Moreover, margial productio costs per to of steel are more or less product idepedet, ad ISMs have little pricig latitude o accout of itese competitio. The ISM is the preferred supplier of steel to several automobile maufacturers, who sig aual cotracts for custom-made products ad require reliable delivery withi the bouds specified by the cotract. Such cotracts, which accout for early 60% of the ISM s aual productio (i tos), are reewed if the ISM maitais or exceeds the specified delivery performace stadards. The ISM is cotractually boud to satisfy demad which is dictated by the weekly build schedule of the automakers, makig it ecessary to carry fiished goods ivetory i may istaces. These customers provide a demad forecast, usually o a quarterly basis, which is used to carry out rough-cut capacity requiremets plaig. The quarterly forecast is updated oly if there are market factors that lead to substatial revisios i the build schedule of the buyer. O a weekly basis, the buyers reveal the exact demad for the esuig period. Deviatios from the forecast are commo for a variety of local productio factors that are ot kow to the steel mill. That is, demad ucertaity is ot resolved util late i the process. The cotractual customers do ot ecessarily geerate more profit per to. Primary processes that produce molte steel must remai operatioal roud the clock o accout of the high cost of restartig after a shut dow. Therefore, ISMs remai profitable by carefully maagig the portfolio of cotracts with trasactioal customers, who place oe order at a time ad order from a umber of differet competig steel makers. I this eviromet, accout maagers usually offer a meu of prices ad fixed lead times up frot to build market share. Note that order bookig, productio plaig, ad order fulfillmet occurs i weekly time buckets, although trasactioal orders arrive cotiuously over time. Trasactioal orders are made to order due to the large umber of ed products i the ISM s product catalog. If a trasactioal order caot be delivered by the promised delivery date, the it is treated as a high-priority order i subsequet plaig periods. Customers are allowed to chage/cacel orders util the ext bookig decisio epoch, i.e., the start of the ext week s productio cycle. We ow tur our attetio to a review of related literature o order selectio ad productio decisios whe customers are differetiated by quoted lead times. Quoted lead times ca idicate both lead times that are fixed i advace ad kow by the firm (as i this paper) ad lead times that are determied dyamically accordig to the state of the system. I the latter approach, order selectio process is exogeous to the firm; see Keskiocak ad Tayur (2004) for a review. Our approach is also differet from Dueyas ad Hopp (995) ad Chatterjee et al. (2002), where customers make the accept/reject decisios based o dyamically quoted lead times. I a closely related work, Carr ad Dueyas (2000) model both accept/reject ad productio cotrol decisios whe the productio facility is a two-class M/M/ queue. While their work is motivated by a applicatio similar to ours, there are several importat differeces. I Carr ad Dueyas (2000), type-2 customers are ot quoted lead times, ad waitig cost is icurred over the etire time their orders remai i the system. I our approach, loger lead times for type-2 customers improve the maufacturer s capacity utilizatio as well as its ability to meet cotractual obligatios. Whereas Carr ad Dueyas show that the optimal policies are characterized by mootoe switchig curves, i our model, the optimal policy must also specify the amout that will be produced ad the umber of type-2 demad that will be accepted. Several recet papers model a maufacturer s productio ad ivetory decisios whe it sells capacity to differet customer classes. A key distictio betwee these papers ad our models is that they do ot model order selectio by the maufacturer. For example, Sobel ad Zhag (200) ad Frak ad Zhag (2003) cosider models i which cotractual customers demad is determiistic ad trasactioal demad is, respectively, either backlogged or results i lost sales. I Cattai et al. (2003), the maufacturer offers custom ad stadard products. The latter ca be produced via a more efficiet stadard capacity, ad therefore the focus of this research is o idetifyig the maufacturer s capacity purchase ad specializatio choices. Dobso ad Yao (2002) formulate a iteger programmig problem that ca be used to compute the best choices
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig Operatios Research 55(2), pp. 367 377, 2007 INFORMS 369 of product offerig, prices, ad MTS/MTO decisios simultaeously. All products are made o a shared productio facility ad demad decreases liearly i both price ad lead time. The key distictio betwee their model ad our approach is that we cosider radom demad, whereas i their model demad occurs at a costat rate. Charsirisakskul et al. (2004) formulate a determiistic mixed-iteger programmig model for the problem of order selectio, due date settig, ad productio schedulig. They carry out a umerical study to compare the relative beefits of due date flexibility ad the flexibility to serve partial orders. 3. Model Formulatio We model the maufacturer s problem as a discrete-time, ifiite-horizo MDP. A separate model is preseted for the MTO ad the MTS-MTO productio modes. Bold-face ad upper-case otatio idicates vectors ad radom variables, respectively (to avoid cofusio with lower case L, we use script l). All parameters are statioary, i.e., they do ot deped o the period idex. Formalizatio of these models requires the followig commo otatio: = oe period productio capacity (iteger valued). l = promised delivery time for trasactioal customers l 2. D j = type-j demad (radom, oegative, ad iteger valued). r j = reveue (cotributio margi) from each type-j demad (assumed oegative). h = ivetory carryig cost rate for type- items. h 2 = tardiess pealty rate for ot meetig a type-2 demad withi l periods. p = per uit lost sales pealty/additioal cost of satisfyig type- demad by spot market/subcotractor purchases. = oe-period discout rate, where 0 < <. We adopt the covetio that the lead time for cotractual orders is oe period ad the miimum lead time for trasactioal orders is two periods. This stems from the fact that demads are realized at the start of a period ad that the earliest that supply ad demad matches ca occur is at the ed of that period. The maufacturer gives higher priority to type- orders whe makig productio decisios because future cotracts are depedet o delivery performace. Usig margial aalysis, the beefit of usig a uit of capacity to produce a type- item is p + r, whereas the worst-case cost is r 2 + h 2 /, which happes whe a type-2 order remais perpetually backlogged as a result of this actio. Therefore, a sufficiet coditio uder which it is optimal to give higher priority to type- orders is r + p r 2 + h 2 /. I some umerical tests we performed, givig higher priority to type- items is optimal so log as p + r >r 2 + h 2, a much weaker coditio. O a differet ote, it is reasoable to expect r 2 r, but we make o such assumptio i our models. However, this relatioship is assumed i our umerical examples. The vector s = s s l deotes the backlog of type-2 orders at the start of a period. For each j<l, s j 0 deotes the umber of type-2 orders that have waited precisely j time periods, whereas s l 0 deotes the umber of type-2 orders that have waited at least l periods. If these orders are ot filled by the ed of the curret period, the the maufacturer will icur a tardiess pealty for each remaiig order i s l. Orders that are already tardy at the start of the curret period icur a tardiess pealty i earlier periods, ad therefore we eed ot keep track of the due dates of orders that are more tha l periods old. Cash ad material flows occur at the ed of each period, whereas operatioal decisios are made at the start of each period. 3.. The MTO Productio Sceario The maufacturer observes s ad realized demads d ad d 2, ad the chooses how may type-2 orders to accept, deoted by a s d d 2 to obtai the followig optimal value fuctio: v s =E r D p D + h 2 s l D + + [( l ] } +E max {r 2 s m +a ) D + + v s a 0 D 2 m= () where + equals max 0, m equals mi m, ad the argumet of a is suppressed for otatioal coveiece. The terms o the first lie above deote cotributio ad lost sales pealty from type- demad, ad the curretperiod tardiess pealty for type-2 orders. The terms o the secod lie cotai the expected beefit from usig leftover capacity for type-2 items ad the optimal future beefit from the ext period forward. Vector s deotes the system state at the ed of the curret period, which is determied accordig to the followig system dyamics equatios: s l = s l + s l 2 D + + ( l ) s j [s = + ] + j D s m ( s [a = D m=j for j = 2 3 l 2 l ) + ] + s m m= ad 3.2. The MTS-MTO Productio Sceario The state of the system is ow described by i s, where i deotes the ivetory level of type- items. The maufacturer makes two decisios: () how may extra type- items to produce (i.e., i excess of mi d i + ), ad (2) how may uits of ew trasactioal demad to accept. Let (2)
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig 370 Operatios Research 55(2), pp. 367 377, 2007 INFORMS v i s deote the optimal value fuctio whe the state is i s. The, [ { v i s = E max r i + x D x y [( l ) ] + r 2 s m + y x m= h i + x D + h 2 s l + x + } p D i+x + + v i+x D + s where D i + x &0 y D 2 ] (3) Key compoets of (3) are as follows. Whe the maufacturer decides to produce x uits of type- items, where d i + x, ad to accept y uits of the realized type-2 demad, where 0 y d 2, it ears r i + x d from type- customers ad r 2 l m= s m +y x from type-2 customers. After supply demad matches occur, the type- ivetory level is i + x d +, whereas the type-2 backlog vector, deoted by s, is determied as follows: s l = s l + s l 2 + x + ( l ) s j [s = + ] + j x s m j=2 3 l 2 ad m=j (4) ( l ) s [y = + ] + x s m m= The ivetory holdig ad tardiess costs are, respectively, h i + x d + ad h 2 s l x +. The cost of type- shortages is p d i + x +. 4. The MTO Productio Model The MDP described i () is computatioally challegig because its state space is the l -fold cross product of oegative itegers, i.e., s 0 0. However, as we show i Theorem i EC.2 of the e- compaio, the value fuctio i () is the sum of well structured fuctios of the partial sums of s j s. The ature of the value fuctio, i tur, results i a relatively simple form of the optimal acceptace policy. Theorem. Let s = s + +s l be the total backlog of type-2 customers at a decisio epoch. The the optimal type-2 order acceptace policy is a threshold policy defied by b d = b + d +, where b is a costat ad d is a realizatio of D.Ifs<b d, the the optimal policy accepts arrivig type-2 orders util either all orders are accepted or the backlog reaches b d, whichever occurs first. No ew type-2orders are accepted if s b d. Why does the order selectio policy deped oly o s ad d? This pheomeo ca be explaied with the help of the followig ituitive argumets. I view of the fact that type- orders have higher priority, a radom amout of capacity (equal to D + ) is available i each future period to meet type-2 orders. Moreover, type-2 orders are served i the order of arrival. Therefore, the beefit from acceptig each additioal type-2 order, which depeds o whe it is served, is a fuctio of the total backlog of type-2 orders ad the exact amout of leftover capacity available i the curret period. The latter is determied by d. 4.. Effect of Model Parameters We study the effect of iput parameters o a maufacturer s expected profit ad acceptace policy through two types of comparisos. This sectio is devoted to stochastic comparisos, whereas 6 presets umerical experimets. Stochastic orders have bee used i may recet studies to facilitate comparisos; see Shaked ad Shathikumar (994) ad Müller ad Stoya (2002) for details. We are cocered here with the cocepts of stochastically ordered demads accordig to the usual, covex, ad icreasig cocave orders, deoted, respectively, by st, cx, ad icv. For radom variables X ad Y, X st Y (respectively, X cx Y or X icv Y )ifef X Ef Y for all icreasig (respectively, covex or icreasig cocave) fuctios f for which the expectatios exist. Proposotio. (a) The optimal expected profit v s, as show i (), is icreasig i r, r 2,, ad decreasig i p, h 2. (b) Suppose that D 2 icv D 2 ad v s deotes the optimal value fuctio with demad D 2. The, v s v s for all s. (c) The optimal threshold b d is icreasig i ad r 2, ad decreasig i h 2 ad. (d) Suppose that D st D. Let b d deote the optimal threshold for the MDP with demad D. The, b d b d. Part (a) above follows from observig the right-had side of (), which cofirms the required effect of each model parameter for a arbitrary order-acceptace policy, ad hece also for the optimal policy. Part (b) is proved i Appedix B. Note that X st Y X icv Y ad Y cx X X icv Y (see Müller ad Stoya 2002 for details). Therefore, part (b) above cofirms the ituitio that the maufacturer will reap greater beefits whe it faces a more cosistet ad larger type-2 demad. At first glace, oe might be tempted to coclude that larger type- demad is also beeficial. This turs out ot to be true i geeral. We preset a couter example i 6. Proofs of parts (c) ad (d) are preseted i Appedix C. 5. The MTS-MTO Productio Model The MDP described i (3) is cosiderably more difficult to aalyze. I fact, we have bee able to obtai limited results that apply oly to the l = 2 case. If l>2, the l-dimesioal optimal value fuctio, v i s, is either cocave or
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig Operatios Research 55(2), pp. 367 377, 2007 INFORMS 37 expressible as a sum of well-structured fuctios. Moreover, eve if oe could compute the optimal actio pair for each state i s ad demads d ad d 2, the resultig l+2 -dimesioal look-up table quickly becomes impractical to store. 5.. Aalysis for l = 2 We show that whe l = 2, v i s is joitly cocave i i s. This helps streamlie the computatio of the optimal policy. Defie v i s = v i s v i s, 2 v i s = v i s v i s, ad 3 v i s = v i s v i s. The, the properties of v i s are as stated i Theorem 2 (see EC.3 of the e-compaio for a proof). Theorem 2. (a) The icremetal beefit of holdig a additioal uit of type- ivetory is bouded by r + p. Also, the icremetal beefit of admittig oe more type-2 order is bouded by r 2. That is, v i s r + p ad 2 v i s r 2. (b) v i s is joitly cocave i i s, which implies v i s is oicreasig i i, 2 v i s is oicreasig i s, ad 3 v i + s+ 3 v i s. Eve though the value fuctio is well behaved, the structure of the optimal policy is quite complex (results are ot show i the iterest of brevity). For each istace of the problem, the optimal actio pair x y depeds o the quadruplet i s d d 2 ; it caot be obtaied from pre-calculated thresholds. 5.2. Heuristics We preset two easy-to-implemet heuristic policies that are cosistet with observed properties of the optimal policy whe l = 2. We also obtai a upper boud. Heuristic H. The first heuristic coverts a problem with l>2 to a equivalet problem with l = 2 ad tardiess pealty rate h 2 = h 2/ l ; the l-dimesioal state is coverted to a equivalet two-dimesioal state via the trasformatio s = l m= s m. That is, the time allowed for deliverig type-2 items is shorteed to two periods, ad the per-uit tardiess pealty is commesurately reduced. The mai advatage of H is the reduced computatioal effort. We store the maximizig actios, x i s d d 2 ad y i s d d 2 i a look-up table ad calculate the lower boudig value fuctio, which we deote as v LB. Heuristic H2. This heuristic specifies two critical umbers: the maximum type- ivetory I ad the maximum workload S, which ca be used to determie the productio ad order acceptace decisios ad the correspodig value fuctio, v LB2. For each give I S pair, type- demad is met up to i +. Ifd <i+, the capacity is used to icrease type- ivetory up to I. After that, ay leftover capacity is used to reduce the backlog of type-2 items. This meas actios x ad y ca be recovered from parameters I ad S via the followig operatios: x i s d d 2 = i + d i + + mi i + d + I i d + + ad y i s d d 2 = S x l m= s m + d 2. We use a two-dimesioal exhaustive search for fidig the optimal I ad S. The Upper Boud (UB). The upper boudig value fuctio is obtaied by settig h 2 = 0. Now, because there are o tardiess costs, the maufacturer should admit all type-2 orders that arrive. It ca be show by iductio that v UB, the correspodig value fuctio, does ot deped o the distributio of curret type-2 backlog by due dates (this result is also ituitive because the iformatio o due dates is redudat if oe does ot charge tardiess pealty). So, i s ca be replaced by i s, where s = l m= s m is a scalar. The optimality equatio for v UB is as follows: v UB i s = E max r i + x D + r 2 s + D 2 x h i + x D + p D i + x + + v UB i + x D + s+ x + D 2 + where x D i + (5) Evaluatio of v UB i s requires o more computatioal effort tha what is eeded to obtai the value fuctio for the problem with l = 2. Whe capacity is isufficiet relative to total demad, acceptig all type-2 demad poses computatioal challeges because s becomes ubouded. I such cases, we obtai the upper boud by assumig ample type-2 demad. Now the maufacturer admits exactly what it ca produce i each period ad carries o backlog. Specifically, s + D 2 x i the first lie of Equatio (5) is replaced with x ad the system state is captured etirely by i. 6. Numerical Experimets I priciple, the optimality equatios i () ad (3) ca be solved via either value iteratio or policy improvemet algorithms. I practice, this is difficult because our model has ifiite states ad such algorithms will require a ifiite umber of operatios at each iteratio. Therefore, we use a approximate value iteratio algorithm, which is adapted from Puterma (994, pp. 239 243). (Key steps of our approach ca be foud i EC.4 of the e-compaio.) I the remaider of this sectio, we preset umerical examples to compare the performace of bouds ad heuristics ad to illustrate maagerial implicatios of our models. 6.. Accuracy of Bouds ad Heuristics Iput parameters (data) used i tests of accuracy of bouds ad heuristics ca be foud i Table. Table 2 shows the various bouds for E v i s. We used the approximate value iteratio algorithm to fid v i s for each state i s ad discrete-evet simulatio to estimate the steady-state probabilities uder H ad H2. Together, these quatities were used to calculate E v i s. The remaiig experimetal desig is as follows.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig 372 Operatios Research 55(2), pp. 367 377, 2007 INFORMS Table. Data used i experimets for testig the accuracy of heuristics ( = 0). Case o. r r 2 p h h 2 ED ED 2 0 20 25 2 5 5 0 9 2 5 20 25 2 5 5 0 9 3 5 20 25 2 5 5 0 9 4 0 5 25 2 5 5 0 9 5 0 30 25 2 5 5 0 9 6 0 20 0 2 5 5 0 9 7 0 20 50 2 5 5 0 9 8 0 20 25 0.2 2 5 5 0 9 9 0 20 25 5 2 5 5 0 9 0 0 20 25 0.5 5 5 0 9 0 20 25 5 5 5 0 9 2 0 20 25 2 3 5 0 9 3 0 20 25 2 9 5 0 9 4 0 20 25 2 5 3 0 9 5 0 20 25 2 5 9 0 9 6 0 20 25 2 5 5 0 8 7 0 20 25 2 5 5 0 95 The base case, labeled Case, is listed i the first row. For each case, we provide the expected profit correspodig to the two heuristics ad the upper boud whe l is either 2, 3, or 5. Note that whe l = 2, the H heuristic is optimal. For H2, we also provide the optimal values of the policy parameters, deoted as I ad S. We systematically vary the iput parameters of our models. Each time a differet parameter is set first lower, ad the higher, tha the base case, with all other parameters remaiig at their base values. The total umber of idividual problem istaces solved is 5. I each case, the percet differeces betwee H ad the upper boud are reported as %, whereas the correspodig quatity for H2 is % 2. The oly exceptio to this is the row correspodig to l = 2, whe i each case, % 2 shows the differece betwee H2 ad the expected optimal (or H) value. The aggregate performace of the two heuristics is as follows. For l = 2, H is optimal ad therefore the correspodig value of % is a measure of the goodess of the upper boud. The average gap i the 7 experimets reported is 2.5% ad the maximum gap is 4.79%. H2 is geerally more accurate ad i each problem istace, % 2 becomes smaller as l icreases. This is particularly satisfyig as we ca expect the heuristic to cotiue to perform well for large l. The average values of % 2 are.55%,.87%, ad 0.6% for l equal to 2, 3, ad 5, respectively; whereas the correspodig maximum values are 4.43%, 3.5%, ad 3.98%. Overall, the average % 2 is.34%. The average % values are.50% ad 3.50% with l = 3 ad l = 5, respectively. The correspodig maximum errors are 2.75% ad 8.53%, respectively. To highlight parametric isights from Table 2, cosider the effect of varyig r 2 ad E D, which is studied i Cases 4 ad 5, ad 2 ad 3, respectively. I Cases 4 ad 5, smaller (larger) r 2 is associated with smaller (respectively, larger) S. I Cases 2 ad 3, larger E D lowers optimal profit. Although we do ot kow the expected optimal value for l = 3 ad 5, both lower ad upper boud with larger E D are smaller tha the lower boud with smaller E D. This example therefore shows that whe the maufacturer is cotractually boud to satisfy type- demad, larger demad from such customers does ot ecessarily improve optimal profits, eve whe >E D. 6.2. Maagerial Isights We observed i Table 2 (Cases 2 ad 3) that larger type- demad ca decrease optimal profits i the MTS- MTO productio eviromet. Here, we focus o the effect of stochastically larger D i the MTO eviromet. I the esuig example, D ad D 2 are Poisso distributed with E D = 4 ad E D 2 = 5. D is also Poisso distributed with E D = 6, which immediately implies that D st D. Other relevat parameters are = 0 9, r = 0, r 2 = 20, p = 25, h 2 = 2, = 0, ad l = 2. With these data, the optimal profits are E ˆv s = 322 ad E v s = 372 5, cofirmig that larger type- demad might ot be beeficial. We also ivestigated the effect of demad variability o E v i s i the MTS-MTO model. For brevity, these results are ot show. We observed that a higher coefficiet of variatio of D (respectively, D 2 ) lowers expected optimal profit ad that a higher variability of D is more detrimetal. This is partly because the maufacturer ca mitigate the effect of variability of D 2 by havig the flexibility to serve type-2 orders over l periods. Figure plots the percet type- demad that is lost due to supply-demad mismatch (left vertical axis) ad expected profit (right vertical axis) as a fuctio of the productio capacity, which is expressed as a proportio of the expected type- demad. It shows that the maufacturer eeds to have sigificatly more capacity tha mea type- demad if shortages must be kept small. The MTO ad the MTS-MTO approaches are very close. I fact, profit differeces are ot distiguishable whe either /ED > 5 or /ED < 0 5. Similar experimets were also performed by varyig E D, rather tha. These experimets showed that higher E D ca be detrimetal, eve whe r >r 2, i situatios where /ED is ot sigificatly more tha oe. I Figure 2, we compare the optio of servig oly type-2 demad with the optio of servig both types of demads i either the MTO mode or the MTS-MTO mode. Data are the same as above, except h = is the oly case cosidered. Note that the optio to serve both types of demad domiates the optio to serve oly type- demad because the maufacturer ca always achieve profits equal to the latter strategy by simply decidig ot to accept ay type-2 orders. However, as Figure 2 shows, it is possible that servig oly trasactioal orders might be more profitable. Note that the MTO ad the MTS-MTO modes geerate similar profits.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig Operatios Research 55(2), pp. 367 377, 2007 INFORMS 373 Table 2. Accuracy of upper boud ad heuristic policies. Case o. l E v i s or v LB % I S v LB2 % 2 v UB 2 407 7 3 42 0 8 382 5 82 455 8 3 424 7 2 8 0 22 49 2 2 58 5 42 0 3 0 0 30 446 9 0 6 2 2 58 3 4 6 0 8 33 5 2 8 206 5 3 75 4 2 64 0 22 70 3 3 09 5 63 8 3 67 0 30 98 0 0 7 3 2 657 0 2 90 0 8 63 3 58 705 3 674 0 86 0 22 668 2 22 5 660 8 2 67 0 30 695 5 0 57 4 2 72 3 3 22 0 7 55 8 43 20 3 86 2 2 0 0 2 84 5 2 6 5 80 4 2 52 0 29 206 9 0 27 5 2 882 2 2 44 0 9 837 8 2 4 928 3 905 2 20 0 24 89 3 95 5 876 7 2 74 0 3 927 9 0 0 6 2 44 4 3 22 0 8 385 8 2 07 459 9 3 429 7 2 0 22 422 6 2 62 5 406 4 3 8 0 3 450 0 68 7 2 405 3 2 65 0 8 376 9 2 06 442 6 3 42 2 50 0 22 43 8 2 03 5 406 7 2 55 0 29 44 0 0 8 2 444 0 03 8 382 8 4 43 458 9 3 449 6 0 64 22 49 5 2 78 5 405 7 3 78 0 30 447 0 0 82 9 2 382 5 4 79 0 8 382 4 0 0 448 8 3 46 4 2 28 0 22 49 3 2 08 5 428 8 40 0 30 446 6 0 5 0 2 48 2 2 65 0 24 402 2 4 455 8 3 439 6 3 0 30 438 7 9 5 408 6 3 35 0 33 454 2 0 2 400 2 3 97 0 6 364 2 2 64 455 8 3 46 8 2 75 0 20 406 5 3 5 5 396 6 4 24 0 27 44 3 0 2 2 297 0 0 30 0 29 296 9 0 00 300 8 3 299 7 0 09 0 34 299 7 0 09 5 297 6 0 25 0 34 300 3 0 04 3 2 00 0 6 5 008 0 0 22 0 8 3 00 0 0 8 5 0 2 0 06 5 853 6 8 53 0 4 973 0 3 98 4 2 084 6 2 0 30 084 6 0 00 096 8 3 088 0 80 0 3 09 8 0 45 5 089 8 0 64 0 30 092 3 0 4 5 2 494 2 0 6 0 3 485 0 62 496 7 3 495 8 0 06 0 5 493 6 0 2 5 489 8 0 46 0 9 496 3 0 02 6 2 703 0 3 76 0 8 688 8 2 06 729 5 3 7 2 59 0 22 709 5 2 8 5 702 4 3 86 0 30 723 5 0 82 7 2 2 88 2 2 75 0 8 2 77 7 68 2 895 6 3 2 853 49 0 22 2 838 5 2 0 5 2 840 0 96 0 30 2 893 8 0 06 I all 7 cases, the value of % i the row correspodig to l = 2 shows the accuracy of the upper boud. I all 7 cases, the value of % 2 i the row correspodig to l = 2 shows the differece betwee H2 ad the expected optimal (or H) value.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig 374 Operatios Research 55(2), pp. 367 377, 2007 INFORMS Figure. Type- lost sales (%) 50 45 40 35 30 Effect of productio capacity, calculated as a proportio of the mea type- demad, o type- lost sales ad optimal profit. 3,000 2,500 2,000,500 25,000 20 Lost sales: MTO Lost sales: MTS-MTO with h = 5 Lost sales: MTS-MTO with h =5 Profit: MTO 500 0 Profit: MTS-MTO with h = Profit: MTS-MTO with h =5 0 5 500 0 0.5.0.5 2.0 2.5 Productio capacity (as a proportio of ED ) Note. Model parameters are = 0 9, r = 0, r 2 = 20, h 2 = 2, E D = E D 2 = 0, l = 2, p = 25, ad h is either or 5. 7. Cocludig Remarks I this paper, we preset two models for a cotract maufacturer who sells productio capacity to two differet types of customers uder differet terms. The maufacturer exercises cotrol over how may trasactioal orders to accept ad o how much extra capacity to allocate to the productio of cotractual items beyod what is eeded to meet demad i the curret period. We show that a threshold policy that depeds oly o the total backlog of trasactioal orders is optimal whe all productio occurs i the MTO mode. For the MTS-MTO mode (i.e., whe the maufacturer may produce cotractual items to stock), a two-critical-umber policy performs well, although the Figure 2. 2,500 2,000 Expected profit Iteractio betwee productio capacity, calculated as a proportio of the mea type- demad, ad optimal profit i differet markets. optimal policy has a complicated structure. I may examples, MTO ad MTS-MTO modes geerate similar profits. A maufacturer ca adapt our approach to quatify the beefits of better egotiated cotracts with type- customers. For example, the maufacturer ca egotiate the maximum capacity that it ca be expected to make available to the cotractual customers i each period. Alteratively, buyer s quatity flexibility ca be costraied, e.g., by defiig a miimum ad a maximum purchase quatity i each period. 8. Electroic Compaio A electroic compaio to this paper is available as part of the olie versio that ca be foud at http://or.joural. iforms.org/ecompaio.html. Appedix Throughout this sectio, we use sq j to deote jm=q s m, ad ad R to deote the sets of iteger ad real umbers, respectively. A real-valued fuctio + R is defied to be cocave if ad oly if s+ s is oicreasig i s. We also defie F as the set of all + R fuctios that are oicreasig ad cocave. Similarly, is the set of fuctios such that if, the () is cocave, ad (2) 0 r 2. It is easy to verify that if F, the. (Coditio () applies because is cocave, ad coditio (2) follows from the fact that 0 0 0 <r 2 Lemma i EC. proves certai properties of fuctios i ad F, ad Theorem i EC.2 establishes a key property of the optimal value fuctio v s for the MTO productio model. These results are used i the proofs preseted below. A. Proof of Theorem From the optimality Equatio () ad Theorem i EC.2, for a fixed realizatio d of type- demad ad a fixed realizatio d 2 of type-2 demad, the optimal actio is obtaied as follows: a s d d 2 = arg max d s l +a a 0 d 2, where d, d s = r 2 s d + + s d + +, ad = lim. The fuctio d is cocave from the fact that d. Next, we defie b d = max b 0 d b d b 0. The, from the cocavity d, the optimal solutio a s d d 2 must be Expected profit,500,000 500 0 500 Serve D 2 oly Serve both D ad D 2 i MTS-MTO mode Serve both D ad D 2 i MTO mode 0.5.0.5 2.0 Productio capacity (as a proportio of ED ) a s d d 2 mi b d s l d 2 if 0 s l <b d = 0 otherwise Note that if d >, d s = s is idepedet of d. We defie s = d s whe d > ad b = max b 0 b b 0. Whe d, wehave d s = r 2 s d + s + d +. Takig the differece betwee d s ad d s whe d, wehave
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig Operatios Research 55(2), pp. 367 377, 2007 INFORMS 375 d s d s = r 2 s d + s + d + r 2 s d s + d + = r 2 s + d + + s + + d + r 2 s d + s + d + = d s + d s (A) Thus, whe d, wehave b d = max b 0 d b d b 0 = max b 0 d b + d b 0 = max b 0 d b d b 0 = b d I the above, the secod equality comes from (A). The last equality follows from the defiitio of b d. Therefore, we have b d = b + d +. B. Proof of Propositio (b) We will show that v s v s for all s. To keep the otatio simple, we preset the argumets with l = 2. The same argumets also exted to situatios whe l>2 with straightforward chages. The proof is by iductio o. For =, we have v s = E r D p D + h 2 s D + + + E r 2 s + D 2 D + v s = E r D p D + h 2 s D + + + E r 2 s + D 2 D + from Equatio (EC2) i EC.2 of the e-compaio because v 0 s = v 0 s = 0. Because s d 2 = r 2 E s + d 2 D + is icreasig ad cocave for all s, it follows that v s v s for all s, so the result holds for =. For a arbitrary >, suppose that v s v s for all s. For a realizatio d of type- demad, we defie d s = r 2 s d + + v s d + +. The d is cocave from Theorem ad part (a) of Lemma (both ca be foud i the e-compaio). Let b d = max b d b + d b 0 (B2) be the optimal accept-up-to level for type-2 demad i period for the problem with demad sequece D 2. A optimal policy for a MDP with demad sequece D 2 prescribes actio ā s d = b d s +. Note that ā s d ad b d do ot deped o D 2 i period. The, v + s = E r D p D + h 2 s D + + + E V s D D 2 (B3) where we defie V s d d 2 = d s +ā s d d 2. Takig the differece betwee V s d d 2 + ad V s d d 2,we obtai V s d d 2 + V s d d 2 d s + d 2 + d s + d 2 if d 2 < ā s d = 0 if d 2 ā s d Whe 0 d 2 < ā s d = b d s +, it follows that s + d 2 < b d, so from (B2), we have that d s + d 2 + d s + d 2 0. A immediate cosequece is that V s d is icreasig. Moreover, V s d is cocave because d is cocave. So, V s d d 2 is a icreasig ad cocave fuctio i d 2 0. By (B2), ā s d does ot deped o the distributio of D 2 i period, so it follows that ā s d is also a optimal actio i state s at period with demad D 2. So, a s d 0, E V s D D 2 = E D s +ā s D D 2 E D s + a s D D 2 Fially, we obtai v + s E r D p D + (B4) h 2 s D + + + E V s D D 2 (B5) = E r D p D + h 2 s D + + + E D s +ā s D D 2 E r D p D + h 2 s D + + + E D s + a s D D 2 (B6) = E r D p D + h 2 s D + + + r 2 s + a s D D 2 D + + v s + a s D D 2 D + + E r D p D + h 2 s D + + + r 2 s + a s D D 2 D + + v s + a s D D 2 D + + = v + s I the above, iequality (B5) follows from (B3) ad the assumptio that D 2 icv D 2. Iequality (B6) follows from (B4) with a s d = a s d, where this particular a s d is a optimal actio i state s, for a realizatio d of type- demad i period with demad sequece D 2. Fially, the last iequality follows from the iductive hypothesis. So, the result holds for +. To complete the proof of the theorem, ote that v v ad v v as. Therefore, the result will hold for the optimal value fuctios of v s ad v s for all s.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig 376 Operatios Research 55(2), pp. 367 377, 2007 INFORMS C. Proof of Propositios (c) ad (d) We show complete details oly of the proof that b d is icreasig i, where d is a realizatio of type- demad. The proofs of sesitivity with respect to r 2, h 2, are similar (therefore omitted). We also show how our argumets help establish the fact that D st D implies b d b d. We cosider two scearios i which oe-period productio capacities are ad, respectively, with. We attach a bar ( ) to deote higher productio capacity; e.g., v deotes the MDP value fuctio with capacity. To keep the otatio simple, we preset the argumets with l = 2. The same argumets also exted to situatios whe l>2 with straightforward chages. The proof relies o the followig three argumets: () v s v s v s v s for all s ad each 0. (2) d s d s d s d s for all s ad each 0. (3) b d b d for each 0. Recall that is the iteratio idex i the value-iteratio algorithm. The fuctio represets the terms withi the maximizatio operator i Equatio (EC2) i EC.2 of the e- compaio. That is, d s = r 2 s d + + v s d + +. Also, b d = max b 0 d b + d b 0 ad v + s = E r D p D + h 2 s D + + + E max a 0 D2 D s + a. We will show that the first argumet is true by iductio o ad for each, if argumet () holds, the it implies that argumets (2) ad (3) also hold. Because the value iteratio eeds to be performed a ifiite umber of times, we arbitrarily iitiate the process by settig v 0 s = v 0 s = 0 for all s 0. Thus, the first argumet above is trivially true for = 0. Usig this termial value fuctio, we observe that 0 d s = r 2 s d +, ad therefore 0 d s 0 d s = r 2 I s d + r 2 I s d + = 0 d s 0 d s (C7) where I is the idicator fuctio. Iequality (C7), together with cocavity of the fuctio d (see part (a) of Lemma i EC. of the e-compaio), implies that b0 d b 0 d. Thus, the three argumets hold whe = 0. Next, we assume that argumet () is true for all t, where 0 is a arbitrary iteger, ad show that this implies that argumets (2) ad (3) must hold for ad argumet () must hold for +. For s, we use the fact that for ay m ad, m = m m +. The, we have d s d s = r 2 s s d + + + v s d + + r 2 s s d + + + v s d + + = r 2 + s d + s d + where we defie x = r 2 x + + v x +, x. Because v is cocave ad v v 0 r 2 (see Theorem i EC.2 of the e-compaio), is cocave. This argumet ca be proved usig steps similar i the proof of part (a), Lemma i EC. of the e-compaio. Here we omit details i the iterest of brevity. Therefore, we have d s d s r 2 + s d + s d + = r 2 s d + + v s d + + r 2 s d + + v s d + + r 2 s d + + v s d + + r 2 s d + + v s d + + = d s d s (C8) The first iequality is a cosequece of the cocavity of ad the fact that s d + s d +. The secod iequality comes from applyig the iductio hypothesis (argumet ()). A immediate cosequece of (C8) is that d b d d b d d b d d b d 0. Therefore, from the defiitio of the optimal acceptace level whe capacity is, wehave b d b d. Now, to complete the proof, we eed oly prove that the first argumet holds for +. From (EC2) i EC.2 of the e-compaio, for every realizatio d of type- demad, we have v d + s v d + s = h 2 I s > d + b d s P D 2 = j d s + j d s + j j=0 + if s b d d s d s if s>b d The remaider of the proof ivolves three mutually exclusive ad collectively exhaustive cases: () s> b d, (2) s b d, ad (3) b d <s b d. (Note that we already proved that b d b d.) Case. s> b d : v d + s v d + s = h 2 I s > d + + d s d s h 2 I s > d + + d s d s = v d + s v d + s The iequality above is a immediate cosequece of the secod argumet ad the fact that.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig Operatios Research 55(2), pp. 367 377, 2007 INFORMS 377 Case 2. s b d : v d + s v d + s = h 2 I s > d + b d s + j=0 h 2 I s > d + b d s + j=0 h 2 I s > d + b d s + j=0 P D 2 = j d s + j d s + j P D 2 = j d s + j d s + j P D 2 = j d s + j d s + j = v d + s v d + s The first iequality above comes from the fact that for b d s<j b d s, d s +j d s +j 0 ad that. The secod iequality follows from the secod iductio hypothesis. Case 3. b d <s b d : v d + s v d + s = h 2 I s > d + b d s + j=0 P D 2 = j d s + j d s + j h 2 I s > d + + 0 h 2 I s > d + + d s d s = v d + s v d + s The iequalities above follow from the observatio that for b d <s b d, d s d s 0 d s d s. Fially, upo takig expectatio, we obtai v + s v + s v + s v + s. We have so far show that the three argumets are, i fact, true for ay. Note that fuctios v ad v coverge, respectively, to v ad v as. Therefore, v s v s v s v s for all s. Now, we ca costruct a series of argumets similar to the argumets leadig up to (C8), but with v (respectively, v) replacig v (respectively, v ) o the right-had side. That is, we ca show d s d s d s d s for all s, a immediate cosequece of which is that b d b d. To see the relatioship betwee the effect of chage i productio capacity ad the effect of chage i D,we assume without loss of geerality that D ad D are costructed o the same probability space (see, e.g., Shaked ad Shathikumar 994, Theorem.A.). That is, d d with probability, where d ad d are realizatios of D ad D, respectively. Note from Equatio (EC2) of the e- compaio, that d affects v s through the term d +. Therefore, d d d + d +. If for a fixed d, we defie = + d d, the ad earlier argumets hold with the differece that ow. Therefore, for each realizatio d of type- demad, a series of argumets, very similar to those preseted above, ca be costructed to prove that d s d s d s d s for all s. Therefore, b d b d. Ackowledgmets This material is based, i part, o work supported by the Natioal Sciece Foudatio uder grat DMI-998872. Ay opiios, fidigs, ad coclusios or recommedatios expressed i this material are those of the authors ad do ot ecessarily reflect the views of the Natioal Sciece Foudatio. The authors are grateful to three aoymous referees, a associate editor, ad area editor Ja A. Va Mieghem for their helpful commets o a earlier versio of this paper. Refereces Carr, S., I. Dueyas. 2000. Optimal admissio cotrol ad sequecig i a make-to-stock/make-to-order productio system. Oper. Res. 48(5) 709 720. Cattai, K., E. Daha, G. Schmidt. 2003. Spacklig: Smoothig maketo-order productio of custom products with make-to-stock productio of stadard items. Workig paper, Uiversity of Califoria, Los Ageles, CA. http://www.aderso.ucla.edu/faculty/ely.daha/ cotet/spacklig.pdf. Charsirisakskul, K., P. M. Griffi, P. Keskiocak. 2004. Order selectio ad schedulig with leadtime flexibility. IIE Tras. 36(7) 676 702. Chatterjee, S., S. A. Slotick, M. J. Sobel. 2002. Delivery guaratees ad the iterdepedece of marketig ad operatios. Productio Oper. Maagemet (3) 393 40. Deto, B., D. Gupta, K. Jawahir. 2003. Maagig icreasig product variety at itegrated steel mills. Iterfaces 33(2) 4 53. Dobso, G., C. A. Yao. 2002. Product offerig, pricig, ad make-tostock/make-to-order decisios with shared capacity. Productio Oper. Maagemet (3) 293 32. Dueyas, I., W. J. Hopp. 995. Quotig customer lead times. Maagemet Sci. 4() 43 57. Frak, K. C., R. Q. Zhag. 2003. Optimal policies for ivetory systems with priority demad classes. Oper. Res. 5(6) 993 002. Keskiocak, P., S. Tayur. 2004. Due date maagemet policies. D. Simchi- Levi, S. D. Wu, Z.-J. (Max) She, eds. Hadbook of Quatitative Supply Chai Aalysis: Modelig i the E-Busiess Era. Kluwer Academic Publishers, Norwell, MA, 485 553. Müller, A., D. Stoya. 2002. Compariso Methods for Stochastic Models ad Risks. Joh Wiley & Sos, Chichester, UK. Puterma, M. L. 994. Markov Decisio Processes: Discrete Stochastic Dyamic Programmig. Joh Wiley & Sos, New York. Shaked, M., J. G. Shathikumar. 994. Stochastic Orders ad Their Applicatios. Academic Press, New York. Sobel, M. J., R. Q. Zhag. 200. Ivetory policies for systems with stochastic ad determiistic demad. Oper. Res. 49()57 62.
OPERATIONS RESEARCH doi 0.287/opre.060.0359ec pp. ec ec5 e-compaio ONLY AVAILABLE IN ELECTRONIC FORM iforms 2007 INFORMS E-Compaio Capacity Maagemet for Cotract Maufacturig by Diwakar Gupta ad Lei Wag, Operatios Research 2007, 55(2) 367 377. This olie compaio cotais the proofs of certai results ad a modified value-iteratio algorithm that were omitted from the mai paper. EC.. Some Fuctios ad TheirProperties Lemma. Suppose that s ad s F. a. For ay oegative iteger x, let g + R be defied as g s = r 2 s x + s x +. The, g s. b. For ay oegative iteger j, let g 2 + R be defied as g 2 s = max a 0 j s +a. The, g 2 s. c. For ay oegative ad iteger-valued radom variable X, which is idepedet of s, let g 3 s = E s X +. The, g 3 s F. d. Let g 4 + R be defied as g 4 s = s + s. The, g 4 s. e. For each 0 < <, ad F. Proof of Lemma. Part (a): We shall prove cocavity of g s whe s R +. Because + R +, it immediately follows that cocavity should hold whe g is defied over iteger poits. Let g s = r 2 s r 2 s x + + s x + = r 2 s +f f 2 s, where f u = r 2 u+ u ad f 2 u = u x +. It is easy to see that f 2 is covex. I view of the fact that is cocave ad 0 r 2, it follows that f is oicreasig ad cocave. Furthermore, the compositio of a covex fuctio with a oicreasig cocave fuctio is cocave. Therefore, g is cocave, beig the sum of cocave fuctios. Next, cosider g g 0 = r 2 x + x + 0. Ifx = 0, the right-had side equals 0 r 2. Otherwise, it equals r 2 + 0 0 = r 2. Therefore, g g 0 r 2 ad the proof of part (a) is complete. Part (b): Upo performig a simple variable trasform, we see that s 0, g 2 s = max s t s+j t. Let t = arg max t t 0 be a oegative maximizer of the fuctio. The, from the cocavity of, it follows that s + j if s + j t g 2 s = t if s<t <s+ j s if t s If 0 s<t j, wehaveg 2 s + g 2 s = s ++j s +j s +j s +j = g 2 s g 2 s. The precedig iequality comes from the fact that is cocave. Similarly, whe t j s t, it follows from the defiitio of t that g 2 s t = g 2 s ad t g 2 s +. Therefore, g 2 s + g 2 s 0 g 2 s g 2 s. Fially, whe s>t, we obtai g 2 s + g 2 s = s + s s s = g 2 s g 2 s. Therefore, g 2 s + g 2 s is oicreasig i s ad g 2 s is cocave. I order to establish that g 2 g 2 0 0, we eed to cosider three differet cases i a parallel set of argumets. For + j t,wehaveg 2 g 2 0 = + j j 0, where the iequality follows from the cocavity of. For 0 <t < + j, g 2 g 2 0 = t t = 0 0. The previous iequality holds because is cocave ad t > 0. Fially, whe t = 0, g 2 g 2 0 = 0. Part (c): For each realizatio x of the radom variable X, let g3 x deote the value of the fuctio g 3. The, g3 x is cocave beig the compositio of a covex fuctio s x + with a decreasig cocave fuctio. Furthermore, g3 x F because is a decreasig fuctio. I view of the fact that X is idepedet of s, cocavity ad mootoicity are preserved upo takig expectatios. Therefore, g 3 F. ec
ec2 Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig pp. ec ec5; suppl. to Oper. Res. 55(2) 367 377, 2007 INFORMS Part (d): The cocavity of g 4 follows from the fact that the sum of cocave fuctios is cocave. Furthermore, g 4 g 4 0 = 0 + 0 0 r 2. Therefore, g 4. Part (e): Cocavity is preserved upo multiplyig with a positive costat ad because 0, 0 0 r 2. Therefore, ad F. EC.2. The Structure of the Optimal Value Fuctio for the MTO Productio Model Theorem. The optimal value fuctio v s ca be writte as a sum of fuctios i the sets ad F. Specifically, ( l ) ( l v s = f s l + f 2 s l + s l 2 + +f l s m + s m ) (EC) m= m= where, ad f f 2 f l F. Proof of Theorem. The proof is by iductio, where we use the value iteratio algorithm to establish the ature of v s for each. The value fuctio at the + th iteratio is determied recursively by the followig expressio: v + s = E r D p D + h 2 s l D + + + E max r 2 s l + a D + + v s where a 0 D 2 (EC2) where s = s s 2 s l, which is determied i (2), ad v 0 s is set equal to zero for every s. Recall that s l = l m= s m. The termial value fuctio ca be chose arbitrarily because the problem has a ifiite horizo. First, ote that the desired structure holds trivially for = 0 by settig f 0 = f 0 2 = =f 0 l = 0 = 0 for all s. Next, assume that the structure of the value fuctio is as show i (EC) for all t. I particular, for the th iteratio, we have v s = f s l + f 2 s l + s l 2 + +f l sl + s l (EC3) where the fuctios f F ad for all s. The proof is completed by showig that this structure also holds for +. To prove that the result holds for +, we first rearrage (EC2) as s l = s l + s l 2 D + + l m= l m=j s m = sl j D + + for j = 2 3 l 2 ad s m = a + sl D + + (EC4) Next, we substitute from (EC3) ito (EC2), simplify usig (EC4), ad use the followig trasformatios: f s = f s for i = l, ad ˆ s = s + f s. The resultig expressio is i i v + s = E r D p D + h 2 s l D + + + E max r 2 s l + a D + + ˆ s l + a D + + a 0 D 2 + + f 2 s l + s l 2 + s l 3 D + + + f s l + s l 2 D + + From Lemma, it follows that f f 2 f l F ad ˆ. Let f + s l = E r D p D + h 2 s l D + + f + i s l l i = E f i s l l i D + + i = 2 3 l It is straightforward to verify that f + is decreasig cocave i s l ad therefore belogs to F. The fact that fuctios f + i F for i = 2 3 l follow from part (c) of Lemma upo choosig the radom l
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig pp. ec ec5; suppl. to Oper. Res. 55(2) 367 377, 2007 INFORMS ec3 variable X to be D + ad, i each argumet, settig the fuctio to be f i. Subsequet substitutio ad simplificatio yields v + s = f + s l + f + 2 s l + s l 2 + +f + l s l + E max r 2 s l + a D + + ˆ s l + a D + + where a 0 D 2 Next, for a realizatio d of demad D, we defie d s l + a = r 2 s l + a d + + ˆ s l + a d + + + s l = E max D s l + a where a 0 D 2 The fuctio d follows from Part (a) of Lemma upo choosig x to be d + ad the fuctio to be ˆ. The fuctio + comes from Part (b) of Lemma ad the fact that the properties of the fuctios i are preserved upo takig expectatios with respect to idepedet radom variables. Fially, we obtai v + s = f + s l + f + 2 s l + s l 2 + +f + l = f + s l + f + 2 s l + s l 2 + +f + l s l + E max s l D s l + a where a 0 D 2 + + s l (EC5) Therefore, the iductio hypothesis holds for +. Moreover, because v s v s as, v s possesses the properties of each v s. EC.3. Proof of Theorem 2 The optimality Equatio (3) i 3.2 simplifies as follows for l = 2: v i s = r 2 i ped + E max r + p r 2 i + x + r 2 s + x + y h i + x D + h 2 s + x + r +p i+x D + r 2 s+x+y + + v i+x D + s+x+y + where x D i + y 0 D 2 (EC6) The proof is by iductio, where we use the value iteratio algorithm to establish the ature of v i s for each. The algorithm begis by settig v 0 i s = 0 for every state i s. The, the value fuctio at the + th iteratio is determied by the followig expressio: = r 2 i ped + E max r + p r 2 i + x + r 2 s + x + y h i + x D + h 2 s + x + r + p i + x D + r 2 s + x + y + + v i + x D + s+ x + y + ] D i + x 0 y D 2 (EC7) Perform a variable trasform with a = i + x, ad b = s + x + y ad let C d d 2 i s deote the set a b i + d i + a i +, s i + a b s i + a + d 2. Furthermore, defie the followig otatio: D a b = r + p a D + r 2 b + + v a D + b + f a b = r + p r 2 a + r 2 b h a D + + D a b J i s a b = f a b h 2 s + a i + V D D 2 i s = max J i s a b where a b C D D 2 i s (EC8) The, we ca write v + i s compactly as follows: v + i s = r 2 i ped + E V D D 2 i s. I our model, the argumets of the fuctios D ad v are itegers. Because the domai of a cocave fuctio must be a covex set, for the purpose of this proof, we assume i s ad a b R + 2. If cocavity holds over this evelopig covex set, it should also hold for the same fuctios whe they are defied over iteger poits. Note that Theorem 2 holds trivially for = 0 whe we set v 0 i s = 0 for all i s. Assume the followig iductive hypotheses hold for all i s ad t :
ec4 Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig pp. ec ec5; suppl. to Oper. Res. 55(2) 367 377, 2007 INFORMS. v i s is joitly cocave i i s. 2. v i 2 s v i s i 2 i r + p i 2 i, ad v i s 2 v i s s 2 s r 2 s 2 s. To complete the proof, we will first show that hypothesis () also holds for +. For every realizatio d of demad D, let g d R + 2 R + 2 be defied by g d a b = a d + b +, ad let d R + 2 R be defied by d a b = r + p a r 2 b + v a b. Note that d is the compositio of fuctios g d ad d, i.e., d a b = d g d a b. It is easy to see that g d is covex i a b. Let a = a + a 2 ad b = b + b 2, where 0, a, a 2, b, ad b 2 R +. It follows that g d a b g d a b + g d a 2 b 2. By hypothesis (2), d is oicreasig i a b. The, we have d g d a b d g d a b + g d a 2 b 2 = d a d + + a 2 d + b + + b 2 + = r + p a d + + a 2 d + r 2 b + + b 2 + + v a d + + a 2 d + b + + b 2 + r + p a d + + a 2 d + r 2 b + + b 2 + + v a d + b + + v a 2 d + b 2 + = d g d a b + d g d a 2 b 2 (EC9) Iequality (EC9) comes from hypothesis (). Hece, d is joitly cocave. Thus, fuctios f ad J are cocave (because sums of cocave fuctios are cocave). Fially, by Propositio B-4, p. 525, of Heyma ad Sobel (984), V d d 2 is cocave. The cocavity of v + i s follows from the fact that takig expectatio over D D 2 preserves cocavity. Next, we will show v + i s 2 v + i s s 2 s r 2 usig a sample path argumet. The fact that v + i 2 s v + i s i 2 i r + p ca be argued i a similar way (therefore is omitted). Suppose for each pair of realizatios d ad d 2 of demads D ad D 2, the optimal actios i state i s 2 are xd d 2 i s 2 ad yd d 2 i s 2, ad the realized optimal value is v d d 2 + i s 2. Note that xd d 2 i s 2 ad yd d 2 i s 2 are also feasible actios i state i s. Let v d d 2 + i s be the value of the system i state i s operated uder the actios xd d 2 i s 2 ad yd d 2 i s 2 (all subsequet decisios are made optimally). Takig the differece betwee v d d 2 + i s 2 ad v d d 2 + i s, by (EC7), if s + xd d 2 i s 2 + yd d 2 i s 2, v d d 2 + i s 2 v d d 2 + i s = h 2 s 2 + x d d 2 i s 2 + + h 2 s + x d d 2 i s 2 + + v i + x d d 2 i s 2 d + s 2 + x d d 2 i s 2 + y d d 2 i s 2 v i + x d d 2 i s 2 d + s + x d d 2 i s 2 + y d d 2 i s 2 s 2 s r 2 <r 2 s 2 s The first iequality above comes from hypothesis (2) ad h 2 > 0, ad the secod from 0 < <, s 2 s.if >s 2 + xd d 2 i s 2 + yd d 2 i s 2, v d d 2 + i s 2 v d d 2 + i s = r 2 s 2 s.ifs + xd d 2 i s 2 + yd d 2 i s 2 < s 2 + xd d 2 i s 2 + yd d 2 i s 2, the we have v d d 2 + i s 2 v d d 2 + i s = r 2 h 2 s 2 + x d d 2 i s 2 + r 2 s + x d d 2 i s 2 + y d d 2 i s 2 + v i + x d d 2 i s 2 d + s 2 + x d d 2 i s 2 + y d d 2 i s 2 v i + x d d 2 i s 2 d + 0 r 2 s + x d d 2 i s 2 + y d d 2 i s 2 + s 2 + x d d 2 i s 2 + y d d 2 i s 2 r 2 r 2 s 2 s The first iequality above comes from hypothesis (2) ad h 2 > 0, ad the secod from 0 < <. That is, for each pair of d ad d 2, v d d 2 + i s 2 v d d 2 + i s r 2 s 2 s. Therefore, upo takig expectatio, we obtai v + i s 2 E v D D 2 + i s r 2 s 2 s. Recall that v + i s is the optimal value of beig i state i s, so it follows that v + i s E v D D 2 + i s. The, v + i s 2 v + i s v + i s 2 E v D D 2 + i s r 2 s 2 s. The argumets above establish that the iductio hypotheses () ad (2) hold for +. Note that Part a. of Theorem 2 is oly a special case of hypothesis (2) whe choosig s 2 = s ad s = s. Hece, the theorem holds from the fact that v i s v i s as.
Gupta ad Wag: Capacity Maagemet for Cotract Maufacturig pp. ec ec5; suppl. to Oper. Res. 55(2) 367 377, 2007 INFORMS ec5 EC.4. The Modified Value Iteratio Algorithm I what follows, we describe a approximate value iteratio algorithm, which is used to solve the optimality Equatios () ad (3) recursively. Additioal details ca be foud i Puterma (994). Step. Iitializatio: Set = 0, v 0 s = max r r 2 / (respectively, v 0 i s = r dˆ + r 2 /, ad set tolerace levels ad 2. Trucate the umber of type- demad i each period at dˆ, where dˆ is a iteger that satisfies the iequality d ˆ j=0 P D = j. (We set = 0 6 i the experimets reported i the mai paper.) Choose a large N that results i the trucated state space as described below. The umber of type-2 orders backlogged is trucated such that 0 s j N for each j. I additio, i the MTS-MTO model, the ivetory level of type- items is trucated as follows: 0 i N. Let T S deote the trucated state space. The, T S = 0 N 0 N } {{ } l -times T S = 0 N 0 N } {{ } l -times for the MTO model ad for the MTS-MTO model. Step 2. Trucate the umber of type-2 demad i each period at dˆ 2, where dˆ 2 is a iteger that satisfies the iequality d ˆ 2 j=0 P D 2 = j. I the MTO model, for each s T S, each d 0 dˆ, ad each d 2 0 dˆ 2, choose a s d d 2 to satisfy a s d d 2 = arg max r 2 s l + a d + + v s where a 0 d 2 where s is determied by (2). If state s exceeds the vector N, the v s = max r r 2 /. Computatioal effort for this step ca be reduced by makig use of the fact that a s d d 2 = b + d + s l + d 2 for all s, where b is the optimal threshold whe d > at iteratio. I the MTS-MTO model, for each i s T S, each d 0 dˆ ad each d 2 0 dˆ 2, choose x i s d d 2 y i s d d 2 to satisfy x i s d d 2 y i s d d 2 = arg max r i + x d + r 2 s l + y x h i + x d + h 2 s l + x + p d i + x + + v i + x d + s where d i + x 0 y d 2 where s is determied by (4). If state i s exceeds the vector N, the v i s = r dˆ + r 2 /. With l = 2, computatioal effort for this step ca be reduced by performig a variable trasform with a = i + x, b = s + x + y ad let C d d 2 i s deote the set a b i + d i + a i + s i + a b s i + a + d 2. We have a i s d d 2 b i s d d 2 = arg max { f a b h 2 s + a i + a b C d d 2 } i s where f is defied i (EC8). By the cocavity of f, a sequetial maximizatio process ca be used to fid a i s d d 2 ad b i s d d 2 quickly. Step 3. Icremet by ad update v s (respectively, v i s ) by usig the maximizig actios a s d d 2 (respectively, x i s d d 2 ad y i s d d 2 ) for each d 0 dˆ ad each d 2 0 dˆ 2. Step 4. If v s v s < 2 for each s T S (respectively, if v i s v i s < 2 for each i s T S ) the stop; else go to Step 2. The covergece tolerace 2 is chose to strike a balace betwee accuracy ad the umber of computatios eeded to achieve covergece. We use sufficietly large N ( 50 i our umerical examples with a typical mea demad of 5). Let v N deote the value fuctio obtaied from the above algorithm. By Theorem 6.0.9, p. 243, of Puterma (994), it is straightforward to verify that v N coverges mootoically to v (i () ad (3)) as N. Refereces See refereces list i the mai paper. Heyma, D. P., M. J. Sobel. 984. Stochastic Models i Operatios Research: Volume II. McGraw-Hill, New York.