A Make-to-Stock System with Multiple Customer Classes and Batch Ordering

OPERATIONS RESEARCH Vol. 56, No. 5, September October 28, pp. 1312 132 ssn 3-364X essn 1526-5463 8 565 1312 nforms do 1.1287/opre.18.549 28 INFORMS TECHNICAL NOTE A Make-to-Stock System wth Multple Customer Classes and Batch Orderng Boray Huang Department of Industral and Systems Engneerng, Natonal Unversty of Sngapore, Sngapore 117576, sehb@nus.edu.sg Seyed M. R. Iravan Department of Industral Engneerng and Management Scences, Northwestern Unversty, Evanston, Illnos 628, ravan@ems.northwestern.edu Ths paper examnes the mpact of customer order szes on a make-to-stock system wth multple demand classes. We frst characterze the manufacturer s optmal producton and ratonng polces when the demand s nonuntary and lost f unsatsfed. We also nvestgate the optmal polces of a backorder system wth two demand classes and fxed order szes. Through a numercal study, we show the effects of batch orders on the manufacturer s nventory cost as well as on the beneft of optmal stock ratonng. It s shown that batch orderng may reduce the manufacturer s overall cost f carefully ntroduced n a frst-come-frst-served (FCFS) system. Wth the same effectve demand rates, the customers order szes also have a strong mpact on the beneft of optmal stock ratonng. Subject classfcatons: batch orderng; make-to-stock; stock ratonng. Area of revew: Manufacturng, Servce, and Supply Chan Operatons. Hstory: Receved October 24; revsons receved July 26, June 27; accepted July 27. 1. Introducton Ths paper nvestgates the mpact of customers batch orderng behavor on nventory management wth multple demand classes. In tradtonal producton/nventory lterature, the mpact of customer order szes s usually studed under a sngle demand class. When all customers are of the same mportance, t s well known that customers batch orderng behavor has a negatve mpact on nventory control because of the bullwhp effect (see Lee et al. 1997a, b; Cachon 1999). When the customers are of dfferent values to the manufacturer, a stock-ratonng polcy can be used to mprove the manufacturer s proft margns. That s, when the manufacturer does not have suffcent nventory on hand, a certan porton of nventory s reserved to satsfy the orders from more-valuable customers, and the orders from less-valuable customers may be turned down. Most exstng lterature on stock ratonng assumes ether an uncapactated system or untary customer demands (.e., each customer requests one unt at a tme). In uncapactated systems, the stock-ratonng problem s known to be very complcated, and therefore most exstng research focuses on presumed or heurstc polces (see Avv and Federgruen 1998, Chen et al. 21, Frank et al. 23). On the other hand, most studes of capactated productonnventory systems assume untary demands. Therefore, the manufacturer s ratonng decson reduces to acceptng or rejectng an arrvng order. Ha (1997a) frst studes the stock-ratonng problem n a capactated system wth lost sales. The threshold-type polces are shown to be optmal for producton and ratonng decsons. Ha (1997b) then provdes structural results for a two-demand-class system wth backorders. de Vércourt et al. (21, 22) extend Ha s model n (1997b) to n demand classes and demonstrate that the beneft of nventory poolng can be realzed only f the stock s effcently allocated. Other extensons can be found n Deshpande et al. (23), Gayon et al. (24), and Axsäter et al. (24). When the customers orders are nonuntary, the problem of ratonng quantty arses. That s, when an order of a demand class arrves, the suppler/manufacturer has to decde whether the entre order should be rejected, partally satsfed, or fully flled wth ts on-hand nventory. The mpact of customers batch orderng behavor on the beneft of optmal stock ratonng s stll an open queston to our best knowledge. In ths paper, we frst characterze the optmal producton and ratonng decsons for a lost sale system and a backorder system under nonuntary demand. Then, through a numercal study, we examne the effects of customer order szes on the beneft of optmal stock ratonng. Even wthout stock ratonng (.e., usng the frst-come-frst-served (FCFS) polcy to fll all customers orders), we fnd that the manufacturer s overall cost can sometmes be lowered by slghtly encouragng batch orderng from the less-mportant customers. Our result provdes 1312

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng Operatons Research 56(5), pp. 1312 132, 28 INFORMS 1313 a dfferent pont of vew to marketers who usually provde more-valuable customers wth quantty dscounts, and to nventory managers who thnk of batch orderng as a pure harm. Ths paper s organzed as follows. Secton 2 ntroduces our multclass lost sale model and characterzes the optmal producton and ratonng polces. Secton 3 shows some results for FCFS systems. Secton 4 explores a backorder case wth two demand classes and fxed order szes. The numercal study s performed n 5, and 6 concludes the paper. 2. Optmal Polces: The Multclass Lost Sale Model Consder a manufacturer (capactated suppler) who produces a sngle product to nventory. The producton capacty s unts per unt tme. There s a convex, nondecreasng, and nonnegatve holdng cost hx per unt tme when the manufacturer s nventory level s x (x Z +, where Z + represents the set of nonnegatve ntegers 1 2). Suppose that there are N dfferent demand classes, whch arrve at the manufacturer as ndependent Posson processes wth rates ( = 1 2N). Each customer of class requests D ( = 1 2N) unts of the product. Let D N =1 be random varables that are mutually ndependent among dfferent demand classes and among dfferent customers n the same class. Wthout loss of generalty, we assume that there s an upper bound M for the quantty requested by each customer of class ( = 1 2N). The probablty that an arrvng customer of demand class requres k tems s PrD = k = p k (k = 1 2M and = 1 2N). The customer s order can be partally fulflled, and the unmet part s lost wth a unt shortfall cost c for class- demand. We assume that c s nondecreasng n (.e., c 1 c N ). Note that when D = 1 for all = 1 2N, our problem reduces to a specal case of untary demand studed n Ha (1997a). At any tme t, the manufacturer can decde to produce and add the fnshed product to ts on-hand nventory, or not to produce and stay dle. When a customer demand arrves, the manufacturer has to decde the number of unts to be allocated from ts on-hand nventory. Defne the state varable xt as the manufacturer s nventory level at tme t. A manufacturer s control polcy descrbes the acton at tme t gven the system state xt. We also assume exponentally dstrbuted producton tmes, and thus, as n de Vércourt et al. (22), restrct our analyss to Markovan polces because the optmal polcy les n ths class (Bertsekas 1995). Ha (1997b) presents cases n whch the exponental producton tme s a reasonable approxmaton n practce. Let A x = A x A 1k xm 1 k=1 A Nk xm N k=1 be the set of actons under polcy. A x s the control acton assocated wth the producton decson. A kx s the control acton when an order n sze of k from demand class arrves = 1 2N. More specfcally, when xt = xx, and lettng k x = mnk x, the control actons are { not to produce A x = 1 to produce A k x = z to allocate z unts from the nventory to an arrvng class- order n sze of k, z k x The problem can be formulated as a Markov decson process (MDP). Let L t be the accumulated lost sales (n unts) for demand class up to tme t. In addton, let 1 be the dscount factor. Our objectve s to fnd the optmal control polcy that mnmzes the manufacturer s dscounted total cost over an nfnte horzon. That s, we try to solve the followng problem: [ mn E x e t hxt dt + N c =1 ] e t dl t where E x denotes the expectaton over tme, gven the ntal system state x and polcy. Wthout losng generalty, we can scale the parameters and let + N =1 + = 1. Followng Lppman (1975), the optmalty equaton of the MDP s fx= hx + mnf x f x + 1 M N + mn f x z + k zc (1) =1 p k zk x k=1 where fx s the optmal dscounted cost under the ntal system state x. The frst mnmzaton on the rghthand sde of (1) represents the manufacturer s producton decson. It s optmal to produce f fx>fx+ 1, and stay dle otherwse. The second mnmzaton corresponds to the manufacturer s optmal ratonng decsons. When a customer of class- arrves and requests k tems, t s optmal for the manufacturer to allocate z xk tems and pay the lost sale cost for the unmet part, where z xk = arg mn zk x f x z + k zc. In Ha s (1997a) untary demand case, M 1 = M 2 = k = 1, so z xk = or1. Defne the dfference operator D x of any functon on Z + as D x x = x + 1 x. The followng theorem shows that, when the order szes are not untary, the optmal producton and ratonng polces are thresholdtype base-stock and stock-reservaton polces, respectvely. The proofs of Theorem 1 and all other theorems n ths paper are presented n Onlne Appendx A, unless otherwse ndcated. An electronc companon to ths paper s avalable as part of the onlne verson that can be found at http://or.journal.nforms.org/. Theorem 1. (a) There exsts an optmal statonary producton and stock-ratonng polcy.

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng 1314 Operatons Research 56(5), pp. 1312 132, 28 INFORMS (b) A base-stock produce-up-to polcy wth a threshold base-stock level Ss optmal for the manufacturer s producton decson. That s, there exsts a crtcal stock level S such that { f x S A x = 1 f x<s S = mnx D x fxx Z + (c) A stock-reservaton polcy s optmal for stock ratonng. There exst reserve-stock levels r 1 r 2 r N such that the manufacturer s optmal ratonng decson s to make the postallocaton nventory level as close to the correspondng reserve-stock level as possble,.e., A k x = z xk = mnk x r + r = mnx D x fx c x Z + where = 1 2N, a + = maxa and k s a natural number. (d) The reserve-stock levels are nondecreasng n, and S rn r 2 r 1 =. Fgure 1 shows the optmal stock-ratonng decson when a class- order n sze of k unts arrves. If the manufacturer has suffcent nventory (.e., x r + k), the order should be fully satsfed. If the manufacturer has nsuffcent nventory (.e., x r ), the entre order should be rejected. When the manufacturer s nventory s slghtly hgher than the reserve-stock level r, the order should be partally flled so that at least r unts are stll left n the nventory after allocaton. We would lke to note that because t s practcally mpossble and never optmal to keep nfnte unts at the manufacturer s nventory, we can set an upper bound on xt. The long-run average cost optmalty can then be obtaned by lettng the dscount factor go to zero (see Weber and Stdham 1987, Ha 2). Fgure 1. Before ratonng The optmal ratonng decsons when a class- order n sze of k unts arrves. Left: x r + k then z xk = k; Mddle: r + k> x>r then z xk = x r ; Rght: x r then z xk =. Inventory Inventory Inventory S S S S S S x k x x r* r* r* r* r* r* r* x After ratonng Before ratonng After ratonng Before ratonng After ratonng Theorem 1 shows that the optmal polcy s a multthreshold polcy wth thresholds r1 r 2 r N S.We develop an algorthm that results n the exact steady-state probabltes of the system under any multple-threshold polcy such as the optmal polcy. These steady-state probabltes can then be used to obtan the total average costs of the correspondng multthreshold polcy. Let ˆ be a multthreshold polcy wth N + 1 thresholds ˆr 1 ˆr 2 ˆr N Ŝ under whch the producton and the ratonng decsons are { f x Ŝ A ˆ x = 1 f x<ŝ A ˆ k x = mnk x r + Wthout loss of generalty, we let S ˆr N ˆr 2 ˆr 1 =. 1 The case of Ŝ = s trval because t means no nventory would be kept and all demands are lost. When Ŝ>, defne M j = k=j p k Let q = be a seres of real numbers, and defne Ŝ= as the steady-state probabltes for states 1Ŝ under the polcy ˆ. Note that any state x>ŝ s transent under ˆ. The followng algorthm yelds Ŝ=. 2.1. Exact Algorthm for Steady-State Analyss of the Multclass Lost Sale Model Step 1. Let q n = for all n>ŝ and n Z +. Step 2. Let qŝ be an arbtrary strctly postve real number (e.g., qŝ = 1). Step 3. Startng from state n = Ŝ, calculate q n 1 for state n 1 through the followng formula: M N q n 1 = 1 n> ˆr j q n 1+j (2) =1 j=1 where { 1 f X s true 1 X = f X s false As a result, we can fnd qŝqŝ 1 q 1 q recursvely. Step 4. (Normalzaton) The steady-state probabltes n Ŝn= can be obtaned by n = q n Ŝ = q When the steady-state probabltes n Ŝn= are obtaned usng the above algorthm, the long-run average cost per unt tme TC ˆ under the polcy ˆ can then be calculated as ( M Ŝ N TC ˆ = hn + c p k k n r + ) + n n= =1 k=1 The justfcaton of the algorthm s presented n Onlne Appendx C.1.

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng Operatons Research 56(5), pp. 1312 132, 28 INFORMS 1315 3. Some Results for the FCFS Stock-Ratonng Polcy We defne the FCFS polcy by the acton set A FCFS x = A FCFS xan FCFS x, where { not to produce A FCFS x = 1 to produce A FCFS x = d x to allocate d x unts to an arrvng class- order of sze d = 1 2N and x s the manufacturer s on-hand nventory. Lettng z xk = k x, the optmalty equaton s f FCFS x = hx + mnf FCFS x f FCFS x + 1 M N + p k f FCFS z z xk +c k z xk =1 k=1 (3) It s nterestng to note that, wth multclass batch demands, the optmal cost functon f FCFS s not always convex n x. However, when the demand szes are dentcally dstrbuted (.e., PrD = k = p k, k = 1 2M, for all = 1N), we can rewrte the optmalty equaton (3) as f FCFS x = hx + mnf FCFS x f FCFS x + 1 M + p k f FCFS x z xk + ck z xk (4) k=1 where = N =1 and c = N =1 c /. The followng theorem shows that the base-stock polcy s optmal for the manufacturer s producton decsons. Theorem 2. When the customer demand szes are ndependently and dentcally dstrbuted (..d.), and the manufacturer apples an FCFS stock-ratonng polcy: (a) There exsts an optmal statonary producton polcy. (b) The base-stock produce-up-to polcy s optmal for the manufacturer s producton decson. (c) The lost sale costs c and the demand rates affect the optmal cost functon and the optmal producton decsons only f c or change. As long as c and reman the same, any change n c, or any change n would not change the optmal costs and decsons. In other words, when the order szes are..d., the dfferent demand classes can be aggregated wth an equvalent (aggregated) arrval rate and an equvalent lost sale cost rate c. The base-stock polcy s also optmal for the manufacturer s producton decson. Ths smple rule of demand aggregaton may not work when the order szes are not..d. Another nterestng result n systems under the FCFS polcy s presented n Theorem 3 for a two-demand-class case wth determnstc order szes. The theorem states that the FCFS polcy and the optmal polcy are asymptotcally equvalent n terms of the manufacturer s long-run average cost when one of the customers order szes becomes very large (.e., approaches nfnty). The asymptotc analyss and the proof of Theorem 3 are presented n Onlne Appendx B.1. Theorem 3. Suppose that there are two demand classes wth determnstc order szes D = d = 1 2. If the optmal polcy has a fnte base-stock level and the effectve demand rates eff = d = 1 2 stay the same, we have TC opt = TC FCFS when d ( = 1 or 2). 4. Two Demand Classes wth Fxed Order Szes and Backorders The analyss of stock ratonng n systems wth backorders s usually consdered more dffcult than ts counterpart wth lost sales (see Ha 1997b and de Vércourt et al. 21, 22). In ths secton, we explore a smple backorder case wth two demand classes and fxed (determnstc) order szes d 1 and d 2. At any tme, the manufacturer can decde whether to produce and allocate the fnshed unt to ts on-hand nventory, to produce and allocate the fnshed unt to fll the backorders, or not to produce and stay dle. When an order arrves, the manufacturer must decde how many on-hand unts should be allocated to the order, and the unsatsfed part of the order wll be backordered. There s a lnear holdng cost h per unt per unt tme for the manufacturer s on-hand nventory. There also exst lnear backorder cost rates b 1 and b 2 per backorder unt per unt tme for class-1 and class-2 demands, respectvely. We let b 1 >b 2 ; thus, an order from demand class-1 should always be satsfed f possble. Furthermore, both on-hand nventory and class-1 backorders cannot occur at the same tme, but t s possble that both on-hand nventory and class-2 backorders coexst (see de Vércourt et al. 21, 22). Parallel to Ha (1997b), we can then defne the followng system state varables xt and yt: xt = xt + xt where xt + s the on-hand nventory level at tme t, and xt s the number of class-1 backorders at tme t. yt = number of class-2 backorders at tme t, where xt +, xt, and yt are all nonnegatve ntegers. The control actons A x y A 1 x y A 2 x y assocated wth a polcy can be defned as follows: not to produce 1 to produce and assgn the fnshed unt A x y = to a class-1 backorder when x<, or to produce and assgn the fnshed unt to on-hand nventory when x, 2 to produce and assgn the fnshed unt to a class-2 backorder when y>, or to produce and assgn the fnshed unt to on-hand nventory when y =

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng 1316 Operatons Research 56(5), pp. 1312 132, 28 INFORMS A 1 x y = d 1 x + A 2 x y = a to allocate (d 1 x + ) unts from the on-hand nventory to an arrvng class-1 order of sze d 1. to allocate a unts from the on-hand nventory to an arrvng class-2 order of sze d 2,ad 2 x +, where A x y represents the producton control under polcy at state xt yt = x y, and A x y ( = 1 2) s the stock-ratonng decson under polcy when a class- order arrves at state x y. Smlar to the prevous secton, we look for the optmal polcy to solve the followng mnmzaton problem over an nfnte horzon: [ ] e t hxt + + b 1 xt + b 2 ytdt mn E x y E x y s the expectaton over tme, gven the ntal system state (x y) and the polcy. Agan, we let + 2 =1 + = 1. The new optmalty equaton for the backorder case becomes 2 gxy = cxy + mngx y gx + 1 y gx y 1 + 1 gx d 1 y + 2 mn gx a y + d 2 a (5) ad 2 x + where gxy s the optmal dscounted cost wth the ntal state x y, and cxy s the nstantaneous cost functon at the state x y, whch s defned by { hx + + b 1 x + b 2 y when y cxy = (6) + when y< Now defne the dfference operators D x, D y, and D xy of any functon on Z 2 as D x x y = x + 1y x y D y x y = x y + 1 x y D xy x y = x + 1y x y 1 The followng theorem characterzes the optmal producton and stock-ratonng decsons: Theorem 4. (a) There exsts an optmal statonary polcy. (b) A two-threshold polcy s optmal for the manufacturer s producton decson. Specfcally, there are two thresholds S and R, where S = mnx D x gx> R = mnx D xy gx1> Gven a system state x y, the manufacturer s optmal producton decson s f S x A x y = 1 f x<r 2 f R x<s where S R. (c) When a class-2 customer arrves and requests d 2 tems, t s optmal to allocate A 2 x y = d 2 x R + unts from the on-hand nventory to satsfy the request. The unflled part of the order s backordered. A 2 x y s nondecreasng n x but ndependent of y. Theorem 4 shows that we can extend the optmalty of the multlevel ratonng (ML) polcy of the untary demand case (see de Vércourt et al. 22) to the cases when the customers order szes are larger than one: there exst two thresholds S and R that are ndependent of the system state. The optmal producton polcy s as follows. At any tme, f x S, t s optmal to stop producton and reman dle. Thus, S s the manufacturer s base-stock level under the optmal producton polcy. On the other hand, f R x<s, t s optmal to produce and assgn the fnshed unt to class-2 backorders f there are any, or to the onhand nventory f there are no class-2 backorders. When x<r, t s optmal to produce and assgn the fnshed unt to class-1 backorders f x<, or to the on-hand nventory f x<r. Smlar to the lost sale case, the optmal stock ratonng polcy s also of a threshold type when a less valuable order comes: the manufacturer should allocate the on-hand nventory so that the postallocaton nventory level can be as close to the threshold R as possble. In the lost sale case, the convexty of the cost functon s the core property that determnes the structure of the optmal polcy. As shown n the proof of Theorem 4 for the backorder case, the submodularty/supermodularty of the cost functon are also crucal to the structure of the optmal polcy, whch dramatcally ncreases the complexty of the analyss n the backorder case. Because the optmal polcy n the backorder case s stll of the threshold type, t s nterestng to check whether the recursve algorthm developed by de Vércourt et al. (22) for the untary demand can be extended to systems wth batch demands. When there are two demand classes wth untary order szes, de Vércourt et al. s algorthm starts wth a sngle-class subproblem that solves an M/M/1 make-to-stock queue wth the nstantaneous cost functon cx = h + b 2 x + + b 1 b 2 x. The optmal base-stock level of the subproblem then becomes the optmal stockratonng level R of the man problem. Havng R, the optmal base-stock level S s then found by applyng the closed-form solutons of M/M/1 queues. We develop a heurstc algorthm to extend de Vércourt et al. s dea to the batch demand case, n whch the optmal thresholds are estmated. Heurstc Algorthm for Estmatng the Optmal Producton and Stock-Ratonng Thresholds Step 1. Solve a sngle-class M d1 /M/1 make-to-stock queue wth a holdng cost rate h + b 2 and a backorder cost

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng Operatons Research 56(5), pp. 1312 132, 28 INFORMS 1317 rate b 1 b 2. Fnd the optmal base-stock level and let t be R. The problem can be solved by searchng for the nonnegatve mnmzer of the followng cost functon g 1 R, where R g 1 R = h + b 1 1 R> + b 1 b 2 =1 [( )( ) ] d1 + 1 1 d 1 R R Z + (7) 2 1 d 1 R = are the steady-state probabltes that can be obtaned n the followng recursve method startng wth = R: f >R 1 1d 1 f = R = (8) d 1 1 +j f <R j=1 Thus, R = arg mng 1 R R Z +. Let g 1 = g 1 R be the optmal long-run average cost of the subproblem. Step 2. Search for the mnmzer of the cost functon g 2 S, where ( g 2 S = 1 1 S> R S = R+1 )g 1 + h + b 21 S> R + b 2 [ 1 d 1 + d 2 1 + 2d 2 + d 2 2 2 1 d 1 2 d 2 S = R+1 ] S (9) Note that n (9), the steady-state probabltes are needed only when S> R. When S> R, probabltes can be obtaned n the followng recursve method: f >S 1 1d 1 + 2 d 2 f = S = (1) 1 d 1 j=1 +j + 2 d 2 +j j=1 f R <S Let Ŝ = arg mng 2 S S R S Z +. Then, Ŝ R are the heurstc s estmates of S R. Note that when d 1 = d 2 = 1, our heurstc algorthm s exactly the same as the algorthm proposed by de Vércourt et al. (22), and thus Ŝ R = S R. For justfcatons and nterpretatons of the heurstc algorthm, please refer to Onlne Appendx C.2. We perform an extensve numercal study wth 6,4 examples to nvestgate the effectveness of the heurstc algorthm. 3 The performance measure PR s defned as PR = TC heu TC opt TC opt 1% where TC heu s the system s long-run average cost per unt tme usng the heurstc thresholds Ŝ R, and TC opt s the system s optmal long-run average cost usng the optmal thresholds S R. We fnd that the heurstc algorthm works well n general, wth the average value of PR around 3%. However, there are some cases where the heurstc algorthm does not fnd good estmates of the thresholds. We observed n 654 (out of 6,4) cases of our numercal study that, when eff 1 /eff 2 was relatvely small (.e., smaller than one) and was relatvely large (.e., larger than.9), the error PR was greater than 1%. Note that the essence of de Vércourt et al. s (22) recursve algorthm les n the property that, gven a stock-ratonng level R, the subset of the system space = xt xt R can be treated as an ndependent sngle-class make-to-stock M/M/1 queue. When the system enters the subset, t starts wth the state xt = R. However, f the order szes (especally d 1 ) are not untary, the system may enter the subset wth any startng state of xt = R R 1R d 1 + 1 +. The probabltes of the startng state depend on the lmtng probabltes of the states xt = R + d 1 R+ d 1 1R+ 1, whch are outsde the subset. The connecton across the ratonng threshold n the batch demand cases makes the closed-form solutons very complcated. It also makes our heurstc algorthm, whch treats the subset as an ndependent M d 1 /M/1 queue, fal to provde an accurate estmate for the system s true performance n some cases. In those cases, the heurstc algorthm assgns a hgher probablty to the state xt = R because t s the startng state of the subproblem. As a result, the stockout probablty n the heurstc algorthm s lower than t should be, resultng n an underestmated overall cost of the subset, and consequently, a lower-than-optmal base-stock level Ŝ. The error becomes crtcal when eff 1 /eff 2 s small (.e., smaller than one) and s relatvely large (.e., larger than.9) because the system would stay n the subset for a longer perod of tme. 5. Numercal Study In ths numercal study, we consder a lost sale system wth two demand classes (N = 2). The order sze of each demand class s constant,.e., D = d = 1 2. As assumed n 2, we let c 1 >c 2. We also assume a lnear nventory holdng cost where hx = hx. Consder the followng system parameters: () effectve demand rates: eff = d for = 1 2; () lost sale cost rato: = c 1 /c 2 ; () traffc ntensty of the system = eff 1 + eff 2 /; and (v) relatve holdng cost rate: h = h/ eff 1 c 1 + eff 2 c 2. The ranges of the above parameters are chosen to be consstent wth those n Ha (1997a). If a polcy s chosen as the benchmark, we evaluate the performance of the optmal polcy and polcy through ther long-run total average costs per unt tme TC opt and TC, respectvely.

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng 1318 Operatons Research 56(5), pp. 1312 132, 28 INFORMS Fgure 2. Average cost per unt tme 7 65 6 55 5 45 4 35 3 25 The effect of order szes. Left: on the total average cost, and Rght: on the beneft of the optmal polcy CR FCFS, compared wth the FCFS polcy ( eff 1 = 1, eff 2 = 1, = 6, = 18, h = 1). Cost reducton by optmal ratonng Cost reducton by optmal ratonng d 1 = n, d 2 = 1, FCFS d 1 = n, d 2 = 1, optmal d 1 = 1, d 2 = n, FCFS d 1 = 1, d 2 = n, optmal 2 5 1 15 2 25 3 Order sze n Cost reducton CR (%) 35 3 25 2 15 1 5 d 1 = n, d 2 = 1 d 1 = 1, d 2 = n 5 1 15 2 25 3 Order sze n The values of TC opt and TC are obtaned from the successve approxmaton wth an error bound.1%. The convergence of the successve approxmaton s shown n Sennott (1999). Defne cost reducton CR that measures the cost effectveness of the optmal polcy compared wth that of the benchmark polcy as CR = TC TC opt 1% TC The larger CR s, the more benefcal t s to apply the optmal polcy nstead of the benchmark polcy. Our frst benchmark polcy s the optmal base-stock polcy wth FCFS stock allocaton defned n 3. Let S FCFS be the optmal base-stock level. S FCFS can be obtaned from the successve approxmaton. Under batch demands, we observe that the mpacts of the effectve demand rates eff, the lost sale cost rato, and traffc ntensty are smlar to the results n the untary demand case of Ha (1997a). Therefore, n ths paper we only focus on the mpact of order szes. A representatve example s chosen n Fgure 2, whch shows the mpact of the order szes on the total average costs TC opt and TC FCFS, and on the beneft of optmal stock ratonng CR FCFS. In addton, the left part of Fgure 3 shows the mpact on S, S FCFS and r2 = r. Note that when we ncrease one of the order szes (d ) n these fgures, we keep eff the same and the other order sze d j equals one ( j). In Fgure 2, we observe that both TC opt and TC FCFS ncrease wth d 1. The beneft of mplementng the optmal ratonng polcy over the FCFS polcy (CR FCFS ) may frst ncrease, but eventually decreases wth d 1 as d 1 /d 2 becomes very large. Ths s because the manufacturer usually has to rase ts nventory to deal wth large orders from more valuable customers. Under the FCFS polcy, the manufacturer does not reserve stock for class-1 demands, so to avod a large lost sale penalty a much hgher basestock level s needed. As a result, the total nventory cost s hgher under the FCFS polcy and CR FCFS may frst ncrease wth d 1. However, when d 1 /d 2 s very large, the class-1 orders seldom arrve (because eff 1 remans the Fgure 3. The effect of order szes. Left: on the base-stock levels and the reserve-stock levels, and Rght: on the beneft of the optmal stock ratonng polcy, compared wth Ha s (1997a) polcy. Base-stock level S and reserve-stock level r 6 5 4 3 2 1 d 1 = n, d 2 = 1, FCFS (S) d 1 = n, d 2 = 1, optmal (S) d 1 = n, d 2 = 1, optmal (r) d 1 = 1, d 2 = n, FCFS (S) d 1 = 1, d 2 = n, optmal (S) d 1 = 1, d 2 = n, optmal (r) 5 1 15 2 25 3 Order sze n Cost reducton CR (%) 25 2 15 1 5 d 1 = n, d 2 = 1 d 1 = 1, d 2 = n 5 1 15 2 25 3 Order sze n

Huang and Iravan: A Make-to-Stock System wth Multple Customer Classes and Batch Orderng Operatons Research 56(5), pp. 1312 132, 28 INFORMS 1319 same). It may not be optmal to rase the nventory wthout a lmt. As Theorem 3 depcts, CR FCFS converges to zero when d 1 wth a fnte base-stock level. When d 2 ncreases, the class-2 customers order less frequently, but wth more unts n each order. It s nterestng to note from the left part of Fgure 2 that, under the FCFS polcy, the total average cost may decrease wth d 2, especally when and are large. Our explanaton s that, wth nfrequent orders from demand class-2, class-1 orders are easer to be flled wth the same base-stock level, resultng n an overall savng n lost sale penalty. The base-stock level can even be lowered (see the left part of Fgure 3) wthout sgnfcantly deteroratng the fulfllment of class-1 orders. As a result, the total cost TC FCFS may decrease wth d 2 when d 2 /d 1 s not very large. We have also proved n Theorem 3 that CR FCFS wll decrease to zero when the class-2 order sze d 2 goes to nfnty. Wth the same effectve demand rates, customers order szes have a strong mpact on the beneft of optmal stock ratonng. For example, n the rght part of Fgure 2, when d 1 ncreases from 1 to 5, CR FCFS ncreases from 25.68% to 3.7%. It then decreases dramatcally to 1.39% when d 1 = 3. We also study the mpact of the order sze varablty on CR FCFS n Onlne Appendx B.2, whch shows that a large varablty n class-1 order szes wll deterorate the beneft of optmal stock ratonng, but the order sze varablty n class-2 demands has lttle mpact on the beneft of optmal stock ratonng. The rght part of Fgure 3 nvestgates the mpact of the order szes on CR Ha, where the benchmark s the polcy proposed by Ha (1997a). Specfcally speakng, we frst use Ha s model of untary demands wth arrval rates = eff ( = 1 2), and obtan the optmal base-stock level S Ha, as well as the optmal reserve-stock level r Ha. We then computed the manufacturer s average cost f S Ha and r Ha are used as the base-stock level and the reserve-stock level n the correspondng batch demand systems under the same effectve demand rates eff = d. We fnd that when d 1 = 1butd 2 > 1, the performance of Ha s polcy remans close to optmal regardless of the batch sze d 2 (note that eff 2 stays the same). However, when d 2 = 1 and class-1 customers order n batches, the manufacturer may save up to 21% n cost f t uses the optmal polcy nstead of Ha s polcy. 6. Conclusons Ths paper studes the optmal producton and ratonng polces n a make-to-stock system where dfferent classes of customers order n batches. A lost sale case wth multple demand classes and a backorder case wth two demand classes are analyzed. We characterzed the optmal polces as threshold-type polces. Through the numercal study on the lost sale case, we have also llustrated that the customers order szes can sgnfcantly affect the beneft of optmal stock ratonng. As ndcated n the ntroducton, batch orderng s usually regarded as a negatve factor n nventory management. Ths s true when all customers are of the same economcal mportance to the manufacturer. However, when the customers are of dfferent mportance and stock ratonng s not allowed by regulatons or by contracts (whch s common n many retal busnesses), encouragng less-valuable customers to order n large batches may lower the manufacturer s overall cost. When supplers are allowed to raton ther nventory for dfferent demand classes, they should also be aware of the mpact of batch demands on the cost effectveness of optmal stock ratonng. When less-valuable customers have larger order szes but order less frequently, the need for stock ratonng decreases. On the other hand, when a quantty dscount s ntroduced to more valuable customers, and therefore they order n large batches, the optmal stock ratonng may become more benefcal. Nevertheless, the beneft of optmal stock ratonng decreases rapdly f ether order sze becomes very large. As a result, when adoptng stock ratonng as an operatons strategy, managers should take the customers orderng behavors and the company s marketng strateges nto consderaton to correctly dentfy the beneft of stock ratonng. Future research on the ntegrated solutons for both nventory management and marketng strateges s thus hghly expected. 7. Electronc Companon An electronc companon to ths paper s avalable as part of the onlne verson that can be found at http://or.journal. nforms.org/. Endnotes 1. If ˆr 1 ˆr 2 ˆr N are not monotoncally ncreasng, we can rearrange the demand classes to make ˆr 1 ˆr 2 ˆr N nondecreasng. 2. Note that the orgnal system state space has a boundary y =. We extend the system state space to Z 2 and let cxy =+ when y<. The operators n ths secton and n Onlne Appendx A.3 are also defned on Z 2. For a detaled dscusson, see Ha (1997b) and Glassman and Yao (1994). 3. 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