Dual-Channel Warehouse and Inventory Management with Stochastic Demand

Transcription

1 Unversty of Wndsor Scholarshp at UWndsor Mechancal, Automotve & Materals Engneerng Publcatons Department of Mechancal, Automotve & Materals Engneerng Wnter Dual-Channel Warehouse and Inventory Management wth Stochastc Demand Fawzat Alawneh Guoqng Zhang Follow ths and addtonal works at: Part of the Busness Admnstraton, Management, and Operatons Commons, Busness Analytcs Commons, E-Commerce Commons, Industral Engneerng Commons, and the Operatonal Research Commons Recommended Ctaton Alawneh, Fawzat and Zhang, Guoqng. (2018). Dual-Channel Warehouse and Inventory Management wth Stochastc Demand. Dual- Channel Warehouse and Inventory Management wth Stochastc Demand, 112, Ths Artcle s brought to you for free and open access by the Department of Mechancal, Automotve & Materals Engneerng at Scholarshp at UWndsor. It has been accepted for ncluson n Mechancal, Automotve & Materals Engneerng Publcatons by an authorzed admnstrator of Scholarshp at UWndsor. For more nformaton, please contact scholarshp@uwndsor.ca.

2 Dual-channel Warehouse and Inventory Management wth Stochastc Demand Fawzat Alawneh and Guoqng Zhang* Supply Chan and Logstcs Optmzaton Research Centre, Department of Mechancal, Automotve & Materals Engneerng, Unversty of Wndsor, Wndsor, Ontaro, Canada *Correspondng author. Emal: Abstract Ths study examnes the nventory polcy for the emergng dual-channel warehouse, whch has a unque structure where the warehouse s dvded nto two areas: one for fulfllng onlne orders and the other for storng products and fulfllng offlne orders. A mult-tem nventory model was developed consderng the warehouse capacty constrant, demand, and lead tme uncertanty. Soluton methods are provded for both unform and normal dstrbutons. Adoptng the proposed nventory polcy for a dual warehouse s cost effectve and adds flexblty to the warehouse and supply chan. The study also offers manageral nsghts on some crtcal ssues faced by companes operatng n a dual-channel context. Keywords: dual-channel warehouse; onlne fulfllment; nventory; uncertanty 1. Introducton Onlne sales have experenced a sgnfcant growth n recent years (Wu, 2015). The total e- commerce sales n the Unted States reached $341.8 bllon n 2015, whch s a 14.8% ncrease from 2014 (U.S. Department of Commerce). It s beleved that ths ncrease was because many frms upgraded ther sngle-channel, offlne sales busness models to dual-channel clcks-andmortar models, whch ntegrate both onlne and offlne sales, durng that tme. Moreover, t has been predcted that such growth n onlne sales wll contnue: web-nfluenced sales are expected to grow annually by 6% between 2015 and 2020 (Wu, 2015). Studes have shown that n 2008, 94% of the best fnancally performng frms were dual-channel sales frms (Klcourse and Rowen, 2008). The emergence of dual-channel frms was manly drven by the expanson n nternet use 1

3 and the advances n nformaton and manufacturng technologes provdng compettve advantage to the supply chan (Gunasekaran et al., 2017). Frms ntroducng onlne sales are facng many challenges n terms of logstcs and delvery processes, such as large volumes of very small orders, short delvery lead tmes, flexble delvery (for example, nghttme and even 24-h shppng), and the pckng and packng process for sngle unt orders, n addton to the usual challenges of the conventonal busness. Warehouses or dstrbuton centers must be ready to prepare orders comng from both offlne stores and onlne shoppers. The conventonal warehouse desgned for physcal stores and delvery does not work under a dual-channel busness envronment. For example, warehouse workers cannot use the same pckng patterns for onlne orders as for physcal shoppers (Master, 2015). Warehouses operatng n the current dgtal era of e-commerce must have the all-purpose nfrastructure, whch s capable of sharng nformaton, beng nterconnected, and handlng dfferent orders from dfferent customer segments wth dfferent features such as dverse order szes and delvery lead tmes (McCrea, 2017; Graves, 2012). Two common strateges for warehouses or fulfllment process n the dual-channel busness envronment are the decentralzed and centralzed polces. A frm wth a decentralzed warehouse polcy establshes a dedcated e-fulfllment warehouse and has separate warehouses where each sales channel has separate nventory, operaton, and commercal teams. In many stuatons, usng a decentralzed polcy for all channels n dual-channel strateges results n neffcency (Bendoly, 2004; Zhang et al., 2010; Hübner et al., 2015). Despte the current profts of these frms, they lack nter-channel coordnaton, whch leads to long-term neffcency and consumer confuson (Zhang et al., 2010). The strategy of usng a centralzed warehouse,.e., one ntegrated warehouse or several warehouses clustered n the same locaton, to serve both onlne and offlne orders for a regon has recently ganed popularty and s the most common organzatonal structure for dual-channel markets (Agatz et al., 2008; Hübner et al., 2015; Hübner et al., 2016). The strategy s growth n popularty s owng to the advantages that have been perceved by the frms adoptng t. Such frms nclude the Internatonal Busness Machnes Corporaton, Hewlett-Packard, Whrlpool Corporaton, Poneer Corporaton, Hamlton Beach, and Nke (Huang et al., 2012; Zhang and Tan, 2014; L et al., 2015; Panda et al., 2015; Xao and Sh, 2016). The advantages of ths structure nclude reducng the faclty cost by buldng an ntegrated warehouse, reducng warehouse space 2

4 and nventory requred for both channels, ncreasng the coordnatng ablty and flexblty of fulfllng both onlne and offlne orders, and ncreasng the servce levels. One of the challenges n runnng the dual-channel warehouse s how to organze the warehouse and manage nventory to fulfll both onlne and offlne (retaler) orders, where the orders from dfferent channels have dfferent features. Two mportant dfferences are the order sze and order tme. Typcal onlne orders are placed at random tmes and are usually of small szes, whle typcal offlne orders are placed at scheduled tmes and are usually of large szes (Agatz et al., 2008). Those dfferences affect the warehouse structure and operaton. Many frms wth dual-channel dstrbuton systems have dffculty on developng an effectve nventory polcy to reach an optmal channel performance. One of the key ssues they face s decdng on the optmal order quantty and reorder pont when a new sales channel s ntroduced. Moreover, they need to consder both capacty constrants and uncertan demands (of both offlne and onlne channels). New streams of research have recently commenced studyng dual-channel supply chans. One stream has focused on the competton and coordnaton that arse between sales channels (Hua and L, 2008; Lu and Lu, 2015; Ln, 2016; Matsu, 2016; Wang et al., 2016; Chen and Chen, 2017). Another stream has studed the challengng logstcs and processes of fulfllng onlne orders once they have been placed (De Koster, 2003; Tetteh and Xu, 2014). Research has also been centered on prce and servce nteracton between channels (Yao and Lu, 2005; Ryan et al., 2013; Panda et al., 2015; Rodríguez and Aydn, 2015; Lu et al., 2016; Xao and Sh, 2016; Yan et al., 2016; Gr et al., 2017; Matsu, 2017), and onlne order fulfllment processes (Agatz et al., 2008; Mahar et al., 2009). Inventory management n dual-channel supply chans has also been explored (Khouja, 2003; Yao et al., 2009; Zhang and Tan, 2014; Zhao et al., 2016). However, none of the emergng research streams has examned nventory management n a jont warehouse whle consderng the operatons and capacty of the warehouse. Therefore, ths study examnes the nventory polces for jont warehouse or dstrbuton systems, called dual-channel warehouse n dual-channel busness. As an mportant part of logstcs, the warehouse plays a crtcal role n fulfllng the demands from both channels. The dual-channel warehouse has a unque structure: the warehouse s separated nto two areas, one for fulfllng onlne orders and the other for storng products and fulfllng offlne orders, as shown n Fgure 1(c) (The detals wll be explaned n Secton 3). A warehouse wth such a structure s utlzed by retalers, manufacturers, or thrd-party logstcs (3PLs), who use a centralzed warehouse for 3

5 fulfllng both onlne and offlne orders. We also have observed a couple of dual-channel warehouses of retalers or 3PLs n both Chna and Canada. A smlar structure can be found n e- commerce frms that only have onlne customers (Xu, 2005), as shown n Fgure 1(b). Fgure 1 shows the dfference between a dual-channel warehouse and a conventonal retaler warehouse or an e-commerce warehouse. As shown, the dual-channel warehouse has two areas that fulfll the onlne and retaler orders. The focus of our study s to analyze the structure of the dual-channel warehouse and determne mult-tem nventory polcy (Q, R) for both areas, takng nto account the warehouse capacty, demand, and lead tme uncertanty so that the total cost of the dual-channel warehouse would be mnmzed. Storage area, offlne orders area Storage area Onlne orders pckng area Storage area, offlne orders area Onlne orders pckng area 3PL 3PL Retaler Onlne Retaler Onlne Consumer Consumer Consumer (a) Product flow (b) (c) Informaton flow Fgure 1. (a) (b) Sngle-channel warehouses and (c) dual-channel warehouse Desgnng a sutable warehouse structure for a centralzed warehouse polcy s crtcal for warehouse operatons to prepare orders from both onlne and offlne shoppers. Logstcs vewpont ndcate that t s common to fnd modern warehouse layouts dvded nto dfferent areas for each customer platform (Master, 2015). One of the best warehouse practces for 2017 s to develop allpurpose facltes that can "talk" to one another, handle small orders, medum orders, and large 4

6 orders, and perform all functons n a very accurate manner (McCrea, 2017). A dual-channel warehouse that ntroduces a new area for e-fulfllment process provdes an effcent and practcal structure to connect two warehouse areas for centralzed warehouse polcy. Usually, for heavy or bulky tems such as refrgerators and large furnture, a dedcated e-fulfllment warehouse s a better choce because t has a low-cost effcency n movng those tems frequently n dfferent areas of a warehouse. For most tems n electroncs, department stores, and even grocery stores, a dualchannel warehouse can be a good opton because the added dedcated e-fulfllment area can be desgned to provde an effcent and flexble soluton for hgh volume of small orders, such as lowdensty warehouse, low nventory, specal equpment or structure, and long operaton tme. Two dual-channel warehouses observed n Chna and Canada are used for electronc products and grocery store tems respectvely. Ths study contrbutes to the exstng lterature on warehouse management n several ways. Frst, t s the frst work to analyze the structure of the emergng dual-channel warehouses and develop a structure related to the nventory polcy for such warehouses. Second, t develops a mathematcal model that determnes the mult-tem product nventory polcy for the two areas n ntegrated dual-channel warehouses, mnmzng ther total expected cost. The constrants of warehouse space and uncertan demands are also consdered. Thrd, t provdes closed-form solutons for nstances wthout a warehouse space constrant as well as a soluton algorthm for the case wth the warehouse space constrant. Furthermore, the proposed soluton can be used to evaluate the performance of two-echelon dual-channel warehouse systems by comparng the total system costs for dfferent warehouse structures and evaluatng the effects of addng a new sales channel. To the best of our knowledge, ths study s the frst work to address the nventory polces of the emergng dual-channel warehouses wth a unque structure, although, there have been several studes on nventory polces of a dual-channel supply chan. The rest of ths paper s dvded nto sx sectons. Secton 2 provdes a comprehensve lterature revew of nventory management and dual-channel warehouses n a dual-channel supply chan. Secton 3 provdes the aforementoned mathematcal model. Next, approaches to the soluton are presented n Secton 4. An extenson to correlated demands s dscussed n Secton 5, whle the numercal results and analyss are presented n Secton 6. Fnally, Secton 7 summarzes the conclusons and dscusses potental and relevant future research. 5

7 2. Lterature revew Ths study s related to two streams of lterature that have examned dual-channel supply chans: nventory management n dual-channel supply chans and warehouse operatons, layout desgns, and capacty management n dual-channel warehouses. A lterature revew of each of these topcs can be found below. Inventory management n dual-channel supply chans Varous forms of nventory management have been studed n the dual-channel supply chan lterature. Chang and Monahan (2005) proposed what may be descrbed as one of the frst models that studed nventory polcy n a two-echelon dual-channel supply chan that receves demands from dfferent customer segments. They assumed that the nventory was stored n both the manufacturer s warehouse to satsfy onlne demand and n retal stores to satsfy offlne demand. They developed a stock-based nventory control strategy to mnmze the system s operatng cost by consderng the nventory holdng and lost sales costs. The model developed by Temoury et al. (2008) s consdered an extenson to that by Chang and Monahan (2005). The former s man contrbutons nclude the separaton of both channels lost sale costs and the development of two soluton algorthms. One algorthm was based on the smulated annealng method, and the other algorthm was based on the best neghborhood concept. Takahash et al. (2011) consdered setup costs for both order producton and order delvery. They proposed an nventory control strategy wth the objectve of mnmzng nventory holdng costs, lost sales costs, as well as producton and delvery costs. They calculated the total cost usng Markov analyss to hghlght the performance of ther proposed nventory control polcy. Boyac (2005) also furthered research on dual-channel supply chans nventory management when he nvestgated the nventory levels of a retaler and a manufacturer wth doublemargnalzaton. The author found that as double margnalzaton ncreased, the manufacturer tended to overstock whle the retaler tended to be out of stock. Addtonally, Geng and Mallk (2007) studed nventory competton between a drect onlne channel owned by a manufacturer and an offlne retal channel. They clamed that the proft of a dual-channel supply chan would ncrease as the capacty ncreases. Furthermore, Hosenna et al. (2013) nvestgated the competton that arose between channels; they based ther system on a Stackelberg game. They analyzed the nventory level and ts relatonshp to producton costs and wholesale prces. 6

8 Moreover, Schneder and Klabjan (2013) studed dual-channel revenue management by analyzng the condtons and effects of offerng channel-specfc prces. They also nspected the necessary condtons for optmal nventory control polces of dual-channel sales wth channel-dependent sale prces. Swamnathan and Tayur (2003) descrbed the major adjustments necessary for a conventonal supply chan to cope wth e-commerce fulfllment processes. After a comprehensve lterature revew, they concluded that channel ntegraton n a dual-channel supply chan ncreases proft, reduces nventory, and enhances customer servce. However, the models studed n ther paper prmarly focused on electronc commerce. Hence, dual-channel operatons and ther nterdependences have not been dscussed. Another sgnfcant revew of supply chan management n an electronc commerce envronment was conducted by Agatz et al. (2008). They focused on the dstrbuton network desgn, warehouse layout, nventory, and capacty management topcs. The authors dvded the dual-channel fulfllment process nto ntegrated fulfllment (usng one warehouse to fulfll the demand of dfferent sales channels) and dedcated fulfllment (usng a dedcated warehouse for dfferent channels). Ths dvson was based on ther lterature survey. Integrated fulfllment s the most common network among dual-channel frms. Zhao et al. (2016) suggested a new nventory strategy called onlne-to-offlne strategy. They consdered a dual-channel supply chan wth one manufacturer and one retaler. They also proposed a centralzed and decentralzed nventory model wth and wthout lateral transshpment. The decson varables n ther model were the nventory level for the store and transshpment prce; however, no orderng or holdng costs were consdered. They demonstrated the exstence of a unque Nash equlbrum of the nventory order levels n the dual channel and an optmal transshpment prce to maxmze the proft of the entre supply chan. However, they nether consdered the dual-channel warehouse nor the orderng and holdng costs. Zhang and Tan (2014) studed a dual-channel supply chan wth one manufacturer, whch sells products through a drect channel and a retaler. They constructed a sngle-perod proft-sharng model between the manufacturers and retalers. The decson varables were the nventory levels of the drect and retaler's channel wth a retaler servce constrant. Nonetheless, they nether consdered the dualchannel warehouse nor the operatonal costs. Yao et al. (2009) studed a dual-channel supply chan comprsng one manufacturer and one retaler. They studed a centralzed nventory strategy, the Stackelberg nventory strategy, and 3PL e-tal operaton strategy. They proposed a sngle-perod 7

9 model to obtan the nventory level for the manufacturer and for the retaler that maxmzes the expected proft. However, they dd not deal wth the dual-channel warehouse n terms of structure or at the operatonal level. Khouja (2003) proposed a 3-stage suppler manufacturer customer supply chan model. They employed a perodc revew nventory polcy and defned nventory coordnaton mechansms such as cycle tme and number of orders. Nonetheless, they dd not consder the dual-channel warehouse, ts structure, or operatons. Revewng the nventory management research stream, we found that the (Q, R) polcy s extensvely used n the lterature. Many of the recently publshed artcles have consdered the (Q, R) polcy (Sarkara et al., 2015). The advanced nventory management systems and the reduced cost of rado frequency dentfcaton technology have made the contnuous revew nventory control polcy (Q, R) a very attractve approach. In the modelng process, the annual orderng cost, annual holdng cost, annual backorderng cost, or annual lost sales cost are consdered subject to some servce constrant, whch s typcally the fll rate. Generally, t s dffcult to obtan a closedform soluton, and a well-known teratve algorthm s used to obtan the optmal order quanttes. Ths has led to the use of many heurstcs or approxmaton approaches n solvng the model. As observed, all the revewed studes above dd not consder the dual-channel supply chan nventory strateges n the context of a dual-channel dstrbuton system. They allocated onlne demand to the manufacturer warehouse wthout studyng the mplcatons that onlne fulfllment capablty has for the dual-channel warehouse structure and operatons. Addtonally, they dd not consder the dual-channel warehouse structure, operatons, or capactes. Fnally, they consdered determnstc lead tmes. Ths study flls these research gaps by examnng the nventory strateges for a dual-channel supply chan whle consderng the dual-channel warehouse structure, operatons, space constrant, stochastc demand, and lead tme. It combnes the research felds of dual-channel warehouse operatons, structure desgns, and capacty management as well. Warehouse operatons and management n dual-channel supply chans The lterature on dual-channel warehouse operatons demonstrates the mportance of pckng processes, partcularly wth regard to drect channel fulfllment processes. Hübner et al. (2015) revewed the operaton structures of mult-channel retalng, ncludng network desgn, nventory management, warehouse operatons, and capacty management. They dscussed the structures and challenges n mult-channel warehouse operatons. They concluded that the man drver n mult- 8

10 channel operatons was an effcent ntegraton of warehouse operatons. They provded nterestng nsghts on mult-channel operatons. However, ther fndngs were based on a lterature survey, and the analyss they presented was not based on an applcaton of the model to a real case study or numercal analyss. Allgor et al. (2003) studed e-retalng settngs and the effects they had on conventonal nventory models. The authors dvded warehouses nto two areas: a deep storage area and a low storage pckng area. They proposed a mult-tem, two-stage perodc revew model (R, T). A heurstc-based algorthm was proposed as a soluton approach. Xu (2005) presented a perodc revew nventory model for a sngle-channel e-talor order fulfllment process consderng warehouse space. To optmze warehouse operatons, the warehouse was dvded nto two areas. One of these areas had a low densty for order pckng and the other had a hgh densty for stockng tems and replenshng the center s pckng area usng a perodc revew nventory control polcy. They consdered a stochastc demand; however, they assumed a determnstc lead tme. Ths study dffers from that of Allgor et al. (2003) and Xu (2005) n the followng two aspects: frst, ths study consders the dual-channel supply chan wth both onlne and offlne demands whle the references dealt wth a sngle channel only,.e., e-talor supply chan; second, the proposed model n ths study s based on a contnuous revew nventory polcy (Q, R) and specfcally consders warehouse structure, operatons, and capactes, whle the references proposed a perodc revew model (R, T). The smlarty between our studes and those n the references s the dvson of the warehouse nto two stage areas. Related to the dual-channel warehouse n terms of dvson of space, the forward-reserve problem has already been modeled n prevous studes. Hackman and Rosenblatt (1990) developed a model to determne whch tems to assgn to the automated storage and retreval system (AS/RS), where the warehouse was dvded nto two areas: AS/RS area and the area for manual or semautomated materal handlng system. Instead of decdng whch area each tem should be placed n, ths study decdes the nventory polcy for each tem, and both areas have all tems to serve onlne and offlne orders. Barthold and Hackman (2008) nvestgated how to allocate a forward pck area n a dstrbuton center. The dual-channel warehouse n ths study offers delvery operatons n both areas. The prevous works nvestgated the forward-reserve problem wth a sngle-channel and determnstc demand, whle no orderng and backordered costs were consdered. 9

11 It s noted that the e-commerce ndustry has been usng the mult-channel warehouse for several years, but only a couple of artcles dscussng such warehouse can be found n the lterature, such as that by Hübner et al. (2015). Furthermore, none of those artcles provded quanttatve analyss for the mult-channel warehouse. A comprehensve lterature revew ndcates that some mathematcal nventory management models have been proposed for dual-channel supply chans; however, there s a lack of research that nvestgates the warehouse structure, operatons, and capacty n a dual-channel context. Some artcles have addressed the warehouse operatons and capacty management of sngle-channel warehouses, but they have not addressed these n a dualchannel context. Therefore, to the best of our knowledge, nventory management, warehouse structure, operatons, and capacty management have not been harmonzed for an ntegrated model n a dual-channel context. 3. Mathematcal model formulaton 3.1 Problem statement The man objectves of a manufacturer s warehouse are to ncrease space utlzaton, reduce operaton cost, and fulfll orders quckly and relably. These objectves are usually conflctng. To obtan hgh space utlzaton, we need to store tems n a hgh-densty storage area such as pallets or hgh beam storage systems. Meanwhle, effcent order pckng for onlne orders, whch are usually of small szes, requres the pcker to have full access to the stored tems, whch means that they need to be dsplayed n low-densty storage areas such as racks or stands. At the same tme, to provde a hgh level of servce, the warehouse needs to have an optmal nventory level for each tem. We consder the emergng dual-channel warehouse to fulfll both onlne and offlne orders. To optmze the operaton, the structure desgn of the dual-channel warehouse reflects the dfferent features of the two dfferent orders: the warehouse s dvded nto two storage areas wth dfferent nventory levels. One area, called Stage 1 area, s usually for pckng tems that are dsplayed on shelves or stands, packng, and shppng small sze onlne customer orders, whle the other area, called Stage 2 area, s for deep storage, to store nventory, replensh Stage 1, and fulfll offlne retaler s large sze orders. Orders from the suppler or the manufacturer wll usually come n pallets and be stored frst n Stage 2 area. Together, the areas form a two-echelon seral nventory control system, whch s shown n Fgure 2. 10

12 Our goal s to develop a decson support tool for the operatonal and strategc decson related to the dual-channel warehouse wth both onlne and offlne fulfllment capablty. On the operatonal level, we ntend to assst n determnng the optmal nventory level, tem flow between the deep storage area and onlne pckng area, as well as the replenshment frequency of both areas. On the strategc level, we wll analyze the effect of the warehouse structure and space reserved for the onlne pckng area on the total operatng cost. Warehouse (dashed lne) wth onlne fulfllment capablty Q 2 Manufacturer s faclty R 2 Stage 2: Storage area, offlne orders area Q 1 R 1 Stage 1: Onlne orders pckng area Retalers Manufacturer s webpage Product flow Informaton flow Offlne consumers Onlne consumers Fgure 2. Dual-channel warehouse wth onlne fulfllment capablty 3.2 Notatons and assumptons Notatons The notatons used n developng the mathematcal model are gven as follows: : Item ndex j: Stage ndex, where j = 1 for warehouse area dedcated to satsfyng onlne demand (onlne pckng area), and j = 2 for warehouse area dedcated to satsfyng both retal and dedcated onlne area demands 11

13 L j : Length of lead tme for tem n stage j (random varable) D j : Expected annual demand for tem n stage j h j : Holdng cost per unt tme for tem at stage j b j : Backorder cost per unt for tem at stage j A j : Orderng cost per order for tem at stage j x j : Demand durng lead tme (DDLT, random varable) for tem n stage j f(x j ): Probablty densty functon of lead-tme demand for tem at stage j γ j : Storage space requred by a stock keepng unt n stage j α: Mnmum requred probablty that total order quanttes wll be wthn warehouse space S: Avalable space of the entre warehouse Decson varables Q 2 : Order quantty for tem n Stage 2 Q 1 : Order quantty for tem n Stage 1 R 2 : Reorder pont when new order s placed for tem n Stage 2 R 1 : Reorder pont when new order s placed for tem n Stage Assumptons and prelmnary analyss 1) The demand rate per unt tme (day or week) durng lead tme s a random varable wth a mean of μ dj and standard devaton of σ dj. We assumed that the demand standard devaton s very small relatve to the mean demand; therefore, the probablty of negatve demand s neglgble (Lee, 2005; Zhang et al., 2006). 2) The lead tme L j s a random varable wth a mean of μ Lj and a standard devaton of σ Lj. 3) If the DDLT for tem n stage j s n a stuaton where the demand and lead tme are normally dstrbuted and statstcally ndependent, then the mean and standard devaton of the DDLT are μ xj = μ Lj μ dj and σ xj = μ Lj σ 2 dj + μ 2 dj σ 2 Lj. In the stuaton where there s a fxed lead tme, μ xj = L j μ dj and σ xj = L j σ 2 dj. (1) (2) In the stuaton where there s a unform dstrbuton of the demand and lead tme, the demand jont dstrbuton functon s defned as 12

14 f(x j ) = 1 (d Mj d mj )(t M j t mj ). Moreover, the mean of the DDLT s μ xj = (d M j + d mj )(t Mj + t mj ), 4 and the standard devaton of the DDLT s σ xj = (d M j d mj ) 2 (t Mj t mj ) 2 +3(d Mj +d mj ) 2 (t Mj t mj ) 2 +3(d Mj d mj ) 2 (t Mj +t mj ) 2, 144 (3) (4) (5) where (t m j, t M j ) are respectvely the lower and upper lmts of the unform lead tme demand dstrbuton, and (d mj, d Mj ) are the lower and upper lmts of the unform demand dstrbuton respectvely (Das and Hanaoka, 2014). In the retal envronment, where the demand per perod s normally large, the normal dstrbuton s an approprate modelng choce (Hadley and Whtn, 1963; Slver and Peterson, 1985), partcularly f we have suffcent hstorcal data from whch the mean and the standard devaton can be drawn. However, a unform dstrbuton s commonly used for new tems n stuatons where such hstorcal data s not avalable (Wanke, 2008). Usually, the warehouse serves many retalers va the offlne channel. The ntegrated offlne demand s large and thus, t can be assumed to reasonably follow the normal dstrbuton or the unform dstrbuton. The unform and normal dstrbutons are both typcally used to descrbe uncertan demands/lead tme. Our model proposed n the next secton s ndependent of the probablty dstrbuton unless t s contnuous, and works for other probablty dstrbutons such as the exponental dstrbuton. However, solvng the problem, partcularly those wth closed-form solutons, depends on the dfferent dstrbutons. 4) After conductng a lterature revew on the dual-channel demand structure, we found that the demand s categorzed wthn two streams. In the frst stream, the demand of each channel s treated as an ndependent random varable. The total system demand s the aggregaton of both channel demands (Alpteknoglu and Tang, 2005; Lee, 2005; Abdul-Jalbar et al., 2006; Sefert et al., 2006; Zhang et al., 2006; Bchescu and Fry, 2009). In the second stream, the demand s correlated, and the total system demand, whch follows a specfc dstrbuton, s known. Then t s splt between the ndvdual channels (Lppman and McCardle, 2004; Tsay and Agrawal, 2004; Chang and Monahan, 2005; Yao et al., 2005). 13

15 In our proposed model, we consdered both cases of ndependent and correlated demand. Addtonally, regardless of the demand structure, we have assumed that customer channel loyalty β j ranges between 0 100%. Ths means that wth 100% channel loyalty, sales are lost n stuatons where there s a sales channel absence. We assumed that onlne and retaler demand s ndependent (the assumpton s relaxed n Secton 5). Consequently, as an llustratve example, the sngle-tem (we dropped the ndex for smplcty) system demand s gven as follows: Stage 2 demand wll be the aggregaton of the onlne and offlne demand,.e., D 2 = D r + D d and the demand at Stage 1 s D 1 = D d. In the case where we have a sngle-retaler channel, Stage 2 demand wll be the retaler demand plus the percentage of customers wllng to swtch from the onlne channel,.e., D 2 = D r + β 1 D d. In cases where there s only an onlne channel, Stage 2 demand wll be the aggregaton of the onlne demand plus the percentage of customers wllng to swtch from the retaler channel: D 2 = D d + β 2 D r. Stage 1 demand s gven by the followng: D 1 = D d where there s a dual sales channel, D 1 = 0 where there s only a retaler channel, D 1 = D d + β 2 D r where there s only an onlne channel. 5) Ths study employs a contnuous revew nventory control polcy, also known as the (Q, R) polcy. Such a polcy s also used extensvely n the exstng lterature, such as n artcles by Khouj and Stylanou (2009) and Sarkara et al. (2015). 6) A demand that cannot be mmedately satsfed by the nventory s backordered wth a penalty cost (Hadley and Whtn, 1963; Nahmas, 2013). Ths s more common when dealng wth onlne demand as onlne orders have more flexble delvery tmes than offlne orders. 7) Each stage (each area n the warehouse) has a reorder pont that corresponds to an nstallaton nventory for that stage. The reorder pont s equal to the expected DDLT plus the safety stock, whch s a functon of stock-out probablty durng lead tme. Stage 1 receves nternal shpments from Stage 2, whle Stage 2 receves shpments from the suppler. 8) The orders do not cross, because a sngle suppler s used or one outstandng order s assumed. 14

16 3.3 Mathematcal models The problem s to determne the nventory polcy for both stages n the dual-channel warehouse so that the total expected cost s mnmzed, subject to the warehouse capacty lmt. The formulaton of the problem s gven as follows. The objectve of the problem s to mnmze the annual total expected cost, denoted as C(Q 2, R 2, Q 1, R 1 ), whch comprses orderng, holdng, and shortage costs. For a gven nventory polcy (Q j, R j ), the average nventory level for Stage 1 s the average cycle nventory plus the safety nventory, approxmately expressed as Q1/2 + R1 μ x1, where R1 μ x1 s the safety stock. The approxmaton on the average nventory s reasonable for many real cases and s wdely used n textbooks and n the lterature (De Bodt and Graves, 1985; Yano, 1985; Zpkn, 1986; Ghalebsaz-Jedd et al., 2004; Khouja and Stylanou, 2009; Nahmas, 2013; Fattah et al., 2015). Smlarly, the average nventory level for Stage 2 s approxmately expressed as Q2/2 + R2 μ x2. Thus, the annual total expected cost s formulated as follows wth respect to the decson varables Q 2, R 2, Q 1, R 1. Objectve: Mn the total expect cost C(Q 2, R 2, Q 1, R 1 ) = A 2D 2 + A 1D 1 + Q 1 Q 2 + h 1 [( Q 1 2 ) + (R 1 μ x1 )] + b 1D 1 [ (x Q 1 R 1 ) f(x 1 ) dx 1 ]. 1 R 1 h 2 [( Q 2 2 ) + (R 2 μ x2 )] + b 2D 2 [ (x Q 2 R 2 ) f(x 2 ) dx 2 ] 2 R 2 The frst and second terms of the objectve functon (6) refer to the annual orderng cost, whch s the order cost multpled by the number of cycles. The thrd and fourth terms refer to the annual approxmated holdng cost. The ffth and sxth terms represent the annual backorder cost, whch s equal to the backorder cost multpled by the expected number of shortages per cycle. We consder the warehouse capacty constrant. Because of uncertan demand, we set the probablty that the total smultaneous tems nventory wthn the warehouse space when the order s receved wll not be smaller than α. Then we have the followng constrants: P[( γ 2 (Q 2 + R 2 x 2 ) + γ 1 (Q 1 + R 1 x 1 )) S] α, (7) (6) 15

17 R j, Q j 0, j. (8) The space constrant (2) can be wrtten as P [ γ 2 x 2 + γ 1 x 1 (γ 2 (Q 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S] α, whch can be reformulated as (γ 2 (Q 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S + μ Y + z 1 α σ Y, where Y = γ j x j, μ Y = γ j μ j, and σ 2 Y = γ 2 j σ j, j j and z 1 α s the value of the cumulatve probablty dstrbuton of the demand at pont 1 α (Ghalebsaz-Jedd et al., 2004). A varant of the above constrant can be appled to ether Stage 1 or Stage 2 n case we have a separate warehouse space lmt. If the warehouse space constrant s appled to ether area, we obtan the followng: For Stage 1, the constrant wll be γ 1 (Q 1 + R 1 ) S 1 + μ Y1 + z 1 α σ Y1, 2 = where μ Y1 = γ 1 μ 1, σ Y1 j γ 2 1 σ 1, and S 1 s the area dedcated for Stage 1. Meanwhle, f the space constrant s appled to Stage 2, we obtan γ 2 (Q 2 + R 2 ) S 2 + μ Y2 + z 1 α σ Y2, 2 = where μ Y2 = γ 2 μ 2, σ Y2 γ 2 2 σ 2, and S 2 s the area dedcated for Stage 2. The model formulated usng (6), (8), and (10), denoted as problem (P), s a constraned nonlnear program, where t s dffcult to fnd a closed-form soluton. A detaled soluton approach s dscussed n the next secton. (9) (10) (11) (12) (13) 4. Soluton Before ntroducng the soluton approach, we defne the expected shortage per cycle (ESC) and cycle servce level (CSL). Slver and Peterson (1985) defned the ESC for the sngle-stage case. We extended the ESC to the dual-stage case as follows: 16

18 ESC(R j ) = (x j R j ) f(x j )dx j, R j (14) R CSL: j f(x j )dx j. 0 The constraned nonlnear problem gven s a convex problem, whch s descrbed by the followng theorem. Theorem 1: The nonlnear programmng problem (P) s convex. Proof. Please see Appendx A. Because problem P s a convex nonlnear program, ths mples that the soluton of the problem (P) s unque and satsfes the necessary Karush Kuhn Tucker (KKT) condtons. We consder a Lagrange functon L(Q 2, R 2, Q 1, R 1, θ) = A 2D 2 + A 1D 1 + h Q 2 Q 2 [( Q ) + (R 2 μ x2 )] + h 1 [( Q 1 2 ) + (R 1 μ x1 )] + b 1D 1 [ (x Q 1 R 1 ) f(x 1 )dx 1 ] 1 R 1 + b 2D 2 [ (x Q 2 R 2 ) f(x 2 )dx 2 ] 2 R 2 (15) + θ [ (γ 2 (Q 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S μ Y z 1 α ], (16) where θ s the Lagrange multpler for the space constrant. Then we can fnd the optmal soluton va the followng KKT frst-order condtons: From L Q j = 0, we obtan A jd j Q j 2 Rearrange to obtan Q j = 2D j(a j +b j ESC(R j )). h j +2γ j θ + h j b 1D 1 2 [ (x 2 Q 1 R 1 ) f(x j R 1 )dx 1 ] + γ j θ = 0. 1 (17) (18) From L R j = 0, 17

19 we obtan h j + b jd j Q j [f(x j )dx j ] + γ j θ = 0. (19) Rearrange to obtan f(x j )dx j R j We also have = (h j + γ jθ)q j b j D j. (20) L = j γ j(q θ j + R j ) S μ Y z 1 α σ y 0 and (21) R j, Q j, θ 0, j. (22) If (18) s substtuted nto (20), we obtan f(x j )dx R j = j (h j +γ j θ) 2D j (A j +b j ESC(R j )) h j +2γ j θ b j D j. Squarng both sdes and arrangng, we obtan 2 [ f(x j )dx R j ] 2 bj D 2 j j = (h j + γ j θ) 2 [ 2D j(a j +b j ESC(R j )) ]. Rearrangng the above equaton, we obtan h j +2γ j θ b j D j (1 CSL(R j )) 2 2(h j + (h j + 1)γ j θ)esc(r j ) 2(h j+(h j +1)γ j θ)a j b j = 0. (23) (24) (25) We wll dscuss the soluton approaches for both unform and normal demand dstrbutons. For each dstrbuton, we also nvestgate two stuatons: wth and wthout warehouse space constrants (or nactve constrant). We dscuss the problem wthout constrant because we can develop closed-form solutons for the stuaton, whch may occur n practce. 4.1 Unform dstrbuton of demand and lead tme Ths secton provdes the soluton when the demand and lead tme follow a unform dstrbuton. The use of unform demand s a common approach n the case of new products whenever one does not have suffcent hstorcal data to obtan the parameters of the probablty densty functon of the demand or lead tme (e.g., the normal dstrbuton mean and standard 18

20 devaton) (Wanke, 2008; Das and Hanaoka, 2014) Unform dstrbuton and determnstc lead tme wthout space constrant Assume that the demand follows the unform dstrbuton (0, U j ); then f(x j )dx j = (1 R j ), U j R j and (x j R j )f(x j )dx j = U j 2 R j R j + R 2 j. 2U j If (26) and (27) are substtuted nto (25), then b j D j (1 2R j U j + R 2 j 2 U ) 2h j ( U j j 2 R j + R 2 j ) ( 2h ja j ) = 0. 2U j b j (26) (27) (28) Rearrangng the above equaton, we obtan ( b j U j 2 h j U j ) R j 2 (2h j 2b jd j U j ) R j + (b j D j h j U j 2h ja j b j ) = 0. (29) The result s a quadratc equaton wth one unknown, R j. Then we can determne the optmal reorder pont for each stage: R j = (2h j 2b jd j ) ± (2h U j 2b 2 jd j ) 4 j U ( b j 2 j U h j j U ) (b j D j h j U j 2h ja j ) j b j 2 ( b j U j 2 h j U j ) Wth R j calculated above, we can determne the optmal order quantty Q j usng (18).. (30) Unform dstrbuton and stochastc lead tme wthout space constrant In the case of a stochastc demand and stochastc lead tme, an ntegraton should be obtaned usng the jont dstrbuton functon of two random varables. If the demand by unt tme follows the unform dstrbuton U~ (0, d M ) and the lead tme U~ (0, t M ), then R j f(x j )dx j = = 1 [ R j (d Mj t Mj ) (1 + ln (d M t j M j ))], R j and (31) 19

21 (x j R j )f(x j )dx = R j 1 (2d Mj t Mj ) [ t 2 M j (d 2 M j 2 R j 2 2 ) R 2 j ln ( d M j t Mj )] t Mj R j (32) R j R j [1 ( (1 + ln (d Mj t Mj ) (d M j t Mj ))]. R j When (31) and (32) are substtuted nto (25), then 2 R j b j D j [(1 ( (d Mj t Mj ) (1 + ln (d M t j M j ))))] R j 2 1 2h j [( (2d Mj t Mj ) ) (t M j 2 (d 2 M j R j 2 2 t ) R j 2 ln ( d M j t Mj )) Mj R j R j R j (1 ( (d Mj t Mj ) (1 + ln (d M t j M j )))] 2h ja j = 0. b j R j (33) Equaton (33) s nonlnear wth the sngle varable of reorder pont R j, whch can be solved usng an Excel spreadsheet, or usng an advanced math program, such as Matlab. Wth the calculated optmal reorder pont, we can determne the optmal order quantty Q j usng (18) for ths case Unform dstrbuton wth space constrant When there s a warehouse space constrant, we can determne the optmal soluton by solvng the dual problem of the Lagrangan functon gven n (16): Max θ Mn L(Q 2, R 2, Q 1, R 1, θ). Actually, we can solve the problem frst wthout consderng the warehouse constrant through equatons (30) or (33), and then check the constrant (10). If the constrant s satsfed, then we determne the optmal soluton for the orgnal problem. Otherwse, we can use ether a subgradent method or bsecton search to solve the Lagrangan dual problem. Because the problem s convex, there s a unque soluton. In ths case, based on (21), we have j γ j (Q j + R j ) S μ Y z 1 α σ y = 0. (34) 20

22 For a gven value of θ, Q j and R j can be calculated usng (30) or (33); then they can be substtuted nto equaton (34). Ths reduces the problem to a soluton for one equaton wth one unknown θ: g(θ) = γ j (Q ~ j + R~ j) S μ Y z 1 α = 0. j As there s one varable and soluton unqueness, we can use the bsecton search method to determne the soluton. Therefore, f there are two dstnct values of θ 1 and θ 2, such that g(θ 1 ) and g(θ 2 ) < 0, satsfyng ths condton s suffcent to allow usng any onedmensonal search technque to solve (30). The followng algorthm s thus proposed. 1. Let θ 1 = 0 and let θ 2 be the smallest number, such that g(θ 2 ) < Let Q 1 ~, R 1 ~ be the soluton when θ = θ 1, and let Q 2 ~, R 2 ~ be the soluton when θ = θ Let θ = θ 1+θ 1 2 and solve for Q ~ and R ~ ; fnd g(θ). 4. If g(θ) > 0, then θ 1 = θ, Q 1 ~ = Q ~, and R 1 ~ = R ~ ; f g(θ) < 0, then θ 2 = θ, Q 2 ~ = Q ~, and R 2 ~ = R ~. 5. If (g(θ 1 ) g(θ 2 )) < ε g, then stop. Otherwse, go to 3. (35) 4.2 Normal dstrbuton demand and lead tme In stuatons where suffcent hstorcal data are avalable, the normal probablty dstrbuton for the demand and lead tme can be generally estmated. Usng the formulas presented n assumpton 3, we can calculate the mean and standard devaton of the DDLT for determnstc or stochastc lead tme. In the next sectons, we wll dscuss the soluton methodology when space constrant s actve or nactve Normal dstrbuton wthout space constrant Gven that R j = μ xj + kσ xj, the expected shortage per cycle can be formulated as a functon of the safety factor k, as presented by Kundu and Chakrabart (2012). In stuatons where there s a sngle channel, the proposed formula may be extended to consder two-echelon dual-channel stuatons. If ESC(R j ) = σ x j 2 ( 1 + k j 2 k j ), (36) 21

23 then the Lagrange functon for the ndependent demand s L(Q j, k j, θ) = A jd j j Q j + h j (( Q j 2 ) + k jσ xj ) + b jd j Q j ( σ x j 2 ( 1 + k j 2 k j )) + θ [ γ j (Q j + μ xj + k j σ xj ) S μ Y z 1 α ]. j (37) Usng the necessary KKT condtons for mnmzaton problems, we obtan L = 0, A jd j Q 2 j Q + h b j T j ( σ x j 2 ( 1 + k j 2 k j )) j j θγ j = 0. Q j (38) Ths leads to σx j Q j = 2D j[a j +b j ( 2 ( 1+k j 2 k j ))], h j +2γ j θ (39) k j L = 0, h k j σ xj + b jd j σ j 2Q xj j k [ ( j If we substtute (39) nto (40), we have 1 )] + θγ j σ(x) j = 0. (40) b j D j σ xj 2 2D j [A j + b j ( 2 ( 1 + k j 2 k j ))] h j + 2γ j θ k j σ xj k ( j ( + h j σ xj 1)) + γ jσ xj θ = 0. (41) As the warehouse space constrant s not actve, θ = 0; the remander s one equaton wth one unknown. We may solve for k j and consequently fnd Q j and R j Normal dstrbuton wth space constrant When the warehouse space constrant s actve, we can apply the soluton approach presented n Secton Smlar to the KKT condtons on Lagrangan multpler wth a unform dstrbuton, we have 22

24 L θ = (γ 2(Q 2 + σ x2 k 2 ) + γ 1 (Q 1 + σ x1 k 1 )) S μ Y z 1 α 0. (42) Wth the bsecton search method n Secton 4.1.3, we can obtan the soluton. 5. Extenson to correlated demands In ths secton, we extend the model to the stuaton where the demands from the two stages are correlated. We assume that the total demand D s known and follows a specfc dstrbuton. To determne the Stage 2 and Stage 1 demand, we defne a channel demand splt factor φ, where the onlne demand = φd and retaler demand = (1 φ)d (Yao et al., 2009). In ths case, Stage 2 demand wll be as follows: D2 = D where there s a dual sales channel; D2 = (1 φd) + β 1 (φd) where there s only a retaler channel; D2 = φd + β 2 (1 φ) D where there s only an onlne channel. Stage 1 demand wll be D1 = φd where there s a dual sales channel; D1 = 0 where there s only a retaler channel; D1 = φd + β 2 (1 φ) D where there s only an onlne channel. The model gven by (1) and (2) s changed wth the followng new objectve functon: C(Q 2, R 2, Q 1, R 1 ) S.T. = A 2D + A 1φ D + h Q 2 Q 2 [( Q ) + (R 2 μ x2 )] + h 1 [( Q 1 2 ) + (R 1 μ x1 )] + b 1D 1 [ (x Q 1 R 1 ) f(x 1 ) dx 1 ]. 1 R 1 + b 2D 2 [ (x Q 2 R 2 ) f(x 2 ) dx 2 ] 2 R 2 (43) (γ 2 (Q 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S + μ Y + z 1 α σ Y. Applyng the soluton approach presented n Secton 4, we obtan (44) 23

25 L(Q 2, R 2, Q 1, R 1, θ) = A 2D + A 1φ D + h Q 2 Q 2 [( Q ) + (R 2 μ x2 )] + h 1 [( Q 1 2 ) + (R 1 μ x1 )] + b 1D 1 [ (x Q 1 R 1 ) f(x 1 ) dx 1 ] 1 R 1 + b 2D 2 [ (x Q 2 R 2 ) f(x 2 ) dx 2 ] 2 R 2 + θ [ (γ 2 (Q 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S μ Y z 1 α ]. Usng the necessary KKT condtons for mnmzaton problems, we obtan b 2 D (1 CSL(R 2 )) 2 2(h 2 + (h 2 + 1)γ 2 θ)esc(r 2 ) 2(h 2 + (h 2 + 1)γ 2 θ)a 2 b 2 = 0, and b 1 φ D (1 CSL(R 1 )) 2 2(h 1 + (h 1 + 1)γ 1 θ)esc(r 1 ) 2(h 1+(h 1 +1)γ 1 θ)a 1 b 1 = 0, L = (γ 2(Q θ 2 + R 2 ) + γ 1 (Q 1 + R 1 )) S μ Y z 1 α σ y 0. The soluton methodology dscussed for the ndependent demand model can be used to solve the correlated demand model for unform and normal demands. (45) (46) (47) (48) 6. Numercal examples and results In ths secton, we present numercal examples to verfy the model and soluton methods and to show the results for dfferent demand dstrbutons and the effects of demand features, warehouse space, and channel preference. 6.1 Model parameters The parameters used for the experment are based on the followng observatons: γ 1 > γ 2 : γ represents the storage requrements n the warehouse per tem. The assumpton s based on the fact that the space requred for each unt stored on pallets n Stage 2 s less than that n Stage 1, where tems are usually stored n low-densty storage systems such as stands or racks to facltate the ndvdual tem pckng process. 24

26 D 2 > D 1 : D represents the demand. Offlne demand s usually hgher than onlne demand and the order sze for an offlne channel demand s larger than that for an onlne channel. A 2 > A 1 : A represents the orderng cost. The orderng process for Stage 1 ams to replensh tems for Stage 2, whle the replenshment for Stage 2 requres orderng tems from the suppler. Thus, the orderng cost for Stage 2 from the external suppler s hgher. b 2 > b 1 : b represents the backorder cost. The backorder cost for the onlne channel s set to be less than that of the offlne channel. The sze of an onlne order s usually smaller than that of an offlne order, and onlne orders have more flexble delvery tmes than offlne orders (Agatz et al., 2008). Havng a shortage n offlne orders usually results n a hgher penalty based on the contract sgned between the manufacturers and retalers, whle shortage n an onlne order has a lesser economc effect on the manufacturers; therefore, t s reasonable to have a shortage cost for Stage 2 that s hgher than that for Stage 1. h 1 > h 2 : h represents the holdng cost per tem. The holdng cost for the onlne channel s hgher than that for the offlne channel as the requred space to store a unt n the onlne low-densty area s greater than that n the offlne hgh-densty area. 6.2 Numercal examples for ndependent demands We testes seven examples wth dfferent demand dstrbutons and lead tmes for the case where the demands are ndependent. The nput parameters used are gven n Appendx B. Unform dstrbuton demand The frst example s the dual-channel warehouse wth ndependent demands that follow the unform dstrbuton, whle the lead tme s determnstc. Table 1 presents the obtaned soluton for two tems wth a unform dstrbuton demand. For nstance, the order sze for tem 1 s 19,010 unts, whle the reorder pont s 1003 unts. Stage 2 replenshes Stage 1 wth a batch of 335 unts at a reorder pont of 131 unts. The total system cost s $33,566. Table 1. Inventory polcy (Q, R) and cost for Example 1 Order Quantty Reorder pont Total Cost Q R $33,566 Q R Q R Q R

27 Example 2 s the same as Example 1 but wthout the warehouse constrant. In addton, both determnstc and stochastc lead tmes are consdered. Table 2 presents the man parameters and results. The reorder pont wth a stochastc lead tme (more safety stock) has ncreased to cope wth hgher uncertanty. Table 2. Results for Example 2 wth unform demand and stochastc lead tme Input parameters Results (Q, R) d M t M D A B h R Q Total Cost Determnstc $29,809 lead tme Stochastc lead $35,964 tme Normal dstrbuton demand Table 3 presents the soluton for Example 3, whch has a normal dstrbuton demand and determnstc lead tme, but no space constrant. Example 4 s the same as Example 3 except that t has a stochastc lead tme for Stage 2 (note that the lead tme for Stage 1 remans determnstc). As we can observe, the reorder pont for the stochastc case s hgher than that of the determnstc case, and the total cost s ncreased from $5,561 to $6,030 as the nventory holdng cost ncreases because we have to keep more safety stock to cope wth hgher demand varaton. Table 3. Results for Example 3 wth normal dstrbuton demand and determnstc lead tme Order Quantty Reorder Pont Safety Factor Total Cost Q R 11 4 k $5,561 Q R k Q R 21 3 k Q R k Table 4. Results for Example 4 wth normal dstrbuton demand and stochastc lead tme Order Quantty Reorder Pont Safety Factor Total Cost Q R 11 4 k $6,030 Q R k