A SIMULATION FRAMEWORK FOR REAL-TIME MANAGEMENT AND CONTROL OF INVENTORY ROUTING DECISIONS. Shrikant Jarugumilli Scott E. Grasman Sreeram Ramakrishnan

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1 Proceedings of he 006 Winer Simulaion Conference L. F. Perrone, F. P. Wieland, J. Liu, B. G. Lawson, D. M. Nicol, and R. M. Fujimoo, eds. A SIMULATION FRAMEWORK FOR REAL-TIME MANAGEMENT AND CONTROL OF INVENTORY ROUTING DECISIONS Shrian Jarugumilli Sco E. Grasman Sreeram Ramarishnan Dep. of Engineering Managemen and Sysems Engineering Universiy of Missouri Rolla, Rolla, MO 6509, U.S.A. ABSTRACT We consider a logisics newor where a single warehouse disribues a single iem o muliple reailers. Reailers in he newor paricipae in a Vendor Managed Invenory (VMI) program wih he warehouse, where he warehouse is responsible for racing and replenishing he invenory a various reailer locaions. The informaion updae occurs every ime a vehicle reaches a locaion and he decision on he delivery quaniy and he nex locaion o visi is made. For a small increase of locaions in he newor, he sae space for he soluion increases exponenially, maing his problem NP-hard. Thus, we propose a soluion mehodology where in he size of he sae space is reduced a each sage. In his wor, we use simulaion o develop he framewor for he real-ime conrol and managemen of invenory and rouing decisions, given his scenario. INTRODUCTION In an efficien logisic newor he primary objecive is o deliver he righ produc, in he righ quaniy, a he righ ime, while eeping he overall cos a a minimum. A ypical logisics newor consiss of muliple facories, disribuion ceners, wholesalers, reail oules, and cusomers, and requires movemen of maerial/ invenory in he newor o be efficien and economical by maing opimal rouing and invenory decisions. Advances in echnology have forced many companies o change heir business models, and many companies paricipae in VMI iniiaives wih heir suppliers. The angible benefis of successful VMI implemenaion include improved forecasing due o increased visibiliy across he supply chain, reduced invenory levels wih higher invenory urns, and reduced coss (Kleyweg, Nori, and Savelbergh 00; Lee and Whang 99). The reailers also gain remarable efficiencies by reducing errors and socous, while increasing cusomer saisfacion. Hence, i becomes essenial for a supplier o follow an inegraed approach for maing opimal rouing and invenory decisions. Invenory rouing problems (IRP), inegrae he managemen and conrol of invenory and vehicle rouing in a logisics newor. The complexiy of IRP lies in he fac ha here could be several alernaive roues among he various reailer locaions in he newor and a varying quaniy of invenory o be allocaed a each locaion. Inuiively, i is ideal o saisfy he demand a each locaion when requesed, however, real life siuaions are consrained by he vehicle capaciy and flucuaing demand paerns a oher locaions. We refer he readers o he wor of Kleyweg, Nori, and Savelbergh (00) and Kleyweg, Nori, and Savelbergh (00), where he invenory rouing problem is formulaed as a Marov decision process (MDP). Adelman (00 a) and Adelman (00 b) also formulae he IRP conrol problem as a MDP, approximaing fuure coss of curren acions using opimal dual prices from a linear programming model. Also, he readers are referred o he wors of Campbell and Savelsbergh (00 a), Campbell and Savelsbergh (00 b) and Campbell and Savelsbergh (00 c) for he various approaches aen o formulae and solve he IRP. In his paper, we consider a siuaion where one warehouse replenishes invenory a muliple reailers using a single vehicle. The vehicle is scheduled o visi all he locaions where he invenory level is below he reorder poin a any ime during he visi. The demand paern a each locaion is sochasic and saionary; hence causing a variaion in he acual and prediced invenory a some or all locaions. During he visi of he vehicle o he planned locaions here may be an addiion of oher locaions where he invenory is below he reorder poin. To accommodae every new locaion ha eners a roue or o caer o he flucuaing demands a he locaions on he curren roue delivery quaniy needs o be varied and/or he vehicle is reroued. Afer visiing each locaion he decision on which /06/$ IEEE 5

2 Jarugumilli, Grasman, and Ramarishnan locaion o visi nex and how much o deliver a he presen locaion are aen. Finally, he vehicle reurns bac o he warehouse when he invenory on he vehicle is exhaused or so low ha i is no desirable o service any more locaions on he roue. We formulae he sequenial decision problem as a sochasic dynamic program considering he sequenial decision maing. The soluion space for his ype of he problems is exremely large maing he problem NP-hard; however, we modify he A* algorihm in order o find good heurisic soluions. We develop a simulaion framewor o validae he heurisic performance for he real-ime managemen and conrol of invenory rouing decisions. The remainder of he paper is organized as follows. Secion presens problem definiion, followed by secion, which presens he modified A * algorihm. The numerical analysis is presened in secion, resuls and discussions in secion 5, followed by he conclusion in Secion 6. PROBLEM DEFINITION In his paper, we focus on he relaionship beween a single warehouse and muliple reailers. The reailers paricipae in a Vendor Managed Invenory program, where he wholesaler is responsible for racing and replenishing he invenory in a imely manner. We consider he demand a he reailer locaions o be sochasic and saionary.. Noaion Prior o describing he model formulaion, a lis noaion used hroughou he paper is provided below: Newor represenaion: K =0 represens he warehouse =,,.K represen he reailer locaions c ij = cos of operaing beween locaions i and j d ij = disance beween locaions i and j ij = ravel ime from locaions i o j Invenory modeling parameers for locaion a = seup cos for each order p = shorage cos per uni h = holding cos of invenory w = uni cos Φ λ (d) = normal disribuion of demand during ime period λ θ = expeced demand during lead ime σ = sandard deviaion of demand during leadime (R,s,S) = he periodic review model where R-review inerval, s-reorder poin, S-order upo quaniy Cos Funcions F() Toal cos of operaion from he warehouse hrough node G() Cos of reaching he node from he warehouse H() Cos of reaching he warehouse from node F (j) Toal cos of operaion from he warehouse hrough node in he modified A* algorihm G (j) Cos of reaching he node from he warehouse in he modified A* algorihm H (j) Cos of reaching he warehouse from node in he modified A* algorihm Sae noaion T = node visiaion couner () = index of he h visied node I = acual invenory a locaion upon arrival a he h node Ĩ = prediced invenory a locaion upon arrival a he h node Q = invenory on vehicle upon arrival a he h node Decision variables D = invenory delivered o locaion while a he h node x ij = binary variable associaed wih selecion of he roue 0, if () = i and ( + ) = j, oherwise. Problem Descripion In his problem, we consider a single produc disribued from a single disribuion cener o wholesaler locaions. Locaions are indexed = 0,,,, K, where = 0 represens he disribuion faciliy. Invenory on a single disribuion vehicle, while a any locaion, is expressed as Q and changes as he vehicle delivers o various locaions according o: + Q = Q D( ), () where D () represens he invenory delivered a he locaion while a node and () is he index of he inerval beween each decision epoch. Upon reaching zero invenory, or a level where i migh no be desirable o service any oher addiional locaions, he vehicle reurns o he disribuion cener prior o saring a new our; any unvisied locaions are unsaisfied and may incur penalies. The invenory rouing processing is described in Figure.. We choose a periodic review model, (R,s,S) o deermine invenory conrol parameers for each locaion as i is consisen wih he sochasic naure associaed wih he invenory rouing problem. A power approximaion, proposed by Ehrhard (979) and Ehrhard and Mosier (9) is used o deermine opimal values of he reorder poin, s, 6

3 Jarugumilli, Grasman, and Ramarishnan he order up o level, S as shown in equaion () and (). Each ime he vehicle reaches a locaion he invenory a all he locaions is reviewed, hence he review of invenory occurs each ime a locaion is visied. The reorder poin is se as a hreshold value o assess he criicaliy of servicing a locaion. S =. ( θ ).9 a h σ + θ s = θ σ.06. 9z z Where Sar z w = S. σ b Obain iniial invenory levels a each locaion Generae pariioned se of locaions Generae iniial feasible delivery quaniy and rouing Se =+ Updae invenory levels and pariioned se Deermine D( ) : 0 D( ) Q { x } ( ) j + Q = Q ) D( Q + = 0 =0 Reurn o disribuion cener Figure.. Invenory/Rouing Processing Flowchar (Adaped from Jarugumilli and Grasman (006)) The invenory a each locaion is reviewed each period. If he invenory level falls below s an order is placed a he disribuion cener in order o replenish he invenory o S. Wihin his framewor, here are wo decision epochs:. generaion of iniial feasible (opimal) delivery quaniies and rouing, and () (). deerminaion of feasible delivery quaniy a locaion while a he h node D () and he decision on he nex node o be visied while a he h node {x ()j }based on updaed informaion. The firs decision may be made using exising vehicle and invenory rouing lieraure. The second decision deals wih real-ime conrol and is presened in Secion. The cos of operaions in he enire newor is he sum of invenory holding and shorage coss and ransporaion coss. Generally he holding coss are defined as he cos of holding he invenory unil i is consumed. The shorage coss are defined as he losses ha he wholesaler incurs due o shorage of an iem and are generally higher han he holding coss. The ransporaion coss are defined as he coss incurred o ravel beween wo or more locaions; his is direcly proporional o he disance beween he locaions. We formulae he problem as a sochasic dynamic program. This approach is adoped considering he sequenial decision maing. The presen sae of he sysem deermines he allowable acions and he possible saes o which he sysem can ransiion. The presen formulaion involves he following componens. The curren sae of he sysem a any insan is defined as: he curren and he prediced invenory a all he locaions in he newor, he curren locaion of he vehicle in he newor and he amoun of invenory on he vehicle. Depending on he curren sae he decisions are made and he sysem ransiions ino he fuure sae. The acion space for any given sae is he se of all decisions ha can be aen wihou violaing he vehicle and reailer capaciy consrains and he rouing consrains. The consrain ha he reailer capaciy is no violaed is expressed as I + D () <S. The consrains for he vehicle capaciy can be expressed as Q Q +, where =,, T. The objecive is o minimize he sysem wide coss considering he various saes wihou violaing he capaciy and rouing consrains. We se up he problem as a recursive funcion, where he oal cos is he sum of coss incurred a each sae ransiion; hus: ~ + + ~ f (,,, ) = min ( ) + +,,, ) ( ) ( I Q I I Q c j Y D f j I j j j ` 0 D Q D( ) + I = + Y D h ( D I ) d Φ ( ) ( ) d = 0 ( ) ( d) + + b ( d ( D I )) Φ λ ( d) ( ) d = D + I ( ) () λ (5) Equaion (), adoped from Jarugumilli and Grasman (006), provides he invenory conrol policy cos componen of he dynamic program, including invenory carrying cos and shorage cos associaed wih he delivery quaniy. The policy cos is calculaed using equaion (5). 7

4 Jarugumilli, Grasman, and Ramarishnan A * ALGORITHM In his secion, we describe he soluion mehodology for he sochasic dynamic program. In he firs phase we deermine he iniial invenory conrol parameers using an (R,s,S) model as discussed earlier in Secion. The second phase involves he use of A* algorihm o dynamically deermine he deliver quaniies and rouing each locaion. In his secion, we describe he A* for he sochasic dynamic program, described in Equaion. Solving his class of problem is complex because of he large number of soluion saes o which he sysem can ransi ino. The complexiy of he problem is reduced o he linear form and a varian of he A* algorihm is proposed o dynamically deermine he deliver quaniies and rouing. The A* algorihm is a graph search algorihm ha finds a pah from a given iniial node o a given goal node. The algorihm calculaes he cos of ravel from an iniial node o he final node as he sum he cos of reaching he inermediae node from he iniial node and he esimaed cos of reaching he final node from he inermediae node. Hence, he cos, F(), is expressed as: F()= G() + H() (6) Where G() he cos of raveling from he iniial node o he inermediae node, H() is he esimaed cos of raveling from he node o he final node. From he Equaion 6, i is eviden ha he algorihm can be effecively used for sequenial decision maing problems lie he dynamic program presened in he Equaion. G() calculaes he cos of ravel o he presen locaion, and H() calculaes he cos of ravel for he forward recursion, i.e., from he presen locaion o he final desinaion. The problem formulaion in Equaion (), considers minimizing he oal cos in he newor, which is comprised of he ravel cos and invenory cos in he newor. The cos componens G(j) and H(j) in he original form used for he A* algorihm consider only he ransporaion cos. In order o accoun for he invenory cos we slighly modify G() and H(). Now, G (j) is defined as he sum of he cos of raveling from he iniial node o he presen node and he cos of invenory for all he locaions covered on roue from he iniial node o he presen node. Similarly, he modified H (j) is he sum of he esimaed cos of ravel from he presen node o he final node and he esimaed cos of invenory for all locaions o be covered on roue o he final node (which in our case is he warehouse). A any given poin in ime G (j) can be deermined as he sum of he coss incurred o reach he curren locaion, bu H (j) is sum of he cos of servicing he remaining locaions on he pariioned se, which is subjec o change in he fuure based on he elemens of he pariioned se. We calculae he coss F (j) of geing o boh he locaions based on Equaions 7 and. H (j)= G (j)= min G ( j) + c( ) + ( Y D ) j ( ) T T min ( c + ) ) + + ( ( )( c( T )(0) Y D ) ( ) j = = Iniially, all he locaions are in he closed lis. While he vehicle is a a locaion, he updaed invenory informaion from all he locaions in he newor is obained and he pariioned se is updaed wih all locaions where he invenory I < s. All locaions in he pariioned se are added o he open lis. F (j) is hen calculaed for all he locaions and he one wih he minimum F (j) is visied. On reaching he locaion, he invenory is again updaed. If he sum of requiremens a all he locaions is less han he ruc capaciy, hen he pre-allocaed invenory is delivered a he curren locaion. If he sum of invenory requiremens a he locaions is greaer han he invenory on he ruc, adjusmens may be made o he invenory allocaion and he rouing decisions. There are several ways of allocaing he invenory and visiing fuure locaions in he newor, causing an increase in he number of saes o which he sysem can ransiion. To reduce he size of sae space, we pu a resricion on he adjusmen ha can be made o he invenory allocaion (Jarugumilli and Grasman 005) a a paricular locaion. A each locaion, here is a possibiliy of adjusing he delivery quaniies o hree levels: (7) (). Deliver he originally allocaed delivery quaniy,. Deliver he difference of he original delivery quaniy and he sum of requiremens in he fuure nodes, and,. Deliver he updaed delivery quaniy. The invenory o be delivered a a locaion is deermined by calculaing he oal cos F (j). We calculae G (j) considering he above menioned hree levels of invenory for he curren locaion. Nex we calculae he cos of servicing he fuure locaions H (j), based on he various levels of invenory on he ruc resuling from he amoun delivered a he curren locaion and he possible roue change. The prioriy is given o he locaions where he cos of servicing is he highes if no serviced, i.e. o locaions wih a low F (j). Afer delivering he invenory a he curren locaion, he curren locaion is added o he closed lis and decision on he locaion o be visied nex is aen based on F (j). Inuiively, if a locaion is no serviced a he righ ime, his will resul in a higher value of H (j), resuling in a higher value of F (j), forcing he vehicle o visi he paricular locaion o eep he coss low. Hence, he minimum cos F (j) is chosen and he corresponding invenory level o be delivered a he curren locaion and he nex locaion o be visied is chosen. A every locaion ha is visied

5 Jarugumilli, Grasman, and Ramarishnan hese calculaions are made unil he invenory on he ruc reduces o a level where visiing a locaion is no longer desirable or all he elemens in he pariioned se are visied. The algorihm is abulaed in Table.. Table.: Modified A* Algorihm Add he saring node (warehouse) o he open lis Repea he following: a. Loo for he lowes value F (j) cos node on he open lis b. Swich i o he closed lis c. For each of he locaions o be visied - chec if he invenory levels are greaer han s or if i is on he closed lis, ignore i. oherwise, do he following: - if i isn on he open lis, add i o he open lis. Mae he curren node he paren of his new node. Record F (j), G (j), and H (j) coss associaed wih he node. - if i is on he open lis already, chec o see if his pah o ha node is beer, using G (j) as a measure. A lower value of G (j) means ha he presen pah is beer and he amoun of invenory delivered is opimal. If so, change he paren node o he curren node and recalculae he values of G (j) and F (j) values of he node. If he open lis is mainained by sored F (j) scores, he lis is resored o accoun for he change. d. Sop when: - add he arge node o he closed lis, in which case he pah and he delivery quaniy has been deermined, or - fail o find he arge node, and he open lis in empy. In his case, here is no pah and no opimal delivery volume. Save he pah and he delivery quaniies, woring bacwards from he arge node, go form each node o is paren node unil he iniial node is reached. NUMERICAL ANALYSIS In his secion, we describe he design of experimens used for validaion of he model. We presen he numerical analysis o es a series of hypoheses relaed o he effec of facor levels on he cos of operaions in he newor.. Objecives of he numerical analysis The main objecives of he numerical analysis are o sudy he following:. Perform saisical analysis of he cos savings in he newor,. Analyze he effec of he demand disribuion, and invenory conrol coss on he overall cos of operaions, and,. Discuss he scalabiliy of he heurisic o handle problems of larger size.. Design of Experimens For he experimenal design, hree imporan design facors were idenified: he demand paern, he raio of he shorage cos o he holding cos, and he ransporaion cos. The demand a each of he reailer locaions is a common source of randomness in he sysem, and based on he demand, he invenory conrol parameers (s, S ) for each of he locaions are calculaed. The invenory levels a each locaion ac as a rigger for adjusing he delivery quaniies and he roues in order o minimize he oal invenory and ransporaion coss. Also, he shorage coss and he ransporaion cos have a significan effec on he cos savings. The hree facors and respecive levels are abulaed in Table.. Table.. Experimenal Design Facor Name Noaion Levels Demand θ / σ Normal Uniform a/b.5..5 Shorage cos vs. p / h Holding cos Transporaion Cos Γ The full-facorial design for hese facors and levels requires a 6 x x, i.e. a 96 rial experimen. Each rial consiss of en replicaions; of 50 simulaed periods... Daa Inpu The simulaion model uses he following inpu daa:. Demand disribuion a each reailer locaion.. Invenory conrol parameers for each reailer locaion based on he periodic review model.. Invenory holding and shorage cos a each locaion.. Capaciy of he delivery vehicle. 5. Cos of raveling from one locaion o he oher. 6. Invenory levels a each of he reailer locaions. The demand a each of he reailer locaions is he common source of randomness in he sysem. In his model, he demand is considered o be saionary and sochasic, following a normal disribuion. Based on he de- 9

6 Jarugumilli, Grasman, and Ramarishnan mand, he invenory conrol parameers for each of he locaions are calculaed. Invenory levels a each of he locaions vary wih he demand each period, maing i a random funcion. The invenory levels a each locaion acs as a rigger based on which he delivery quaniies and he roues are adjused, so ha he oal invenory and ransporaion coss are minimum... Model Verificaion and Validaion The iniial verificaion of he model included he following seps adoped from Law and Kelon (00):. he model was programmed and debugged in seps. o es each program pah he debugger was used exensively.. model oupu resuls were checed for reasonableness. model summary saisics for he values generaed from he inpu probabiliy disribuions were compared o ha wih a newor wih fixed roue and delivery quaniy. The process of validaion of his scenario is a difficul process. We have compared our assumpions and consrains wih he exising lieraure for logical correcness and accuracy. Differen daa ses for he same randomness were esed for homogeneiy and merged only if appropriae. We esed all he probabiliy disribuion for he correcness.. Oupu Analysis In his secion, we presen he oupu of he numerical analysis menioned in he previous secion. We will presen and es hypoheses dealing wih he impac of individual design facors on he experimens and he ineracions among various facors, based on he resuls he resuls of he 96 rials. The mean percenage, savings, is he response variable, is he mean value of he percenage savings for 0 replicaions of he same rial. The 96 rials were conduced for hree values of θ in case of he normal disribuion and hree values of a in case of he uniform disribuion... Main Effecs In his subsecion, we es he hypoheses o chec he significance of he levels of he individual facors. The hypoheses esing are done for he facors: demand, shorage cos and he ransporaion cos. Figure.-. show he main effec plos a differen values of mean and a for he normal and uniform disribuion respecively. Hypohesis : H 0 : Levels of facor for Demand does no change he mean percenage saving. H a : Levels of facor for Demand changes he mean percenage saving. H 0 : μ = μ = μ ; H a : μ μ ; μ μ ; Tes for he alernae hypoheses was carried ou separaely. The resuls of he paired -es conduced for he hree levels of demand for normal and uniform disribuion. The p- values for all he facors < 0.05, H 0 is rejeced and H a is acceped. Hypohesis : H 0 : Levels of facor Shorage Coss does no change he mean percenage saving. H a : Levels of facor Shorage Coss changes he mean percenage saving. H 0 : μ = μ = μ =μ ; H a : μ μ ; μ μ ; μ μ ; Tes for he alernae hypoheses was carried ou separaely. The resuls of he paired -es conduced for facors shorage cos for normal and uniform disribuion. The p-values for all he facors < 0.05, H 0 is rejeced and H a is acceped. Hypohesis : H 0 : Levels of facor Transporaion Coss does no change he mean percenage saving. H a : Levels of facor Transporaion Coss changes he mean percenage saving. H 0 : μ = μ = μ = μ ; H a : μ μ ; μ μ ; μ μ ; Tes for he alernae hypoheses was carried ou separaely. The resuls of he paired -es conduced for ransporaion cos for normal and uniform disribuion. The p-values for all he facors < 0.05, H 0 is rejeced and H a is acceped... Two Facor Ineracion In his subsecion, we es he hypoheses o chec he significance of he levels of he ineracion beween wo facors. The hypoheses esing are done for he ineracion beween he facors: demand, shorage cos and he ransporaion cos. Figures. -. show he ineracion plos beween he facors. Hypohesis : H 0 : Change in he levels of facors Demand and Shorage cos does no aler he mean percenage saving. H a : Change in he levels of facors Demand and Shorage cos alers he mean percenage saving. 90

7 Jarugumilli, Grasman, and Ramarishnan The resuls of he wo-way ANOVA es conduced for facors demand (for normal and uniform disribuion) and shorage cos. The p values for all he facors > 0.05, H 0 is no rejeced for he normal demand. The p values for all he facors < 0.05, H 0 is rejeced and H a is acceped for uniform demand. Hypohesis 5: H 0 : Change in he levels of facors Demand and ransporaion cos does no aler he mean percenage saving. H a : Change in he levels of facors Demand and ransporaion cos alers he mean percenage saving. The wo-way ANOVA es conduced for facors Demand and Transporaion Cos. The p values for all he facors > 0.05, H 0 is no rejeced. Hypohesis 6: H 0 : Change in he levels of facors shorage cos and ransporaion cos does no aler he mean percenage saving. H a : Change in he levels of facors shorage cos and ransporaion cos alers he mean percenage saving. The wo-way ANOVA es conduced for facors shorage cos and ransporaion cos. The p values for all he facors > 0.05, H 0 is no rejeced. 5 DISCUSSION OF RESULTS The analysis performed on he daa ses wih normal and uniform disribuions can be summarized as:. The shorage cos is a significan facor in he sysem, and he locaions wih higher shorage coss mus be on he prioriy of service.. The demand is a significan facor in he sysem, variabiliy increases, he mean percenage saving reduces.. The ransporaion cos is a significan facor in he sysem.. For change in he levels of facors of demand and he shorage coss, here is a change in he mean percenage savings only for cases where he variances are high. In his sudy, he daa se for normal demand does no reflec his fac mainly due o he lesser variaion in he demand paern. This concep is well refleced in he uniform disribuion daa se. 5. In his sudy, here is no ineracion observed beween he ransporaion cos and he demand. 6. Again, here was ineracion observed beween he ransporaion cos and he shorage cos. The resuls show ha here can be some subsanial savings achieved by he implemenaion of he connecive echnology in he logisics newor, which will enable he real-ime informaion exchange beween he vehicle and he various locaions in he newor. The use of connecive echnology, will aid in he real-ime decision maing on dynamically deermining he delivery quaniies and rerouing of he vehicle. Mean of Mean Percenage Saving Normal Demand Mean= 0 Normal Demand Shorage Cos Transporaion Cos Figure. Main Effec Plos for Normal Demand, Mean=0 Mean of Mean Percenage Saving Uniform Demand a = 0 Uniform Demand Transporaion Cos Shorage Cos Figure. Main Effec Plos for Uniform Demand, a =0 Normal Demand Normal Demand Mean =0 Shorage Cos T ransporaion Cos Normal Demand Shorage Cos Transporaion Cos Figure. Ineracion Plos for Normal Demand, Mean=0 0 0 Uniform Demand Uniform Demand a =0 Shorage Cos Transporaion Cos 0 Uniform Demand Shorage Cos Transporaion Cos Figure. Ineracion Plos for Uniform Demand, a =0 6 CONCLUSION In his paper, a mehodology for dynamic conrol of a invenory and rouing decisions in a logisics newor is pre- 9

8 Jarugumilli, Grasman, and Ramarishnan sened. Using a vendor managed invenory approach; we compare he cos of operaions in newors enabled by he connecive echnologies wih newors no implemening such echnology. The problem has been se up as a sochasic dynamic program. A* algorihm is used o solve his problem and o bring ou he cos benefi associaed wih he dynamic conrol of invenory and rouing decisions. We have presened an exensive numerical analysis o es he efficiency of he mehodology. This model should be considered along wih he se up cos associaed wih ordering each ime an order is placed. The model may be exended o more complex sysems, such as sysems wih muliple vehicles or muliple disribuion ceners. In hese siuaions, he invenory rouing problem may require pariioning algorihms, in order o deal wih naure of he vehicle rouing problem. Oher exensions include curren advances in VRP and IRP, such as anicipaory roue selecion probabilisic rouing, or delivery windows. Seing of dynamic hreshold values for each of he locaions will help in more opimized operaions in he logisics newor. This model can be exended o include use of muliple vehicles in he newor, he disribuion of muliple producs and considering he disribuion of perishable invenory. REFERENCES Adelman, D. 00. Price-direced replenishmen of subses: mehodology and is applicaion o invenory rouing. Manufacuring Service Operaion Managemen 5(): -7 Adelman, D. 00. A price-direced approach o sochasic invenory/rouing. Operaions Research 5(): 99-5 Campbell, A.M., and M. W. P. Savelbergh. 00. A decomposiion approach for he invenory- rouing problem. Transporaion Science (): -50 Campbell, A.M., and M. W. P. Savelbergh. 00. Delivery volume opimizaion. Transporaion Science (): 0- Campbell, A.M., and M. W. P. Savelbergh. 00. Efficien inserion heurisics for vehicle rouing and scheduling problems. Transporaion Science (): 69-7 Ehrhard, R The power approximaion for compuing (s,s) invenory policies. Managemen Science 5(): Ehrhard, R., and C. Mosier. 9. A revision of he power approximaion for compuing (s,s) policies. Managemen Science 0(5): 6-6 Jarugumilli, S., and S. E. Grasman An inegraed invenory and rouing decision framewor in onewarehouse muli- reailer sysems using RFID echnology, In Proceedings of he Inernaional Conference on Operaions Research Applicaions in Infrasrucure Developmen Jarugumilli, S., and S. E. Grasman RFID-enabled invenory rouing problems. In. J. Manufacuring Technology and Managemen. To Appear in he Special Issue on Connecive Technologies and Their Impac on Manufacuring and Logisics Kleyweg, A., V. Nori, and M. W. P. Savelbergh. 00. The sochasic invenory rouing problem wih direc deliveries. Transporaion Science 6(): 9- Kleyweg, A., V. Nori, and M. W. P. Savelbergh. 00. Dynamic programming approximaions for a sochasic invenory rouing problem. Transporaion Science (): -70 Law, A. M., and D. W. Kelon. 00. Simulaion Modeling and Analysis. rd Ediion, New Yor, McGraw-Hill Lee, H. L., and S. Whang. 99. Informaion sharing in a supply chain. Research paper No. 59, Research Papers Series, Graduae School of Business, Sanford Universiy AUTHOR BIOGRAPHIES SHRIKANT JARUGUMILLI is a graduae suden currenly woring as a graduae research assisan in he Deparmen of Engineering Managemen and Sysems Engineering a Universiy of Missouri, Rolla. He has a B.E. (Indusrial Engineering and Managemen) from Visvesvaraya Technological Universiy, India. His research ineress are invenory rouing problems and dynamic programming. He is a member of IIE and INFORMS. His address is <sjarugumilli@gmail.com>. SCOTT GRASMAN is an Assisan Professor of Engineering Managemen and Sysems Engineering a he Universiy of Missouri-Rolla. Dr. Grasman received his B.S.E., M.S.E., and Ph.D. degrees in Indusrial and Operaions Engineering from he Universiy of Michigan. His primary research ineress relae o he applicaion of quaniaive models o manufacuring and service sysems, focusing on he design and developmen of supply chain and logisics. He has received research funding from, among ohers, Naional Science Foundaion, US Deparmen of Sae, and SAP America. His address is <grasmans@umr.edu>. SREERAM RAMAKRISHNAN is an assisan professor of Engineering Managemen and Sysems Engineering in he Universiy of Missouri- Rolla. He has a Ph.D. in Indusrial Engineering from Pennsylvania Sae Universiy, a M.S. (Indusrial Engineering) from Binghamon Universiy, and a B.Tech. (Mechanical) degree from Universiy of Kerala, India. His research ineress include disribued sysems modeling, simulaion-based conrol, and modeling conrol sysems. He is a member of IIE and ASEE. His e- mail address is <sreeram@umr.edu>. 9