Transportation Research Part C

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1 Transportaton Research Part C 29 (2013) Contents lsts avalable at ScVerse ScenceDrect Transportaton Research Part C jornal homepage: A secre and effcent nventory management system for dsasters Eren Erman Ozgven, Kaan Ozbay Cvl & Envronmental Engneerng Department, Rtgers Unversty, NJ 08854, USA artcle nfo abstract Artcle hstory: Receved 1 Jne 2010 Receved n revsed form 14 Jne 2011 Accepted 30 Agst 2011 Keywords: Dsaster plannng and logstcs Hmantaran ad Stochastc nventory control Emergency management RFID technologes An effcent hmantaran nventory control model and emergency logstcs system plays a crcal role n mantanng relable flow of vtal spples to the vctms located n the shelters and mnmzng the mpacts of the nforeseen dsrptons that can occr. Ths system shold not only allow the effcent sage and dstrbton of emergency spples bt shold also offer the ablty to be ntegrated wth emergng ITS technologes sch as Rado Freqency Identfcaton Devces (RFIDs) for commodty trackng and logstcs. Ths paper proposes a comprehensve methodology for the development of a hmantaran emergency management framework based on the real-tme trackng of emergency spples and demands throgh the se of RFID technology ntegrated wth a mlt-commodty stochastc hmantaran nventory management model (MC-SHIC). Frst, logstcs and management aspects of RFID technologes n the context of the emergency dsaster relef framework are dscssed. Then, MC-SHIC model proposed to determne the optmal emergency nventory levels to prevent possble dsrptons at the mnmal cost s presented. The solton of the model wth several senstvty analyses obtaned sng the pleps algorthm s presented and dscssed. Realzng that actal emergency nventory levels can devate from optmal vales drng the actal dsaster relef perod de to the possble stochastc dsrptons sch as flctatng demand for vtal spples n the shelters, a comprehensve on-lne nventory control framework s proposed to mnmze mpacts of these nforeseen dsrptons, or at least to address the problem at hand as fast as possble. Wthn ths methodology, we obtan an approxmaton of the MC-SHIC model sng a smltaneos pertrbaton stochastc approxmaton (SPSA) based fnctonal approxmator, and compare the performance of these algorthms for solvng the new nconstraned optmzaton problem. Fnally, proposed model-free on-lne control methodology s dscssed sng examples to nderstand the effcency and practcalty of both algorthms n terms of comptatonal tmes and accracy of reslts. Pblshed by Elsever Ltd. 1. Introdcton The most mportant challenge of emergency dsaster management and logstcs s redcng the harm a dsaster cases to ts vctms. A sgnfcant component of ths challenge s to be able to satsfy the vtal needs of the people located n the emergency shelters, sch as the sperdome shelter at New Orleans sed after the hrrcane Katrna. Ths reqres dsaster planners and engneers to fnd a way to redce the ncertantes assocated wth the emergency operatons, to calclate and compare the possble expected costs of delvery and consmpton processes throghot these operatons, and therefore to manage the avalablty and dstrbton of vtal resorces. Natonal Research Concl (NRC) of the Natonal Academes report (2007) mentoned the two mportant problems that we face for the nstable dsaster statons: the complexty ntrodced by the Correspondng athor. Tel.: E-mal addresses: ozgvene@rc.rtgers.ed (E.E. Ozgven), kaan@rc.rtgers.ed (K. Ozbay) X/$ - see front matter Pblshed by Elsever Ltd. do: /j.trc

2 172 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) dynamcs of the emergency response operatons and the complexty ntrodced by the nformaton ncertanty n the stochastc delvery and consmpton processes de to the flctatng demand and possble dsrptons n the transportaton network. Gven ths pctre of the dsaster management and emergency response, t s obvos that nformaton technologes (IT) n terms of Intellgent Transportaton Systems (ITS) provde capabltes that can help the planners and engneers grasp the dynamc and stochastc realtes of a dsaster more clearly and help them formlate better decsons more qckly. ITS technologes can address the aforementoned challenges, whch reqres both the carefl trackng of vtal resorces sppled to the shelters, and also a methodology to ncorporate that trackng data nto a dsaster framework where plannng and on-lne management phases are ntegrated. RFID technologes, among other ITS technologes, appear to have crtcal mportance for the emergency nventory operatons n terms of resorce trackng and logstcs. Natonal Research Concl of the Natonal Academes (2007) stated that the nvestments on technologes lke RFID taggng sed to manage massve spply chans lke those of Walmart, FedEx and IKEA shold be expanded, and they shold be pervasvely mplemented for dsaster management operatons. In terms of plannng, the most mportant sse s to focs on mantanng a reserve capacty as a bffer for the vtal spples n the aftermath of a dsaster. A recent stdy by Holgn-Veras et al. (2007) stated that the lack of an effcent hmantaran nventory control model cased major negatve conseqences for the dsaster vctms after Hrrcane Katrna. Moreover, Brode et al. (2006) ndcated that the srvval needs of the vctms were not effectvely satsfed n the shelters after the hrrcane Katrna. In Hat, more than 3 mllon people were n need of emergency spples, and n Japan, 350,000 people were reported to become homeless and were stayng n shelters n the aftermath of the two devastatng earthqakes. Clearly, the ndertlzaton and neffcent control of desperately needed resorces can pt the health and welfare of the srvvors n jeopardy. Therefore, there s a need for the development and analyss of a hmantaran nventory management model developed pror to the occrrence of a natral or man-made catastrophe, whch can determne the safety stock level that wll prevent possble dsrptons at a mnmal cost. However, jst determnng these safety stocks beforehand s not satsfactory for the stochastc dsaster condtons. That s, apart from beng prepared for a dsaster, on-lne response drng the dsaster or drng the recovery perod of the dsaster s also crcal. Ths s becase the ntal plannng polcy may not adeqately captre the stochastc real lfe statons de to the needs of srvvors can drastcally change drng the dsaster relef operatons, resorces avalable to prodce new prodcts may get scarce de to ncreased consmpton or redced spply levels, and the damaged transportaton system mght not be able to sstan the flow of emergency commodtes to the shelters. Therefore, an emergency nventory management system shold nclde an on-lne operatonal strategy to mnmze mpacts of these nforeseen dsrptons, or at least to address the problem at hand as fast as possble. The real-tme management of the preparedness actvtes wll reveal the actal performance of the response, recovery and mtgaton processes after the dsaster. As a reslt, to have a robst emergency management system, the off-lne plannng and prodcton polces shold be ntegrated wth an on-lne nventory management strategy to accont for the expectatons of flctatng demand for vtal spples (food, water, medcal spples, etc.) and the dsrptons n the transportaton network drng the dsaster relef perod. Ths stdy s motvated by creatng sch an ntegrated control strategy to ensre the safety and contnty of the flow of the emergency spples n the aftermath of a dsaster. Fg. 1 ndcates the proposed strategy as a real-tme on-lne feedback control scheme. In ths artcle, we make for mportant contrbtons that can be stated as follows: 1. Unlke prevos approaches n the lteratre, proposed model s desgned to sccessflly address the crtcal and strategcal factors assocated wth the hmantaran relef operatons to ensre the contnty of the npredctable stochastc delvery and consmpton processes. Moreover, flexblty of the proposed nventory management/control model allows the applcaton of or mathematcal model to any extreme event, accontng for the flexblty measres of hmantaran relef operatons, namely the ablty to change the otpt levels and varetes of prodcts, and the flexblty of delvery tmes, defned by Beamon and Balck (2008). 2. To obtan an effcent and realstc off-lne dsaster management plan, the frst step s to have a practcal yet robst nventory management strategy. Ths strategy shold be closely nterconnected wth a real-tme complex mlt-commodty problem that can be mplemented and sed as a bass for on-lne decson makng. Therefore, we model the off-lne Fg. 1. Comprehensve feedback-based emergency nventory control strategy.

3 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) plannng model as a mlt-commodty and mlt-sppler stochastc hmantaran nventory control model (MC-SHIC). MC-SHIC s a major modfcaton of the lmted sngle-commodty approach proposed n Ozbay and Ozgven (2007) whch s not applcable to the type of problem stded n the paper. MC-SHIC model proposed n ths paper wll ad n adeqately respondng to a dsaster or a hmantaran crss focsng on the nterdependency between the commodtes, transportaton of jont resorces, lmtatons on emergency spples, vtal/pershable commodtes, mlt-spplers and cross-shppng possbltes. We propose a practcal and easly applcable solton methodology, namely pleps algorthm (Prékopa, 1990), to solve ths mlt-commodty stochastc hmantaran nventory control (MC-SHIC) model so that a decson maker/planner can se the reslts to decde on the nventory levels for the emergency commodtes needed n the shelters. We solve the problem for a range of demand vales to observe the changes n the safety stocks whch gves the planner a self-controllng mechansm to determne these safety stock vales for the emergency spples more accrately. 3. We se a smltaneos pertrbaton stochastc approxmaton (SPSA) based on-lne fncton approxmator to be able to control the hghly stochastc and ncertan condtons drng the extreme events. The target or desred levels of nventory for the fnctonal approxmator s calclated sng MC-SHIC model, and the fnctonal approxmator s traned to obtan the optmal vales n the presence of nosy data generated by ncorporatng the random -consmpton and delverydstrbtons of emergency commodtes throgh the MC-SHIC model. 4. Or ltmate goal n ntegratng the two models descrbed above s to develop a model-free closed loop feedback based nventory control strategy that can be effectvely appled to dfferent knds of extreme events drng emergency relef perods. Ths model-free feedback strategy reqres real-lfe data abot the nventory levels, flow of emergency vehcles and spples as npt. RFID technologes sed to keep track of the dynamc changes n many commercal nventory systems (sch as Walmart, FedEx, IKEA and Procter& Gamble, nstallng RFID systems n ther dstrbton centers and trackng commodtes comng from spplers, have reported satsfactory progress wth ther mplementaton programs) can be sed to collect ths trackng nformaton n real-tme wth the mnmm cost and nfrastrctre. It s mportant to note that the pont of the stdy s to focs on the needs of the vctms located at the shelters. Therefore, evacaton operatons are not n the scope of or stdy. We do not ndertake a transportaton network analyss, rather we focs on a resorce management sse namely, plannng and management of the safety stock levels of the emergency nventores. However, the transportaton of commodtes by the se of emergency vehcles and mlt-sppler/cross-shpment optons are also consdered n the paper. 2. Lteratre revew There are several stdes n the lteratre focsng on dfferent aspects of the emergency nventory plannng and management for dsasters. Table 1 gves the overvew of these stdes wth respect to dfferent areas of applcatons. The mportant pont s that there s not mch work done motvated by ntegratng both the off-lne and on-lne control strateges Off-lne emergency management stdes Spply chan logstcs and nventory control stdes Blanco and Goentzel (2006) revewed the hmantaran spply chan stdes defnng hmantaran spply chans as the spply chans spportng response to dsasters and/or hmantaran crses. She (2007) stated the challenges of emergency logstcs management n the context of dsaster relef and response operatons. Ergn et al. (2007) worked on the hmantaran relef logstcs ncldng the desgn of spply chan network, transportaton control, and demand management. Ozdamar et al. (2004) presented a mathematcal model for emergency logstcs plannng drng dsasters. Smlarly, L and Tang (2008) proposed the Artfcal Emergency-Logstcs-Plannng System (AELPS) to help government and dsaster relef organzatons prepare for and manage severe dsasters. Tomaszewsk et al. (2006) desgned a Geocollaboratve Web Portal (GWP) applcaton to provde a common and nttve nterface throgh whch asynchronos, geocollaboratve actvtes can be condcted n spport of hmantaran relef logstcs operatons. Chang and Hseh (2007) developed a flood emergency logstcs database for geographc nformaton system (GIS) by sng the data processng and network analyss fnctons of the geographc nformaton system, dsaster predcton sbsystem and resce resorce sbsystem. Fredrch et al. (2000) sggested a mathematcal model allowng for calclatng an optmzed resorce schedle for assgnng resorces n space and tme to the mpact areas of the dsaster. Ray (1987) developed a sngle-commodty, mlt-modal network flow model on a capactated network over a mlt-perod plannng horzon to mnmze the sm of all costs ncrred drng the transport and storage of food ad n West Afrca. Tzeng et al. (2007) created a mlt-objectve mathematcal model for desgnng relef delvery systems. As the emprcal data reqred to valdate the hmantaran spply chan models s not always readly avalable, the applcablty of these theoretcal models becomes the weakest lnk of the lteratre. Wth the adeqate data, Stochastc Hngaran Inventory Control model of Ozbay and Ozgven (2007) can be effectvely appled to real lfe statons. Ozgven and Ozbay (2009) sggested soltons to enhance ths model sng the concept of p-level effcent ponts. Yang and Federgren (2006) developed dsaster plannng models focsng on selectng spplers by analyzng a plannng model coverng ncertan

4 174 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Table 1 Comparatve lteratre revew for emergency nventory management stdes. Off-lne emergency management stdes Spply chan logstcs and nventory control Transportaton and network On-lne emergency management stdes Ray (1987) Ozbay and Ozgven (2007), Haghan and Oh (1996) Me (2002) Ozgven and Ozbay (2009) Sheral et al. (1991) Y and Ozdamar (2007) Oh and Haghan (1997) Bran et al. (2003) Fredrch et al. (2000) Chang et al. (2007) Barbarosogl et al. (2002) Rask et al. (2004) Mrray-Tte and Mahmassan (2003) Holgn-Veras et al. Ozdamar et al. (2004) Wlmot and Me (2004) (2007, 2008) Ozdamar et al. (2004) L and Tang (2008) Ltman (2006) Dnbar and Desa (2005) Barbarosogl and Arda (2004) Beamon and Balck (2008) Tzeng et al. (2007) Gadh and Prabh (2006) Kongsomsaksakl et al. (2005) Ozdamar and Y (2008) Y and Kmar (2007) Mnyard (2007) Blanco and Goentzel (2006) Yshmto et al. (2009) Han et al. (2007) Henderson (2007) Yang and Federgren (2006) Lee et al. (2009) Yazc and Ozbay (2007) Wagh (2000) Beamon and Kotleba (2006) Wdener and Horner (2010) Afshar and Haghan (2008) Troy et al. (2007) Tomaszewsk et al. (2006) Ln et al. (2010) Ben-Tal et al. (2009) L et al. (2007) She (2007) De Leew et al. (2009) Yao et al. (2009) Natonal Research Concl of the Natonal Academes (2007) Ergn et al. (2007) Destro and Holgn-Veras (2011) Xe and Trnqst (2009) Noal et al. (2009) Tzeng et al. (2007) Alcada-Almeda et al. (2009) Chatfeld et al. (2010) Chang and Hseh (2007) Shen et al. (2009) demand from mltple spplers. Chang et al. (2007) formlated the flood emergency logstcs problem as a GIS-based stochastc programmng model ncldng ncertanty whereas Haghan and Oh (1996) formlated and solved a mltcommodty, mlt-modal network flow model to mnmze the nmber of casaltes and to maxmze the effcency of the resce operatons. Oh and Haghan (1997) tested ths optmzaton model extensvely wth two herstcs. Barbarosogl and Arda (2004) extended the model of Haghan and Oh (1996) nto a two-stage stochastc programmng model where they nclded ncertantes de to the estmaton of needs of frst-ad goods, vlnerablty of resorce spplers and srvvablty of the dsaster regon rotes. Beamon and Kotleba (2006) developed an nventory control model determnng optmal order qanttes and reorder ponts for a long-term emergency response. Holgn-Veras et al. (2007) proposed a synthess of prelmnary sggestons for the mprovement of emergency logstcs sses, and Holgn-Veras et al. (2008) analyzed the reqests for crtcal spples after Hrrcane Katrna by obtanng the temporal dstrbton of reqests ndcatng the demand and mportance of dfferent commodtes. Yshmto et al. (2009) developed a Vorono dagram-based algorthm to locate a fnte nmber of facltes to provde a qck response tme for the dsaster relef. Sheral et al. (1991) developed a model solvng the nonlnear mxed-nteger programmng problem for the selecton of shelter locatons among potental alternatves n a manner feasble to avalable resorces. For the flood evacaton plans, Kongsomsaksakl et al. (2005) presented a b-level programmng methodology for locatng and allocatng the shelters wth capacty constrants. Ozdamar and Y (2008) sed greedy neghborhood search for maxmzng the servce for avalable emergency vehcles to dsaster-areas srvvors, and Y and Ozdamar (2007) descrbed a locaton-dstrbton model for coordnatng logstcs spport and evacaton operatons n dsaster response actvtes. Y and Kmar (2007) presented an ant colony optmzaton (ACO) method to solve the logstcs problem arsng n dsaster relef actvtes whereas De Leew et al. (2009) developed a framework of logstcs aspects for flood emergency plannng. Ln et al. (2010) proposed a nteger programmng based logstcs model for dsaster relef operatons, and appled the model on a smlated earthqake scenaro whereas Mrray-Tte and Mahmassan (2003) focsed on the trp estmatons and hosehold behavor n evacaton condtons throgh a seres of two lnear nteger programs Transportaton and network related stdes Barbarosogl et al. (2002) focsed on sng helcopters for delverng spples and savng vctms drng the dsasters. Ozdamar et al. (2004) presented a mathematcal model for emergency logstcs plannng drng dsasters where they ntegrated the mlt-perod mlt-commodty network flow problem and the vehcle rotng problem. Shen et al. (2009) developed a b-level model to determne the locatons of safety shelters n a transportaton network. Afshar and Haghan (2008) worked on a smlaton-optmzaton framework to fnd the SO solton for an emergency evacaton plannng problem. Xe and Trnqst (2009) focsed on the macroscopc problem of the network contraflow desgn to maxmze the effcency of emergency vehcle assgnment, and sed the Lagrangan relaxaton and tab search methods to solve ths problem. Tzeng et al. (2007) created a mlt-objectve mathematcal model for desgnng relef delvery systems, and they presented an emprcal case to demonstrate the feasblty and effectveness of ths model. Han et al. (2007) presented a spply locaton selecton and rotng problem where any spply sorce cold also send a commodty to a snk to take advantage of the mltple spply sorces. Alcada-Almeda et al. (2009) worked on a GIS-based decson spport system wth a mlt-objectve model to locate emergency shelters and to dentfy evacaton rotes.

5 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Yazc and Ozbay (2007) addressed the locaton and effectveness of hrrcane evacaton shelters wth a stochastc programmng model ncldng probablstc constrants. Ben-Tal et al. (2009) and Yao et al. (2009) developed robst lnear programmng models amng to provde a robst and tractable framework for the evacaton management n a large-scale network. Lee et al. (2009) condcted a smlaton-based analyss to obtan the dstrbton of emergency relef spples, and Wdener and Horner (2010) proposed a herarchcal capactated-medan methodology to model a hrrcane-based dsaster relef dstrbton. Destro and Holgn-Veras (2011) worked on a dfferent aspect of the materal convergence problem for the dsasters (movement of goods to the dsaster ste) and performed an econometrc modelng of the donatons made n the case of Hrrcane Katrna On-lne emergency management stdes In practce, for the on-lne emergency management, the se of ntellgent transportaton systems (ITS) n terms of nformaton technologes (IT), whch covers both comptng and commncatons capabltes to enhance the management of natral and hman-made dsasters, reqres specal attenton. Mnyard (2007) stated that a crtcal component of evacaton plannng reled on the nformaton system that can track people and assets from the plannng stage throgh regstraton, evacaton, shelterng and re-poplaton phases. Noal et al. (2009) stated that deaths and njres as well as economc losses cold be effcently redced by tmely and effectvely response to the dsasters. Troy et al. (2007) presented the reslts of a plot stdy on mplementng a commnty-based resorce database throgh the collaboraton wth local Amercan Red Cross chapters and pblc and prvate commnty organzatons. Wagh (2000) provded a concse ntrodcton to emergency management and ts vared components coverng the hstory and evolton of emergency management and ts organzaton nto governmental and regonal systems. One of the most mportant nformaton technologes, n terms of relevance for ths stdy, s the RFID systems whch can provde the dynamc change n the nventory levels, flow of emergency vehcles and spples and even people on the move. The RFID lteratre on emergency management and dsaster response s not very wde. Gadh and Prabh (2006) presented some basc nformaton abot RFID technologes wth spplyng the news from Msssspp s Harrson Conty where health offcals mplanted RFID chps on Hrrcane Katrna vctms. Chatfeld et al. (2010) made an analyss of the lteratre on exstng RFID technology benefts. Henderson (2007) dscssed and elaborated on the se of RFID systems to track commodtes ncldng emergency spples, n near real tme, flowng n and ot of the regon. Theoretcally, model predctve control (MPC) strateges serve a wde range of control methods that can be sed n spply chan management of emergency evacaton. Bran et al. (2003) showed that an MPC framework can handle demand networks effcently wth robst management. Rask et al. (2004) worked on an mplementaton of a better cost fncton n nventory control and spply chan management problems, whereas Dnbar and Desa (2005) demonstrated an applcaton of nonlnear MPC strateges. L et al. (2007) proposed a model reference adaptve control based real-tme traffc management framework for the evacaton ncldng both dynamc network modelng and adaptve control theory technqes. The sage of neral networks n the emergency management area s lmted. Me (2002) performed a detaled analyss of artfcal neral network models and developed alternatve trp generaton models for hrrcane evacaton movement sng logstc regresson and neral networks. Wth ths dea, Wlmot and Me (2004) estmated models wth varos forms of neral networks and tested the models wth a data set of evacaton behavor collected n sothwest Losana followng Hrrcane Andrew. It s clear that there are a wde range of stdes for the emergency management and relef operatons. When ncertanty or probablstc sses are consdered, the nmber of stdes decreases. There are also a few stdes related drectly to the determnaton of the optmm stock of emergency spples ndcatng the need for effcent nventory management and emergency plannng models for dsasters that nclde stochastc featres of the real lfe statons. On the other hand, on-lne emergency management s stll a very open area to work on. There are a few stdes that consder the real-tme management of logstcs n the lteratre. Therefore, the man focs of ths paper s to provde a secre and effcent hmantaran nventory management system whch shold be robst to the dsrptons that wll possbly occr n the nventory control and transportaton systems drng/after dsasters. 3. Proposed emergency management framework Or objectve n ths paper s to develop a realstc control model of the tme-dependent nventory plannng and management problem for the development and mplementaton of effcent pre and post-dsaster plans. For ths prpose, we need a robst and effcent procrement system wth an adeqate off-lne plannng strategy and a carefl on-lne nventory management polcy. However, t s hghly mportant to nderstand that hgh level of ncertanty drng hmantaran relef operatons makes the resorce trackng n the aftermath of these events a crcal reqrement. Wth the contnos poston reportng by the possble se of Global Postonng Systems (GPS) and satellte connectons, RFID technologes can provde the dynamc change n the nventory levels, flow of emergency vehcles and spples. Ths data can be fed nto MC-SHIC model to obtan realstc delvery and consmpton dstrbtons to be sed drng the on-lne feedback control. Therefore, the reslts of a possble RFID applcaton wll not only help nderstandng the pros and cons of the emergency response for the dsasters bt t wll also help to valdate and verfy or model. Wth ths prpose, before ntrodcng the off-lne and on-lne methodologes, we wll frst stdy the logstcs aspects of RFID devces n detal n the proceedng sectons.

6 176 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Rado Freqency Identfcaton Systems (RFID) technologes If hstorcal data becomes avalable to obtan the real response of the modeled emergency system, then t s possble to se standard optmzaton algorthms and therefore to obtan a controller that can effcently create hstorcal npts, controls, and otpts. Ths, as an approach for the real-world condtons, t s possble to se RFID systems to track the n and ot flow of emergency spples. An example applcaton of RFID for dsaster management focsng on the logstcs and transportaton aspects of the problem can be seen n Fg. 2. RFID technologes have three man advantages for emergency operatons: 1. Trackng: Ths fncton of RFID works on dentfyng the moton of any emergency spply. The dea ncldes not only the real tme poston trackng (flow of commodtes) bt also trackng the moton throgh the entry and ext ponts. 2. Identfcaton, sensng and athentcaton: Ths can be sed for dentfyng the vctms of the dsaster to have accrate nformaton. Most mportantly, for the patents n the dsaster regon, ths wll help to accrately respond to the needs of the patents. It can also be sed drng the trackng procedre of the commodtes sch as medcne and blood to be able to make a secre applcaton to the vctms. 3. Atomatc data collecton and transfer: Ths one s manly sed for mnmzng the data entry and collecton errors, redcng the nformaton processng tme and nventory management problems. On the other hand, a thorogh lteratre revew on the RFID systems revealed the followng barrers and obstacles for the mplementaton phase: 1. Interferences/relablty/envronment: Dsaster management s concerned wth the envronments that are ntrnscally ncertan and nstable. Therefore, the lack of standardzaton of the RFID protocols at the hardware and software levels may case severe problems. Moreover, RFID technology s affected by the presence of metal, lqds, organc matter, and nterference from other rado-freqency (RF) sorces. These materals case RF energy absorpton, reflecton, mlt-path, RF sgnal shadng, sgnal boncng and skn effects. 2. New technology problems: The adopton of the technology has been hampered by standards. Ths makes t very dffclt to deploy nteroperable soltons. Both the development and deployment of many promsng technologes sch as RFID systems are rsky and costly compared wth the opportnty presented by the commercal market for these technologes today. 3. Costs: The ndstry has been watng for the cost of the RFID technology to come down. Moreover, mportant sorces of fnds are typcally only avalable once a dsaster has been declared and mst also be spent n a short wndow of tme. 4. Operatonal/capacty/manageral/knowledge sses: Dsaster management organzatons often lack the resorces to acqre valable capabltes. In most agences wth dsaster management responsbltes, there s no one charged specfcally wth trackng RFID technology, dentfyng promsng technologes, ntegratng them nto operatons, or nteractng wth RFID vendors. 5. Cltral and ethcal concerns: For RFID chps beng sed on cadavers or vctms, there are many oppostons de to cltral, relgos, socal and ethcal concerns. 6. Prvacy, secrty, data ntegrty and legal sses: RFID systems sed for emergency management reqre specal secrty and prvacy n order to mantan the feasblty of the flow of goods and people, and to avod the msse of the data. The applcaton of the technology s always bond to legal reqrements that shold be flflled, and no personal and confdental nformaton shold be transmtted va tags. 7. Local sses: Decsons regardng RFID shold be made jontly by local mncpaltes and organzatons that mst work together n dsasters, whch s most of the tme one of the most mportant drawbacks n transportaton related works. Careflly stdyng the lteratre of RFID systems, ths stdy ndcates the ways for how to overcome the drawbacks aforementoned: 1. Researchers are workng on a new RFID technology whch s based on software-defned rado (Islam et al., 2009). Ths strategy allows nnovatons n the physcal layer that wll mnmze the effects of most of the challenges lsted n nterferences/relablty/envronment. 2. Operatonal/capacty/manageral problems shold be solved by establshng control (moble and permanent) centers all over the regon. Wth ths way, t s possble to mantan an effcent management and trackng of emergency commodtes. Ths wll allow the offcals and planners to make sre that basc srvval needs of the vctms are effectvely satsfed. It wll also be possble to provde a capablty to see and track the emergency vehcles as they move throghot the dsaster regon. Ths wll clearly help to determne the degradaton of the transportaton network as some of these vehcles may need alternatve rotng to reach ther destnatons. 3. Edcatng the offcals and personnel s a mst to effcently respond to the nventory management and transportaton problems drng the aftermath of a dsaster. The personnel can be traned and the new technology and knowledge related problems can be solved by the help of prvate companes that are famlar wth the technology.

7 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 2. RFID enabled procrement of evacaton demand n shelters. 4. The goals of the federal and state agences, the US Army, and the prvate companes shold be to create a force strctre to meet the severe statons (nsffcent commodty spply, dsrptons n the roadway network) that can occr after a dsaster. Ths force shold be the key to solve the local problems, cltral, ethcal and relgos concerns, and prvacy, data ntegrty and legal sses. Mostly, as the RFID systems are vlnerable to compromse and tags can be removed easly, a legal force wth the spport of army wold be a necessty drng or after a dsaster. 5. Cost and mplementaton problems can be handled by a cost-jstfed ncorporaton of the federal/state agences as well as the prvate companes sch as RFID vendors. Consderng the lves of people and the satsfacton of ther basc srvval needs, and the ablty to pnpont what s gong wrong n the system n terms of nventory management and handlng the emergency spples, t s obvos that the advantages of the RFID systems shold be taken nto accont serosly n spte of the hgh cost Implementaton wthn the emergency management framework Henderson (2007) stated that RFID systems cold be effcently sed by US Army Battle Command Sstanment Spport System (BCS3) to track commodtes ncldng emergency spples, n near-real tme, flowng n and ot of the regon. The dea was to establsh an atomaton network that has fxed and moble nterrogators to montor RFID devces onlne, on a map-based compter screen throgh lve trackng tools and n-transt vsblty. Wth ths motvaton, we defne or RFID-based emergency management and relef system as the process of synchronzng the crtcal spply and transportaton network to facltate the sstanable dsaster trackng, and to provde an effcent dstrbton of flow n spport of the for stages of dsaster relef efforts defned by Henderson (2007). These stages are revsed and adapted to or emergency management system based on a tme framework (Fg. 3). These emergency relef framework stages are defned as follows: 1. Stage 1 (pre-dsaster operatons): Ths s probably one of the most mportant stages of the whole process. It begns by establshng a lnk between dfferent levels of government, local organzatons, mltary, cvlan agences, and prvate sector. Ths wll help to constrct a workable and exectable spport plan. Then, avalable assets n the dsaster regon shold be dentfed and notfed. Moreover, tests and exercses shold be condcted to valdate and verfy the mplementaton of the plan. 2. Stage 2 (ntal dsaster operatons): Snce the coordnaton between dfferent agences s obtaned n the prevos stage, Stage 2 makes se of ths as the preparedness of the emergency spples for delvery and consmpton. Ths reqres the dentfcaton of possble dstrbtors and nventores for se, the estmaton of the emergency commodtes stocked n Fg. 3. For stages of RFID-based emergency management system for dsasters (adapted from Henderson, 2007.

8 178 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) these nventores, and the establshment of methodologes to assst the personnel workng n the commodty flow management. The MC-SHIC model, therefore, wll be extremely sefl n ths stage as t wll enable s to determne the ntal safety stocks n the nventores. 3. Stage 3 (sstanment of dsaster operatons): Ths s bascally the onlne emergency management step where the offcals manage the commodty flow and dstrbte the commodtes across the shelters. Therefore, t reqres a carefl predcton of the safety stocks and workforce, and the contnos capablty of n-transt vsblty of the emergency commodtes wthn the transportaton network. Ths wll be provded by a RFID mplementaton wthn the system to forecast and report the reqrements of the vctms n the shelters, and to spport forward demands. Or proposed on-lne method wll make se of the RFID data n ths stage. 4. Stage 4 (closre of dsaster operatons): Ths stage bascally ends the relef spport procedres lke the management of vtal spply flow and personnel. The data collected and lessons learned shold be provded for ftre reference. In the proceedng sectons, we wll ntrodce these methodologes that can be sccessflly ntegrated wth the RFID trackng of vtal commodtes. 4. Off-lne plannng: stochastc hmantaran nventory management Ltman (2006) states that, after the hrrcane Katrna, cvl organzatons were not allowed to transport and delver the vtal spples to the shelters de to the mltary and safety reasons. Ths shows the navalablty of a sffcent transportaton scheme for the emergency commodtes to the shelters; therefore, the model shold attempt to have the maxmm beneft whle determnng the stock levels of mltple commodtes and transportng them at the mnmm cost. Therefore, ths paper ams to develop a mlt-commodty stochastc hmantaran nventory control model (MC-SHIC) wth the followng concerns that happen commonly whle the emergency operatons are gong on drng/after a dsaster: There may be necesstes on the stock levels/storage of some commodtes n the emergency nventores drectly affectng and lmtng the stock of other commodtes. Transportaton of jont resorces,.e., food and water, s a major concern as the emergency vehcles spplyng the shelters may be lmted de to the dsrptons n the transportaton network and the navalablty of the personnel and vehcles. There may be lmted resorces of vtal/pershable commodtes sch as medcne, whch can sbsttte each other, where a mlt-commodty approach s needed. Mlt-spplers may be needed for the same commodty, whch can also be solved consderng the same type of commodty comng from dfferent spplers. We model the off-lne plannng model as a mlt-commodty and mlt-sppler stochastc hmantaran nventory control model (MC-SHIC), as an extenson of the lmted sngle-commodty approach of Ozbay and Ozgven (2007) that s not applcable to the type of problem stded n the paper. The mathematcal descrpton of the MC-SHIC model n the context of hmantaran logstcs, based on the Hngaran Inventory Control Model, ntrodced by Prékopa (2006), wll be presented n the followng sectons Mathematcal backgrond In or model, we assme that delveres, fxed and desgnated by n, take place accordng to some random process at dscrete tmes wthn a fnte tme nterval [0,T]. These random tmes have jont probablty dstrbtons the same as that of n random ponts chosen ndependently from the nterval [s;s + T] accordng to a nform dstrbton. A mnmal amont, d, s delvered wth each delvery n. If the total amont of delvery s D for some spport, 2 U, U beng a fnte set of spports, there s also a random amont of delvery obtaned by choosng a random sample of sze n 1, from a poplaton nformly dstrbted n the nterval [0,1 nd]. The consmpton process s defned smlarly, wth parameters C for some, 2 U, as the total amont of consmpton, c as the mnmal amont of consmpton, and s as the nmber of consmpton tmes. We assme that delvery and consmpton processes are ndependent, and there wll be a sperscrpt l for each commodty. The decson varables n the model are m ðlþ, the addtonal safety stock for each commodty l, l =1...r, and for each, 2 U, and M (l), the storage capacty of each commodty l, l =1...r. We have an ntal safety stock n the nterval [0,T] and to satsfy the needs of the vctms located n the shelters n terms of the commodtes, we are tryng to fnd the optmm addtonal safety stocks, m ðlþ vales and the optmm storage capactes M (l). We approxmate the jont dstrbton of the random consmpton and delvery varables sng an approxmate mltvarate normal dstrbton wth the random varable, W ðlþ, for each commodty l, l =1...r, for =1,..., n, and for each spport, 2 U. Therefore, W ðlþ0 s smply represent the vales of the probablty dstrbton of the commodtes n terms of consmpton mns delvery for any tme step: where W ðlþ ¼ cðlþ þ Y ðlþ 1 þ...þ YðlÞ n dðlþ X ðlþ 1... XðlÞ n ; l ¼ 1;...; r; ¼ 1;...; n for each ; 2 U ð1þ

9 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Y ðlþ X ðlþ : random varable of consmpton dstrbton, l = 1,..., r; = 1,..., n, : random varable of delvery dstrbton, l = 1,..., r; = 1,..., n, d (l) : mnmm amont of stock sppled wth a delvery, l =1,..., r; =1,..., n, c (l) : mnmal amont of commodtes consmed wth a delvery, l =1,..., r; =1,..., n. The delvery and consmpton vectors are r-component random vectors on dscrete spports { 2 U}, where U s a fnte set. The correspondng probabltes for each {D,C, 2 U} are gven by p. The expectatons, varances and elements of the covarance matrx for the random varable W ðlþ ; l ¼ 1;...; r; ¼ 1;...; n; 2 U are calclated as follows followng the normal approxmaton gdelnes gven n Prékopa (2006).!! l ðlþ ¼ cðlþ d ðlþ þ C ðlþ k ðlþ ncðlþ D ðlþ V ðlþ þ 1 h ðlþ ndðlþ ; ð2þ L ðlþ þ 1 Here, 2 2 ¼ C ðlþ k ðlþ r ðlþ ncðlþ G ðlþ ¼ DðlÞ ndðlþ V ðlþ þ 1! þ D ðlþ V ðlþ þ 2 h ðlþ ndðlþ L ðlþ þ 1! 2 1 ; ð3þ L ðlþ þ h ðlþ h ðlþ f h ðlþ B C 2 k 2 A C ðlþ L ðlþ k ðlþ f k ðlþ B C 2 A; ð4þ ðlþ þ 1 L þ 2 V ðlþ ðlþ þ 1 V þ 2 v: sample from nformly dstrbted poplaton n [0,C nc] for each, 2 U, x: sample from nformly dstrbted poplaton n [0,D nd] for each, 2 U, V: sample sze of v, and L: sample sze of x, k : postve ntegers selected randomly from the sample y, =1,..., n 1; f 6 n 1, h,f : postve ntegers selected randomly from the sample x, =1,..., n 1; f 6 n 1, G ðlþ : represents any element of the covarance matrx, WðlÞ Objectve fncton In or model, the objectve cost fncton conssts of the ndvdal costs lsted below: Cost of storage, g (l) : It s mportant to consder the storage costs for each commodty l, l =1...r, snce the occrrence of a dsaster s not known a pror. Cost of srpls, q þðlþ : Ths s ncrred f there s more nventory than demand. It can be modeled as a fxed cost or as a step fncton that allows very low or no cost for a certan srpls level, and then a steep ncrease for hgher levels of srpls. Cost of shortage, q ðlþ : Ths s ncrred f there s not sffcent nventory to satsfy the demand of the vctms. Ths s the most mportant cost component and chosen more than srpls, as n the case of dsasters, shortage of vtal spples cased many losses of lfe. It can also be modeled as fxed or varable lke the cost of srpls. Cost of adjstment, f (l) : Ths cost s a fncton of the addtonal safety stock. Sppose that we have an ntal amont of safety stock, however we need more to satsfy the probablty constrant for each commodty. Ths adjstment can be de to the nexpected factors sch as the strength of a dsaster, ncreased nmber of people who are affected and need help, etc. Of corse, ths has to be penalzed. The objectve fncton ncldes the demands for the consmpton of vtal commodtes mltpled by ther correspondng probabltes. Ths allows s to calclate the total cost where the hghest and lowest demands have the lowest probabltes accordng to a pre-determned dscretzed normal dstrbton. The total cost of a severe dsaster may be hgher than others de to the addtonal safety stocks reqred, however, the probablty assocated wth ths hgh demand wll be smaller than lower levels of demand closer to the mean (normal dstrbton assmpton). Ths self-controllng mechansm gves the analysts the chance to determne the safety stocks more accrately Constrants Or model takes nto accont the stochastctes of the consmpton and delvery processes gven the hghly stochastc natre of the problem doman before/after a dsaster nlke the relatvely predctable condtons of or daly lves n terms of probablstc constrants. As the probablstc constrants ensre the mnmal dsrpton of the commodtes n the shelters wth a gven probablty, the sm of ntal stocks and delveres has to be greater than or eqal to the consmpton for any tme step. By replacng W ðlþ s wth ther expectatons and varances of the approxmate normal dstrbton, or probablstc constrant s defned as

10 180 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) P W ðlþ 6 mðlþ þ m ðlþ P 1 e ) Yr l¼1 U mðlþ þ m ðlþ r ðlþ l ðlþ ; G ðlþ! P 1 e: The choce of jont constrants or ndvdal chance constrants s a matter of model formlaton and the dynamcs of the problem. It may be meanngless to employ a sngle-commodty analyss when the safety stocks n or problem do affect each other, or when there are mlt-spplers for the same commodty. Other constrants n the MC-SHIC model are the capacty constrants. At any tme step, the ntal safety stock pls the optmal addtonal stock mst be smaller than the storage capacty for that commodty, and the sm of storage capactes for each commodty (a (l), space occped by each commodty l, l =1...r, mltpled by the storage capacty for each commodty, M (l) ) mst be smaller than the overall capacty, M MC-SHIC formlaton Usng the mathematcal nformaton gven n the prevos secton, the overall mlt-commodty stochastc programmng (MC-SHIC) problem becomes as follows: ( "!! #!) mn Sbject to Yr Xr l¼1 l¼1 g ðlþ ðm ðlþ Þþ 1 T U mðlþ þ m ðlþ r ðlþ X 2U l ðlþ p f ðlþ m ðlþ þ X n ¼1 ; ¼ 1;...; n 1; G ðlþ q ðlþ m ðlþ þ Xn ¼1! P 1 e m ðlþ þ m ðlþ 6 MðlÞ ; m ðlþ P 0; 2 U; l ¼ 1;...; r X r l¼1 a ðlþ M ðlþ 6 M: Z 1 q þðlþ þ q ðlþ m ðlþ þm ðlþ 1 U z lðlþ r ðlþ Hngaran nventory control model, on whch or MC-SHIC model s based, was shown to be a convex nonlnear programmng problem by Prékopa (2006) assmng that the cost fnctons g (l), f (l), q +(l), q (l) are convex. Drng the analyss, g (l), q +(l), q (l) are taken as fxed costs whereas f (l) s selected as a lnear fncton of the addtonal safety stock ensrng the convexty. The convexty of the feasble soltons s a conseqence of the general theorems on mltvarate logconcave measres ntrodced n Prékopa (1971) Proposed solton approach sng the pleps algorthm Solton of MC-SHIC model reqres a complex approach where p-level effcent ponts (pleps) method, developed by Prékopa (1990), s proposed pleps method (Prékopa, 1990) dz ð5þ Defnton. Let n be a random varable, Z be the set of all possble vales of n, and F be ts CDF. A vector z 2 Z s a plep (p-level effcent pont) of the dstrbton of n f F(z) P p, p 2 [0,1], and there s no y sch that y 2 Z, y < z and F(y) P p. Explanaton. F s the cmlatve dstrbton fncton of the r-dmensonal random vector n =(n 1,...,n r ) T. That s, F(z)=P(n 6 z). For the r-dmensonal vector z =(z 1,...,z r ) T, y < z means y q 6 z q for q =1,..., r, and y q < z q for at least one q. Theorem. Let z (j),j=1,..., N be the pleps for the probablty dstrbton n Z = Z 1 x...xz r, and let F(Z) be the CDF of the dscrete random vector n. Then,! Pðz 0 P nþ P p f and only f P z 0 2 [N fzjz P z ðjþ g : ð6þ j¼1 pleps methodology provdes dscretzed set of ponts, whch gves the lower bond of a specfc probablty dstrbton (Prékopa, 2003). These are sed to create the determnstc eqvalent of the probablstc constrants, and they assre that the constrants wll satsfy the gven relablty level p =1 e. However, frst of all, before applyng pleps method, contnos dstrbton fnctons n or model have to be converted nto approxmate dscrete dstrbtons (Noyan and Prékopa, 2006). For that prpose, for each demand, 2 U, we approxmate the random varable W ðlþ by a dscrete varable nðlþ wth possble vales - ðlþ 1 < -ðlþ 2 <...< -ðlþ L, where the dstrbton fncton s: P m ðlþ þ m ðlþ P WðlÞ ) U W ðlþ F W ðlþ - ðlþ # ¼ ( F ) - ðlþ n ðlþ # ; # ¼ 1;...; L 1 ; l ¼ 1;...; r: ð7þ 1; # ¼ L

11 where - ðlþ 1 < -ðlþ 2 <...< -ðlþ L h are chosen to be eqdstant on some nterval 0; BðlÞ where F ðlþ W B ðlþ ¼ 1 f for a prescrbed small tolerance f. Here, B ðlþ s the selected pper bondary of the nterval of -ðlþ vales for each commodty l, l =1...r, and for each, 2 U. Usng ths process, we obtan the followng probablstc constrant n the form of mltplcaton of the cmlatve dstrbton fnctons of n ðlþ : Y r l¼1 F n ðlþ - ðlþ # P 1 e; # ¼ 1;...; L 1; l ¼ 1;...; r for each ; 2 U: on ts entre doman. Dr- That s, we dscretze or contnos cmlatve dstrbton fncton assocated wth W ðlþ ng the analyss, we try to keep a sbstantal amont of accracy whle choosng the possble vales of n ðlþ h nterval 0; B ðlþ selectng B ðlþ E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) n the and N accordngly. Wth ths dea, we try to preserve most of the nformaton related wth the orgnal fncton n the dscretzed one. The mportant pont s that we focs on the pper regons of the dstrbton fncton that contrbte to the calclaton of pleps wth respect to the selected dsrpton probablty. Fnally, wth the se of ths methodology, the relaxaton for the mlt-commodty problem (MC-SHIC) s performed as follows: Orgnal Problem Dsjnctve Programmng Relaxed Dsjnctve mn f m ðlþ Problem Programmng Problem sbject to mn f m ðlþ mn f m ðlþ Q r l¼1 F n ðlþ - ðlþ # P 1 e sbject to sbject to m ðlþ M ðlþ 6 m ðlþ m ðlþ þ m ðlþ P z ðjþ m ðlþ þ m ðlþ P PN k j z ðjþ j¼1 m ðlþ P 0; 2 U; l ¼ 1;...; r; for at least one j ¼ 1;...; N PN j¼1 k j ¼ 1 # ¼ 1;...; L 1 m ðlþ M ðlþ 6 m ðlþ m ðlþ M ðlþ 6 m ðlþ m ðlþ P r a ðlþ M ðlþ P 0; 2 U; l ¼ 1;...; r m ðlþ ; k P 0; 2 U; l ¼ 1;...; r 6 M P r a ðlþ M ðlþ P 6 M r a ðlþ M ðlþ 6 M ¼1 ¼1 The llstratve explanaton for the relaxaton process for L = 4 can be seen n Fg. 4. ¼ Prékopa Vzvar Badcs algorthm (Prékopa, 2010) Solvng the nonlnear MC-SHIC problem wth an exact solton technqe reqres extensve programmng and optmzaton knowledge. pleps method, however, serves as a practcal and easly applcable approxmate method so that a decson maker/planner can se the reslts to decde on the nventory levels for the emergency commodtes needed at the shelters. For ths prpose, the Prékopa-Vzvar-Badcs algorthm wll be sed to generate the plep sets for the relaxed dsjnctve programmng problem. The algorthm s based on the dscrete random varable n ðlþ where Z = Z 1x...xZ r, s the prodct set contanng the spport of n, the vector of the dscrete random varable n ðlþ. For the sake of llstraton, we consder the r-dmensonal vector as n =(n 1,...,n r ) T. The algorthm s as follows: Step 0. Intalze k 0. Step 1. Determne z 1;j1 ;...; z r;jr sch that z 1;j1 z 2;j2.. z r;jr ¼ arg mnfyjfðy; z 2;k2 þ1;...; z r;krþ1þ P 1 eg ¼ arg mnfyjfðz 1;j1 ; y; z 3;k3 þ1...; z r;krþ1þ P 1 eg ¼ arg mnfyjfðz 1;j1 ;;...; z r 1;jr 1 ; yþ P 1 eg and let E fz 1;j1 ;...; z r;jr g. Step 2. Let k k +1.Ifj 1 + k > k 1 + 1, then go to Step 4. Otherwse, go to Step 3. Step 3. Enmerate all the pleps of the fncton Fðz 1;j1 þk; yþ; y 2 R r 1 and domnate at least one element n E (y domnates z f y P z, y z). If H s the set of the remanng pleps, then let E E [ H. Go to Step 2. Step 4. Stop, E s the set of all pleps of the CDF F(z)=P(n 6 z).

12 182 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 4. pleps method llstraton for relaxng the formlaton Example nmercal stdy for the MC-SHIC model Frst, a base case two-commodty scenaro s created where ndependent dstrbtons of medcne and MRE s (mealready-to-eat) are consdered. Dependng on the severty of the dsaster, there are ten dfferent demand scenaros gven n an ascendng order, where demand type 1 (=1) represents the lowest demand (best case), and demand type 10 (=10) s the hghest demand (worst case). The demand for MRE s starts lower than the demand for medcne, bt ncreases more rapdly as the severty of the dsaster ncreases. For the base case scenaro, the followng vales are chosen: e s 0.05, f s 0.01, N s 100 and B ðlþ s 100 for all cases. Nmber of delveres n the tme nterval (a month) s n =4. Amonts of ntal safety stock, m (1) and m (2) are 20 and 20 nts, respectvely. q +(1) = 1.2/nt, q +(2) = 1/nt, and q (1) = 120/nt, q (2) = 100/nt. d (1), d (2), c (1) and c (2) are calclated for each commodty separately. Cost of adjstment fncton s selected as f(x)= 2x. Costs of storage for each good are 1/nt and 1.2/nt, respectvely. Total storage capacty s 100 nts. Spaces occped by commodtes are 1/nt and 1.2/nt, respectvely. Expected total delvery and expected total consmpton vales are taken as D 1 ¼½10; 12; 14; 16; 18; 20; 22; 24; 26; 28Š C 1 ¼½30; 32; 34; 36; 38; 40; 42; 44; 46; 48Š : D 2 ¼½0; 4; 8; 12; 16; 20; 24; 28; 32; 36Š C 2 ¼½20; 24; 28; 32; 36; 40; 44; 48; 52; 56Š Probabltes of the dscrete spports of consmpton and delvery vales are p ¼½0:03; 0:06; 0:10; 0:14; 0:17; 0:17; 0:14; 0:10; 0:06; 0:03Š: The reslts are gven n Table 2 where 96 pleps are sed for the demand scenaros. To satsfy the needs of the vctms 95% of the tme, ntal safety stock mst be at least more than 71% of the total expected consmpton for the medcal spples. That s, havng more than 71% of the commodtes n the nventory before the dsaster wll prevent dsrpton 95% of the tme. For the MRE, the ntal stock mst be more than 75% of the total expected consmpton. Ths ndcates the mportance of correctly determnng the ntal safety stocks n the nventory as havng ths percentage of stocks wll prevent dsrpton 95% of the tme. When the total expected consmpton s eqal to the ntal safety stock, the model fnds the optmal addtonal safety stock vale as zero. Moreover, t may not be possble to make 4 delveres n a gven perod of tme, ths addtonal amont of safety stock has to ncrease Senstvty analyss wthn the emergency management framework Change n the hrrcane category Sppose that a category 3 hrrcane affects New Orleans wth the strength of the hrrcane Katrna when t frst ht the coast regon of the US east coast. Wth ths scenaro, terran may be flooded wth possble dsrptons n the traffc network. People leavng n the flood areas are gathered nto shelters. We assme that 1000 people are gathered n the New Orleans sperdome. 2 MRE s and 1/2 medcne are assmed to be gven per person per day n average (as n Lee et al., 2009). Dependng on the strength of the hrrcane, there are fve demand levels gven n an ascendng order, = 1 represents the lowest

13 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Table 2 Reslts for the two-commodty nmercal example. Demand =1 =2 =3 =4 =5 =6 =7 =8 =9 =10 Intal stock (medcne) Optmal addtonal stock (medcne) Total ntal stock (medcne) Total expected consmpton (medcne) Proporton of ntal safety stock to total expected consmpton (medcne) (%) Intal stock (MRE) Optmal addtonal stock (MRE) Total ntal stock (MRE) Total expected consmpton (MRE) Proporton of ntal safety stock to total expected consmpton (MRE) (%) demand whereas = 5 s the hghest. Demand for MRE s ncreases more rapdly than the medcne as the severty of the hrrcane ncreases. For the base case, the followng vales are chosen: e s 0.1, f s 0.01, N s 40, and B ðlþ s 200 for MRE s and 1000 for medcne. Nmber of delveres n the tme nterval (a month) s n =4. m (1) and m (2) are 500 and 2000 nts, respectvely. q +(1) = 1.2/nt, q +(2) = 1/nt, and q (1) = 120/nt, q (2) = 100/nt. d (1), d (2), c (1) and c (2) are calclated for each commodty and scenaro separately. Cost of adjstment fncton s selected as f(x)= 2x. Costs of storage are 5/nt and 1/nt, respectvely. Total storage capacty s 50,000 nts. Spaces occped by commodtes are 1/nt. Expected total delvery and consmpton vales are D 1 ¼½250; 300; 350; 400; 450Š C 1 ¼½750; 800; 850; 900; 950Š D 2 ¼½0; 400; 800; 1200; 1600Š C 2 ¼½2000; 2400; 2800; 3200; 3600Š : Probabltes of the dscrete spports of consmpton and delveres are p ¼½0:08; 0:25; 0:34; 0:25; 0:08Š: The delvery and consmpton vales are sed to calclate the mean, varance, and covarance matrx for the dstrbton of the vtal spples. The reslts are gven n Table 3 where 75 pleps are sed. To satsfy the needs of the vctms 90% of the tme, ntal stock mst be more than 69% of the total expected consmpton for medcne. However, for MRE s, the ntal stock mst be more than 75% of the total expected consmpton to prevent dsrpton 90% of the tme. When the consmpton s eqal to the ntal safety stock of MRE s, the model fnds the optmal addtonal safety stock as 0 MRE s. It s possble for a hrrcane to ntensfy n strength n a short tme ncreasng the nmber of vctms located n the shelters and creatng more demand. In 2005, hrrcane Katrna rapdly ntensfed after enterng the Glf, growng from categores 3 5 n jst 9 h. Ths rapd growth was de to the storm s movement over the nsally warm waters of the loop crrent, whch ncreased wnd speeds. For a category 5 hrrcane, floodng cases major damage to many strctres and roadways. Massve evacaton of resdental areas wll be reqred. Therefore, we assme that 10,000 people are gathered n the New Orleans sperdome. Agan, we are focsng on medcne and MRE s. Note that the daly demand for MRE s can have hgh varances dependng on the tme-dependent mpact of the dsaster. Fve demand scenaros are analyzed as before. All other parameters beng kept same as the base case (other than these vales: m (1) and m (2) are 5000 and 20,000, respectvely, N s selected as 100, and B ðlþ s 5000 for MRE s and 25,000 for medcne), we expect an ncrease n the total delvery and consmpton vales at the sperdome: D 1 ¼½5000; 5500; 6000; 6500; 7000Š C 1 ¼½10000; 10500; 11000; 11500; 12000Š D 2 ¼½20000; 24000; 28000; 32000; 36000Š C 2 ¼½40000; 44000; 48000; 52000; 56000Š : Wth these vales, we change the mean, varance and the covarance matrces of the dstrbton of the vtal spples. Especally for MRE s, the dstrbton s changed sbstantally to accont for the severty of the category 5 hrrcane. To satsfy the needs of the vctms 90% of the tme, ntal stock mst be more than 67% of the total consmpton for medcne (Table 4). That s, havng more than 67% of the commodtes n the nventory before the dsaster wll prevent dsrpton 90% of the tme. However, for MRE s, the ntal stock mst be more than 73% of the total consmpton. On Agst 28, 2005, the Losana Natonal Gard delvered seven trckloads of MRE s to the sperdome, enogh to spply 15,000 people

14 184 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Table 3 Reslts for the two-commodty analyss of category 3 hrrcane. Demand =1 =2 =3 =4 =5 Intal stock (medcne) Optmal addtonal stock (medcne) Total ntal stock (medcne) Total expected consmpton (medcne) Proporton of ntal safety stock to total expected consmpton (medcne) (%) Intal stock (MRE) Optmal addtonal stock (MRE) Total ntal stock (MRE) Total expected consmpton (MRE) Proporton of ntal safety stock to total expected consmpton (MRE) (%) Total cost 1786 Table 4 Reslts for the two-commodty analyss of category 5 hrrcane. Demand =1 =2 =3 =4 =5 Intal stock (medcne) Optmal addtonal stock (medcne) Total ntal stock (medcne) Total expected consmpton (medcne) 10,000 10,500 11,000 11,500 12,000 Proporton of ntal safety stock to total expected consmpton (medcne) (%) Intal stock (MRE) 20,000 20,000 20,000 20,000 20,000 Optmal addtonal stock (MRE) 10,000 13,000 15,000 19,000 22,000 Total ntal stock (MRE) 30,000 33,000 35,000 39,000 42,000 Total expected consmpton (MRE) 40,000 44,000 48,000 52,000 56,000 Proporton of ntal safety stock to total expected consmpton (MRE) (%) Total cost 4192 for three days whch was not enogh for the 20,000 vctms located n the sperdome. 20,000 vctms can lve wth 2 MRE s per day, whch makes the total expected consmpton amont as 40,000 MRE s (Ths s or demand = 1 n Table 4). However, the sppled amont of MRE s s 30,000 (15,000 people wth 2 MRE s per day). Here, as the cost optmal reslt, the model gves an ntal stock of 30,000 MRE s as the ntal safety stock, and an addtonal 10,000 MRE s to be delvered afterwards. As the strength of the hrrcane ncreases, the demand for vtal spples ncreases n a short amont of tme. Moreover, the demand for MRE s ncreases more rapdly than t s for the medcne. The model reacts to ths noton by ncreasng the addtonal safety stock vales more rapdly for MRE s (Fg. 5). The most severe condton of the category 5 hrrcane havng the largest demand reqres the largest safety stocks for both commodtes Dsrptons n the transportaton network As mentoned n the prevos secton, Drng Katrna, Losana Natonal Gard delvered seven trckloads of MRE s, enogh to spply 15,000 people bt that was not enogh for the 20,000 vctms located n the sperdome. Ths ndcates the mportance of determnng the ntal safety stocks wth respect to the nmber of delveres. In the aftermath of dsasters, changes n the nmber of delveres manly de to ncreased demand, redced spply and possble transportaton system dsrptons are expected. For a severe hrrcane, a majorty of the roadways may be flooded, so a lmted delvery process may be gong on. Hence, we start wth a worst case scenaro where the vehcles can arrve at the shelter once for the whole tme perod (.e., a month). Gradally, the transportaton dsrptons end, and the nmber of vehcles servng the sperdome ncreases. Ths can be vewed as Hrrcane Katrna slowng down as the wnd speed decreases. In Fg. 6, we start wth 1 delvery per perod, and gradally ncrease the nmber of delveres to 9. As the nmber of delveres ncreases, the addtonal amonts of safety stock vales decrease. Ths s a logcal behavor snce ths ncrease leads to lower addtonal stock vales to satsfy the probablty constrant. For the category 3 hrrcane wth lower consmpton needs, when the nmber of delveres reaches 8 vehcles drng the whole tme nterval, there s no need for addtonal safety stock for the medcne and MRE s. Hence, the ntal safety stock s large enogh to satsfy the evacee demand for the total perod. As the dsaster strength ncreases, the ntal safety stock appears to be nsffcent, and a hge amont of addtonal safety stock s needed when the nmber of delveres s lmted de to damaged/congested roadways. Total cost also decreases wth lower levels of addtonal safety stocks. Ths s becase, f the system s hghly stochastc, the probablstc constrants are satsfed only f addtonal safety stock s hgh. Ths wll mpose hgher costs. However, f the ntal safety stock s too low, the problem becomes nfeasble. Ths, MC-SHIC model acheves a balance between satsfyng the probablty constrants and keepng the cost low.

15 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 5. Changes n the addtonal safety stock vales verss changes n the demand scenaros for the hrrcane. Fg. 6. Changes n the addtonal safety stock vales for medcne (a) for MRE s (b) changes n total cost (c) for dfferent nmber of delveres Mlt-spplers or cross-shppng In the dsaster relef perod, there may not be enogh commodtes that can be dstrbted to the shelter from the sal sppler, or delveres may not be possble de to the damaged traffc network. Therefore, to satsfy the needs of

16 186 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 7. Changes n the addtonal safety stock vales for MRE s va mlt-spplers. the vctms, emergency orders may be needed from other spplers whch wll ncr more costs. It s also possble that de to the flctatons n the delveres and consmptons, some shelter nventores may experence shortages whle some others experence srpls. Rather than havng delveres from the sppler, n some cases, delveres from shelter to shelter may be needed between close shelters sch as the New Orleans sperdome and conventon center. That s, f MRE s stored are more than enogh for the vctms n the conventon center, and f a shortage s experenced n the sperdome, these extra MRE s can be shpped to the conventon center to accont for that shortage. Althogh the mlt-spplers or cross-shppng optons are vable when there s a sdden notce of demand ncrease, t does not mean that these alternatves are cost effectve. Emergency orders are mostly expensve as the orderng charges are rased nnecessarly, and therefore not preferred ntl an extreme case happens. The model reacts wth a smlar noton to ths costly order from Sppler 2, and ncreases the amont of addtonal safety stock nstead of the commodtes beng delvered by Sppler 2 (Fg. 7). Ths backp plan wll make the planner aware of the rsk nvolved n stockng the emergency nventory commodtes Vtal/pershable commodtes Ths secton ncldes a senstvty analyss ndcatng where sngle-commodty and mlt-commodty approaches are sefl drng the emergency management operatons. Frst of all, some vtal spples may need extra attenton drng the emergency relef operatons. For example, f a large nmber of njred are broght to a shelter n an emergency after a dsaster, medcne wll be mmedately needed. After Katrna, n the sperdome at New Orleans, 32% of the srveyed vctms experenced njres where 13% of these njres appeared to be seros (Brode et al., 2006). Frst, we assme that the percentage of the njred people ncreases de to the ncdents cased by the category 5 hrrcane condtons (the demand for medcne ncreases). Therefore, medcne becomes the most mportant commodty to be sppled mmedately and the analyss s condcted wth a sngle-commodty approach. Ths dea can also be sed for pershable commodtes and commodtes wth shelf lves, or se by dates. Therefore, both the emergency management of sch commodtes and modelng for extra nventory levels to reflect losses de to shelf lfe expraton shold be factored nto the nventory control model. The reslts are compared wth the two-commodty analyss performed before. The total cost for ths analyss s larger than the snglecommodty analyss (Table 5). Ths s becase we are focsng on a sngle-commodty rather than two-commodtes. However, snce the shortage costs for the medcne are hgher, the cost s sbstantally hgh relatve to a less mportant commodty. De to the hgh costs, the model does not allow a hgh safety stock for the medcne and the resltng total base stock vales are approxmately 10% lower than two-commodty case. The decson of whch method to se (jont constrants or ndvdal constrants) depends on the prortes of the dsaster plannng process where the planner shold be carefl abot the reqrements of the decson-makng process. If the focs s on a sngle-commodty, or f the pershablty s or man concern, t s sefl to condct a sngle-commodty analyss (.e., medcne). In ths case, cost becomes hgher de to the stockng reqrements and shortage possbltes. If those concerns are mnmal, and f we are jst concerned wth the qanttes of the safety stocks, a mlt-commodty analyss s reqred. Drng the emergency condtons n the aftermath of a dsaster, transportaton of jont resorces,.e., food and water, s also a major concern as the emergency vehcles spplyng the shelters may be lmted de to the dsrptons n the transportaton network and navalablty of the personnel and vehcles. Moreover, relatonshps may exst between the prodct demand streams sch as food tems or medcal spples leadng to the sbsttton possbltes drng a shortage. In these cases, a mlt-commodty analyss wold be logcal.

17 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Table 5 Comparson for sngle and two-commodty analyss for medcne. Demand =1 =2 =3 =4 =5 Sngle-commodty analyss Intal stock Optmal addtonal stock Total ntal stock Total expected consmpton 10,000 10,500 11,000 11,500 12,000 Proporton of ntal safety stock to total expected consmpton (%) Total cost 2974 Two-commodty analyss Intal stock Optmal addtonal stock Total ntal stock Total expected consmpton 10,000 10,500 11,000 11,500 12,000 Proporton of ntal safety stock to total expected consmpton (%) Total cost Next step n emergency management: an on-lne model 5.1. On-lne control methodology There s a crcal need for the development and analyss of a hmantaran nventory management model sch as MC-SHIC model, developed pror to the occrrence of a natral or man-made catastrophe, whch can determne the safety stock level that wll prevent possble dsrptons at a mnmal cost. However, an emergency nventory management system shold also nclde an on-lne operatonal strategy to mnmze mpacts of the nforeseen dsrptons, or at least to address the problem at hand as fast as possble. The real-tme management of the preparedness actvtes wll reveal the actal performance of the response, recovery and mtgaton processes after the dsaster. Therefore, the problem s to be able to have an on-lne nventory control methodology wthn or framework. As stated by Pao et al. (1992), determnng the controllers for nonlnear systems s extremely dffclt, even n the determnstc settngs where the eqatons that govern the system dynamcs are flly known. Smlarly, wthn or complex nventory management model MC-SHIC, t s mpossble to determne the control law needed sng exstng adaptve control procedres. Ths provdes the motvaton for developng an on-lne nventory control procedre that does not reqre a model or that reqres mnmal nformaton for the nderlyng nventory management system. One way to deal wth ths problem s to se a fncton approxmaton method to represent the controller tself. The neral network s traned wth the data obtaned from MC-SHIC model and sed to approxmate the control vector whch wold normally be calclated for systems for whch a control law can be obtaned. The overall proposed on-lne control methodology can be seen n Fg. 8. In or model, y k+1 are the nventory levels of emergency commodtes where k represents the tme step. We have three basc goals n or model: to adapt or tran the fnctonal approxmator wth the random consmpton and delvery dstrbtons of the MC-SHIC model, to approxmate the control vector k sng the weght vector h k that belongs to the neral network, to reach the desred target nventory levels by penalzng the devaton between the otpt nventory vales y k+1 and the target nventory vales where k represents any tme step (possbly a day). Under perfect condtons, whch clearly do not exst drng or after the dsasters, these levels can be sed repeatedly to spply the basc needs of the dsaster vctms located n the shelters as long as there are no dsrptons n the system. One of the major challenges here s that t s dffclt to control and reglate the system gven the stochastcty n the natre of nderlyng spply and consmpton processes. When the system parameters (not the strctre of the model) are nknown, adaptve control procedres can be sed wth a control law estmatng these parameters sng data. However, these methods are lmted as they need some sort of system eqatons for the model. For the complex dsaster process, on the other hand, these closed form eqatons descrbng the system dynamcs are generally nknown, makng t mpossble to determne the control law. For ths prpose, we propose to se a fnctonal approxmator (FA) to deal wth the nosy data and to obtan an acceptable control methodology wthot havng the knowledge of process dynamcs for the nonlnear stochastc emergency relef system. The data for the tranng of FA s generated by the MC-SHIC model sng the probablstc consmpton and delvery dstrbtons of emergency commodtes, and after that, non-addtve nose s ntrodced nto the system to better smlate the extreme complexty expected nder real-world emergency condtons. We are gong to test the proposed framework on a system that ncldes the key components, namely the MC-SHIC model and the fncton approxmator, whch s the neral network. We se the followng eqaton as the tranng data for the neral network where nonaddtve nose (error) s ntrodced nto the system at every tme step k:

18 188 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 8. RFID-based emergency management system wth the MC-SHIC model and a fncton approxmator for the controller. y kþ1 ¼ Ky k þ!w k þ Ew k ð8þ where K: nose matrx for the crrent nventory level, where any element of K s a normally dstrbted random nose selected between [-10, 10].!: nose matrx for the stochastc consmpton and delvery, whch we select as the dagonal matrx, wth the dagonal elements eqal to 1. W k : MC-SHIC parameter representng the stochastc Consmpton-Delvery dstrbton at tme step k (a randomly selected element wth Eq. (1)). E: mltplcaton matrx, where any element of E s ky k k. ky k k: the Ecldean norm of the nventory level y k, at tme step k. w k : an ndependent scalar of Bernoll ±0.5 process. Ths type of model s nterestng as the nose enters the system n a mltplcatve and very nstable way, rather than jst smply addng the nose to the crrent nventory levels. Target seqence s also obtaned from the MC-SHIC model accordngly wth the data created n Eq. (11). Here, RFID trackng data s the key to obtan a relable target seqence as prevos observatons wll make the overall process for obtanng the data more relable Development of the fncton approxmator Fnctonal approxmators sch as neral networks, can be sed to deal wth the problems of controllng and reglatng the stochastc systems wth nknown nonlnear dynamcs becase they can be sed to approxmate the nderlyng controllng system wthot the need to constrct a separate model for the nknown process dynamcs. Neral networks have been known to be exceptonally sccessfl on learnng complex behavors from observatons. Moreover, the behavor learned by the neral network can be adjsted effectvely wth the learnng algorthms. These sses generate the bass for consderng the se of neral networks for the adaptve control problem gven n ths paper. Neral networks have been sed for smlar prposes for dfferent problems n transportaton and traffc engneerng felds. Several examples of neral network stdes nclde ther applcatons to the feedback based ramp meterng strateges (Zhan and Rtche, 1997), and traffc control systems (Nakatsj and Kak, 1991; Papageorgo et al., 1984). The neral network consdered n ths paper has a fxed nmber of layers and nodes. The varaton of the network comes from the pdates of the nderlyng parameters at every teraton. These nderlyng parameters for neral networks are the connecton weghts between the npts, hdden layers, and otpts, and the bas vales nclded n the hdden layers and otpts. Tranng process of the neral network reqres the estmaton of these weghts towards the optmal nventory level vales. The weght estmaton process, commonly called tranng task, solves an nconstraned nonlnear mnmzaton problem where the optmal vales are calclated for the varables mnmzng the objectve fncton. In or control system, the neral network has no nformaton abot the analytcal strctre (MC-SHIC model) generatng the measrements (wthot pror nformaton abot the strctre of the system). Moreover, n or neral network, assocated wth the control vector k for the controller, there s an nderlyng connecton weght vector h k whch s to be traned for optmalty. The overall methodology can be seen n Fg. 9. We frst optmally calclate the nventory levels for emergency commodtes sng the MC-SHIC model. Then, the neral network s traned to reach the optmm nventory levels sng the tranng data and target nventory levels obtaned from MC-SHIC model based on these calclatons. Ths leads to an on-lne approxmate controller that can qckly come p wth optmal level of emergency spples rather than solvng the MC-SHIC problem on-lne n real-tme. For the neral network, fndng the optmal control strategy at a tme step k s eqvalent to mn L k ðh k Þ

19 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) where h k 2 R p s a p-dmensonal vector of the weghts, and p s not a fncton of k. The optmal vales of h k, therefore, wll normally lead to an optmal control vector k, leadng to the optmal vales of otpts. The objectve fncton sed n ths paper s a loss fncton based on the expected vale (E[]) of the qadratc trackng error loss fncton: h L k ðh k Þ¼E ðy kþ1 t kþ1 Þ T A k ðy kþ1 t kþ1 Þþ T k B k where A k, B k are postve sem-defnte matrces ndcatng the relatve weght of the devatons from the target vales, and the cost assocated wth larger vales of the controls. Therefore, or nconstraned nonlnear optmzaton problem becomes: h mn L k ðh k Þ¼ ðy kþ1 t kþ1 Þ T A k ðy kþ1 t kþ1 Þþ T k B k ð9þ where h k 2 R p, p s not a fncton of k. The mnmzaton of L k (h k ) actally reqres, for each k, a mnmzng solton h k to: g k ðh k k k ¼ k k cannot be compted, t s not generally possble to calclate g k (h k ), and the standard gradent descent type algorthms are not very effectve and feasble n solvng or problem. Hence, we consder the stochastc approxmaton algorthm that has the standard recrsve form of ^h k ¼ ^h k 1 a k g k ð^h k 1 Þ where the gan seqence {a k } satsfes the well-known condtons (Kshner and Yn, 1997): ð10þ P 1 k¼1 P 1 k¼1 a k ¼1 a 2 k < 1 Here, g k ð^h k Þ s the estmate of the gradent g k ðh k k, and ^h k s the estmate of the weght vector calclated at any teraton solvng the nconstraned optmzaton problem. As no control law or system eqatons can be acheved governng the control of or dsaster relef nventory system sng MC-SHIC model, the gradent of the objectve fncton sed n standard optmzaton algorthms s not avalable to be able to calclate ths parameter vector. Therefore, we propose the smltaneos pertrbaton stochastc approxmaton (SPSA) method, frst proposed by Spall (1992), to fnd the estmate of the gradent gven n Eq. (9) as SPSA only reqres two measrements of the system rather than ts fll fnctonal form. That s, f there s a devaton from the target levels, we se the on-lne SPSA-based fncton approxmator to restore the nventory levels to optmal vales. Ths type of approxmaton s appled by Spall and Crston (1997) and Spall (1998) on wastewater treatment. The performance of the proposed methodology s hghly dependent on the specfc propertes of or problem sch as the sbstantal amont of nose n the observatons, whch s hghly possble wthn the emergency relef operatons. Therefore, we compare the performance of or mplementaton by ntrodcng nose nto the system to evalate the performance of SPSA and Levenberg-Marqardt (LM) algorthms whle searchng for the optmm gradent descent. LM algorthm (Levenberg, 1944; Marqardt, 1963) s a good choce for comparson snce t appears to be the fastest method (p to ten to one hndred tmes faster than the standard backpropagaton algorthms) for tranng moderate-szed feed forward neral networks. SPSA, on the other hand, has been sccessflly mplemented for optmzaton n transportaton network analyss problems (Ozgven and Ozbay, 2008). Both algorthms are sed to estmate the weght vector ncldng the bas vales for the neral network at each teraton. In or model, SPSA algorthm ses the form n Eq. (9) where ^g k ð^h k Þ represents the smltaneos pertrbaton approxmaton to g k ð^h k Þ. The th component of ^g k ð^h k Þ, =1,..., p, s calclated as: ^g k ð^h k 1 Þ¼ b L ðþþ k b L ð Þ k 2c k D k ð11þ Here, b L ðþ k : estmated vales of L k ð^h k 1 c k D k Þ sng the observed y ðþ and t kþ1 k. ^h k : weght vector sed to approxmate (±) k n the neral network, where ^h k ¼ ^h k 1 c k D k, and D k =(D k1, D k2,...,d kp ) T, wth the {D k } ndependent, bonded, symmetrcally dstrbted random varables "k,, dentcally dstrbted at each k, wth E D 2 k nformly bonded "k,. {D k } cannot be nform or normal. y ðþ kþ1 : nventory level state vales based on the model.

20 190 E.E. Ozgven, K. Ozbay / Transportaton Research Part C 29 (2013) Fg. 9. Overall methodology for tranng the neral network sng MC-SHIC model. {a k } and {c k } are called gan seqences, and they are seqences of postve nmbers. In a settng where the dynamcs and objectve fncton are changng, t s best to pck constant coeffcents as a k = a, c k = c "k so as to satsfy the traceablty of the tme-varyng solton h k (Spall, 1998). The power of SPSA comes from the fact that only two nventory level measrements are needed at any teraton to estmate the gradent drecton. Ths s n contrast to other standard fnte dfference stochastc approxmaton methods whch wold need 2p measrements. Moreover, the ntal selecton of weght vector ^h 0 s also very mportant for SPSA for whch random ntalzaton s always possble. If there s any pror nformaton abot the system, t s meanngfl to select an ntal vector accordngly to obtan more precse controls. Whle sng Eq. (9), we also create a nomnal state at every tme step provdng a measre of how the estmaton procedre s performng as t reles on the pdated weght estmate n the neral network. The nomnal state creaton s desrable especally at the very early teratons to montor the tranng performance of