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1 Publié par : Publised by: Publicación de la: Édition électronique : Electronic publising: Edición electrónica: Disponible sur Internet : Available on Internet Disponible por Internet : Faculté des sciences de l administration 2325, rue de la Terrasse Pavillon Palasis-Prince, Université Laval Québec (Québec) Canada G1V 0A6 Tél. P. Tel. : (418) Télec. Fax : (418) Marylène Paradis Vice-décanat à la recerce Faculté des sciences de l administration ttp://www4.fsa.ulaval.ca/cms/site/fsa/accueil/recerc e/publications/documentsdetravail rd@fsa.ulaval.ca DOCUMENT DE TRAVAIL Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Walid KLIBI Soumia ICHOUA Alain MARTEL Document de travail également publié par le Centre interuniversitaire de recerce sur les réseaux d entreprise, la logistique et le transport, sous le numéro CIRRELT Version originale : Original manuscript: Version original: Série électronique mise à our : On-line publication updated : Seria electrónica, puesta al dia ISBN

2 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Walid Klibi 1,2, *, Soumia Icoua 1,3 and Alain Martel 1,4 1 Interuniversity Researc Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) 2 Operations Management and Information Systems Department, BEM-Management Scool, Bordeaux, France 3 College of Business, Embry-Riddle Aeronautical University, Florida, USA 4 Département Opérations et systèmes de décision, Faculté des sciences de l administration, Université Laval, Québec, Canada ABSTRACT Tis paper studies te strategic problem of designing emergency supply networks to support disaster relief over a planning orizon. Tis problem addresses decisions on te location and number of distribution centers needed, teir capacity, and te quantity of eac emergency item to keep in stock in time. To tackle te problem, a scenario based approac is proposed involving tree pases: disasters scenario generation, design generation and design evaluation. Disasters are modeled as stocastic processes and a Monte Carlo procedure is derived to generate plausible catastropic scenarios. Based on tis detailed representation of disasters, a multi-pase modeling framework is proposed to design te emergency supply network. Te two-stage stocastic programming formulation proposed is solved using a sample average approximation metod. Tis scenario based solution approac is tested wit a case inspired from real-world data to generate plausible scenarios, to produce a set of alternative designs and ten to evaluate tem on a set of performance measures in order to select te best design. Keywords. Humanitarian Logistics, Relief Network Design, Risk Modeling, Multi-azards, Stocastic Programming, Monte-Carlo Scenarios. * Corresponding autor: Walid.Klibi@bem.edu Copyrigt Walid Klibi, Soumia Icoua and Alain Martel, 2013

3 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 1. Introduction In te last decade, te world as noticed an escalating trend in te number of natural disasters. In 2010, 385 natural disasters were reported worldwide killing more tan 297,000 persons, affecting over 217 million oters and causing US$ billion in economic damages (Gua- Sapir et al., 2011). Despite te alarming effects of tese disasters on global economy, mitigating teir impacts on uman lives remains te maor concern. Hence, wen a maor disaster strikes, te callenge is to deliver te appropriate emergency supplies in sufficient quantities exactly wen and were tey are needed. Tus, te efficiency of logistics operations tat account for 80% of relief operations (Van Wassenove, 2006) is crucial in order to insure a good responsiveness wen disasters occur. To ensure an adequate and timely response, umanitarian organizations managing relief aid in disaster-prone regions typically preposition different emergency supplies at some distribution centers (DCs) in tese regions. Prepositioning decisions include te number and locations of te permanent DCs needed; teir capacity and te quantity of eac item tey sould old. Tese decisions are constrained by budget limitations tat restrict pre-disaster expenses dedicated to establising DCs and olding inventories. Tus te callenge is to find a good trade-off between maximizing demand coverage and minimizing setup and operating costs. Te literature on umanitarian logistics, including pre-disaster preparedness and post-disaster responsiveness, as grown in te past two decades (Atlay and Green (2006), Simpson and Hancock (2009) and Icoua (2011)). However, tere is still a lack of modeling and solution approaces tat can efficiently address te callenging operating conditions of an emergency scene (i.e., making urgent decisions trougout te disaster lifecycle in a igly dynamic risky environment were information is scarce). Te contribution of tis paper is treefold. First, to caracterize potential treats, we propose a risk modeling approac based on time-space stocastic processes tat builds on te general framework proposed in Klibi and Martel (2012). Hence, in addition to modeling disasters of multiple types, tese multi-azards unfold dynamically over time, leading to dynamic-stocastic scenarios. Eac of tese scenarios consists of a cronological list of multi-azards caracterized by an occurrence time as well as te impact on demand and availability of prepositioned supplies at DCs and at supply sources. Second, to account for te particularities of disaster evolution trougout its main stages (i.e., deployment, sustainment and recovery), a multi-pase modeling framework based on stocastic pase-dependent demands is proposed to tackle te prepositioning problem. Tird, wile most papers in te literature of umanitarian relief networks address twoecelon DC/Demand distribution systems, we propose a tree-ecelon system tat also considers suppliers (i.e., pre-selected vulnerable vendors wit limited resources and a backup source). Te intent is to replicate real-life emergency settings as practiced for instance by Wal-Mart response during Hurricane Katrina in or by Home depot contingency plans during Hurricane Gustav in Te remainder of te paper is organized as follows. Section 2 provides a non-exaustive review of existing work related to pre-disaster preparedness. Section 3 describes te problem context. Section 4 presents te proposed risk modeling approac wic depicts ow multi-azards unfold randomly in time over a planning orizon and in space over te geograpical region under consideration. 1 Source: Wal-Mart at Forefront of Hurricane Relief (Te Wasington Post, 2005/09/05) 2 Source: money.cnn.com (Home Depot's urricane plan - Fortune, 2010) 2

4 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Section 5 presents te stocastic multi-pase matematical program developed for te prepositioning problem. Section 6 describes te solution approac used to solve te prepositioning problem and to evaluate te performance of te relief network designs produced. Section 7 reports computational results obtained on a set of plausible scenarios inspired from real-world data. Finally, section 8 summarizes our findings and proposes future researc avenues. 2. Literature Review In te last two decades, te field of umanitarian logistics as gained increasing attention. Interesting survey papers on pre-disaster preparedness and post-disaster responsiveness include Atlay and Green (2006), Simpson and Hancock (2009) and Icoua (2011). Tis section solely presents a non-exaustive review of existing work related to pre-disaster preparedness. Many papers witin tis researc line are dedicated to pre-positioning problems formulated as variants of facility location models (Daskin, 1995 and Snyder, 2006). In te following, te literature on pre-disaster preparedness is classified in two maor categories, depending on weter te modeling approac is deterministic or stocastic. Witin eac category, a classification is made based on te type of disruptions modeled: demand, capacity (complete/partial loss of pre-positioned supplies) and transportation network (lane perturbations) disruptions. It is wort noticing tat complete loss of capacity or facility failure as been relatively well studied compared to partial loss of capacity. Snyder et al. (2006) present a set of facility location models for designing reliable/fortified networks to facility failures. 2.1 Deterministic Models Papers tat belong to tis category eiter assume tat problem inputs suc as impacted populations demand size and state of te transportation networks are all known wit certainty or use single-point estimates for tese inputs. In te former case, temporary facilities must be located at te beginning of te disaster and may need to be relocated at different time periods during te response duration. Te problem is operational rater tan strategic. It is tus assumed tat necessary information can be obtained and updated eac time location-relocation decisions need to be made. Papers tat use deterministic inputs often address disaster impact on demand only (Dekle et al. (2005), Tzeng et al. (2007), and Halper and Ragavan (2011)). In Nolz et al. (2011), transportation network disruption is also addressed. In te case of single-point estimates, istoric data are used to forecast expected values of te inputs (Iakovou et al. (1996) and Akkial (2006)) or input values corresponding to te worst or best case (Gormez et al. (2011)). Tese values are ten used as parameters in deterministic models. Wen establising temporary facilities to respond to a disaster, it is reasonable to use deterministic values for te problem inputs, provided a timely accessibility to information needed. Te latter is affordable nowadays wit te advances in information tecnology. However, wen addressing te strategic decision of establising permanent facilities to prepare for future disasters over a long time planning orizon, problem information is uncertain due to te time lag. Tus, te use of single point estimates is likely to lead to poor network performance. Stocastic modeling approaces are aimed at overcoming tis weakness. 3

5 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 2.2 Stocastic Models Tis line of researc as sougt to develop robust modeling and solution approaces tat account for te randomness in some problem inputs suc as demand and availability of transportation. For relatively simple cases (e.g. a single facility or demand point), continuous probability distributions are used. Wen more realistic cases are considered, scenario-based approaces are favored to limit te possible outcomes to a finite set of plausible futures. In te following, we briefly review some of te work tat belongs to tese two categories. Modeling Demand Disruption Only Continuous probability distributions were used in Lodree and Taskin (2008a,b) to address inventory control problems aimed at finding optimal level of emergency supplies to preposition at a known facility. Murali et al. (2011) assume tat demand of medicine follows a log-normal distribution wen addressing te problem of prepositioning medicine in capacitated facilities. Tey propose a cance constrained programming model wic maximizes coverage wile guarantying an upper-bound on te probability of violating demand constraints. Cang et al. (2007) examine te problem of prepositioning rescue resources in preparation for floods. Tey propose two stocastic programming models to minimize total expected set up, transportation and equipment costs. Teir models are solved using a sample average approximation scema. Duran et al. (2011) also use a scenario-based approac for a similar problem. It is assumed tat demand for relief supplies can be met from prepositioned quantities and suppliers. Te proposed mixed integer programming model minimizes te expected average response time assuming a limit on total inventory allowed and on te number of wareouses to open. A scenario-based approac is also used in Balcik and Beamon (2008) wo propose a variant of te maximal covering location model wit obective to maximize covered demand under pre-disaster and post-disaster budget constraints. However, in teir model scenarios involve a single event (i.e., a single demand location and amount). Finally, a scenario-based approac was also applied for designing a global supply network to support te umanitarian assistance, peacekeeping and peace enforcement missions of te Canadian Armed Forces in Martel et al., (2012). A scenario generation procedure is proposed to generate worldwide disasters and conflicts over a planning orizon, to determine if tese give rise to a mission and, if so, to specify product demands and returns at specific locations during te mission deployment, sustainment and redeployment pases. Modeling Demand and Capacity/Transportation Network Disruption In Ukkusuri and Yusimoto (2008), te problem of pre-positioning supplies is modeled as a location-routing problem were some pre-selected transportation links ave a known probability of failure. Te obective is to maximize te probability to reac all demand points. Te proposed integer programming model integrates a procedure for finding te most reliable pats. A continuous probability distribution is also used in Campbell and Jones (2011) to model demand wen considering te problem of locating supply points and deciding on teir levels of inventory given tat eac of tese as a known probability of being destroyed. Te autors determine te optimal stocking quantity and te total expected costs associated wit delivering to a demand point from a single supply point. Te results are ten used in a euristic wic address te case of multiple demand and supply points. However, it is assumed tat eac demand point is served by a single facility. 4

6 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Jia et al. (2007) develop a scenario-based approac to solve different variants of facility location models for te prepositioning of supplies in large-scale emergencies. Te models use a set of discrete scenarios to represent te likeliood tat a certain emergency situation affects a given demand point. Te models also account for te reduction in service capability of a facility under an emergency scenario. However, in teir illustrative example, te autors do not consider facility disruptions. Mete and Zabinsky (2010) examine te problem of prepositioning medical supplies under demand and transportation disruptions. Six scenarios are used to represent different demand and travel time realizations based on eartquake data in te Seattle area. Te autors propose a two-stage stocastic model wic integrates location and inventory decisions. Te obective is to minimize te expected sum of fixed and transportation costs along wit a penalty of unsatisfied demand. Rawls and Turnquist (2010) model a similar problem as a two-stage stocastic program, solved using a euristic wic combines te L-saped metod and Lagrangian relaxation. Tey consider demand, transportation and capacity disruptions. Historical data from a sample of fifteen urricanes in te souteastern US are used to generate 51 probabilistic scenarios. Rawls and Turnquist (2011), extend te model to include service quality constraints using te notion of reliable sets introduced in Daskin et al. (1997). Tis model is extended furter in Rawls and Turnquist (2012) to account for time-dependent static demand over te deployment pase wic is partitioned into four time periods. Te model also uses time-dependent capacities on flows between origins and destinations to account for transportation link damages. However, capacities on flows are not implemented in te reported experiments. Te literature review reveals tat despite te increased interest in scenario-based approaces for umanitarian relief network design, to te best of our knowledge all existing models are based on static scenarios tat overlook te dynamic-stocastic nature of multi-azard occurrences. Wen compared to generic extreme events modeling frameworks available in te literature (Grossi and Kunreuter, 2005; Banks, 2006), it appears tat te azards caracterization in current relief network design models is too simplistic. In addition, tese models account for te deployment pase only of te disaster relief cycle, terefore ignoring te particular needs of subsequent pases (sustainment, recovery and redeployment). Moreover, emergency distribution networks are often modeled as two-ecelon DC/Demand distribution systems tat ignore contracted vendors and backup sources tat are maor players in real-life umanitarian supply cains. Teir inclusion at te design level is crucial to adequately anticipate te callenging problem of efficient coordination during relief operations (Van Wassenove, 2006). Tis paper is aimed at overcoming tese drawbacks. 3. Problem Context We consider a geograpical region wic is prone to multiple natural disasters over a given discrete time orizon. Tese disasters ave different severity levels and are of different categories e.g. floods, urricanes, eartquakes, etc. Tey may strike random zones witin te region terefore resulting in stocastic demands of different types of emergency supplies (referred to as items in tis text). Some lifesaving items suc as packaged food, water, ice and tarps are required for all disasters. Tese items can be partitioned into two sets: consumable items and durable assets. Eac item type is caracterized by a target response time. Prior to a disaster onset, a umanitarian organization wic manages relief aid in te geograpical region typically preposition different items at some DCs in tat region. Te intent is to ensure an adequate and timely response. 5

7 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac To elp acieving tis goal, eac zone urisdiction also identifies a set of Points of Distribution (PODs) from were items will be directly issued to populations in need. A population-based PODs dispersion over te disaster-prone area is provided to sape adequately te demand zone caracteristics in terms of needs, distance coverage and delays. Prepositioning decisions include te number and locations of DCs needed; teir capacity and te quantity of eac item tey sould old. Tese decisions are constrained by budget limitations tat restrict pre-disaster expenses dedicated to establising DCs and olding inventories. Prepositioned items are used afterward to respond to disasters tat may unfold witin te planning orizon. During a given disaster, contractual transportation services are used to conduct daily distribution of items to affected populations for te disaster response duration. Tis period can typically be divided into four pases: deployment, sustainment, recovery and redeployment. Wile durable assets need to be delivered to PODs only during te deployment pase, consumable items must be sipped to tese PODs during te deployment, sustainment and recovery pases. Deployed reusable items are returned to te DCs during te redeployment pase. Te deployment is caracterized by a caotic setting were demand surge is ig and DCs cannot be replenised. Te duration of tis ectic status corresponds to te deployment lead time (i.e., te time it takes to restore te infrastructure needed to resupply te DCs). Quick relief aid must be provided to affected populations using existing items at DCs. Wen prepositioned supplies are depleted, demand may be satisfied from vendors tat ave been pre-selected on te basis of best value bids. In tis bidding process, vendors specify teir proposed price for eac item, as well as te maximum quantity tey can provide witin pre-establised target times. Prenegotiated contracts are ten awarded to vendors tat meet te best combination of price, availability of stock and timely delivery. A backup source may also be used as last recourse in case supplies acquired from contract vendors are not sufficient to satisfy demand in a timely manner. Tis backup source may represent federal autorities, military support, regional collaborations, etc To better cope wit ig time pressure wic is inerent to te deployment pase, a service-based obective function is needed. Te sustainment pase lasts generally longer tan te deployment and is caracterized by a stationary demand and te availability of functional DC resupply lines. Tus, vendors and te backup source are able to replenis DCs up to teir desired stock levels. During tis less caotic pase, te satisfaction of te demand at te PODs is allowed solely from te DCs and contract vendors. During te recovery pase, demand for relief efforts decreases gradually as te infrastructure and essential services at affected zones recuperate. Operations similar to tose conducted during sustainment must be performed to satisfy tis gradually decreasing demand. During bot te sustainment and recovery pases, te obective is to minimize total procurement and transportation costs. Te redeployment pase starts wen relief efforts are completed and finises wen all assets and unused items are moved back to DCs. It is wort noticing tat umanitarian organizations are generally responsible for operations in te four pases. However, since redeployment operations take place in a stable environment wit no time pressure, we do not consider tis pase explicitly. Te underlying assumption is tat relief operations during te time to recovery are more critical. On te oter and, we assume tat disasters are independent and terefore, since te deployment pase is relatively sort, it is unlikely tat two disaster deployments would ave to be made from a given DC during a deployment lead time. Hence, prepositioned quantities are sufficient to ensure an adequate responsiveness. Figure 1 illustrates te type of relief distribution network considered in tis paper. 6

8 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac In tis figure, te indexes and sets p P, i I, v V and J denote items, PODs, vendors (v = 0 denotes te backup source) and DCs, respectively. Te set K represents possible capacity configurations for DC. Te network consists of tree types of nodes (supply sources, DCs and PODs) and tree types of flows (deployment, sustainment/recovery and resupply). Backup Source (0) Vendor v = 1 v V Vendor v= V p P p P v DC k K =1 k J = J p P Compound Stocastic Hazard Process POD i =1 i = 2 i I i = I Deployment Flows Sustainment-Recovery Flows Figure 1: Te Relief Distribution Network Resupply Flows Wen a disaster its a given zone, its impact on te relief network performance depends on four vulnerability sources, namely vendors, DCs, PODs and transportation lanes. It is assumed tat te back-up source is always available. Disaster impact associated to te four vulnerability sources can be represented by stocastic perturbations tat distress te quantity of supplies tat vendors can provide, capacity levels at DCs, demand levels at PODs and/or travel times on transportation lanes. Furtermore, disaster impact on a given it zone varies witin its life cycle. Hence, during te deployment pase, te storage capacity of a it DC or vendor decreases and remains stagnant during te sustainment pase before it starts ramping up during te recovery pase. Tis could lead during te planning orizon to te temporary unavailability of a subset of DCs and vendors for a given azard. Conversely, demand at a it POD increases during te deployment stage till it reaces a maximum tat is maintained during te sustainment pase. Tis amount ten decreases during te recovery pase. On te oter and, travel time perturbations are likely to be more pronounced during te deployment pase wic is generally more caotic. Terefore, a multi-pase formulation based on stocastic pase-dependent demands, capacities and travel times udiciously represents a disaster scene. Addressing te problem of prepositioning emergency supplies accurately requires a udicious modeling of te disaster process. To tis end, te next section presents a risk modeling approac wic builds on te general framework proposed in Klibi and Martel (2012). 7

9 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 4. Modeling te Disaster Process Te proposed risk modeling approac is based on a compound stocastic process wic represents ow azards unfold randomly in time over te planning orizon, and in space over te geograpical region under consideration. Given a partition of te region into a set of zones, te proposed risk modeling approac depicts te azard process in two steps. Te first step caracterizes azard attributes including occurrence time, intensity and severity. Te second step models azard impact on response duration, items demand as well as ability of DCs and vendors to supply items needed. It is assumed tat transportation lanes are not directly disrupted and tat DCs and vendors disruption is sufficient to capture te perturbation of deliveries in terms of available inventories and transportation. Te resulting stocastic processes are used to generate a set of equiprobable scenarios using Monte-Carlo metods. In te following, te risk modeling approac and te Monte-Carlo procedure are described in details Disaster Attributes We consider a geograpical region prone to disasters in time. Tese disasters are grouped into generic meta-events, called multi-azards (Scawtorn et al., 2006), based on teir similar impact from te perspective of te relief network. Multi-azards tat treaten te region ave similar caracteristics tat depend on te region geological and meteorological features. Tese caracteristics, including intensity, severity and occurrence frequency, are determined from istorical data and expert-based catastropic risk assessments. We assume tat azards occur in te region at te beginning of discrete time periods corresponding to days, witin a long planning orizon T. Te resulting multi-azard arrival process is caracterized in terms of random inter-arrival times wit probability distribution F λ (.), were λ represents te inter-arrival time between two consecutive multi-azards. Using istorical azards data, an exponential distribution wit estimated mean λ is generally te best fit for multi-azards inter-arrival times (Banks, 2006). Additionally, te intensity β of an occurring azard caracterizes its pysical impact. Hazard intensity β is a random variable wic follows a given probability distribution F β (. ). Moreover, azard severity can be measured by te magnitude of relief interventions tat azard triggers. Te extent of tese interventions in terms of emergency items needed and response duration depends on te azard pysical impact. Tus azard severity is a random variable correlated to β. Toug azard presents a treat to te entire region, its severity and impact can vary from one part of te region to anoter. To capture tis variability, te region is partitioned into a set of zones z Z. We assume tat zones z Z uniformly inerit te region azard intensity wile azard severity is generally zone-dependent. Accordingly, azard severity at zone z is a random variable denoted ρ z wic follows a given probability distribution F ρz (β )(. ). In tis paper, azard severity ρ z is modeled as a fraction used to determine te number of abitants wo are likely to stay in zone z following azard (e.g. elderly people, omeless, financially callenged families/individuals, etc ). Finally, to account for future environmental canges, a set of probabilistic evolutionary trends κ K is used. Eac trend κ occurs wit a given probability p κ, κ K. We assume tat tese trends ave an impact on te mean time between multi-azards, but not on teir severity. Terefore, under evolutionary trend κ, a function g κ (. ) is used to superimpose a time pattern on te istorical mean time between azards λ estimated at te beginning of te planning orizon T. 8

10 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Tus, under trend κ, te time between a azard occurring at te beginning of period t T and a subsequent one is an exponentially distributed random variable wit mean λ tκ = g κ (λ, t) Disaster Impact Wen a multi-azard strikes at te beginning of time period t T, its impact on te region is zone-dependent. Tus, a compound propagation process is used to determine te subset Z Zof zones impacted by azard, and its centroid zone, i.e. te main affected zone. Te probability tat zone z is te multi-azard centroid zone wen tere is a it is denoted π z, z Z. Since multi-azard is likely to propagate to oter zones in te vicinity of its centroid, a set of adacent zones z Z are added to Z using conditional propagation probabilities denoted α, zz, Z. Probabilities π z' z z, z Z and α, zz, Zdepend on te granularity of te zones z' z defined. Tey are estimated using azard frequencies derived from istorical data tat are collected over a time interval. Tese probabilities are calculated as follows: n π z z =, z Z n z Z z nz' z and α =, zz, Z z z n, ' z were n z denotes te number of azards recorded for zone z over te time interval considered, and n z z is te number of times zone z was affected wen zone z was it. Multi-azard impacts affect zones z Z as follows: A relief response is needed in all zones z Z for te incident duration. Demand of different items occurs in eac zone z Z. DCs located in zones z Z become unavailable. Supplies become inaccessible from vendors located in zones z Z. Wen a multi-azard strikes, all zones z Z are assumed to ave te same time to recovery θ. Te latter is a random variable correlated to azard intensity β, terefore following a probability distribution F θ β (. ). Since prepositioning of emergency supplies deals wit strategic planning rater tan daily operations, te time to recovery θ is partitioned into tree maor pases: deployment (denoted D), sustainment (denoted S) and recovery (denoted R). Te intent is to capture te particular needs of eac pase wile avoiding operational details tat migt be useless given te igly dynamic-stocastic nature of a disaster scene. Te tree pases were discussed in details in section 3. Te duration of eac pase is denoted by θ τ, τ = D, S, R. Te deployment pase duration θ D is generally sort and is assumed to be fixed for all azards. Tus, it will ereafter be denoted θ D. Wen te time to recovery is longer tan te deployment period, a sustainment pase is subsequently observed, followed by a recovery pase. Te latter is as- ϕ 0,1 of te sustainment duration. Tus, for a given multi-azard, sumed to last a fraction [ ] te sustainment and recovery pase durations are given by: θ S = 1 1+φ max(0, θ θ D ) and θ R = φθ S On te oter and, wen a multi-azard its te region under consideration, a stocastic time and zone dependent demand for items pεp arises in affected zones z Z τ. Let d zpt be te demand for item pεp in zone z Z for period t witin pase τ. 9

11 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Tis demand depends on azard severity ρ z, te population size of zone z wic is known, and te daily need per abitant for product p in period t of pase τ. Te latter is a random variable τ denoted ξ pt wic follows a given probability distribution F ξ pt (. ). To ensure a timely relief, eac zone z Z is supported by a set of PODs i Iz from were emergency items are directly issued to te population. Te zone portion covered by eac POD is predetermined by experts based on zip code aggregation or population density, and te population ν i served by POD i Iz is tus defined. Consequently, for a given multi-azard, te demand of POD i for product p in period t witin pase τ is a random variable, given by d τ and is correlated to random variable ρ zi () wit te following relation d τ ipt zi () v τ i pt ipt = ρ ξ, were z(i) denotes te zone containing POD i. Daily demand during te deployment pase is erratic and terefore, it is ard to estimate its probability distribution. Tus, te deployment pase is considered as a single period of lengt θ D for wic total demand of item pεp at POD i is estimated as follows: d = θ ρ v ξ D D ip zi () i p Were ξ p is an estimate of average daily need per abitant for product pεp. Conversely, daily demand for consumable items during te sustainment pase is fairly stable. S ξ is a stationary stocastic process Let P C be te subset of consumable items in P. Wen { pt} wit an estimated mean ξ p, {d S ipt }, p P C, is also a stationary process. Terefore it is reasonable to assume tat for every day t witin pase S, d S S ipt follows a Normal distribution wit mean μ ip and standard deviation σ S ip given by: μ S ip = ρ z(i) ν i ξ p, σ S S ip = CV p μ ip were CV p is te coefficient of variation of te demand for product p P C. Furtermore, daily demand for relief items decreases during te recovery pase R as te infrastructure at affected zones gets restored. To reflect tis trend, given azard starting date η, te average of te Normal demand of product p P C for day t of te recovery pase is calculated as follows: R μ ipt = μ S S t ip μ ip η + θ, i I z, p P C Hence, starting from a fraction of te sustainment average daily demand μ S ip at te beginning of te recovery pase, μ R ipt decreases till it reaces zero at te end of incident duration θ. Te standard deviation is given by σ R ip = CV p μ R ip, as for te sustainment pase. Figure 2 illustrates te resulting stocastic processes wic describes demand patterns for a given item p trougout te planning orizon. 10

12 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Location and inventory prepositioning decisions T Planning orizon Zone l Zone 2 POD 1. POD I z POD 1. Demand level / item p D dip ( ω) S dipt ( ω) R dipt ( ω). Zone Z POD I z Support Operations after disaster Deployment θ D Sustainment θ S t T Recovery R θ Figure 2- Demand Pattern for an Item p During te Planning Horizon Finally, during te incident duration θ, te set J z of DCs and te set V z of vendors located in an affected zone z Z could be it by azard and terefore become unavailable. Let z() and z(v) denote te azard zone enclosing DC and vendor v, respectively. Conditional attenuation probabilities α z() and α v z(v) are used to determine impacted DCs J and vendors V for a given azard witin te affected zones set Z =, respectively. Tese probabilities depend on te granularity of te zones defined and can be estimated using istorical data and experts assessment of facility exposure. Hereafter, J and V respectively denote te set of DCs and dors tat are available during azard. Given te set V p of vendors tat can provide item p P, te subset V p of available vendors for item p is derived as follows: V p =V V p. We recall tat disasters are assumed to be independent and terefore DC capacities and available quantities at vendors are fully restored before anoter multi-azard its Monte-Carlo Procedure Te Monte-Carlo procedure used to generate a disaster scenario ω is summarized in Figure 3. In tis procedure, u denotes a uniformly distributed pseudo random number generated in te interval [0, 1], and Φ 1 (. ) te inverse of te standardized Normal distribution. After selecting an evolutionary pat, te procedure creates a cronological azard list H ( ω). For eac multi-azard H ( ω), te starting date η and te intensity β are computed. Ten, te procedure generates multi-azard response duration θ along wit pase durations θ τ, τ = S, R. Subsequently, te set Z of affected zones is determined and for eac zone z Z azard severity ρ z is generated. Afterwards, for eac pre-identified POD i I z and for eac item p P, te procedure generates te total deployment demand d D ip. It also produces te daily demands (d τ ipt ) τ=s,r for consumable items during te sustainment and recovery pases. Ten, te sets of available DCs and vendors are identified. Finally, aggregated demands (d τ ip ) τ=s,r over tese two pases are derived and added to obtain total demand over te combined sustainment-recovery (SR) pase. 11

13 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 1) Select an evolutionary pat κ randomly using p κ, κ K 2) Generate a cronological azard list H ( ω) : H ( ω ) = ; t = 0 Wile t T, do: 2.1) Calculate λ tκ = g κ (λ, t) and compute te next azard arrival time t = t + F 1 λ (u), ten set η = t 2.2) Compute te azard intensity β = 1 () u 2.3) Insert azard wit attributes (, ) F β η β cronologically into te list H ( ω) End Wile 3) Generate azards severity and impacts: For all in H ( ω), do: 3.1) Generate azard response duration and derive pase durations: 1 θ = F u and calculate θ τ, τ = S, R given θ D Compute ( ) θ( β ) 3.2) Generate azard locations, azard severity at tese locations and demands at teir PODs: Z =, J = and V = i) Select te azard centroid zone and update Z = Z { z} z = z π < u π, z Z k k z < k+ 1 z k ii) Construct te azard zone propagation set: For all z Z / { z} do: If ( u α z ' z Z = Z z iii) For all z ) ten { } 1 Z do: generate ρ = ( ) z F u ρz ( β ) iv) Generate POD s daily deployment demand: For all z Z, i I z, p P do: d D ip = θ D ρ z(i) ν i ξ p v) Generate POD s daily sustainment and recovery demand: C For all z Z, i I z, p P do: a) Compute μ S ip = ν i ρ z(i) ξ S S p, σ ip = CVpµ ip and generate te daily sustainment demands: d S ipt = μ S ip + Φ 1 (u) σ S ip, t = η + θ D,, η + θ D + θ S b) Compute μ R ipt = μ S S t ip μ ip, σ R R η +θ ipt = CV p μ ipt and generate te daily recovery demands: d R ipt = μ R ip + Φ 1 (u) σ R ip, End For vi) Identify available DCs and vendors: For all z Z, J z : If ( u α z ( ) For all z Z, v V z : If ( u α vz ( v ) End For t = η + θ D + θ S,, η + θ ) ten J = J { } ) ten V = V { v} 12

14 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 4) Aggregate demands over pases S and R: For all p P C, i I z, z Z, H( ω) d SR S ip = η +θ D t η +θ D +θ S R 1 d ipt + η +θ D +θ S t η +θ 1 d ipt End For Figure 3- Monte Carlo Procedure for Scenario ω 5. Modeling te Prepositioning Problem Prior to a azard onset, emergency items must be prepositioned at some DCs to ensure an adequate and timely response. Prepositioning decisions include te number and locations of DCs needed; teir capacity and te quantity of eac emergency item tey will old. DC locations are cosen among a set J of pre-selected sites. Sites pre-selection is based on several criteria including political priorities, risk levels, proximity of igways, etc Moreover, a set of feasible configurations K is associated to eac DC and eac configuration k K is caracterized by a fixed usage cost f k and a capacity a k (expressed in available space volume). A parameter o p is used to convert te available quantity of item p into required space. Prepositioning decisions are constrained by a budget limit b wic restricts pre-disaster expenses dedicated to establising DCs and olding inventories. A olding cost e p is incurred for eac unit of item p eld in inventory at DC. It is wort noticing tat te budget limit b as well as te costs f k and e p are assumed to cover te entire planning orizon. Prepositioning emergency supplies is a strategic decision aimed at designing te relief distribution network before a potential azard strikes. Te designed distribution network is used later to respond to azards tat may unfold witin te planning orizon. Tis involves conducting daily operations aimed at delivering emergency supplies to affected populations and resupplying DCs for azard durations. Hence, design decisions ave a significant impact on daily relief operations. Terefore, anticipating te impact of prepositioning decisions on emergency supplies daily distribution is essential to te enancement of relief efforts. Tis anticipation represents operational decisions tat optimize an estimate of some operational obective function under te approximate operational information available wen design decisions are made. Tis leads to a twostage stocastic programming model were te first stage addresses strategic design decisions wile te second stage tackles operational decisions triggered by a azard occurring witin te planning orizon. Te proposed model is presented below. First Stage (strategic decisions) As sown in section 4, azards unfold over te planning orizon according to complex compound stocastic processes. Tus, a set Ω= { ω} of probabilistic scenarios is used to compute total expected (anticipated) second stage value over te planning orizon. 13

15 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac ( ) Tis expected value, denoted E V( X,Q) ( X,Q ) made, X and Q being vectors of te decision variables X k = 1, if DC is opened wit configuration k K 0, oterwise (i. e., DC is closed) Q p : Quantity of item p to be eld at DC, also depend on te strategic prepositioning decisions and it is given by te expression E ( V( X,Q) ) = p ( ω) V (, ω) ω Ω X,Q, were p ( ω ) is te probability of occurrence of scenario ω, and V ( X,Q,ω) is te optimal second stage value under scenario ω and prepositioning decisions ( X,Q ). Since a scenario ω Ω is a set of plausible H ω =, V ( X,Q, ω) V ( X,Q, ) azards ( ) = stage value under azard ( ω) ( V ( X *,Q * )) H ( ω) = H =and decisions ( ) E is obtained by solving te following matematical program: E * * ( V ( X,Q )) = Min XQ E( V ( X,Q) ) = = p( ω) V ( X,Q, ) subect to f k X k + epqp b J k K p P oq ax p P p p k k k K, ω Ω H ( ω), were V ( X,Q,) is te optimal second X,Q. Consequently, te optimal value J (1) (2) (3) X k 1 J (4) k K X k Q p { 0,1} 0 J, k K J, p P Te obective function (1) minimizes total expected (anticipated) second stage obective function value over te planning orizon. Constraint (2) guarantees tat total fixed costs for establising DCs and olding inventories do not exceed te budget. Constraints (3) require tat inventory can be eld at a DC only if it is opened, in wic case total inventory eld cannot not exceed its capacity. Constraints (4) ensure tat at most one configuration from set K is cosen for DC. Constraints (5) are te binary constraints for location variables. Finally, constraints (6) are te non-negativity constraints for te quantity of eac item eld in eac DC. Second Stage (Anticipated Operational Decisions) A possible anticipation V ( X,Q,ω) of te effect of prepositioning decisions over daily delivery operations can be obtained using an estimate unit cost of transporting an item from DCs to PODs, instead of actual routing costs. Moreover, as discussed in te previous section, azard response duration can also be aggregated into its two maor pases τ = D, SR. (5) (6) 14

16 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Tus, V (, ) V τ X,Q ( X,Q, ) = ω H ( ω) = τ = D, SR, were V τ (X, Q, ) is te optimal value of second stage obective function during pase τ of a azard H ( ω) =. Using DC opening decisions in X te set of available DCs under azard is given by J ( X ). Furtermore, eac azard τ H ( ω) =is caracterized by te set of demands d, i I, z Z, p P, τ = D, SR. During azard time to recovery, demand for item p P may be satisfied from te following sources: 1- Quantities Q, J ( X ), p P prepositioned at opened DCs J ( X ) p 2- A set of preselected vendors V p V 3- A backup source v = 0 used as last recourse in case supplies available from DCs and vendors are not sufficient to ensure a timely response. Hence, te following decision variables are required to formulate te second stage program: Y τ pnn' : quantity of item p delivered from origin n (0, v or ) to destination n (i or ) during pase τ = D, SR of azard. i) Deployment pase To capture te ig time pressure inerent to te deployment pase, a service-based obective function is used. Te intent is to speed up te delivery of eac item to demand points by rewarding te proximity of DCs to PODs, using a set of coverage levels. For eac POD several coverage levels are defined based on te distance between te POD and te DCs and a reward (penalty) is offered to te nearest (fartest) levels. During te deployment pase, demand for item p P may be satisfied wit different coverage levels l Lp. For a multi-azard, let J li ( X ): Set of DCs tat can provide coverage level l Lp to POD i I = I z, i.e. set of lip DCs tat are witin a distance l i ip r from POD i ( ( ) J X = J ( X ) ). z i l li V : Set of vendors tat can deliver item p to POD i witin coverage level l ( V I : Set of PODs tat can be covered from DC ( I vp I ). I : Set of PODs tat can be covered from vendor v for item p ( I S ; Maximum quantity of item p tat can be provided by vendor v. vp Eac coverage level l for item p is associated wit penalties J ulp and vp I ). z Z = V ). ip l lip V u lp incurred wen item p is 0 delivered from a DC or a vendor, respectively. A large penalty u p is also incurred wen demand for item p is satisfied from te backup source. Tese penalties are proportional to te distance between te network nodes in order to underline te criticality of additional delays to deliver emergency items. 15

17 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac In order to encourage deliveries witin lower coverage levels and to ensure tat demand is satisfied from vendors only wen prepositioned supplies are depleted, penalties are specified so J J J tat 0 = u 1p < u 2p < < u Lp p < u V 1p < u V V 2p < < u Lp p < u 0 p, p P. Using tis notation, te second stage program for te deployment pase of disaster is: V D (X, Q, ) = Min p P u lp J D J Y li (X) pi + u V D lp v V Y lip pvi + u 0 D i I l L p p Y p0i (7) subect to D Ypi Qp J ( X ), p P (8) i I i Ivp Y D pvi Ji ( X) S vp Y + Y + Y = d D D D D pi pvi p0i ip v Vip Y D pi, Y D pvi, Y D 0 p0i J ( X ), v v i V, V, p P I, p P i (9) (10) I, p P (11) Obective function (7) minimizes te total penalty associated to satisfying POD demands witin iger coverage levels. It also ensures tat demand is satisfied from available vendors only wen prepositioned supplies are depleted. In addition, te penalties values structure used in te obective function guarantees tat te backup source is used only as a last recourse wit a very large penalty. Constraints (8) ensure tat prepositioned quantities of items at eac available DC are not exceeded. Teir slack variables provide te inventory of item p left in DC at te end of te deployment pase. Constraints (9) guarantee tat maximum vendor supply quantities are not exceeded. Constraints (10) ensure tat te deployment demand is satisfied, eiter from te DCs, te vendors or te backup source. Finally, (11) are non-negativity constraints for te deployment decision variables. ii) Combined sustainment-recovery pases C Recall tat during te sustainment-recovery pase, consumable items p P only need to be sipped to PODs. During tis pase, consumable products are mainly sipped from opened DCs, and quantities sipped must be resupplied from te vendors or te backup source. In oter words, during tese pases, we must ave flow equilibrium at te DCs. Moreover, it is assumed tat te sipping capacity of available DCs is sufficient to support daily trougputs. Furtermore, te start of te sustainment-recovery pases marks te end of te caotic deployment pase and tus te urgent nature of deliveries. In tis context, a cost-based obective function is more suitable tan a service-based one for tese pases. More specifically, te obective for tese pases is to minimize total operating costs, wic include te procurement of items from te backup source and te vendors, as well as te transportation of tese items to te DCs and to te PODs. A unit transportation cost c pi is incurred wen an item p is delivered from DC to POD i. A procurement and transportation cost g pvi is incurred wen an item p is delivered directly from vendor v to POD i. Similarly, a cost g pv is carged wen an item p is purcased and transported from source v (vendors and backup) to DC. 16

18 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Te resulting second-stage program for te sustainment-recovery pases under azard, is written as follows: SR SR SR SR SR V ( XQ,, ) = Min cpiypi + gpviypvi + gpvypv + gp0 Y p0 C (12) p P i I J ( X) v V J ( X) v V subect to J ( X) Y + Y = d SR SR SR pi pvi ip v V p τ SR SR pi pv p0 τ = D, SR i I v V p i I, p C P (13) C Y = Y + Y J ( X ), p P (14) Y SR pi, Y SR pvi, Y SR pv, Y SR p0, Y D 0 pi J ( X ), v V, i I, p C P (15) In tis model, te obective function (12) minimizes total transportation and procurement costs. Constraints (13) ensure tat POD demand of eac consumable item is fully satisfied from available DCs and vendors. Constraints(14) 3 ensure tat at te end of te recovery period, te prepositioned quantities of consumable items used during te tree pases ave been resupplied for all DCs. It is assumed tat all durable assets are reusable and terefore tere is no need to acquire tem at te end of te azard. Finally, constraints (15) are te required non-negativity constraints. It is wort noticing tat models (7)-(11) and (12)-(15) are not separable due to constraints set (14). Integrated Stocastic Programming Formulation Te complete stocastic program to solve is obtained by integrating te previous secondstage deployment and sustainment-recovery models wit te first-stage program, for all azards in all scenarios. To allow adusting te importance of ig coverage versus low operating costs, te obective function is expressed as a weigted sum of tese two criteria. Te weigt associated to te coverage attribute is denoted by γ [0,1]. Denoting te set of all potential disasters by H = ω Ω H ( ω), te program to solve is te following: Min p(ω) ω Ω subect to H ( ω) γ u J D lp Y pi + u V D lp Y pvi + u 0 D p Y p0i + J li v V i I p P l L p lip SR SR SR (1 γ) c pi Y pi + g pvi Y pvi + g pv Y pv + g p0 Y p0 p P C i I J v V J v V SR (16) 3 Tis constraint is a transformation of te following inventory accounting relationsip: D SR SR SR Q = ( Q Y ) + ( Y + Y ) Y p p 0 i I pi v V pv p pi p i I 17

19 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac X f X + e Q b 1 k k p p J k K p P k K p P k p p k k k K (17) J (18) oq ax J (19) Y + Y + Y = d i I, p P, H (20) Ji D D D D pi pvi p0i ip v Vip Y + Y = d C, p P, H (21) SR SR SR pi pvi ip J v V p i I D Ypi Qp i I J D Ypvi Svp i Ivp v V Y = Y + Y J, p τ SR SR pi pv p0 τ = D, SR i I v V p SR Ypi M X c p P i I k { 0,1} k K k, p P, H (22), p P, H (23) C P, H (24) J, H (25) X, Q 0 J, k K, p P (26) p Y SR pi, Y SR pvi 0 Y p0i J, v V,i I C, p P, H (27) Y D pi, Y D pvi, 0 J, v V, i I, p P, H (28) D C 0 i I, p P, H (29) Y, Y 0 SR SR pv p0 J, v V, p C P, H (30) Note tat in tis formulation, te sets J and teir subsets are not dependent on X as in te second-stage models. Tis is because te aim of tis integrated program is to find te optimal value of (X,Q) and terefore X is not known beforeand. Tis is also wy constraints (26) must be added to make sure tat, at te operational level, DCs are used only if tey are opened. Te scalar M in tis constraint is an arbitrarily large number. 6. Solution Approac A maor callenge in solving te matematical program formulated in te previous section comes from te cardinality of te set of plausible scenarios Ω. Wen continuous probability distributions are used (e.g. Normal item demands and exponential azard inter-arrival times), tere is an infinite number of possible scenarios. Tis difficulty can be overcomed by using te Sample Average Approximation (SAA) metod proposed to solve stocastic programs (Sapiro, 2003). In addition, a multicriteria evaluation approac can be used to select te best design among te set of alternative designs produced by te SAA metod. Te scenario-based solution approac proposed to produce and evaluate alternative designs is summarized in Figure 4. It involves tree pases: 1) azards scenarios generation, 2) design generation, and 3) multicriteria evaluation and selection. 18

20 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Designs Generation (using CPLEX-12) n SAA ( Ω m ) : Min (31) s.t (17)-(30) 2 Small samples replications Ω n m, m= 1,..., M Candidate Designs : DC locations Inventory levels ( X, Q ), m= m m 1,..., M ' ( M ' M ) Hazards Scenarios Generation 1 Best Design * * ( X, Q ) Multicriteria Evaluation and Selection Evaluation using models (7)-(11) and (12)-(15) to get: n' ( X,Q ω) τ = = ω Ω τ V m m,, D, SR, m 1,..., M ', Performance measures computation: τ τ ( ( X,Q )) D ( X,Q ) Multicriteria selection ( ) E V m m, V m m, τ = D, SR, m = 1,..., M ' 3 Large evaluation sample n' Ω Figure 4- Solution Approac Te scenario generation pase uses te MonteCarlo procedure presented in Figure 3 to provide disaster scenarios to te design generation and evaluation pases. M independent small n Monte Carlo samples of n scenarios Ω m, m = 1,..., M, are produced for te design generation pase. A larger sample of scenarios Ω n', n' >> n is generated for te design evaluation pase. Te design generation pase uses several samples of scenarios to approximate te proposed matematical program (16)-(30) by SAA models. Tis is acieved by replacing te set of scenarios Ω wit a sample Ω Ω of n scenarios, and approximating te obective function (16) by n (31), to get te following SAA model: Min 1 n n ω Ω H ( ω) γ u J D lp Y pi + u V D lp Y pvi + u 0 D p Y p0i + J li v V i I p P l L p lip SR SR SR (1 γ) c pi Y pi + g pvi Y pvi + g pv Y pv + g p0 Y p0 p P C i I J v V J v V SR (31) n subect to constraints (17)-(30), wit H replaced by H = n ( ω) ω Ω H. As typically done wit tis approac, te SAA models are used to generate several designs (i.e., prepositioning decisions ( XQ, )) using M scenario sample replications. A commercial solver suc as CPLEX is ten employed to solve te SAA model for te M replications to obtain distinct designs ( X, Q ), m= m m 1,..., M ' ( M ' M ). Te design evaluation and selection pase is ten required to evaluate tese designs wit a set of performance measures, and to select te best one. M ' alternative 19

21 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Using second stage models (7)-(11) and (12)-(15), eac design ( Xm, Qm), m= 1,..., M ' is assessed n' independently for eac scenario ω Ω, by computing te deployment and sustainment-recovery τ cost obective function values ( X,Q ) V m m, ω, τ = D, SR. Using tese values a set of performance measures based on sample average and deviation operators can be derived. In our context, for eac design ( Xm, Q m), te evaluation is based first on te two following expected return measures: τ 1 τ E( V ( X m,qm) ) = n ' V ( X m,qm, ω), τ = D, SR ω Ω n' A coerent risk measure (Sapiro, 2008) is also used in te evaluation pase. Since downside deviations from mean values are undesirable, eac design ( Xm, Q m) is assessed wit te following mean semi-deviation measures: τ 1 τ τ D( V ( X m,qm) ) = max n' ( V ( X m,qm, ω) E ( V ( X m,qm) )); 0, τ = D,SR ω Ω n' (33) Given tese evaluations, te selection of te best design among te candidates generated can be done using any multicriteria decision-making metod. It is wort noticing tat te parameters n, n and M need to be carefully adusted as tey greatly impact te solution robustness and quality. Teir calibration is based partly on te solvability of te resulting models, and on estimated statistical optimality gaps (Sapiro, 2003). 7. Computational Results Te experiments reported in tis section are conducted on a test case inspired from realworld data obtained from te Nort Carolina Emergency Management Division (NCEM) and te Federal Emergency Management Agency (FEMA). NCEM actually operates two permanent DCs. In addition, FEMA Incident Support Base at Fort Bragg provides backup assistance in case of maor disasters. Our goal is to caracterize future disasters treatening te region and identify potential new relief network designs to adequately support emergency operations. In te following, te test case is first described. Ten, numerical results are reported and discussed. 7.1 Test Case Te state of Nort Carolina is partitioned into 101 zones tat correspond to counties, eac supported by a predetermined set of PODs. Te number of PODs per county ranges from 1 to 10 leading to a total of 700 PODs, eac wit a population varying between 5,000 and 20,000 abitants. Tree item families are considered: 1) a typical durable asset, i.e., selter tents (p = 1) and two types of consumable items, namely, medical kits (p = 2) and a generic item family composed of water and Meals-Ready-to-Eat (MRE) (p = 3). Tese items can be supplied by 14 potential vendors including te backup source at Fort Bragg. Ten of tese vendors are located outside te state and terefore are not prone to te disaster treats under consideration. Wile te backup source carries all items, eigt vendors supply water and MRE, two provide medical kits and tree oters provide durable assets. Table 1 sows, for eac item, te number of available vendors, teir capacity range during te deployment pase (in number of pallets), and teir unit price range (in dollars per pallet). (32) 20

22 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Te distribution network also includes 10 potential DCs, eac wit two capacity configuration options (k =1, 2) also sown in Table 1, along wit teir associated fixed opening/usage costs range and inventory olding costs. Te latter are expressed as a percentage of items unit price. Te table also provide inbound and outbound transportation costs. DCs k=1 (small) k=2 (large) Fixed costs ($) [90K, 148K] [151K, 232K] Storage capacity (pallets) [770, 1190] [1210, 1860] Inventory olding cost 5% Vendors + Backup source Durable Assets Medical Kits Water & MRE Number Deployment Capacity (pallet) [400, 5000] [700, 1000] [100, 1600] Prices ($/pallet) [1600, 2500] [1250, 3200] [1500, 3800] Transportation Inbound unit cost ($) / Pallet-mile Outbound unit cost ($) / Pallet-mile Coverage levels Level 1 Level 2 Level 3 Maximum Distance (miles) Table 1-Network Data Furtermore, it is assumed tat a total budget of 1M$ is allocated to preparedness over 3 years. It is also assumed tat during te deployment pase, demand at PODs may be satisfied wit 3 different coverage levels: 12, 24 and 48 ours, corresponding to a maximum distance of 200, 400 and 800 miles, respectively. Recall tat, for eac item p, penalty costs are set so J J J tat 0 = u 1p < u 2p < u 3p < u V 1p < u V 2p < u V 0 3p < u p to facilitate delivery witin lower coverage levels (ideally level 1), and to ensure tat demand is satisfied from vendors only wen prepositioned supplies are depleted, te backup source being used as last recourse. In addition, tese penalties are set to account for te degree of item s urgency. It is assumed tat item 2 (medical kits) is more critical, followed by item 1 and ten item 3. Tus, te penalties associated to a coverage level l were obtained by multiplying te truckload (TL) transportation rate associated to te level s maximum distance by two coefficients: δ, te degree of urgency of item p, and δ te priority weigt of supply source J V {0}. Te values of u p u δ p were fixed to 1.5, 2 and 1 for s items 1, 2 and 3, respectively. Te values of δ were fixed to 1, 5 and 10 for DCs, vendors and te backup source, respectively. Te TL transportation rate was estimated to $5.50 per mile, based on trucks capacity. To estimate disaster stocastic processes, we used istorical azards data spanning te years In te following, we sow ow te Monte Carlo procedure in Figure 3 was used to generate plausible future scenarios. In te derived azard arrival process, events occur dynamically over a tree year planning orizon. Te time between two consecutive disasters, estimated at te beginning of te planning orizon, follows an exponential distribution wit a istorical mean of 434 days. To account for future environmental canges, tree equiprobable evolutionary trends κ K ={, 123, } were used. s 21

23 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Te first trend is optimistic: fewer disasters are observed compared to te istorical trend. Te second one is pessimistic and more disasters are observed tan in te past. Finally, te tird trend corresponds to an as-is situation. Under a given evolutionary trend κ, te following linear function is used to derive te mean inter-arrival times in period t: λ = λ(1 + ˆ δ t), were te slope δ is fixed at ˆκ , and , respectively, for κ = 1, 2 and 3. Additionally, since natural disasters are commonly associated wit five levels ranging from very low to very ig, te intensity β of an occurring azard was assumed to follows a discrete Uniform distribution defined on te interval [1, 5]. for a given azard intensity ˆβ, θ( β) and ( ) Since time to recovery θ and severity ρ are random variables correlated to azard intensity, ˆ ρ ˆ β are random variables wit probability distributions specified in Table 2. In tis table, estimates for te parameters associated to time to recovery are given in days and tose associated to severity are given as percentages of impacted zone populations. Tese estimates are based on istorical data and on our discussions wit NCEM personnel. Similarly, te deployment pase lasts 3 days, wile te recovery duration is estimated to account for 20% of its sustainment counterpart. Collected data are also used to estimate te average daily needs for eac product type: a person needs two meals and one gallon of water per day (p = 3); a medical kit (p = 2) satisfies te weekly demand of 4 persons; and 8 persons use a unit of asset (p = 1). Te coefficients of variation for te 3 items are 0.2, 0.2 and 0.1, respectively. Demand for eac item is converted to pallet units and a Log-normal distribution is used to generate te daily demand for items during te sustainment pase. Random Variable β θ ρ Distribution Discrete Uniform Continuous Uniform Continuous Uniform Parameters Intensity level Probability Minimum (days) Maximum (days) tκ Minimum (%) κ Maximum (%) 1) Very low ) Low ) Medium ) Hig ) Very ig Table 2: Specifications of Probability Distributions Used in te Monte Carlo Procedure. z Finally, te data available was used to determine te frequency of disasters in counties and of simultaneous its in nearby counties. Tese frequencies were ten used to estimate centroid zone attenuation probabilities ( π ) and conditional propagation probabilities ( α ). In our experiments, π z ranges from to wile α z z as an average of 0.36 and a coefficient of variation of 0.7. Additionally, 41% of te probabilities α are smaller tan 0.25 wereas 11% are z' z larger tan z' z 22

24 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Conversely, estimating DCs and vendors attenuation probabilities ( α and α ) is callenging due to te lack of istorical data. Hence, in our tests for a given facility (DC or z( ) vz( v) vendor), α equals 0 wen β 2, 0.5 wen β = 3 and 1 wen β 4. z( ) 7.2 Nort Carolina Risk Analysis Tis section examines te risk exposure of Nort Carolina and te extent of relief operations needed in te event of a disaster. Figure 5 illustrates te distribution of te number of azards occurring in te state during a 3 year orizon, for a sample of 1,000 scenarios. Te scenarios are partitioned into 3 categories according to te number of its occurring witin te orizon: mild scenarios if at most one it is observed, serious concern scenarios wen te number of its is between 2 and 4, and worst case scenarios oterwise. Wile Figure 5 sows a very low probability of worst case scenarios, it also sows a relatively low probability of mild scenarios (i.e., 0.3) compared to its serious concern counterpart (i.e., 0.6). Tis indicates tat te region is undeniably exposed to azards. Te left plot in Figure 6 displays te distribution of te number of affected PODs during a scenario and te rigt plot te distribution of affected population. Te figure sows tat tere is a ig probability (i.e., 0.8) of aving more tan 200 affected PODs, and tat more tan 400,000 people are involved in 50% of disasters. 0,25 0,2 Mild scenarios Serious concern scenarios Probability 0,15 0,1 0,05 Worst cases Number of azards during te planning orizon Figure 5- Distribution of te Number of Hazards under a Scenario Figure 6- Distributions of te Number of Affected PODs and Population Involved. 23

25 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Figure 7 illustrates te distribution of water and MRE demand during te deployment pase. It sows tat wile demand is most likely to range between 4,000 and 6,000 pallets, several severe azards require up to 30,000 pallets. Similarly, te rigt plot in Figure 8 depicts te distribution of daily water and MRE demand during te sustainment-recovery pase. For trucks wit a 20 pallets capacity, in 50% of cases more tan 100 truckloads must be delivered during a day. Te left plot of Figure 8 displays te distribution of te sustainment-recovery pase duration. Given tat te deployment pase lasts 3 days, tis figure sows tat te time to recovery can reac 6 weeks. It is over 2 weeks in 20% of disasters and below one week in 50% of cases. Te latter is due to te low probability of a ig intensity azard in te region (see Table 2). 0,25 0,2 Probability 0,15 0,1 0,05 0 Deployment Demand for Water and Meals (in pallets) Figure 7- Distribution of Water and MRE Demand during te Deployment Pase Probability 0,6 0,5 0,4 0,3 0,2 0,1 0 [0,7) [7,14) [14,21) [21,28) [28,35) Sustainement-Recovery Pase Duration (in Days) Probability Figure 8-Distributions of Water-MRE Daily Demand and Sustainment-Recovery Duration Tese statistics reveal te extent and complexity of relief operations in terms of magnitude of needs, duration and scope. Moreover, te availability of supply sources is an important issue in relief operations. Figure 9 illustrates te distribution of te number of unavailable DCs/vendors per azard. It sows a non-negligible probability of a azard itting 2 or more DCs/Vendors, wic aggravates relief distribution operations , , , , , , , , , [0,2.5K) [2.5K,5K) [5K,7.5K) [7.5K,10K) [10K,12.5K) [12.5K,15K) Sustainment-Recovery Demand for Water and Meals (in pallets) 24

26 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac 0,9 0,8 0,7 Mild scenarios Vendors DCs Probability 0,6 0,5 0,4 0,3 Serious concern scenarios 0,2 0,1 0 Worst cases Number of unavailable depots/vendors Figure 9- Distribution of te Number of Unavailable Supply Sources 7.3 Network Design Solution Analysis Te aim of tis section is to sow ow to select te best design for te case studied, based on te solution approac (Figure 4) described in section 6. Tis involves te sampling of several small subsets of scenarios, te generation of te corresponding SAA models and teir resolution using CPLEX, and te computation of a set of evaluation measures in order to compare tem. First, preliminary tests are reported on te SAA models solvability and on te quality of te designs obtained. Next, te set of alternative designs produced using M scenario samples are inspected in terms of DC location, capacity and inventory level decisions. To illustrate te key caracteristics of te relief network structure obtained, a typical design is analysed in dept. Subsequently, a multi-criteria evaluation of te alternative designs produced is presented, and teir performance is discussed. Finally, sensitivity analysis is performed to investigate te impact of varying coverage weigts and te prepositioning budget on te solutions produced. Te SAA models were solved using OPL-CPLEX 12.3, and te experiments reported were performed on a 64-bit server wit a 2.5 GHz Intel XEON processor and 16 GB of RAM. Calibration Tests A significant effort as been made to calibrate te size of te scenario samples used in order to guarantee te solvability of te resulting SAA models and te quality of te designs produced. To determine te best value of n, sample sizes of 30, 50 and 60 were tested 4, eac wit 10 replications. Only SAA models wit n = 30 could be solved to optimality, wereas larger models do not reac te default CPLEX MIP tolerance in a reasonable time. A closer look revealed tat, for tese models, wen approacing optimality, te value of second-stage recourse variables only are modified, and prepositioning decisions remain te same. For tis reason a MIP Relative Tolerance of was used to solve models wit n = 50, 60. Wen observing te variability of te value of obective function (16) between te 10 instances (measured as (max value min value)/average value), it reaces 66% for n = 30, but falls down to about 37% for samples wit 50 and 60 scenarios. 4 Larger scenario samples provide models wit more tan 10 million decision variables and 250,000 constraints wic cannot be solved using CPLEX. 25

27 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Tis could mean tat te value of stocastic solutions (Birge and Louveaux, 2011) for our problem is relatively ig, wic ustify te use of a ig number of sample replications. Te SAA models obtained wit n = 60 are muc more difficult to solve tan tose obtained wit n = 50. However, wen inspecting te designs obtained wit tese two sample sizes, we noticed tat tey are very similar in terms of DCs opening, wic was not te case wit n = 30. For tese reasons, a sample size of n = 50 scenarios was selected as te best trade-off to use in our experiments. Tis produces SAA models wit 7,052,749 decision variables (including 20 binary variables) and 199,555 constraints. Tese models are solved wit OPL-CPLEX 12.3 in 5 ours on average. Te number of azards generated (i.e. te cardinality of H ) ranges between [121, 154], wic explains te complexity of te second-stage program and te variability of te obective function value. Wen generating 10 SAA models (M = 10) and evaluating te resulting designs using a sample of n = 200 scenarios, a statistical gap (Sapiro, 2003) of 2.57% was obtained, wic is relatively low. Remember tat obective function (31) of our SAA models includes two criteria, te minimization of a weigted coverage function and te minimization of expected sustainment and recovery costs, and tat te former is based on subective penalty costs set to improve service during te deployment pase. In addition, te relative importance of tese two criteria is weigted wit te parameter γ (initially set to 0.75) wic is also subectively estimated. Finally, recall tat altoug robustness is an important design criterion, obective function (31) is expressed purely in terms of expected value and does not take dispersion into account explicitly. Given tis, te aim of te second pase of our solution approac (Figure 4) is to generate a number of good alternative designs, wic can subsequently be evaluated using all relevant criteria. Tis set of potential designs is expended by doing a sensitivity analysis for all subective parameters. For all tese reasons, our experiments were done using te following SAA parameter values: n = 50, M = 10 and n = 200. Generated Designs Analysis Te solutions obtained wit te 10 SAA models solved wit a coverage weigt 0.75 γ = and a budget of 1M$ are summarized in Table 3. For eac design, te table provides DC opening and configuration decisions, te number of opened DCs, te total inventory level prepositioned in te network, and also te total DCs fixed costs and inventory olding costs incurred. In terms of selected locations, Table 3 sows tat 8 alternative structures were obtained, eac involving 5 locations, wic is significantly different from te current 2-depot structure. For instance, designs 9 and 10 (similarly, designs 3 and 5) recommend te same locations and configurations but, te prepositioning of te inventories tey propose is sligtly different as sown by te inventory costs. On te oter and, altoug designs 2 and 3 use te same locations, tey suggest different configurations (small/large). As can be seen from te inventory costs reported, te inventory levels for te tree products prepositioned differ sligtly for eac design due to te variability witin te scenarios sample used. Te last column of te table provides statistics on te presence of eac DC in te 10 designs produced. Tis underlines te degree of similarity between te designs obtained and illustrates te importance of Badin, Tarboro and Nas wic are always selected. Tese designs will be examined more torougly in te evaluation pase. 26

28 Prepositioning Emergency Supplies to Support Disaster Relief: A Stocastic Programming Approac Design 1 Design 2 Design 3 Design 4 Design 5 Design 6 Design 7 Design 8 Design 9 Design 10 Opening Costs Number of Opened DCs Inventory Costs Inventory Level DC Location Configuration Decisions Presence (%) 1 Badin % 2 Tarboro % 3 Nas % 4 Nortstar Drive % 5 Duram 1 10% 6 Lincolnton % 7 Carlotte % 8 Leland 1 10% 9 Fletcer % 10 Aberdeen 0% Table 3- Network Structure Decisions Provided by te Design Generation Pase To get a better feel of te nature of te designs produced, Design 1 is illustrated on te Nort Carolina map in Figure 7: te location of te 10 potential DCs is sown and te 5 opened DCs are encircled. Table 4 provides more details on te solution obtained. Its last row gives total item inventory levels as percentages of expected deployment demand. Note tat te two current DC locations used by NCEM are selected in tis design but wit more capacity. Also, capacity utilizations at opened DCs are at teir maximum, wic clearly reflects te preference for a deployment from DCs. Te item percentages in te last row reflect specified priorities (i.e., assets, ten medical kits ten water and MRE). Figure 10- DCs locations for Design 1 on NC Map Table 5 presents, for Design 1, te proportion of total demand satisfied from DCs, vendors and te backup source for eac pase. As expected, during te deployment pase, demand is mostly satisfied from DCs (73%), wile backup source and vendors usage proportions depend on te trade-offs between transportation and recourse costs. In addition, our tests indicate tat demand is fully fulfilled from DCs only for 41.4% of azards. 27