Case Study Based Evaluation of a Stochastic Multi-Commodity Emergency Inventory Management Model

Transcription

1 Case Stdy Based Evalation of a Stochastic Mlti-Commodity Emergency Inventory Management Model Eren Erman Ozgven, M.Sc. Gradate Research Assistant, Department of Civil and Environmental Engineering, Rtgers, The State University of New Jersey, 623 Bowser Road, Piscataway, NJ USA, Tel: (732) ozgvene@rci.rtgers.ed Kaan Ozbay, Ph.D. Professor, Department of Civil and Environmental Engineering, Rtgers, The State University of New Jersey, 623 Bowser Road, Piscataway, NJ USA, Tel: (732) kaan@rci.rtgers.ed Word cont: Figres + 6 Tables = 8511 Abstract: 250 Sbmission Date: November 15, 2011 Paper Sbmitted for Presentation and Pblication at the Transportation Research Board s 91 th Annal, Washington, D.C., 2012 (Revised 11/15/11)

2 ABSTRACT Over the last three decades, disasters worldwide claimed more than 3 million lives and adversely affected lives of at least 1 billion people (1). Emergency disaster management has emerged as a vital tool to redce the harm and alleviate the sffering these disasters case to its victims. A significant task of planners involved in the emergency disaster management is the ability to plan for and satisfy the vital needs of the people located in the emergency shelters, sch as the Sperdome shelter at New Orleans. This task reqires to find a way to redce the ncertainties associated with the emergency operations, and to estimate the possible expected costs of delivery and consmption of vital spplies throghot these operations. This paper attempts to address these isses by applying a case stdy based approach to demonstrate the seflness of sing a mlti-commodity stochastic hmanitarian inventory control model while estimating the minimm safety stock levels of the emergency inventories. First, emergency inventory management problem is discssed and previos emergency inventory stdies are reviewed to identify the need for a stochastic mlti-commodity emergency inventory management model. After introdcing the mathematical formlation for the model, it is applied to a nmber of realistic case stdies bilt based on the experiences in recent major disasters, sch as Katrina. The paper is conclded by a smmary of lessons learned for the model when applied on a wide range of scenarios drawn from real-life experiences, and sed to create emergency inventory management strategies for different types of disasters. 2

3 INTRODUCTION A recent National Research Concil of the National Academies report (2) mentioned two important problems faced for the nstable disaster sitations: the complexity introdced by the dynamics of the emergency response operations and the information ncertainty de to the flctating demand and possible disrptions in the transportation network. Unfortnately, the threats posed by these disaster problems may likely be even worse in the ftre de to the following reasons: Poplation growth in the areas vlnerable to disasters, i.e., along coastal areas, or near the dangeros falts, Rapid growth in terms of technology and indstrialization especially in the developing contries. According to Noji (1), we will experience many natral hazards dring the next decade, specifically abot 1 million thnderstorms, a thosand floods, damaging earthqakes, wildfires, hrricanes, tsnamis, droghts and volcanic erptions. Based on this information, the most important challenge of emergency disaster management and logistics against these disasters is apparently redcing the harm and alleviating the sffering a disaster cases to its victims. A significant component of the emergency disaster logistics is to satisfy the vital needs of the people located in the emergency shelters, sch as the Sperdome shelter at New Orleans sed after the hrricane Katrina. A stdy by Holgin-veras et al. (3) stated that the lack of an efficient hmanitarian inventory control model cased major negative conseqences for the disaster victims after Hrricane Katrina. Therefore, disaster planners and engineers shold find a way to redce the ncertainties associated with the emergency operations, to calclate and compare the possible expected costs of delivery and consmption processes throghot these operations, and to manage the availability and distribtion of vital resorces. With these reqirements, preparedness for disasters becomes extremely important where stocking inventory of vital spplies to cope with a disaster becomes a mst in order to facilitate the rapid mobilization of all available resorces dring the emergency operations. In terms of planning, it is crcial to focs on maintaining a reserve capacity as a bffer for the vital spplies in the aftermath of a disaster. With this motivation, there is a need for the development and case based analysis of a hmanitarian inventory management model developed prior to the occrrence of a natral or man-made catastrophe, which can determine the safety stock levels for different scenarios that will prevent possible disrptions at a minimal cost, as proposed in this paper. As it will become obvios from the literatre review section of this paper, there is a very large nmber of stdies dealing with different aspects of the emergency management and relief operations. The main focs of this paper is to demonstrate the importance of an efficient hmanitarian inventory management system throgh the se of realistic case stdies bilt sing information available from past disasters. It is clear that a hmanitarian inventory management system shold be robst with respect to the disrptions to minimize the impact of the severity of a particlar disaster on the spply and consmption of vital commodities. Or ltimate goal is to se realistic case stdies to emphasize this simple yet very important reqirement. 3

4 CASE BASED EMERGENCY INVENTORY MANAGEMENT Natral or man-made disasters are extreme, nexpected events often occrring with little or no warning. These catastrophic events make all of s actely aware of or vlnerabilities to disasters. Table 1 shows the distribtion of these disasters by type all arond the world and the nmber of people affected by different disaster types for the time period taken from the Emergency Disaster Database as provided by the Centre for Research on the Epidemiology of Disasters (4). Here, affected people indicate the people reqiring immediate assistance dring a period of emergency. Note that the difference in characteristics of these disasters make it necessary for a model to make it possible to perform case stdies possibly sing information available from past disasters. TABLE 1 Smmary Statistics for Disasters by Type ( ) (4) Disaster Type Natral Disasters Percentage Nmber of Share of Disasters Disasters Nmber of People Killed by Disaster Nmber of People Affected by Disaster Flood % 6,936,146 3,332,422,054 Storm % 1,379, ,852,155 Epidemics % 10,047,798 50,554,210 Earthqake (Inclding Tsnami) % 2,586, ,980,947 Droght % 11,808,692 2,101,143,086 Mass Movement % 60,419 13,765,583 Wild Fires % 4,025 6,085,821 Volcano % 97,765 5,357,731 Insect Infestation % 0 553,240 Extreme Temperatre % 172, ,660,357 Total % 33,098,833 6,675,410,903 Disaster Type Technological Disasters Percentage Share of Disasters Nmber of Disasters Nmber of People Killed by Disaster Nmber of People Affected by Disaster Indstrial Accident % 53,238 4,891,565 Transport Accident % 61,323 3,451,858 Miscellaneos Events (Inclding Man-made % 213, ,601 Attacks) Total % 328,176 8,572,023 Moreover, the difficlty in handling the emergency operations differs for each type of disaster increasing as one goes from localized and slow onset disasters to dispersed and sdden onset disasters. Therefore, an efficient inventory management system shold be capable of handling different types of disasters with distinct characteristics (localized or dispersed, slow or fast). With this motivation, regarding these different characteristics, we will introdce the case based emergency inventory 4

5 management problem within a disaster framework. First of all, there are a nmber of objectives soght by the planners, researchers and engineers to obtain an efficient emergency inventory management methodology. These objectives mainly differ from the commercial inventory management since disaster inventory management is niqe and significantly different from classical inventory and manfactring framework. Figre 1 illstrates or overall problem with objective fnction and its constraints within the case based disaster management concept. Space Constraints Disrption Constraints Maximization of the available vital spplies for the victims Minimization of sffering and maximization of chance of srvival for the victims Case Based Objectives Cost minimization (storage, shortage, srpls, etc.) Ensring the flexibility of the inventory operations in the presence of ncertain demand and spply, and dynamically changing environment FIGURE 1 Overview of the Case Based Emergency Inventory Management Problem LITERATURE REVIEW In this work, we focs on the emergency inventory management where commodity delivery and consmptions, shelter location and allocation, network logistics for the commodities revolve arond it. Table 2 shows these stdies in a chronological order as well as their relationships with respect to each other. Inventory Control and Allocation This is the most critical component of the overall emergency inventory management process. There are several stdies that try to find optimal soltions to these problems in the literatre. These stdies inclde Van Wyk (5), Taskin and Lodree (6), Rawls and Trnqist (7), and Lodree (8) where the focs is on the optimal pre-stock of emergency spplies. Determining the optimal locations of shelter and inventory locations, and their proximities to each other also reqires a significant effort dring the application of mathematical programming techniqes for the disaster operations. Among these stdies, Yshimito et al. (19), Jia et al. (10), Dran et al. (11), and Balcik and Beamon (12) worked on the facility location problem for hmanitarian relief chains trying to determine the nmber and locations of the distribtion centers in the relief network. 5

6 TABLE 2 Comparative Literatre Review for Emergency Inventory Management Field Stdies Description of Work Inventory Control Distribtion and Transportation Inventory Control with Distribtion and Transportation Van Wyk (2010), Taskin and Lodree (2010), Rawl and Trnqist (2010), Lodree (2011) Jia et al. (2007), Balcik and Beamon (2008), Yshimito et al. (2009), Dran et al. (2010) Haghani and Oh (1996), Barbarosogl and Arda (2004), Yang and Federgren (2006), Chang et al. (2007), Tzeng et al. (2007), Lin et al. (2010), Fiedrich et al. (2010) Barbarosogl et al. (2002), Ozdamar et al. (2004), Yi and Kmar (2007), Ozdamar and Yi (2008) Beamon and Kotleba (2006), Ozbay and Ozgven (2007), Jaller et al. (2008), Mete and Zabinsky (2010), Ozgven and Ozbay (2011) Sherali et al. (1991), Kongsomsaksakl et al. (2005), Dessoky et al. (2006), Han et al. (2007), Yi and Ozdamar (2007), Shen et al. (2009) Prepositioning of Vital Spplies Allocation of Shelters and Inventories Transportation of Vital Spplies Emergency/Resce Vehicle Operations Prepositioning and Transportation of Vital Spplies integrated Allocation of Shelters and Inventories and Transportation of Vital Spplies integrated Distribtion/Transportation Another important fnction of the emergency inventory system shold be not only to set p centers that stock inventories of spplies needed, bt also to receive and distribte them. Several emergency planning models have been developed to focs on the delivery of the vital prodcts dring the disasters. Haghani and Oh (13) and Barbarosogl and Arda (14) stdied on the transportation of many different goods to minimize the nmber of casalties and to maximize the efficiency of the resce operations. Other similar models developed inclde Lin et al. (15), Fiedrich et al. (16), Yang and Federgren (17), Chang et al. (18), and Tzeng et al. (19). There are also several stdies that specifically focsed on the emergency/resce vehicle operations sch as Ozdamar et al. (20), Ozdamar and Yi (21), Yi and Kmar (22), and Barbarosogl et al. (23) where they presented mathematical models for emergency logistics planning dring disasters. Inventory Control and Allocation Integrated with Distribtion/Transportation Hmanitarian inventory control models integrated with an effective distribtion methodology are needed to aid in adeqately responding to a disaster or a hmanitarian crisis. Several researchers recognized this isse sch as Beamon and Kotleba (24) who developed an inventory control model determining optimal order qantities and reorder points for a long-term emergency relief response, and Ozbay and Ozgven (25) and Ozgven and Ozbay (26) who worked on a sb-problem of the general hmanitarian emergency response problem to determine the safety stock while preventing disrptions 6

7 at a minimal cost. A similar approach was proposed for fixed lifetime commodities by Jaller et al. (27) as a cost minimization stochastic inventory model for disaster planning. Other stdies inclde Dessoky et al. (28), Yi and Ozdamar (29), Kongsomsaksakl et al. (30), Shen et al. (31), Han et al. (32), and Sherali et al. (33) where they focsed on the selection of shelter locations among potential alternatives. It is clear that there are a wide range of stdies for the emergency management and relief operations whereas case stdy based emergency inventory management is still a very open area to work on. There are a few stdies that consider specifically case stdies for extreme events in the literatre, sch as Mete and Zabinsky (34). However, their focs was only on medical spplies and their distribtion, and they did not consider the case stdies that will be performed in this paper. Therefore, the aim of this paper is to provide an efficient hmanitarian inventory management system which shold be robst to the disrptions, to stdy real-life case stdies based on this model inclding the affects of the severity of the disaster, critical consmption and delivery distribtions, and nmber of deliveries, and to come p with inventory management strategies for different types of disasters. Methodology In this section, we will introdce the proposed case based inventory management model. This case stdy based approach reqires certain steps inclding the creation of demand and spply distribtions and parameter selection which were methodologically introdced in Figre 2. Using these steps, planners and decision makers are able to perform case stdies based on real life experiences from previos disasters. 7

8 Objectives & Constraints Case Based Planning Model Create the consmption and delivery distribtions Determine the model parameters and base stock Create the base case No Is the objective fnction minimized based on these vales? No Prékopa-Vizvari-Badics Algorithm Yes Design case stdies No Change in Disaster Strength Change in the Distribtions Is the objective fnction minimized based on new vales? Change in the Nmber of Deliveries High Skewness Rn the model Inventory Strategy FIGURE 2 Overall Methodology for the Case Based Inventory Management Model 8

9 We will first mathematically describe the case stdy based model in the context of hmanitarian logistics. Or aim is to find the amont of safety stock in the disaster inventories with a probability 1, so that independent delivery and consmption processes go on withot disrption at minimm cost. For instance, if the vale of, the probability of disrption, is 0.1, the aim is that disrption will not occr 90% of the time. In or model, we assme that deliveries, fixed and designated by n, take place according to some random process at discrete times within a finite time interval [0, T ]. These random times have joint probability distribtions the same as that of n random points chosen independently from the interval [ ; T] according to a niform distribtion. A minimal amont,, is delivered with each delivery n. If the total amont of delivery is D for some spport, U, U being a finite set of spports, there is also a random amont of delivery obtained by choosing a random sample of size n 1, from a poplation niformly distribted in the interval [0,1 n ]. The consmption process is defined similarly, with parameters C for some, U, as the total amont of consmption, as the minimal amont of consmption, and s as the nmber of consmption times. We assme that delivery and consmption processes are independent, and there will be a sperscript l for each commodity. The decision variables in the model () are m l, the additional safety stock for each commodity l, l 1... r, and for each, U, and M, the storage capacity of each commodity l, l 1... r. We have an initial safety stock in the interval [0, T] and to satisfy the needs of the victims located in the shelters in terms of the commodities with a probability 1, we are trying to find the ( )* optimm additional safety stocks, l ( l)* m vales and the optimm storage capacities M. We approximate the joint distribtion of the random consmption and delivery variables sing an approximate mltivariate normal distribtion with the random () variable, W l for each commodity l 1,..., r, for each delivery i 1,..., n, and for each (), U (35). Therefore, W l represent the vales of the probability distribtion of the commodities in terms of consmption mins delivery for any time step: ( ) ( ) ( ) ( ) ( ) ( ) ( ) W l l l 1... l l l 1... l i Y Yn i X X n, l 1,..., r, i 1,..., n for each, U (1) The expectations, variances and elements of the covariance matrix for the random variable W, l 1,..., r, i 1,..., n for each, U are calclated as follows following the normal approximation gidelines given in (35). ( l) ( l) ( l) ( l) ( l) ( l) ( l) ki ( l) ( l) hi i i C n ( l) D n ( l) (2) V 1 L 1 Here, k 1 h 1 V 1V 2 L 1 L 2 ( l) ( l) i ( l) ( l) ( l) ( l) ( l) ( l) ( l) i ( l) ( l) C n D n ( l) ( l) ( l) ( l) ( l) ( l) 2 h 2 ( ) ( ) i ( hf h ) i k ( ) ( ) i ( k f k ) l l l l i () D n 2 C n (4) 2 ( l) ( l) ( l) ( l) Li 1 Li 2 Vi 1 Vi 2 (3) 9

10 D :total delivery, l 1,..., r, for each, U, () C l :total consmption, l 1,..., r, for each, U, : A minimal amont delivered with each delivery n, : A minimal amont consmed at each time step, v :sample from niformly distribted poplation in [0, C n ] for each, U, x :sample from niformly distribted poplation in [0, D n ] for each, U, V :sample size of v, and L :sample size of x, k :positive integers selected randomly from the sample y, i 1,..., n 1;1 i f n 1, i h :positive integers selected randomly from the sample x, i 1,..., n 1;1 i f n 1, i, f (): represents any element of the covariance matrix,. () ( W l ) With this information, an overview of the model that shows the inpts and otpts can be seen as follows: Inpts n : Nmber of deliveries l m : Initial safety stock D : Total amont of delivery l C : Total amont of consmption l l W : Approximate normal distribtion variable of the random consmption and delivery distribtions l l ( l) ( l) g, f, q, q : Associated costs M : Total capacity : Probability of disrption SHIC Model Otpts l m : Additional amont of safety stock reqired to satisfy the needs for the vital spplies l M : Storage capacity for each commodity Objective Fnction The objective cost fnction is the sm of individal costs listed below: Cost of Storage, g : It is obvios that there is a cost for storing each commodity l, l 1,..., r. In case of disaster operations, it is important to consider storage costs since the occrrence of a disaster is not known a priori. Cost of Srpls, q i : This is incrred if there is more inventory than demand. It can be modeled as a fixed cost or a step fnction. Cost of Shortage, q i : This cost is incrred if there is not sfficient inventory to satisfy the demand of the evacees. This is the most important cost component, as the shortage of vital spplies can case loss of life. Cost of Adjstment, f : This cost is incrred by the natre of the two-stage model. Sppose we have an initial amont of safety stock, bt to satisfy the probability constraint, we need more. This adjstment can be de to the 10

11 nexpected factors sch as the strength of a disaster, increased nmber of people who are affected and need help, etc. The objective fnction incldes the demands for the consmption of vital commodities mltiplied by their corresponding probabilities. This allows s to calclate the total cost where the highest and lowest demands have the lowest probabilities according to a pre-determined discretized normal distribtion. The total cost of a severe disaster may be higher than others de to the additional safety stocks reqired, however, the probability associated with this high demand will be smaller than lower levels of demand closer to the mean. This self-controlling mechanism gives the analysts the chance to determine the safety stocks more accrately. Constraints There are two types of constraints in the model: The probabilistic constraints ensre the minimal disrption of the commodities in the shelters with a given probability, therefore the sm of initial stocks and deliveries has to be greater than or eqal to the consmption for any time step. By replacing W 's with their expectations and variances of the approximate normal distribtion, or probabilistic constraint is defined as r ( l) ( l) ( l) ( l) ( l) ( l) m m ( l) PW m m 1, () W 1 l l1 (5) Other constraints in the model are the capacity constraints. At any time step, the initial safety stock pls the optimal additional stock mst be smaller than the storage capacity for that commodity, and the sm of storage capacities for each commodity mst be smaller than the overall capacity, M. With this information, the formlation of the model is as follows: r n ( ) ( ) 1 l l ( l) ( l) ( l) ( l) z min g ( M ) p f ( m ) ( qi qi ) 1 dz l1 T U i1 ( l) ( l) m m Sbject to r l1 ( l) ( l) ( l) m m, i 1,..., n1, ( ) 1 m m M, U, l 1,..., r ( l) ( l) m 0, U, l 1,..., r r ( l) ( l) a ( M ) M l1 Prékopa-Vizvari-Badics Algorithm Solving this nonlinear problem with an exact soltion techniqe reqires extensive programming and optimization knowledge. Therefore, we will first convert the continos distribtion fnctions into approximate discrete distribtions (37), then introdce the Prékopa-Vizvari-Badics Algorithm to solve this problem (36). For that () prpose, for each spport, U, we approximate the random variable W l by a (6) 11

12 () discrete variable l ( l) ( l) ( l) with possible vales L, where the distribtion fnction is: ( l )( ), 1,..., 1 ( l) ( l) ( l) ( l) ( l) F L P( m m W ) W F ( l )( ), l 1,, r. W 1, L (7) ( l) ( l) ( l) where L are chosen to be eqidistant on some interval [0, B ] where () F ( B ) 1 for a prescribed small tolerance. Here, B l is the selected pper ( l ) W () bondary of the interval of l i vales for each commodity l, l 1... r, and for each, U. Using this process, we obtain the following probabilistic constraint in the form () of mltiplication of the cmlative distribtion fnctions of l : r F ( l )( ) 1, 1,..., L 1, l 1,, r for each, U. l1 (8) That is, we discretize or continos cmlative distribtion fnction associated with W on its entire domain. Dring the analysis, we try to keep a sbstantial amont of accracy while choosing the possible vales of in the interval [0, B ] () selecting B l and N accordingly. After this step, the algorithm can be sed efficiently which is based on creating p-level efficient points (pleps) (37) to create the deterministic eqivalent of the probabilistic constraints, and they assre that the constraints will satisfy the given reliability level p 1. The algorithm is based on the () discrete random variable l where Z Z1 x xz r, is the prodct set containing the spport of, the vector of the discrete random variable. For the sake of illstration, we consider the r-dimensional vector as ( 1, ) T r. The algorithm is as follows: Step 0. Initialize k 0. Step 1. Determine z,..., z sch that 1, j1 r, j r 1, j 2, k 1 r, k r and let E z1, z, z arg min y F y, z,..., z 1 z arg min y F z, y, z..., z 1 2, j 1, j 3, k 1 r, k 1 z arg min y F z,,..., z, y 1 r, j 1, j r1, j,...,. j1 r j r 1 r1 r Step 2. Let k k 1. If j1 k k1 1, then go to Step 4. Otherwise, go to Step 3. Step 3. Enmerate all the pleps of the fnction j k r F z1,, y, y R by r 1 1 and eliminate those which dominate at least one element in E ( y dominates z if y z, y z ). If H is the set of the remaining pleps, then let E E H. Go to Step 2. 12

13 Step 4. Stop, E is the set of all pleps of the CDF F( z) P( z). CASE STUDIES In this section, we work on several case stdies based on the available information for real life experiences from the disasters sch as Hrricane Katrina. Before introdcing the case stdies, it is important to nderstand the natre of the Hrricane Katrina formed on Agst 23, 2005, and crossed sothern Florida, casing deaths and flooding before strengthening rapidly in the Glf of Mexico. The storm was a category 3 storm on the morning of Monday, Agst 29 in sotheast Loisiana. Then, it rapidly intensified after entering the Glf, growing to a category 5 hrricane in jst nine hors. This rapid growth was de to the storm's movement over the "nsally warm" waters of the Loop Crrent, which increased wind speeds (38). While focsing on Katrina, we will consider the New Orleans Sperdome shelter that has been sed for sheltering prposes dring hrricanes Georges (1998), Ivan (2004), and Katrina (2005) as well. As reported in (38), despite these previos periods of emergency se, as Katrina approached the city, officials had still not stockpiled enogh generator fel, food, and other spplies to handle the needs of the thosands of people who wold be seeking refge there. First, approximately 9000 residents spent the night in the Sperdome as Katrina came ashore with the strength of category 3 hrricane. According to (38), Bennett Landrenea, Adjtant General for the Loisiana National Gard, said that the nmber of people taking shelter in the Sperdome rose to arond as the strength of the hrricane increased and search&resce teams broght more people to the Sperdome from areas hit hard. On Agst 28, 2005, the Loisiana National Gard delivered seven trckloads of MRE's (Meal-Ready-to-Eat) to the sperdome, enogh to spply people for three days which was not enogh for the victims located in the sperdome (38). This real life information obtained for Hrricane Katrina constittes the basis of or case stdies in the following sections. The detailed parameter information for these case stdies is given in Table 3. 13

14 Case Stdy Base Case (Category 3, the first hit by Hrricane Katrina) Increased Strength (Category 5, Hrricane Katrina at the peak of its strength) Changes in the Mean of the Commodity Distribtion Changes in the Variance of the Commodity Distribtion Changes in the Skewness of the Commodity Distribtion Changes in the Nmber of Deliveries TABLE 3 Parameter Information for the Case Stdies ( indicates a change in the parameter whereas - means no change) Distribtion Total Total Mean Variance Kappa Demand Consmption Parameters Nmber of Deliveries Cost of Storage Cost of Srpls Objective Fnction Cost of Shortage Cost of Adjstment N Bl Case Stdy 1 (Base Case): Change in the Severity of the Disaster We are first focsing on the early stages of the localized, slow on-set disaster, namely hrricane Katrina, when it hit the coast region of the U.S. east coast where its strength was category 3. Under this scenario, we assme that people are gathered in the New Orleans sperdome where 2 MRE s and ½ medicine are assmed to be given per person per day in average (39). Depending on the strength of the hrricane, there are five demand levels given in an ascending order, 1 represents the lowest demand whereas 5 is the highest. Demand for MRE's increases more rapidly than the medicine as the severity of the hrricane increases. For this case, the following vales are chosen: is 0.1, is 0.01, N is 100 and B is 1000 for MRE's and for medicine. Nmber of deliveries in the time interval (i.e., a day) is n 4. 14

15 Amonts of initial safety stock, (1) m and (2) m are 5000 and nits, respectively. (1) q (2) 1.2 / nit, q (1) 1.0 / nit, and q (2) 120 / nit, q 100 / nit. (1) (2) (1),, and (2) are calclated for each commodity and scenario separately. Cost of adjstment fnction is selected as f ( x) 2x. Costs of storage for each commodity are 5 /nit and 1/nit, respectively. Total storage capacity is nits. (1) Spaces occpied by commodities are selected as a (2) 2 / nit, a 1.0 / nit, Expected total delivery and expected total consmption vales are taken as: 1 1 D [2500,3000,3500, 4000, 4500] C [7500,8000,8500,9000,9500]. 2 2 D [0, 4000,8000,12000,16000] C [20000, 24000, 28000,32000,36000] These vales are created by the 2 MRE s and ½ medicine assmption given in (38) based on the vital spplies delivered dring the emergency operations condcted by Loisiana National Gard. Probabilities of the discrete spports of consmption and delivery vales are p [0.08,0.25,0.34,0.25,0.08]. Note that these spports represent the demands for the consmption of commodities mltiplied by their probabilities in the objective fnction. Delivery and consmption vales are sed to calclate the mean, variance, and covariance matrix for the distribtion of the vital spplies. The reslts are given in Table 4. For medicine, to satisfy the needs of the victims 90% of the time, initial stock mst be more than 69% of the total expected consmption for medicine. However, for MRE's, the initial stock mst be more than 75% of the total expected consmption to prevent disrption 90% of the time. When the consmption is eqal to the initial safety stock of MRE's, the model finds the optimal additional safety stock as 0 MRE's. TABLE 4 Reslts for the Two-Commodity Analysis of Category 3 Hrricane Demand =1 =2 =3 =4 =5 Initial Stock (Medicine) Optimal Additional Stock(Medicine) Total Initial Stock(Medicine) Total Expected Consmption (Medicine) Proportion of Initial Safety Stock to Total Expected Consmption (Medicine) 70.0% 68.8% 67.7% 69.4% 68.4% Initial Stock (MRE's) Optimal Additional Stock(MRE's) Total Initial Stock(MRE's) Total Expected Consmption (MRE's) Proportion of Initial Safety Stock to Total Expected Consmption (MRE's) 100.0% 87.5% 78.6% 75.0% 75.0% Total Cost

16 It is possible for a hrricane to intensify in strength in a short period of time increasing the nmber of victims located in the shelters and creating more demand as in the case of Hrricane Katrina. For a category 5 hrricane, we assme that people are gathered in the New Orleans sperdome. Again, we are focsing on medicine and MRE's with five demand scenarios. All other parameters being kept same as the base case (other than these vales: N is selected as 500, and B is 5000 for MRE's and for medicine), we expect an increase in the total delivery and consmption vales at the sperdome: D 1 1 D [5000,5500, 6000, 6500, 7000] C [10000,10500,11000,11500,12000]. [20000, 24000, 28000,32000,36000] C [40000, 44000, 48000,52000,56000] 2 2 These vales are again created by the 2 MRE s and ½ medicine assmption given in (39) based on the vital spplies delivered dring the emergency operations condcted by Loisiana National Gard. Especially for MRE's, the demand (consmption) is changed sbstantially to accont for the severity of the category 5 hrricane (Table 5). As observed from Table 5, having more than 67% of the commodities in the inventory before the disaster will prevent disrption 90% of the time for medicine. However, for MRE's, the initial stock mst be more than 73% of the total consmption victims can live with 2 MRE's per day, which makes the total expected consmption amont as MRE's (This is or demand 1 in Table 5). However, the spplied amont of MRE's is (10000 people with 2 MRE's per day). Here, as the cost optimal reslt, the model gives an initial stock of MRE's as the initial safety stock, and an additional MRE's to be delivered afterwards. TABLE 5 Reslts for the Two-Commodity Analysis of Category 5 Hrricane Demand =1 =2 =3 =4 =5 Initial Stock (Medicine) Optimal Additional Stock(Medicine) Total Initial Stock(Medicine) Total Expected Consmption (Medicine) Proportion of Initial Safety Stock to Total Expected Consmption (Medicine) 67.5% 69.0% 68.2% 67.4% 68.8% Initial Stock (MRE's) Optimal Additional Stock(MRE's) Total Initial Stock(MRE's) Total Expected Consmption (MRE's) Proportion of Initial Safety Stock to Total Expected Consmption (MRE's) 75.0% 75.0% 72.9% 75.0% 75.0% Total Cost Case Stdy 2: Changes in the Consmption and Delivery Distribtions Changes in consmption and delivery distribtions have tmost importance in determining the behavior of the model as or model can be solved for different distribtions that represent the level of npredictability related to the level of severity of 16

17 the disaster. These reslts will lead to different stock levels that can be sed by planners and decision makers to decide on different strategies dring the planning process. Therefore, in this section, we gradally increase the skewness, mean and standard deviation vectors of the random variable, W, and observe the behavior of or model according to different distribtions for which parameter information is given in Table 6. The reslts can be seen in Figre 3. As seen, the model behaves accordingly by increasing the safety stock levels with the increase in the expected consmption, or with a possible decrease in the nmber of deliveries. Figre 3 indicates that the change in the variances also create an increase in the additional inventory stocks. Given the stochastic conditions in the aftermath of a disaster, this shows that or model reqires more initial safety stocks to accont for the high variances in consmption and delivery distribtion of vital spplies. These high variances may be high for the sdden, on-set disasters at first, therefore the planner shold be aware of the conseqences of not having the optimal initial safety stocks. On the other hand, they may vary with time for a slow on-set disaster sch as famine, where response will reqire a thorogh stdy of different consmption and delivery distribtions. 17

18 TABLE 6 Parameter Selection for Different Distribtions of MRE's and Medicine (Info: Category 3 Vale/Category 5 Vale) Mean Change Standard Deviation Change Skewness Change Mean Change Standard Deviation Change Skewness Change MRE's Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 250/ /2000 0/0 Distribtion 1 375/ /2000 0/0 Distribtion 2 563/ /2000 0/0 Distribtion 3 844/ /2000 0/0 Distribtion / /2000 0/0 Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 500/ /2000 0/0 Distribtion 1 500/ /6500 0/0 Distribtion 2 500/ /9000 0/0 Distribtion 3 500/ / /0 Distribtion 4 500/ / /0 Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 500/ / /-0.6 Distribtion / / /-1.0 Distribtion / / /-1.4 MEDICINE Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 100/ /750 0/0 Distribtion 1 150/ /750 0/0 Distribtion 2 200/ /750 0/0 Distribtion 3 250/ /750 0/0 Distribtion 4 300/ /750 0/0 Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 100/ /750 0/0 Distribtion 1 100/ /1250 0/0 Distribtion 2 100/ /2150 0/0 Distribtion 3 100/ /3050 0/0 Distribtion 4 100/ /4250 0/0 Distribtion Mean Standard Deviation Skewness (Kappa) Base Case Stdy 100/ / /-0.5 Distribtion 1 300/ / /-0.8 Distribtion 2 500/ / /

19 (a) (b) FIGURE 3 Changes in Additional Safety Stock Vales for MRE's (Figre 3.a) for Medicine (Figre 3.b) with respect to Distribtion Changes 19

20 Moreover, the distribtions for emergency spplies can be highly skewed especially at the early periods of the disaster. This will most likely to happen for sdden on-set type of disasters sch as the devastating Japan earthqake, where the initial need for vital spplies was enormos. To investigate this problem, we work on the skewnormal distribtion given by the following eqation (40): ( z; ) 2 ( z) ( z) (9) where () z and () z denotes the probability density and cmlative distribtion fnctions of the standard normal distribtion N (0,1). The parameter is the skewness parameter that reglates the skewness between,, and 0 means the distribtion N (0,1). With this approach, we increase the left skewness of or distribtion by changing and observe the behavior of the model in Figre 4. When there is a very large demand of emergency spplies at the beginning periods of the disaster creating a left-skewed distribtion, it is appropriate to se this concept to determine the initial inventory stocks given the distribtions in Table 6. Figre 4 shows that additional safety stock vales tend to increase as the left skewness of or distribtion increases to accont for the hge consmption demand coming from the victims located at the shelters (a) 20

21 (b) FIGURE 4 Changes in Additional Safety Stock Vales for MRE's (Figre 4.a) for Medicine (Figre 4.b) with respect to Skewness Changes Case Stdy 3: Disrptions to the Transportation Network Dring Katrina, Loisiana National Gard delivered seven trckloads of MRE's, enogh to spply people bt that was not enogh for the victims located in the sperdome (38). Therefore, in the aftermath of disasters, changes in the nmber of deliveries mainly de to the characteristics of the disaster (dispersed/localized), increased demand, redced spply and possible transportation system disrptions are expected. In Figre 5, we start a worst case scenario with no deliveries per period (i.e., a day), and gradally increase the nmber of deliveries to 9 deliveries per period (i.e. 9 trcks per day). At first, when there are no deliveries, the whole amont of emergency spplies needed shold be stocked as bffer. Gradally, disrptions to the transportation network end, and as the nmber of deliveries increases, the additional amonts of safety stock vales decrease. This is a logical assmption since this increase leads to lower additional stock vales to satisfy the probability constraint of or model in Eqation (6). For the category 3 case with lower consmption needs, when the nmber of deliveries reaches 8 vehicles dring the whole time interval, there is no need for additional safety stock for the medicine and MRE s. Hence, the initial safety stock is large enogh to satisfy the evacee demand for the total time period. As the disaster strength increases, the initial safety stock appears to be insfficient, and higher additional safety stock is needed with a high cost when the nmber of deliveries is limited de to damaged/congested roadways. 21

22 (a) (b) 22

23 (c) FIGURE 5 Changes in Additional Safety Stock Vales for Medicine (Figre 6.a), for MRE's (Figre 6.b) and Changes in Total cost (Figre 6.c) for Different Nmber of Deliveries CONCLUSIONS De to the highly probabilistic natre of demands and spplies in the aftermath of disasters, probabilistic inventory control models are better sited for the development of realistic plans. However, there are not many practical models in the literatre for which the convergence properties can be analytically shown de to the modeling and soltion complexities. In this paper, we propose to se a hmanitarian inventory control methodology as the base model for or case stdies which has been proven to reach optimality (35). Unlike previos approaches in the literatre, this case stdy based model is designed to sccessflly address the critical and strategic factors associated with the hmanitarian relief operations to ensre the continity of the npredictable stochastic delivery and consmption processes. Reslts of the proposed mlti-commodity stochastic hmanitarian inventory control model when applied on varios case stdies is encoraging in the sense that it provides a basis for the realistic analysis of risk involved in planning for the stock levels whereas the flexibility of the model allows the application of the mathematical formlation to any extreme event. The practical and easily applicable Prékopa-Vizvari-Badics algorithm is sed to solve the problem so that the case stdy based approach can easily be incorporated into disaster relief plans, and planners can become aware of the varios potentials for the serios inventory related problems that can occr at the shelters, as well as their probabilities. By concentrating on high risk scenarios, the planner can take recovery actions to ensre the best allocation strategy of the vital resorces. Several realistic case stdies bilt sing information available from past disasters are stdied sing the model addressing the following important isses: Change in the severity of the disaster, Disrptions in the transportation network, 23

24 Change in the consmption and delivery distribtions, and skewness. Based on these case stdies, the proposed model can be sed to examine the differences in emergency inventory strategies for different types of disasters as follows: Dispersed, sdden on-set disasters: This type of disaster is the most difficlt one in terms of emergency response since it is spread over an area and therefore might affect a large region, and happens very fast, sch as the 2004 Indian tsnami. As a reslt, at the very beginning, the variances of the consmption and delivery distribtions will be very high. Here, an analysis based on the changes in mean and variance from Case Stdy 2 (changes in the consmption and delivery distribtions) will be crcial. Therefore, the planner shold be aware of the conseqences of not having the optimal safety stocks in the emergency inventories initially. The strategy shold be keeping more inventory safety stock initially to accont for the high demand in the early aftermath of a disaster, which can be obtained by high variance and mean distribtions in Figre 3. Althogh skewness can also be sed as an alternative strategy, that concept will be more sefl for localized, sdden on-set disasters. Dispersed, slow on-set disasters: This type of disaster might also spread and affects a large region, bt it reqires a longer time period than the sdden disasters. Examples inclde the 2009 Avian fl epidemic, or famines in Africa. In this case, the variances and needs vary with time, where response will reqire a thorogh stdy of different consmption and delivery patterns. Based on the analysis reslts of Case stdy 3 (disrptions to the transportation network), it can be conclded that, for this type of disasters, there is not necessarily a need for a hge initial safety stock, bt depending on the nmber of deliveries that can be made to the shelters, safety stocks for vital spplies shold be adjsted in the emergency inventories sing the distribtions in Figre 3. Ths, if the transportation/spply systems are robst enogh to spport higher level of deliveries, the athorities can afford to maintain lower levels of safety stocks. This will redce the cost of emergency preparedness withot compromising the wellness of evacees. Localized, slow on-set disasters: This type of disaster affects a specific location rather than a large region, and it takes a long time period as the severity of the disaster increases gradally, an example is the infamos 2005 Hrricane Katrina. As extensively stdied in Case Stdy 1 (Change in the severity of the disaster) of this paper, this type reqires being able to respond to the disaster disrptions focsing on the different consmption and delivery distribtions and the change in the severity of the disaster. As it is localized and slow, it will be easier to respond in terms of adjsting the safety stocks than the other types of disasters. Localized, sdden on-set disasters: This type of disaster also affects a specific location rather than a large region, bt it happens fast sch as the 2010 Haiti or 2011 Japan earthqake where the initial need for vital spplies was enormos. In this case, the distribtions for emergency spplies can be highly skewed especially at the early periods of the disaster. The skewness based analysis reslts of the Case Stdy 2 (changes in the consmption and delivery distribtions) show that, as an emergency inventory strategy for this case, it is appropriate to se this 24

25 analysis to determine the initial inventory stocks given the distribtions in Table 6 when there is a hge demand of emergency spplies at the beginning periods of the disaster creating the need for a left-skewed distribtion." As we aim to develop a realistic time-dependent inventory planning and management model for the development of efficient pre-disaster plans, the planning and operational aspects of the stdy shold be given the same importance. Therefore, a better model incorporating these policies in real-time can be obtained sing feedback inventory control strategies. Moreover, to bild even more realistic and comprehensive case stdies and test or model sing these, there is a need for better data collection from real life cases. This type of stdy will show whether or not this model works properly and efficiently with regard to realistic disaster cases. Explicit addition of commodity transportation cost into the model is also an interesting area of ftre work since it will reqire a distance based clstering model that controls the location and allocation of the shelters and emergency inventories, and the flow of spplies from these inventories to shelters. REFERENCES 1. Noji, E.K., The natre of disaster: General characteristics and pblic health effects, The Pblic Health Conseqences of Disasters, edited by Eric K. Noji, Oxford University Press, 3-20, National Research Concil of the National Academies, Improving disaster management: The role of IT in mitigation, preparedness, response, and recovery, National Academies Press, Washington D.C., USA, Holgin-Veras, J., Perez, N., Ukksri, S., Wachtendorf, T., and Brown, B., Emergency logistics isses affecting the response to Katrina: A synthesis and preliminary sggestions for improvement, Transportation Research Record, 2022, 76-82, EM-DAT, The International Disaster Database, Center for Research on the Epidemiology of Disasters (CRED), 5. Van Wyk, E., Strategic Inventory Management for Disaster Relief, BS Thesis, University of Pretoria, Soth Africa, Taskin, S., Lodree, E.J. Jr., Inventory decisions for emergency spplies based on hrricane cont predictions, International Jornal of Prodction Economics, 126, 66-75, Rawls, C.G., and Trnqist, M.A., Pre-positioning of emergency spplies for disaster response, Transportation Research Part B, 44, , Lodree, Jr., E.J., Pre-storm emergency spplies inventory planning, Jornal of Hmanitarian Logistics and Spply Chain Management, 1(1), 50-77, Yshimito, W.F., Jaller, M., and Ukksri, S., Facility location on disasters: A voronoi based heristic algorithm with an application to hrricane Katrina, Presented at TRB s 88th Annal, Washington D.C., USA, Jia, H., Ordonez, F., and Dessoky, M., A modeling framework for facility location of medical services for large-scale emergencies, IIE Transactions, 39, 41-55, Dran, S., Gtierrez, M.A., and Keskinocak, P., Pre-positioning of emergency items worldwide for CARE International, Interfaces, 41(3), , Balcik, B., and Beamon B.M., Facility location in hmanitarian relief, International Jornal of Logistics: Research and Applications, 11(2), ,