Stochastic Vehicle Routing Problem for Large-scale Emergencies

Transcription

1 Stochastic Vehicle Routing Problem for Large-scale Emergencies Zhihong Shen Maged Dessouky Fernando Ordóñez November 30, 2005 Abstract In this paper, we are interested in routing vehicles to service large-scale emergencies, such as natural disasters and terrorist attacks. We first analyze the characteristics of the routing problem for large-scale emergencies. We then survey the stochastic Vehicle Routing Problems (VRPs) and some variants of VRPs that are relevant to emergency response. We propose and discuss a mathematical formulation for this problem and a two-stage solution approach for it. Lastly, in an illustrative example of an anthrax terrorist attack, we demonstrate the effectiveness of our formulation in developing robust routes. 1 Introduction Large-scale emergencies (or major emergencies) are defined as any event or occurrence that overwhelms local emergency responders, which severely impacts the operation of normal life, and has the potential to cause substantial casualties and property damage. Examples are natural disasters (earthquake, hurricane, flooding, etc.) and terrorist attacks, like September 11 th, Careful and systematic pre-planning, as well as efficient and professional execution in responding to a large-scale emergency, can save many lives. Rational policies and procedures applied to emergency response could maximize the effectiveness of the scarce resources available in relation to the overwhelming demands. A key ingredient in an effective response to an emergency is the prompt availability of necessary supplies at emergency sites. Given the challenges of delivering massive supplies in a short time period to dispersed demand areas, operations research models could play an important role in addressing 1

2 and optimizing the logistical problems in this complex distribution process. Larson et al. (2005a and 2005b) conducted a detailed analysis based on well-known and recent largescale emergencies. They emphasized the need for quantitative, model-oriented methods provided in the operations research field to evaluate and guide the operational strategies and actions in response to major emergencies. In certain emergency settings, medication or antidotes must be applied within a specified time limit from the occurrence of the event to maximize their effectiveness, and in some situations, to save lives. Traditional pharmaceutical supply chains are no longer adequate to push a huge volume of medical supplies to the affected population within a short time frame. Usually the local caches are only enough to serve the first responders. The majority of the supplies would be from the national repository, which is called the SNS (Strategic National Stockpile) which includes pre-positioned push packages, and larger managed inventory available from manufacturers. The SNS is administrated by the Centers of Disease Control and Prevention in the United States. The SNS has large quantities of life-saving pharmaceuticals and medical material to be used in cases of emergencies. If there is a public health emergency severe enough to cause local supplies to run out, the SNS supplies can be accessed. Initial medical supplies from the push packages will be delivered to any state in the U.S. within 12 hours of federal and local authorization. This initial delivery can be followed by supplies from the managed inventories if needed. It is each state s responsibility to have plans to receive and distribute SNS medicine and medical supplies to local communities as quickly as possible. In this work, we are concerned with the distribution process after the SNS supplies have been received at the local level. We strive to establish efficient dispatching plans and policies to help save lives through a timely delivery. Specifically, we formulate a Vehicle Routing Problem (VRP) to determine the optimal set of routes to be performed by a fleet of vehicles to serve a given set of customers. Routing of vehicles in response to large-scale emergencies must consider the unique features of such large-scale emergencies. The most important characteristic is the stochastic nature of the problem, such as unpredictable demands and highly uncertain travel times due to possible congestion caused by the emergency event or destruction of the physical road network. The VRP with such random parameters is referred to in the literature as a Stochastic VRP (SVRP). In addition, the problem might consider: allowing for split-delivery, using multiple depots, demands with time-windows, etc. The most common objective of VRPs is to minimize the total travel time or to minimize the number of vehicles used. In an emergency situation, demands sometimes directly represent the saving of lives. Hence, minimizing unsatisfied demand (saving more lives) outweighs any travel or operational cost. When we consider the problem of routing inventory in response to large-scale emergencies, both routes and delivery quantities are important in creating an effective dispatching plan. 2

3 These two aspects of the solution are related to the following uses of efficient inventory routing plans for a large-scale emergency: First, from a planning perspective, efficient routes can be used by agencies in mock trial runs of an emergency event to pre-design the routing policies and provide training opportunities. Second, from an operational perspective, the models can be used to update the pre-designed routes and promptly determine exact delivery quantities to dispensing or treatment sites to generate real-time near-optimal dispatching solutions after an emergency situation has happened. Therefore, we use a two-stage solution approach to solve this problem. In the first stage, we solve a proposed CCP (Chance Constrained Programming) model well in advance of any possible emergency to generate pre-planned routes, which can be used for mock trial runs and training purposes. In the second stage, a linear program model is executed to generate the delivery requirements with the given routes from the first stage at the time of the emergency when more information is available. The second stage algorithm must be able to run fast and find an efficient solution in order to quickly respond to the emergency requirements. The purpose of this article is to analyze the problem of routing vehicles in response to a large-scale emergency and discuss its relation to well-known problems in the literature; review the literature on the SVRP, some variants of the VRP related to emergency response, routing/dispatching problems in regular emergencies and logistic planning for large-scale emergencies; formulate a mathematical model of the VRP that is applicable for a specific largescale emergency scenario; present a two-stage solution approach which is applicable for both pre-planning and operational purposes; demonstrate the effectiveness of the model and solution approach in developing robust routes on an illustrative example representing an anthrax attack. This work focuses on the modeling and solution framework for the VRP in response to a large-scale emergency. The development of efficient algorithms to solve the models in the solution framework are a subject for future research. The rest of this paper is organized as follows. In section 2, we present characteristics of the SVRP for large-scale emergencies. Section 3 reviews relevant literature, including stochastic vehicle routing problems, variants of the deterministic VRP related to emergency response, vehicle routing/dispatching problems for regular emergencies and logistic planning for large-scale emergencies. Section 4 proposes the mathematical model and section 5 presents a two-stage solution approach. 3

4 We present some illustrative computational results in section 6 and, finally, conclude in section 7. 2 Characteristics of the SVRP for Large-scale Emergencies The distinguishing characteristics of large-scale emergencies are high demand for supplies, low frequency of occurrence, and high uncertainty in many aspects (e.g. when and where they happen). This makes it virtually impossible to predict the occurrence and impact of large-scale emergencies. As mentioned in the previous section, in this setting, an analysis of a vehicle routing/dispatching problem provides a method to plan for the first responder s reaction to the emergency and, in the event an emergency occurs, a methodology to quickly adjust dispatching policy to provide an efficient response. Hence, to be effective in this capacity, the routing model must represent important aspects of a large-scale emergency. The high demand and low frequency of large-scale emergencies lead to situations where local first responders are overwhelmed and local resources to meet the demand are scarce. In this situation we consider as the objective of the routing problem to minimize the unsatisfied demand. In fact, unmet demand in an emergency situation can result in loss of life, an impact that outweighs more common VRP objectives such as the travel time or number of vehicles used. The initial supplies in push packages will typically not be sufficient to cover all demands in the recommended response times. Priorities can be given to these demands to maximize the survivability ratio in the affected region. An additional aspect important in meeting the demand in a large-scale emergency is the response time. Since the delivery of supplies to the population within a time-frame makes an appreciable health difference in a large-scale emergency, it is natural to associate a time window with each demand, to model situations in which a late delivery directly leads to loss of life, such as in an anthrax attack. In addition to minimizing unmet demand and considering time windows, in this work we consider a VRP with the following properties relevant to large-scale emergencies: Stochastic demand: The highly unpredictable nature of large-scale emergencies leads to many stochastic elements in the problem. Among all the uncertain parameters, stochastic demand is one of the most important. At a given demand point (e.g. a neighborhood block), the quantity of required emergency supplies (antidotes, protective equipments, medication, etc.) is often proportional to the size of the population and/or numbers of casualties. The casualties exposures, or demands 4

5 among worried well are hard to accurately predict for an overwhelming emergency occurrence, like a dirty bomb attack. Stochastic travel time: In an emergency situation, the traffic conditions can become highly uncertain, due to unpredictability in people s behavior. At the same time, the emergency event itself may directly affect road conditions. For example, an earthquake may destroy a segment of the freeway and render it impassable. In less severe situations, some lanes may be destroyed and traffic will become snarled. Therefore, the travel times between points are stochastic. Split delivery: At some demand points which are seriously affected, the demand quantity most probably cannot be satisfied by a single truckload. Hence we allow for split delivery. This requires a demand point being visited more than once if the demand quantity exceeds the load capacity of available vehicles. Multi-depot: The VRP for large-scale emergencies is a multi-depot problem since there are usually several local stocks dispersed in the region to ensure the overall serviceability. However unlike the normal multi-depot VRP, which restricts the vehicle s return to the same depot that it departed from, we allow the vehicle to return to any depot with enough supplies to fill up and continue serving other demand points. 3 Literature Review In this part, we review several well-known problems in the literature which are relevant to characteristics of the SVRP for large-scale emergencies. We mainly focus on their relevance to our problem. This section is organized as follow: the general SVRP problem is briefly introduced and reviewed. Two specific topics, SVRP with stochastic customers and demands and SVRP with stochastic travel time and service time, follow the general discussion. Then, two major variants of the VRP: split delivery and multi-depot, which are closely related to a large-scale emergency context, are examined. Lastly, the vehicle routing/dispatching problems for regular emergencies and logistic planning problems for large-scale emergencies are reviewed. 3.1 Stochastic VRPs The classical Vehicle Routing Problem (VRP) determines the optimal set of routes used by a fleet of vehicles to serve a given set of customers on a predefined graph G = (V, A), where V = {v 1, v 2,..., v n } is a set of vertices and A {(v i, v j ) : i j, v i, v j V } is the arc set. A comprehensive overview of the Vehicle Routing Problem can be found in Toth and Vigo 5

6 (2002), which discusses problem formulations, solution techniques, important variants and applications. The Stochastic VRP (SVRP) arises whenever some parameters of the VRP are random. Common examples are uncertain customers and demands, and stochastic travel times or service times. The SVRP differs from its deterministic counterpart: several fundamental properties of the deterministic VRPs no longer hold in the stochastic case, and solution methodologies are considerably more complicated. A general review on the SVRP appeared in Gendreau et al. (1996), and is the subject of this and next two subsections. The Stochastic Vehicle Routing Problem can be broadly classified based on the following criteria: Uncertainty in the problem: The uncertainty in the Vehicle Routing Problem can be shown in different aspects. It is the mostly used classification scheme. It is usually divided into VRP with stochastic customers (VRPSC), VRP with stochastic demands (VRPSD), VRP with stochastic travel time (VRPSTT) and VRP with stochastic service time (VRPSST). Modeling method: The modeling method can also be the criterion to classify SVRPs. Stochastic VRPs can be cast into a stochastic programming framework, which is further divided into chance constrained program (CCP) and stochastic program with recourse (SPR). Another approach to model the SVRP is to view it as a Markov decision process. Some more recent modeling approaches include neurodynamic programming/reinforcement learning methodology. Solution techniques: Different solution techniques are the direct result of different modeling methods and the nature of the model. They usually broadly fall into two categories: exact methods and heuristic methods. The exact solution methods successfully applied to the SVRP include branch and bound, branch and cut, integer L-shape method and generalized dynamic programming. Numerous heuristics have been used to solve the SVRP, e.g. the savings algorithm, tabu search, etc. 3.2 SVRPs with Stochastic Customers and Demands The VRP has stochastic demands (VRPSD) when the demands at the individual delivery (pickup) locations behave as random variables. The first proposed algorithm for the VRPSD by Tillman (1969) was based on Clarke and Wright (1964) savings algorithm. Another early major contribution on the VRPSD comes from Stewart and Golden (1983), who applied the chance constrained programming and recourse methods to model the problem. Later on, Dror (1986) illustrated the impact that the direction of a designed route can have on the expected cost. A major contribution to the study of the VRPSD comes from Bertsimas (1988, 1992). This work illustrated the a priori method with different recourse policies (re-optimization is allowed) to solve the VRPSD and derived several 6

7 bounds, asymptotic results and other theoretical properties. Bertsimas and Simchi-Levi (1996) surveyed the development in the VRPSD with an emphasis on the insights gained and on the algorithms proposed. Besides the conventional stochastic programming framework, a Markov decision process for single stage and multistage stochastic models were introduced to investigate the VRPSD in Dror et al. (1989) and Dror (1993). More recently, a re-optimization type routing policy for the VRPSD was introduced by Secomandi (2001). The VRP with stochastic customers (VRPSC), in which customers with deterministic demands and a probability p i of being present, and the VRP with stochastic customers and demands (VRPSCD), which combines the VRPSC and VRPSD, first appeared in the literature of Jézéquel (1985), Jaillet (1987) and Jaillet and Odoni (1988). Bertsimas (1988) gave a more systematic analysis and presented several properties, bounds and heuristics. Gendreau et al. (1995, 1996) proposed the first exact solution: L-shaped method and a meta-heuristic: tabu search for the VRPSCD. Another research direction of the VRP with stochastic demands or customers incorporates a dynamically changing environment, where the demands vary over time. Bertsimas and Van Ryzin (1991, 1993) pioneered this work and named it dynamic traveling repairman problem (DTRP). They applied queuing theory to solve it and analyzed several dispatching policies for both light and heavy traffic. Based on their results, Papastavrou (1996) and Swihart and Papastavrou (1999) defined and analyzed a new routing policy using the branching process. The multiple scenario approach (MSA) (Bent and Hentenryck, 2004) and waiting strategies (Branke et al., 2005) have also been introduced for this problem recently. 3.3 SVRPs with Stochastic Travel Time and Service Time Compared with stochastic customers and demands, the research on the stochastic travel time and service time problem of the TSP and VRP have received less attention. The VRP with stochastic travel time (VRPSTT) describes the uncertain environment of the road traffic condition. Kao (1978) first proposed heuristics based on dynamic programming and implicit enumeration for the TSP with stochastic travel time (TSPSTT). Carraway, Morin and Moskowitz (1989) used a generalized dynamic programming methodology to solve the TSPSTT. Laporte, Louveaux and Mercure (1992) performed systematic research on the VRP with stochastic service and travel time (VRPSSTT). They proposed three models for the VRPSSTT: chance constrained model, 3-index recourse model and 2-index recourse model and presented a general branch-and-cut algorithm for all 3 models. The VRPSSTT model was applied to a banking context and an adaptation of the savings algorithm was used in the work of Lambert, Laporte and Louveaux (1993). Jula et al. (2005) developed a procedure to estimate the arrival time to the nodes in the presence of hard time windows. 7

8 In addition, they used these estimates embedded in a dynamic programming algorithm to determine the optimal routes. The VRP with stochastic service time is faced by a company who receives calls (failure reports) from customers and dispatches technicians to customer sites. This problem was introduced and formulated by Hadjiconstantinou and Roberts (2002). They used a two stage recourse model and a paired tree search algorithm to solve it. 3.4 Split Delivery The classical VRP problem makes the assumption that every customer is visited only once by a single vehicle. Dror and Trudeau (1989, 1990) investigated a relaxation of the VRP in which this assumption is removed, i.e., customer demand can be split between several vehicles. This variant of the VRP is called the Split Delivery Vehicle Routing Problem (SDVRP). A heuristic algorithm for the SDVRP was proposed in their work and it showed that allowing split deliveries could yield substantial savings, both in the total distance traveled and in the number of vehicles used in the optimal solution. Dror et al. (1994) first formulated the SDVRP as an integer linear program and presented a constraint relaxation branch and bound algorithm. Some particular case and real applications of the SDVRP were studied by Frizzell and Giffin (1995), Mullaseril, Dror and Leung (1997) and Sierksma and Tijssen (1998). Recently, under some mild assumptions, Belenguer, Martinez and Mota (2000) presented a lower bound for the SDVRP; Archetti et al. (2001, to appear) showed the complexity of the problem with different vehicle capacities and proposed a tabu search algorithm. The exact algorithm and the tabu search heuristic (Gendreau et al., 1995, 1996) applied on the VRPSD also can deal with split delivery. 3.5 Multi-depot VRPs The Multi-Depot Vehicle Routing Problem (MDVRP) considers multi-depot and usually assumes each route starts and ends at the same depot. It first appeared in the literature in the late 60 s in Tillman (1969), who proposed a constructive savings heuristic. Further improvement on the savings heuristic for the MDVRP can be found in Tillman (1971, 1972), where a look-ahead procedure and an embedded savings heuristic within a partial enumerative scheme were applied respectively. Another category of the heuristic for the MDVRP is the sweep procedure proposed by Gillett and Miller (1974), which improved an initial solution by sweeping the region along a virtual reference axis. Li and Simchi-Levi (1990) presented a theoretical analysis on the worst-case performance of tour partitioning heuristics for the capacitated MDVRP. Later on, more efficient and complicated meta-heuristics were applied on the MDVRP, which include a multi-phase 8

9 heuristic by Chao, Golden and Wasil (1993) and a tabu search algorithm by Renaud, Laporte and Boctor (1996). In contrast to the abundant work on heuristics, the research on exact algorithms is relatively limited. Laporte and Nobert (1984, 1988) formulated the symmetric and asymmetric MDVRP as integer linear programs and solved them optimally by a branch and bound algorithm. Another contribution to the exact solution method comes from Ribeiro and Soumis (1994), who formulated the MDVRP as a set partitioning problem with side constraints, whose continuous relaxation is amenable to be solved by column generation. Mingozzi and Valletta (2003) described an integer programming formulation and an exact method for the MDVRP. 3.6 Vehicle Routing/Dispatching for Emergencies Emergency service systems (e.g., police, fire and emergency medical) need to dispatch their response units (e.g., police cars, fire engines and ambulances) to service the caller s request. In a regular emergency, the objective is to save lives, so sending response units to arrive on the scene as soon as possible has the highest priority. The emergency request comes in low frequency but is highly unpredictable. In a real-time emergency context, this problem is modeled as a vehicle location/relocation problem. Probabilistic models can avoid the difficulties in static models by reflecting the fact that emergency vehicles are busy only a fraction of the time and become unavailable whenever they service requests. This line of research is pioneered by Larson (1974, 1975), who proposed the hypercube queuing model, which considers the congestion of the system by calculating the steady-state busy fraction of servers on a network. Other approaches include a probabilistic covering model (Ball and Lin, 1993) and dynamic models (Marianov et al., 1995; Brotcorne et al., 2003; Gendreau et al., 2001, 2003). Unlike a regular emergency, large-scale emergencies have their own distinguishing characteristics (Jia et al., 2005). The research on logistic planning for large-scale emergencies, though very limited, does contribute to the analysis of our problem. Rathi (1992) considered supply logistics in conflict or emergency situations. The problem was modeled by three LP formulations that can be solved efficiently. Ozdamar and Ekinci (2004) set up a hybrid deterministic and dynamic system which integrates the multi-commodity network flow problem and the vehicle routing problem. The model is decomposed into two multicommodity network flow problems. The two sub-problems are coupled and can be solved by relaxing arc capacity constraints by Lagrangian relaxation. The model and algorithm are tested on data from the earthquake at Izmit, Turkey in

10 4 A Mathematical Model for the SVRP for Largescale Emergencies In this section, based on the analysis in section 2, we formally define the notation and propose the model. We assume multiple commodity types, multiple depots in this problem, and use a commodity flow model. 4.1 Notation We consider a set K of vehicles, a set C of nodes and a set A of commodities. Sets CD and CS are subsets of C, which represent demand nodes and supply nodes (depots) respectively and have no overlap between them. We denote node 0 as a dummy node, which can be viewed as a virtual central depot with links to each real depot (supply node) and the cost or travel time on these links are zero. The dummy node is used to express the availability of the vehicles. Set RO includes all nodes excluding the dummy node so the relationship between the nodes can be summarized as follows: RO node 0 = C, RO node 0 =, CD CS = RO, CD CS = (where represents the empty set). Indexed on sets K, C and A, we define the following deterministic parameters: n i : w a : c k : e a,i : l a,i : s a,i : initial number of vehicles at supply node (depot) i unit weight of commodity a load capacity of vehicle k earliest service start time for commodity a at node i latest service start time for commodity a at node i amount of commodity a supplied at supply node (depot) i We use M as a large constant to help transform nonlinear terms to linear ones for some constraints. We also let α D and α T represent the confidence level of the chance constraints defining the unmet demand at each node and the arrival time of each vehicle at each node respectively. We consider the following two parameters as stochastic variables, unknown at the time of planning, with the given distribution. τ i,j,k : ζ a,i : time required to traverse arc(i, j) for vehicle k; it is set to for non-existent links amount of commodity of type a needed at node i Finally, four groups of decision variables are defined as follows, indexed on sets K, C, A: 10

11 X i,j,k : Y a,i,j,k : U a,i : T i,k : binary flow variables, equal to 1 if (i, j) is traversed by vehicle k and 0 otherwise amount of commodity type a traversing arc(i, j) using vehicle k amount of unsatisfied demand of commodity type a at node i start of service time at node i provided by vehicle k 4.2 Chance Constrained Model The problem at hand falls into the stochastic programming framework. Since a plausible recourse penalty function for this problem is relatively complicated to formulate and solve, we can enforce that the constraints must hold with high probability, e.g. applying the chance constrained technique to model the problem. We propose a model with the timewindows constraint on the delivery of commodities. Model CCP: minimize U a,i a A i CD subject to: Constraints (1)-(16) The objective of model CCP is to minimize the total unsatisfied demands over all demand points. We explain the constraints in detail as follows. In a commodity flow formulation, the constraints include the vehicle flow part, commodity flow part and links between them as well as the standard nonnegative/binary constraints on the decision variables. The following constraints (1)-(9) characterize the vehicle flows on the path. j CD X 0,i,k n i ( i CS) (1) k K X i,j,k = 1 ( i CS k K) (2) j CD X j,i,k = 1 ( i CS k K) (3) X i,j,k 1 ( i RO) (4) j RO k K X j,i,k 1 ( i RO) (5) j RO k K 11

12 X i,j,k = X j,i,k ( i CD k K) (6) j RO j RO P {τ (T i,k + τ i,j,k T j,k ) (1 X i,j,k )M} 1 α T ( i, j C k K) (7) T i,k = 0 ( i CS k K) (8) T i,k X i,j,k M ( i CD k K) (9) j RO Constraint (1) specifies that the number of vehicles to service must not exceed the available quantity ready at each supply node at the beginning of the planning horizon. The number of vehicles to service is stated by the total number of vehicles flowing from the dummy central depot 0 to the real supply nodes. Constraints (2) and (3) represent each vehicle flow from and back to only one supply depot, which are not necessarily the same. Constraints (4) and (5) state that each demand node must be visited at least once, which also allows for split delivery, e.g. multiple visits for a demand node. Constraint (6) requires that all vehicles who flow into a demand point must flow out of it. The inner part: (T i,k + τ i,j,k T j,k ) (1 X i,j,k )M in constraint (7) guarantees schedule feasibility with respect to time considerations. Chance constraints allow this constraint to be violated by a small possibility α T. The fact that all vehicles leave the depots at time 0 is specified by constraint (8). Constraint (9) sets the visit time to be zero if the vehicle does not service a node. s a,i Y a,i,j,k Y a,j,i,k 0 ( a A, i CS) (10) k K j C j C Y a,0,i,k + Y a,i,0,k = 0 ( a A, i RO, k K) (11) Constraints (10)-(11) state the construction on the commodity flows. Constraint (10) provides the balancing material flow requirement on the supply nodes. Constraint (11) prohibits the commodity flows from and to the dummy node. Since the dummy node is only used as a dummy vehicle depot, which will not accommodate any commodities. X i,j,k c k a A w a Y a,i,j,k ( i, j RO, k K, a A) (12) X i,j,k c k M Y a,i,j,k + Y a,j,i,k j RO a A j RO j RO ( i CD, k K, a A) (13) 12

13 Constraint (12) allows the flow of commodities as long as there is sufficient vehicle capacity. Constraint (13) requires that whenever a vehicle visits a demand node, it must carry some commodities either from or to that node. These two constraints establish the connection between the commodity flow and vehicle flow. e a,i j + (i) X i,j,k T i,k l a,i X i,j,k ( a A, i, j C k K) (14) j + (i) P ζ Y a,j,i,k Y a,i,j,k U a,i ζ a,i 0 1 α D k K j C j C ( a A, i CD)(15) Constraint (14) enforces the time-windows constraints on each demand point. Constraint (15) enforces the balanced material flow requirement for demand nodes from a probability perspective. It states that a small probability for unmet demands at each demand node is allowed but within a given threshold. X i,j,k binary; Y a,i,j,k 0; U a,i 0 T i,k 0; (16) Constraint (16) states the non-negativity properties of the decision variables. This vehicle routing problem considers all properties relevant to large-scale emergencies discussed in section 2. The problem minimizes unmet demand and considers time windows explicitly. The other four major properties of the vehicle routing problem for largescale emergencies described in section 2 have been addressed explicitly in the constraints. Stochastic demands and travel times are reflected in constraints (15) and (7) respectively. Constraints (1)-(3) indicate the multi-depot nature of this problem. Constraints (4)-(6) allow for split delivery at any demand node. This model accommodates the emergency situation where late deliveries could lead to fatalities, e.g. an anthrax attack. Anthrax is an acute infectious disease caused by a spore-forming bacterium. At the earliest sign of the disease, adult patients should be vaccinated and treated with antibiotics. Otherwise, shock and death ensue within 24 to 36 hours. To maximize the life-saving in the anthrax emergency, medications should be sent to the affected people within 24 hours of the onset of symptoms to impact the patient survival. Hence, to minimize the life-loss at any demand area, the medical supplies must arrive within this time-window. At the same time, we intend to minimize the total unmet demands, which indirectly reflect the predicted death toll. Hence model CCP, which enforces time-windows constraints, is well suited for this kind of emergency scenario. 13

14 5 Solution Approach In this section, a two-stage solution approach for our problem is introduced and its pros and cons are discussed. The first stage consists of determining the pre-planned routes by solving the CCP model and is executed well in advance of any possible emergency. These pre-planned routes can be used for mock trial runs and training purposes. The second stage is executed at the time of the emergency when more information becomes known and a fast algorithm is needed to determine the exact delivery requirements. In this stage, a linear program model is employed to determine the delivery requirements. In the planning stage, the CCP model is solved to determine the vehicle flows (the routes) and the commodity flows (delivery quantities) at the same time; however, the solution of the Y a,i,j,k variables (delivery quantities) in the CCP model does not have significant meaning for the final delivery amount since the actual demand quantity is subject to tremendous fluctuations in each emergency scenario. Hence, in the second stage, after the emergency has occurred, at which time the value of the random parameters have been revealed, we solve a linear programming problem to decide the commodity quantity to be carried and delivered by each vehicle based on the determined routes from the first stage CCP. It is true there will be some fluctuation to the transit times and some adjustments to the pre-planned routes may need to be made, but in general there are significant training advantages to having pre-planned routes in place in order to be able to react effectively to the emergency. Since most of the input parameters are known and only the commodity flows are decision variables in the second stage, this optimization model is a deterministic linear programming problem (DLP) with the same basic structure as the CCP model we presented in section 4. But we can see that the vehicle flow X i,j,k and service start time T i,k are no longer decision variables; instead, X i,j,k is the given input from the results of the CCP model. T i,k can be calculated consequently with the given sequence of visiting according to the X i,j,k value. We eliminate those demand points whose T i,k already missed the deadline and minimize the unmet demand by adjusting the commodity flow Y a,i,j,k. We formally show the model as follows: Model DLP: minimize U a,i a A i CD subject to: Constraints (10)-(13) Y a,j,i,k Y a,i,j,k U a,i d a,i 0 ( a A, i CD) (17) k K j C j C Y a,i,j,k 0; U a,i 0; (18) 14

15 The d a,i in constraint (17) is the given demand quantity at each node. This constraint is the deterministic counterpart of constraint (15) in CCP. Constraint (18) shows the non-negativity properties of the decision variables and the DLP is a linear programming problem. In the DLP, we are not guaranteed to obtain the optimal delivery quantities given the routes because the LP assumes instantaneous deliveries. We use the vehicle flow from the CCP solution, which allows for split delivery (e.g. a single demand node could be served by more than one vehicle to fulfill its requirement on the delivery quantity); hence some of the demand nodes may be transshipment points leading to a vehicle dropping a quantity more than necessary at the node, and later on, another vehicle picking up the extra quantity to serve the following nodes on its route. This type of route is most likely generated with capacitated vehicles. Since the LP does not constrain the sequence of visiting a node by multiple vehicles, we assume the simultaneous delivery of the quantities. Hence the optimal solution to the DLP model is an approximation to the actual problem at hand. In reality, an early arriving vehicle that is scheduled for a pickup will have to wait for the other vehicle carrying the commodity to arrive at the transshipment point before it can depart. Although this might affect some routes to meet the deadline at a later node, it is still the simplest, efficient and computational affordable way to solve this problem. In the next section, we demonstrate the efficiency of this approach as opposed to re-solving the integer program in the second stage. We tackle the problem by this two-stage approach because of two main reasons. First, if we solve the entire problem as an integer programming problem after the outbreak, at which time most of the parameters become known, we might be able to obtain the real optimal solution. But this approach may not be computationally feasible due to the time complexity of solving the integer programming model and time is of the essence once the emergency has occurred in order to save lives. Moreover, under an emergency situation, the problem parameters are still subject to change from time to time, so the deterministic solution is not only time-consuming to obtain but also is not robust to any subsequent changes. On contrast, in the two-stage approach, we move the heavy computational load of the IP problem to be solved in advance when we have enough time and computational resources; after the emergency outbreak, only the DLP is required to be solved, whose computational load is light and can be efficiently solved. Second, from a practitioners perspective, we believe that it is of great benefit if the routing plan can be pre-determined (even if it is subject to changes after the emergency occurs) so that the responders can have pragmatic drills in advance and be better prepared to react to the emergency in a real situation. In this two-stage approach, the route plan is relatively stable, unless the expected distributions of the stochastic parameters are changed significantly (for example, some freeways have been destroyed so that the expected travel time between two demand nodes has been dramatically increased). Then, we have to regenerate the routing plan by running the CCP for the new situation. 15

16 6 Illustrative Example In this section, we demonstrate how the proposed CCP model in section 4 and the solution approach in section 5 can be used to solve the VRP for an anthrax emergency scenario. We first describe the input parameters for our example and then we show the result and analyze its quality by comparing it with the result of a deterministic model through simulation experiments. In this example, we consider an anthrax attack has occurred and requires a universal dispensing of prophylactic treatment and/or vaccination (if available) to the population. As we described in section 2, the medical supplies are usually not stored at the local level, during an anthrax emergency the SNS responds as much as possible to the demands at the local areas (we use Los Angeles County in the example). The initial push package supply of material is received at the region s primary distribution warehouse within 12 hours of the incident. Thereafter, followup supplies may be delivered as available from managed inventory locations, with varying delivery times. The push package containers are opened and divided into smaller loads which may take a few hours and are then sent to pre-selected dispensing sites which are determined by solving a related facility location problem (Jia et al., 2005). The problem of routing inventory from the primary distribution warehouse to dispensing sites is illustrated here. In the chance constrained formulation, with the given distribution of the stochastic parameters, the probabilistic constraints can be transformed into deterministic ones in a straightforward manner. We assume a normal distribution for all random variables for illustrative purposes. At the same time, capacitated and uncapacitated vehicles are used respectively in the experiments. Therefore, two categories of test cases are solved and compared. The setting of the input parameters is described in the following subsection. 6.1 Parameter Specification The input parameters can be categorized into three parts: related to demand nodes, supply depots and vehicles. The ten demand points are the pre-selected local dispensing sites which are the direct results from solving a facility location problem (Jia et al., 2005). We use the Euclidean distances between any two of the nodes and assume a symmetric complete graph topology. The mean travel time between any two nodes is proportional to the distance. We can also determine the demand on each selected dispensing site by summing up the population in the areas which are covered by it. The population size is used as the criteria to specify the demand size; for example, 1 container of 100,000- dose anthrax vaccine is needed for every 100,000 people. We show the mean distance between the nodes and the mean demand quantity at each demand node in table 1 and table 2, respectively. As we previously stated, the demand of each dispensing site and 16

17 travel time between the nodes are stochastic variables. We use a normal distribution with the population size and Euclidean distance as the given expected mean value; the standard deviation is set to 20 percent of the mean value. For simplicity, we use a single commodity and a single depot in the example. The Los Angeles Airport (LAX) is used as the distribution warehouse (depot). The base quantity of the total supply at the depot is set to be the summation of the demand quantity at all the demand nodes. We use 80%, 95%, 100%, 110% and 120% of this base quantity as the available supply quantity for different test cases in the computational experiments. Ten dispensing sites are served by three uniform capacity vehicles. We experiment on both the capacitated case and uncapacitated case. For the capacitated case, we set the capacity of each vehicle to 40% of the total demands at all the demand nodes, and 100% of the total demands for the uncapacitated case. Finally, the deadline of the service start time at each demand node is set to be some multiple of the average length of all the edges in the graph times 3.3 (10 demand nodes are served by 3 vehicles; so on average, each vehicle serves 3.3 demand nodes). The experimented multiples are 105%, 120% and 145% of the base length. We summarize the numerical experiments as follows. Two major categories are classified according to the vehicle capacity; within each category, 15 test cases are identified by the different combinations of the deadline on the demand nodes (3 types: 105%, 120%, 145%) and total supply at the depot (5 types: 80%, 95%, 100%, 110%, 120%). depot dpt1 dpt2 dpt3 dpt4 dpt5 dpt6 dpt7 dpt8 dpt9 dpt10 depot dpt dpt dpt dpt dpt dpt dpt dpt dpt dpt Table 1: Euclidean distance between depot and demand points (Unit: Mile) Demand Quantity dpt1 dpt2 dpt3 dpt4 dpt5 dpt6 dpt7 dpt8 dpt9 dpt Table 2: Expected demand quantity at each demand point (Unit: 100,000-dose package ) 17

18 6.2 Solution and Analysis Our first set of experiments test the efficiency and robustness of our chance constrained model in determining pre-planned routes. We compare solutions from our CCP model against routes generated by a deterministic integer program model that uses the mean values of the demand quantity and travel times as problem parameters. The effectiveness of the routes generated by both the CCP model and the deterministic integer program in minimizing the unmet demand are tested on random instances of the problem parameters. For each random instance, a new DLP is solved to determine the delivery quantity keeping the pre-planned routes fixed. The CPLEX solver is used to solve both the deterministic and chance constrained models. We utilize CPLEX 9.0 with default settings on a 3.2 GHz CPU with 2GB RAM and we allow a maximum CPU time of 1 hour for each case. Within one hour of CPU time, some of the test cases can be solved to optimality for both the deterministic and chance constrained models. For each test case which can reach the optimality for both models, we generate 50 random instances of the problem parameters of the demand quantities and travel times. As mentioned above, for each instance the DLP described in section 5 is solved to determine the delivery quantity and total unmet demand, constrained by the deadline and the total available quantity at the depot and using the deterministic routes and chance constrained routes respectively. By observing 50 instances in the simulation instance pool for each test case, we can calculate the average percentage of the unmet demand over the total demand for both the deterministic and chance constrained routes. The results are summarized in the tables below (the a b pair in the left most column represents deadline totalsupply pair as explained in section 6.1). Nor Cap100 Deterministic Chance constrained % 5.16% % 0.06% % 0.06% % % % 0.12% % 0.0 Table 3(a): Average percentage of the unmet demand over the total demand with an uncapacitated fleet (normal distribution of random parameters). 18

19 Nor Cap40 Deterministic Chance constrained % % 0.30% % % % % % 0.0 Table 3(b): Average percentage of the unmet demand over the total demand with a capacitated fleet (normal distribution of random parameters). From the results in the above tables, we can conclude that the chance constrained routes outperform the deterministic ones. We believe that it is because of the conservative nature of the chance constrained model, which leads to similar number of nodes in different routes. In contrast, the deterministic routes are more prone to have an uneven number of nodes between different paths leading to a higher chance of having unmet demand. To further test the quality of the chance constrained routes and our two-stage approach for solving the problem, we compare the final unmet demand solution of the 350 generated random instances from our two-stage approach to an integer program formulation with the value of the demands and travel times equal to the simulation instances as the input parameters. This latter approach can be viewed as solving the entire problem as an integer program in real-time when the problem parameters become known. It serves as a lower bound to our two-stage approach. Again, we use CPLEX to solve this integer program and again a one hour CPU time limit is used. In cases where CPLEX is not able to find an optimal solution within the one hour CPU time, we record the lower bound given by CPLEX. We note that we only need to solve this new integer program for the problem instances in which our two-stage approach has an unmet demand value greater than zero. The table below shows the results of this set of experiments. Nor Cap avg. gap No. of opti Table 4(a): Comparison of 2-stage solution approach (DLP with pre-determined chance constrained routes) to the lower bound of each of the 50 instances with an uncapacitated fleet. Nor Cap avg. gap No. of opti

20 Table 4(b): Comparison of 2-stage solution approach (DLP with pre-determined chance constrained routes) to the lower bound of each of the 50 instances with a capacitated fleet. According to the above tables, by summing up the instances from different test cases, 97.14% and 98.29% of the problem instances reach optimality for the uncapacitated and capacitated categories respectively (note that there are a total of 350 random problem instances in each category); only a small proportion of the instances have some gap with the lower bound. These instances appear in the tighter time-windows condition. The results show that using the pre-planned routes from the CCP model in conjunction with the DLP model is not far from optimality. In addition, this two-stage approach has significant computational advantages over solving the integer programming model in real-time when the emergency has occurred. 7 Future Work and Conclusions The CCP model in this work is a combinatorial optimization problem which prohibits the CPLEX solver from reaching optimality within a reasonable time when the problem size grows larger (more than demand nodes). Therefore, it is natural to develop a fast heuristic algorithm for this model and evaluate its solution on a real-world size problem (200 and above demand nodes) in terms of both quality and performance. The classical constructive and improving heuristics for the standard VRP and some meta-heuristics will be investigated and geared specifically to this problem. We also discuss some topics that we have not incorporated into our models proposed in section 4. We believe that these topics are important for large-scale emergencies setting and will be our ongoing research directions. Vehicle Depots vs. Inventory Depots: In the real situation, the sets of vehicle depots and inventory depots are usually not overlapped. For the normal routing problem, the time or distance used from the vehicle depots to the inventory depots, in most cases, is not contingent for the problem solved, so it is seldom given attention. Nevertheless, for large-scale emergencies, time is always the most crucial issue immediately after the occurrence at which time the vehicles are still in the vehicle depots. Hence, a better routing policy should deploy the vehicles starting from the vehicle depots to the inventory depots to collect commodities and then distribute to the demand points. Deterministic vs. Stochastic Supplies: Supplies are deterministic for most regular routing problems. However, the huge unpredictable impact caused by large- 20

21 scale emergencies might destroy some local stockpile sites, block the supply connection from national or regional stockpile to local sites, etc. Such unpredictability impacts the consequences of the emergencies causing the supplies to also be stochastic variables. Multiple Transportation Modes: Not only is the fleet heterogeneous, but also more than a single transportation mode might be applied in the transportation network for large-scale emergencies. Different transportation modes will result in different cost/travel times on the same arc (e.g. by railway, by truck, by river, by sea, etc.). This implies that there might be more than one arc linking a pair of nodes, each arc representing a different mode. In this paper, we primarily focus on two goals. The first is to review the literatures related with the vehicle routing problem in a large-scale emergency setting, which mainly includes the stochastic VRP, variants of the VRP related with our problem, routing/dispatching applications for regular emergencies and logistic planning for large-scale emergencies. The second goal is to analyze the characteristics of the stochastic vehicle routing problems for large-scale emergencies and propose a chance constrained model. We also present a two-stage approach for solving the problem and some illustrative computational results. Suggestion for ongoing research directions of this topic are finally discussed. Acknowledgment This research was supported by the United States Department of Homeland Security through the Center for Risk and Economic Analysis of Terrorism Events (CREATE), grant number EMW-2004-GR However, any opinions, findings, and conclusions or recommendations in this document are those of the author(s) and do not necessarily reflect views of the U.S. Department of Homeland Security. Also the authors wish to thank Hongzhong Jia, Harry Bowman, Richard Larson and Terry O Sullivan for their valuable input and comments for the improvement of this paper. References [1] C. Archetti, R. Mansini, and M. Speranza. The split delivery vehicle routing problem with small capacity. Technical Report 201, Department of Quantitative Methods, University of Brescia, Italy, [2] C. Archetti and M. Speranza. A direct trip algorithm for the k-split delivery vehicle routing problem. Technical Report 205, Department of Quantitative Methods, University of Brescia, Italy,