HAZMAT Transportation and Security: Survey and Directions for Future Research

Transcription

1 HAZMAT Transportation and Security: Survey and Directions for Future Research James Luedtke Chelsea C White III Department of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA August 1, Motivation The U.S. Department of Transportation (DOT) Office of Hazardous Materials Safety estimates that an average of over 800,000 shipments of hazardous materials (HAZMAT) are made each day in the United States [27]. The DOT also reports that approximately 1.5 billion tons of hazardous materials are shipped each year [26]. The majority of the hazardous material shipped is classified as flammable liquid (e.g. petroleum). Other types of hazardous material include (in decreasing order of annual amount shipped) corrosive materials, gases, flammable solids, toxics and radioactive materials. The DOT also reports statistics on incidents involving hazardous materials shipping and their consequences. For 1999, the DOT reported five fatalities, 217 injured persons, and $23 million in property damage associated with hazardous materials transportation on highways [28]. The above statistics (fortunately) do not include any cases where HAZMAT trucks were intentionally used to attack a vulnerable site. Unfortunately, it is a reality that there is now a risk of such an event occurring and the consequences could be enormous. Whereas HAZMAT incidents occurring as a result of accidents are a risk to people on the road and living near the location where the accident occurs, an explosion or release of hazardous material at a vulnerable site such as a nuclear power plant or large population water source can greatly amplify the amount of people at risk. In the past, HAZMAT routes have only been planned to be sufficiently far away from such sites that if an accident occurred on the route the site would not be affected. But the possibility of a rogue driver whose intent is to attack a vulnerable site makes that approach inadequate. Routing decisions are needed that minimize the probability of a successful attack, and these decisions must be made in conjunction with decisions on how to defend the vulnerable sites. 1

2 We consider a site to be a vulnerable site if either (a) damage done to the site can have significant effects extending beyond the site itself or (b) the site itself contains a very large number of people. Examples of sites that fit definition (a) are a nuclear power plant where a radioactive release could poison residents in a large area near the plant, or a water treatment facility from which poisoned water could reach many residents. An example of a site fitting definition (b) is a sporting event stadium. This is a simplistic version of how we believe hazardous materials routing is currently done. Using [8], states and other regulatory bodies create designated and restricted HAZMAT routes. HAZMAT carriers must travel on designated routes when available and are forbidden to travel on restricted routes. If a state or a location does not establish either of the above, we assume HAZMAT carriers are free to operate without legal restriction. Within the restrictions imposed by the designated and restricted routes, individual HAZMAT carriers can then determine their own routes. We assume that this route determination is done on a least cost basis. 2 Basic Problem Definition Assume the following situation. There is road network, composed of intersections and roads connecting the intersections. A truck carrying HAZMATs is to travel from an origin to a destination. There is a vulnerable site and an interdiction vehicle meant to defend the vulnerable site. We assume that the origin, destination, vulnerable site, and interdiction vehicle are all initially located at different intersections. There is a probability that the truck driver is a rogue, with intent on attacking the vulnerable site. Under this circumstance, we assume that the amount of damage caused to the vulnerable site is given by the function D(n, n ), where n is the intersection where the truck discharges and n is the intersection of the vulnerable site. The objective of a rogue driver is to maximize damage; the objective of the interdiction vehicle is to interdict the truck at a point where the damage is minimized. We assume that the truck discharges immediately upon interdiction. There are two objectives for the problem. The first is to minimize expected damage due to discharge. The second is to minimize cost of transporting the hazardous materials from origin to destination, assuming the driver is a secure driver. The problem is to choose a route from origin to destination that minimizes a weighted sum of these two costs, where we may not know precisely the function D, the probability that the driver is a rogue, and the trade off weight between objective 1 and objective 2. We assume that the driver of the truck reveals himself/herself as a rogue driver if and only if the truck leaves the determined route. Immediately upon leaving the determined route, the interdiction vehicle becomes aware that the driver is a rogue, intending to maximize damage to the vulnerable site. Due to the possibility of placing Global Positioning Systems on HAZMAT 2

3 trucks, we assume that the interdictor will always have the advantage of knowing where the rogue driver is. On the other hand, the rogue driver only knows that the interdictor exists and is trying to prevent him from reaching the target. We assume that at discrete time units (e.g. every 30 seconds) the interdictor and rogue can decide to change their direction or speed. For the rogue this decision is based only on the time and the rogue s current location; for the interdictor the decision is based on the time and current locations of both the interdictor and the rogue. The decisions to be made in the problem are the planned route of the HAZ- MAT truck, the best patrol strategy for the interdictor vehicle, the best evasion strategy for the rogue driver (including where to leave the planned path), and the best pursuit strategy for the interdictor once the rogue has left the path. We assume the routing and interdiction decisions can be made by the same party in order to find globally satisfactory solutions. 3 Extensions The above basic model can be extended to deal with a wider variety of objectives and more realistic operating assumptions. An additional objective that may be considered is to minimize risk due to accidents; this is the objective most commonly considered in the HAZMAT literature. Creating risk equity across the population so that there is no great discrepancy between the people most exposed and least exposed is another objective that can be considered. In addition to the obvious appeal of fairness, enforcing risk equity may have a practical benefit of increasing the chances a selected solution will be accepted by the population. It is likely the case that many different HAZMAT shipments travel near a vulnerable site, and so consideration of multiple shipments in the above model would be a necessary extension. In addition, more than one interdiction vehicle may be necessary to properly defend a site and/or an interdiction vehicle may be able to defend multiple sites. Hence, consideration of multiple interdiction vehicles and multiple vulnerable sites would make the problem more realistic. In the above model of the interdiction problem, we assumed the interdictor had complete and real-time information about the location of the rogue. In reality there may be a delay in this information, reducing its value to the interdictor. In addition, it may be possible for the rogue to disconnect the global positioning unit, in which case information about the rogue s location would be greatly reduced, if known at all. Investigation into the effects of different levels of information on the game would be interesting research and would make the model more realistic. There are at least two reasons why scheduling the times of HAZMAT shipments should be considered in addition to their routing. First, some of the risk measures, especially the accidental risk measures, may vary by time of day. For example, accident probabilities tend to be higher at night. Second, it may be possible to more effectively and efficiently defend a vulnerable site if HAZMAT 3

4 shipments are scheduled so as to prevent multiple shipments from coming near the vulnerable site at approximately the same time. Because randomness is an inherent part of a transportation network, it would be useful to generalize the selection of routes to the selection of routing policies that consider the current information at each decision point. With this generalization, it would be possible to incorporate real-time re-routing, since the policy scenario is really just real-time routing as no a priori route is set. This generalization could greatly improve the safety and efficiency of a HAZMAT transportation network by considering much more information (e.g. weather conditions, traffic, interdictor status) than fixed routes consider. Another perspective on the decisions to be made could be based on the way we believe HAZMAT routing is currently done. We could take the position of a regulating body creating hazardous materials designated and restricted routes. We could assume that we know all the volumes of HAZMATS to be shipped, and that given the restricted and designated routes, these volumes will all be shipped according to least cost on the eligible road network. With these assumptions we could then try to decide on the designated and restricted HAZMAT routes so as to minimize risk while not overly burdening the transporters with excessive costs. 4 Literature Review The problem we consider deals with routing of HAZMAT shipments and interception of an attacking rogue driver. Research on hazardous materials routing is relevant for the first part of the problem and pursuit and evasion research is relevant to the latter. We give a brief summary of research done in both these areas, and also discuss some of the government documents discussing HAZMAT transportation. 4.1 Hazardous Materials In a comprehensive survey of HAZMAT transportation research, List et al. [10] classify the research into the categories of risk analysis, routing/scheduling and facility location. Risk analysis is concerned with the appropriate ways to assess risk, including assessment of incident probabilities and degrees of consequences when incidents occur. Routing and scheduling problems focus on finding appropriate routes according to a variety of competing objectives including cost, some measure(s) of risk, and perhaps even a measure of risk equity. Facility location problems consider similar objectives in locating facilities that ship and receive HAZMATs. The problem we are addressing is most closely related to routing and scheduling so we will restrict our attention to research in this area. Another good survey of HAZMAT logistics can be found in [5]. The authors of that book chapter discuss some relevant risk assessment methods, location models and transport planning models. They also discuss the roles of cost and equity in HAZMAT logistics decisions. 4

5 Sivakumar et al. [25] consider a conditional risk model motivated by their belief that expected consequence (probability of incident times consequence of incident) is an inadequate measure of risk since it does not properly capture increased aversion to the highest consequence incidents. Instead, they assume HAZMATs will be shipped along a set of routes repeatedly until an accident occurs, and they seek to minimize the expected consequence given that the accident has occurred. They include risk equity and cost constraints in their model and use a transformation to change their originally nonlinear model into a linear one. They then propose a column generation and set partitioning approach to solving their model. In a critique of the conditional risk approach, Erkut [4] gives a number of examples in which the conditional risk criteria seems to give obviously inferior solutions, and concludes that it is an inappropriate risk measure. Sherali et al. [23] use an objective similar to Sivakumar et al., but to alleviate the concerns brought up by Erkut, they include constraints on the overall expected value of consequence and accident probability to prevent the model from choosing illogical routes. They then develop a branch-and-bound solution method and perform a case study which includes discussion on how to collect data for the HAZMAT problem. By considering probability of traversing a path without an accident as the indicator of risk, Marianov and Revelle [11] are able to consider a simple linear bi-objective (probability and cost) model for HAZMAT transportation. In addition, by introducing fictitious nodes, their model includes the ability to model increased accident probability at nodes of the transportation network. The major limitation of their model is that solely using probability of an accident as a risk measure does not consider the potentially wide variance in the level of consequence associated with an accident. Motivated by the observation that the arc attributes related to risk are time-varying, Nozick et al. [16] consider scheduling in addition to routing. For example, accident probabilities are higher at night, while the number of people exposed can be lower at night. They heuristically extend a multiple-criteria shortest path algorithm to this case and show through a hypothetical case study that the set of non-dominated solutions created when scheduling is considered is significantly different than the set created when no scheduling is considered. Zografos and Androutsopoulos also consider the routing and scheduling of hazardous materials [31]. They consider the objectives of cost and risk, and assume their risk measure is additive across arcs. They assume that customers must be served within specified time windows and hence model the problem as a bi-objective vehicle routing problem with time windows. They propose a variant of an insertion heuristic to solve the problem in a reasonable amount of time, which they claim performs better than other heuristics for this problem. In [24] Sivakumar and Batta introduce a Lagrangian relaxation with gapclosing solution methodology for solving variance constrained shortest path problems. They claim that this model can be useful for HAZMAT routing problems in which you want to be assured your risk level does not vary too wildly from the expected risk. 5

6 Nembhard and White [13] consider the possibility that path attribute values may not be simply additive across arcs in the path, causing the criterion to be non-order-preserving (NOP). This prevents the use of traditional dynamic programming techniques in the solution method. To deal with this, the same authors develop in [14] an algorithm that works with NOP criterion. Akgün et al. [1] consider various methods, including one they developed, for finding spatially dissimilar paths. This can be useful in the context of HAZMAT routing, in particular when equity is considered as an objective, because it may be desirable to have multiple routes that are significantly different in order to evenly spread the risk. Another topic of interest in HAZMAT routing is the potential for real time re-routing in order to react to changing conditions. Beroggi and Wallace [3] consider a situation in which a dispatcher receives real-time information such as weather and traffic reports and can use that information to re-direct shipments if appropriate. They compare four decision support systems to aid the dispatcher in this decision process, all of which rely on the dispatcher to make subjective decisions about the level of increase in risk due to the real-time events. One of the few papers considering randomness in HAZMAT transportation was written by Wijeratne et al. [29]. They consider a multiobjective routing network in which arc attributes are normal random variables and they assume that all path attributes are a sum of the arc attributes. They develop some simple approximations to standard stochastic dominance based on the moments of the attributes and use these approximations in a deterministic algorithm for the search of non-dominated solutions. We found no HAZMAT research which considers the possibility of rogue drivers attacking a vulnerable site. 4.2 Pursuit and Evasion Pursuit and evasion problems are similar to the rogue driver interdiction problem in that they consider a two-person game in which the goal of one party is to capture the other, while the other party wants to avoid being captured. However, there is an important difference between pursuit and evasion and the problem we consider: in our problem the rogue driver does not simply want to avoid being captured, he/she wants to reach or at least get as close as possible to a target before being captured. An early study of a pursuit evasion problem is given by Ryll-Nardzewski [21]. This study defines a very general framework for pursuit evasion and gives a criterion for existence of a minimax solution. One class of pursuit evasion problem is the herding problem, in which the pursuer tries to herd an evader into a pen. Kachroo et al. discuss a special case of this problem in which the evader moves stochastically and independently of the location of the pursuer [9]. In this special case, stochastic dynamic programming can be used to find the best pursuer strategy to minimize expected time until evader is caught. 6

7 Another problem which has been studied is pursuit and evasion on a grid, with the assumptions that the pursuer can only see across the rows and columns, but the fugitive always knows where the pursuer is. In this framework, research has been done to determine the minimum speed required by the pursuer to guarantee the evader can be caught (the evader s max speed is fixed). Neufeld showed that if the grid is n m, with n m, and the evader has unit maximum speed, then a pursuer with maximum speed greater than 2n/3 can always catch the evader [15]. When there may be multiple pursuers and a single evader travelling on a finite graph, a problem that has been studied is to determine the minimum number of pursuers needed to guarantee a capture of the evader. This problem has been shown to be NP-hard [12]. Some very complicated models of pursuit evasion problems have been made with the pursuer being a missile and the evader being an aircraft. These models allow motion in three dimensions, but have constraints on the evader such as minimum altitude and maximum dynamic pressure. Once again, the objective is for the evader to maximize time before capture. An example of a paper dealing with this type of model is given in [19]. A problem in which the evader has a goal to reach a target is considered in [18]. In this paper, it is assumed that when on the same node, the pursuer catches the evader with a probability less than one. The pursuer and evader start in node one, and in each time unit they can move forward one, backward one, or stay in the same node. The evader wants to reach node n. With this simple structure, the author is able to find an optimal strategy. These are just a few of the many papers on pursuit evasion. A very large bibliography of pursuit evasion differential games published before 1989 is given in [20]. We have attempted to give a flavor for the types of pursuit evasion problems that may be most relevant to the rogue interdiction problem, but a more thorough search is needed. In general, the pursuit evasion problem is a special type of stochastic or Markov game. Although none of the above pursuit evasion problems exactly matches the rogue interdiction problem, it can be formulated as a Markov game in a similar way as the pursuit evasion problem. There are many references on stochastic and Markov games. In [22] Shapley defines a stochastic game, defines a strategy in the game, and proves the existence of an optimal saddlepoint solution. Zachrisson discusses Markov games in [30] and Hordijk and Kallenberg discuss a special case of Markov games which can be solved by linear programming [7]. A couple books which cover topics related to dynamic games of this sort have been written [6], [2]. 4.3 Government Documents The U.S. DOT has published a guide which is available on their web site [8] for states and other regulatory bodies for establishing restricted and designated HAZMAT routes. This guide includes recommendations on how to staff a project for designating HAZMAT routes, how to define alternative routes for 7

8 consideration and how to assess the risk for the alternative routes. The guide also recommends considering other issues such as avoiding sensitive environmental areas and minimizing undue burden to the transportation providers. The guide suggests that it is important to get public input during the process and to publicize the routes afterwards. Finally, an example of the process is given. The U.S. DOT also publishes a biennial report, which can be found online, on the status of HAZMAT transportation in the United States [17]. This report includes a summary of the current HAZMAT regulations, recent safety performance data and information about the programs in place to improve HAZMAT safety. References [1] Vedat Akgün, Erhan Erkut, and Rajan Batta. On finding dissimilar paths. European Journal of Operational Research, 121: , [2] T. Basar and G.J. Olsder. Dynamic Noncooperative Game Theory. Academic, London, [3] Giampiero E. G. Beroggi and William A. Wallace. Operational control of the transportation of hazardous materials: an assessment of alternative decision models. Management Science, 41: , Dec [4] Erhan Erkut. On the credibility of the conditional risk model for routing hazardous materials. Operations Research Letters, 18:49 52, [5] Erhan Erkut and Vedat Verter. Facility location: a survey of applications and methods. Springer, New York, [6] J. Filar and O.J. Vrieze. Competitive Markov Decision Processes: Theory Algorithms and Applications [7] A. Hordijk and L.C.M. Kallenberg. Linear programming and markov games. In Game Theory and Mathematical Economics. Proceedings of the Seminar, pages , [8] National Highway Institute. Highway routing of hazardous materials, guidelines for applying criteria. Technical Report FHWA-HI , U.S. Department of Transportation, Nov Available at [9] Pushkin Kachroo, Samy A. Shedied, and Hugh Vanlandingham. Pursuit evasion: the herding noncooperative dynamic game the stochastic model. IEEE Transactions on Systems, Man, and Cybernetics Part C: Applications and Reviews, 32:37 42, Feb [10] George F. List, Pitu B. Mirchandani, Mark A. Turnquist, and Konstantinos G. Zografos. Modeling and analysis for hazardous materials transportation: Risk analysis, routing/scheduling and facility location. Transportation Science, 25: , May

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10 [25] Raj A. Sivakumar, Rajan Batta, and Mark H. Karwan. A multiple route conditional risk model for transporting hazardous materials. INFOR, pages 20 33, Feb [26] Bureau of Transportation Statistics U.S. Department of Transportation and Census Bureau U.S. Department of Commerce economic census 1997 commodity flow survey, hazardous materials. Washington, DC, Dec Available at [27] Office of Hazardous Materials Safety U.S. Department of Transportation. Hazardous materials shipments. Washington, DC, Oct Available at hazmat.dot.gov. [28] Office of Hazardous Materials Safety U.S. Department of Transportation. Hazardous materials information system database. Washington, DC, Oct [29] Ajith B. Wijeratne, Mark A. Turnquist, and Pitu B. Mirchandani. Multiobjective routing of hazardous materials in stochastic networks. European Journal of Operational Research, 65:33 43, [30] Lars E. Zachrisson. Markov games. In M. Drezner, editor, Advances in Game Theory, pages Princeton University Press, Princeton, NJ, [31] Konstantinos G. Zografos and Konstantinos N. Androutsopoulos. A heuristic algorithm for solving hazardous materials distribution problems. European Journal of Operational Research. Accepted for publication. 10