Optimizing facility use restrictions for the movement of hazardous materials

Transportation Research Part B 44 (2010) 267–281

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Optimizing facility use restrictions for the movement of hazardous materials Yashoda Dadkar a, Linda Nozick a,*, Dean Jones b,1 a b

School of Civil and Environmental Engineering, Cornell University, Hollister Hall, Ithaca, NY 14853, United States Sandia National Laboratories, P.O. Box 5800 MS 1138, Albuquerque, NM 87185-1138, United States

a r t i c l e

i n f o

Article history: Received 3 March 2008 Received in revised form 15 July 2009 Accepted 20 July 2009

Keywords: Hazardous materials Routing Game theory

a b s t r a c t The modeling tools that have been developed over the last 25 years for the identification of routes for hazmat shipments emphasize the tradeoffs between cost minimization to the shipper/carrier and controlling the ‘‘natural” consequences that would stem from an accident. As the terrorist threat has grown, it has become clear that a new perspective, which allows for the representation of the goals and activities of terrorists, must be incorporated into these routing models. Government agencies can determine which specific facilities to restrict for each class of material and for which times of the day and/or week. This paper develops a game–theoretic model of the interactions among government agencies, shippers/carriers and terrorists as a framework for the analysis. It also develops an effective solution procedure for this game. Finally, it illustrates the methodology on a realistic case study.  2009 Elsevier Ltd. All rights reserved.

1. Introduction Approximately 800,000 shipments of hazardous materials (hazmat) move daily through the US transportation system (US Department of Transportation, 1998a,b) and approximately one truck in five on the US highways is carrying some form of hazardous material (US Department of Commerce, 1994). In general, the safety record is excellent, but public sensitivity to the risks associated with hazmat shipments is substantial. The public sector plays a direct role in managing the level of risk through the regulation of the transportation network. Government agencies can determine which specific facilities to restrict (or allow) for each class (or classes) of material and for which times of the day and/or week. Further these regulations may take the form of designations (e.g.: ‘‘preferred highway routes” for radioactive materials – US Department of Transportation, 1992) or prohibitions (e.g.: the State of New York prohibits movements of explosives across the Verrazano Narrows Bridge – US Department of Transportation, 2004). The development of a model to provide guidance to federal, state and local governments regarding which highway facilities might be restricted (for which materials and under which conditions) is a challenging but important activity. This is particularly important in the context of the US Department of Homeland Security’s Advisory System (color-coded ‘‘threat levels”). If threat level ‘‘Orange” or ‘‘Red” is declared, DHS suggests that government agencies consider constraining the transportation system ‘‘as appropriate”, but there is little specific guidance on how to determine what actions might be ‘‘appropriate” (US Department of Homeland Security, 2004). The goal of the model developed in this research is to lead to a better understanding of how to make such determinations.

* Corresponding author. Tel.: +1 607 255 6496; fax: +1 607 255 9004. E-mail addresses: [email protected] (Y. Dadkar), [email protected] (L. Nozick), [email protected] (D. Jones). 1 Tel.: +1 505 284 4886. 0191-2615/$ - see front matter  2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2009.07.006

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Government: Where, when and for which materials should restrictions on facilities be implemented, given beliefs about the likelihood of an attack?

Terrorist: Where an attack should be attempted, given the beliefs about what is in the trucks and where they are likely to be?

Shipper/Carrier: Which routes should be used, given the material transported and beliefs about the likelihood of an attack?

Fig. 1. Game–theoretic problem structure.

This research develops a game–theoretic model of the interactions among government agencies, shippers/carriers and terrorists as a framework for the analysis. Each of the relevant actors makes choices over time and has only partial control over the outcome realized as a result of their choices. For example, government agencies can make decisions to restrict the use of some parts of the transportation network, but the results of those choices are dependent on the actions of the shippers/carriers and the terrorists and conversely these decisions also cause repercussions for the shippers/carriers and the terrorist. The shipper/carrier selects a route given the rules established by government agencies. Beyond honoring government restrictions, the shipper/carrier is concerned about what the terrorist might do and this concern affects the choice of which route(s) to use. However, since the probability of an attack against any single shipment is likely to be very small, the goals of the shipper/carrier are still to move the material from one location to another economically while controlling the level of risk, but now the risks include both ‘‘natural” risks, including the probability of an accident and the effects should an accident occur and ‘‘induced” risks generated by potential terrorist activities. As for the terrorist, we can assume that their interest is in maximizing the damage they can inflict by targeting specific facilities. The choice of the facility is influenced by the route(s) chosen by the shipper/carrier and the regulations adopted by the government. Fig. 1 portrays the overall interactions among the actors in this game–theoretic framework. The inner box describes the relationship between the terrorist and the shipper/carrier. For this relationship, it is assumed that the regulations which indicate when (and for which materials) the shipper/carrier is allowed to use each facility are already specified. The analysis that links these two boxes together is normative. That is, given: (1) the shipper/carrier’s perception about the likelihood of an attack; (2) the choices available to the shipper/carrier and the terrorist; (3) the payoff to each, given their decisions and those made by their opponent and (4) the terrorist’s perception about what is in a given shipment, the analysis identifies the following. First, which routes should the shipper/carrier use (and with what frequency) and second, where should the terrorist attack (and with what frequency). The interactions between the shipper/carrier and the terrorist is modeled as a non-cooperative two-person non-zero sum game with the shipper/carrier wanting to maximize the value of the routes used when there is a known probability of an attack and the terrorist wanting to inflict as much damage as possible with an attack. The outer bigger box represents the influence of the government on those decisions. The government has the opportunity to control the severity of an attack (should one occur) through the use of facility prohibitions. These prohibitions can be sufficiently detailed so as to prohibit specific types of hazmat from certain facilities during specific time periods and to have those prohibitions change based on the perceived likelihood of an attack. These prohibitions bound the consequences of an attack and constrain the choices that the shipper/carrier can make. These recommendations are based on the identification of Nash equilibrium strategies in the inner game between the terrorist and the shipper/carrier, given the prohibitions. This paper makes three key contributions. First, a game is developed between the government, a shipper/carrier of hazardous materials and a terrorist. Second, we show how that game can be used to (1) determine the regulations that the government should consider adopting; (2) how those regulations should change as the probability of an attack increases; (3) the routes the shipper/carrier should consider along with the probability of use and (4) the facilities the terrorist should target along with the probabilities. Finally these ideas are applied to a realistic case study. The next section describes the key literature on which this paper has been developed. The third section describes the formulation. The fourth section develops a heuristic procedure to identify the solution. The fifth section develops a realistic case study and illustrates how the decisions made by the various players change as the probability of an attack changes. The sixth section describes key conclusions and opportunities for future research. 2. Literature review This paper draws on literature in three key areas. The first key area is routing of hazardous materials. While there is extensive literature in this area, we focus on discussion of common routing attributes and path finding in stochastic dynamic networks for brevity. The second area is transportation and terrorism. The third area is the game theory literature of direct

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relevance to this research, including the use of game theory to model terrorist activities and the application of game theory models to the modeling of transportation problems. The remainder of this section describes key research in each area. Shippers/carriers of hazardous materials are commonly concerned with minimizing the economic cost and the risk consequences of a release stemming from an accident when selecting a route or routes. Much of the economic cost to the carrier is proportional to the time taken for the shipment to travel from the origin to the destination. Thus cost minimization can be obtained through minimizing the total travel time. Defining public risk is more complex, but two characteristics – population exposure and accident rate – have gained wide acceptance. The population exposure, depending on the characteristics of the hazardous material to be transported, could be defined as the people in other vehicles that are within some distance of the shipment as it moves from its origin to its destination or it could be expanded to include the population residing within some distance of the shipment during its journey. The accident probability for a route is the probability of an accident happening along the route. Population exposure and accident rate can be combined into a single consequence measure for a route by summing across all links in the route the product of accident rate and exposure. Erkut and Verter (1998), Erkut and Ingolfsson (2000) and Chang et al. (2005), among others, use the same measure in their hazmat routing studies. Neither the travel time nor the consequence measure is deterministic since they both depend on the traffic volumes and activity patterns. Further they vary with the time of the day since characteristics like visibility, traffic volumes and activity patterns vary throughout the day. Using the expected values of the attributes ignores the variation resulting from these two causes and could lead to selection of routes that have acceptable expected values but perform ‘‘poorly” based on when they are actually used (Nozick et al., 1997). Hence, we represent the uncertainty for each attribute by a probability distribution which varies by the time-of-day. Dadkar et al. (2008) and Nielsen et al. (2005) developed K shortest path algorithms for stochastic and dynamic networks. Dadkar et al. (2008) focused on problem instances where the distribution of each link attribute is continuous whereas Nielsen et al. (2005) focused on problem instances where the distribution of the link attributes is discrete with integer values. In the routing of hazardous materials, the objectives of interest produce continuous distributions for link performance. Hence Dadkar et al. (2008) is more relevant to this application. Also, Nielsen et al. (2005) focused on identifying the exact solution to the K shortest paths problem which leads to computational challenges in large networks whereas Dadkar et al. (2008) developed an algorithm that is computationally feasible for large networks. Hence for the purposes of this analysis we use the algorithm in Dadkar et al. (2008). There is substantial interest in research related to transportation and terrorism. To date much of that research has focused on identifying what parts of the transportation system are vulnerable to terrorism and initial ideas of what should be done about those vulnerabilities. For example, see Szyliowicz and Viotti (1997), Chatterjee et al. (2001), Frederickson and LaPorte (2002), and Haimes and Longstaff (2002). Many game theory models have been developed to address a wide variety of problems. For a discussion of other game theory models see Fundenberg and Tirole (1995) or Kreps (1992). The seminal paper by Nash (1951) established the existence of equilibria for finite non-cooperative games but did not develop an algorithm for finding them. Among others, Mangasarian and Stone (1964) designed an algorithm for computing all the equilibria for finite, two-person, non-cooperative, non-zero sum games (also known as bimatrix games). They modeled a bimatrix game as a single integrated mathematical programming formulation and proved that each solution to this formulation is a Nash equilibrium point of that game. There have been several studies that have used game theory to model terrorism. Sandler and Arce (2003) and Sandler and Enders (2003) provide excellent literature reviews. These studies focus on a variety of issues including the effectiveness of the no-negotiation strategy (Lapan and Sandler, 1988 and Selten, 1988), the accommodations which can be reached between a terrorist and a host government (Lee, 1988) and the strategic interdependence between nations as they attempt to combat terrorism (Sandler, 2003). Game theoretic models can be applied to a variety of problems in transportation systems modeling. Hollander and Prashker (2006) provide an excellent review of the various instances in transportation literature that use non-cooperative game theory concepts. While acknowledging the strengths and the usefulness of the game theoretic approach, this paper also outlines drawbacks. Solving games becomes more complicated and involved as the size of the game increases. The structure of the game impacts the ease of solving the game. Generally, the games that have special structure and hence are more easily solved often are substantially less realistic and hence are of limited value for real-world decision-making. Most of the more involved games described by Hollander and Prashker (2006) follow the Stackelberg leadership model (von Stackelberg, 1934). Stackelberg games are played between two players – one of whom is a leader and the other is a follower. The leader makes a decision and the follower sees the outcome of the leader’s move and makes her move. Transportation situations adapt well to Stackelberg games since the government/transportation authorities can be considered the leader because they provide the infrastructure and dictate the rules under which it may be used. Travelers then make use of this infrastructure given the rules in place for its use. Stackelberg games are often expressed as bilevel optimizations for formulation and solution. Bell (2000), Bell and Cassir (2002), and Bell (2004) used a two-person zero-sum non-cooperative game to measure the performance reliability of transport networks in case of single user, multiple users and freight vehicle routing problems, respectively. Bell (2003) sought to identify the most crucial links or nodes with an aim to defining the network’s vulnerability. In each game, the user(s) seeks a least-cost path with a virtual network tester or an ‘‘evil demon” trying to maximize trip cost. To calculate the equilibria of these games, they are solved as maximin problems with the Method of Successive Averages (MSA).

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Biaonco et al. (2009) presented a bilevel formulation focused on risk equity. Both levels corresponded to government agencies – the meta-local authority that aims to minimize the maximum link risk in the whole network and the regional area authority that aims to minimize the total risk over their network. While the focus on equity is significantly different than our focus, the core concepts explored in the modeling are related. Kara and Verter (2004) proposed a bilevel formulation where the government imposes restrictions on the network and the shipper/carriers then choose the routes. Erkut and Gzara (2008) extended this research. They proposed incorporating the transport costs in the government objective along with risk considerations, so that the government can represent a trade off between cost and risk. A key difference in the core problem examined in these papers and the one considered in the presented research is the character of the policy adopted by the government and the shipper/carrier. In case of Kara and Verter (2004) and Erkut and Gzara (2008), the policies adopted by both the government and the shipper/carrier should be static. The only uncertainty represented in these papers is that associated with natural risks (which are independent of the actions of either player). Our analysis focuses on situations for which there is a third player in the form of a malicious terrorist attacker and therefore an analysis that admits randomized policies is likely to be more beneficial to the shipper/carrier. This, in turn, will force a randomized policy to be more advantageous to the terrorist. Since the policy must be implementable it is likely important that the prohibitions enacted by the government be static (or vary only with threat level). Despite these key differences, Kara and Verter (2004) and Erkut and Gzara (2008) create en excellent point of departure for this analysis. 3. Model formulation and illustrative example 3.1. Model formulation The objective of this model is to identify the prohibitions that maximize the benefit to the shipper/carrier while limiting the consequences should an attack occur. We develop this formulation in two steps. The first step describes a non-cooperative two-person non-zero sum game between the shipper/carrier and the terrorist as shown in Dadkar (2008). Then that formulation is extended to include the goals and the influence of the government so that decisions about prohibitions can be made. For the purpose of this formulation, we assume that in case of an attack, only one link is targeted each time an attack is mounted and every attack is successful. We also assume that the probability of an attack (p) is known. p can be interpreted as indirectly reflecting a rate at which attacks are mounted or as a subjective estimate of likelihood at a given time. The estimation of p is likely to be based on intelligence information. Therefore, it will be important to perform an analysis to understand the sensitivity of the recommendations to changes in the estimate for p. This type of analysis is illustrated in Section 5. For a given likelihood of an attack (p), the objective of the model is to identify the prohibitions that should be enacted by the government which lead to advantageous equilibrium solutions between the shipper/carrier and the terrorist. We consider repetitive shipments of an extremely hazardous substance between one origin–destination pair. Suppose there are m route options available between this origin–destination pair for the shipper/carrier and n links of interest to the terrorist. These n links are all the links that appear on at least one route the shipper/carrier is considering. Let A and B be the mxn payoff matrices for the shipper/carrier and the terrorist, respectively. The payoffs to both the players depend on whether the route chosen by the shipper/carrier traverses the link targeted by the terrorist. The payoff matrix for the shipper/carrier can be formulated using probability of attack (p), the utility of each route to the shipper/carrier as well as the consequences of a successful attack to the shipper/carrier. A is the mxn payoff matrix for the shipper/carrier. The matrix entry A(i, j) is the payoff to the shipper/carrier when he uses route i and link j is attacked. If link j is not on the route i, A(i, j) is assumed to be the utility of that route. However, if route i traverses link j, A(i, j) is assumed to be the expected value of the utility of route i in case of no attack and the representative value of damage caused by a successful attack (in this case, the population exposure on the link j when the route i traverses it). The payoff matrix for the terrorist can be formulated using probability of attack (p) and the utility of a successful attack to the terrorist. B is the mxn payoff matrix for the terrorist. The matrix entry B(i, j) is the payoff to the terrorist when he targets link j and route i is used. B(i, j) is assumed to be p times the population exposure on link j if the shipper/carrier uses route i that contains that link and 0 otherwise. Section 3.2 provides a detailed example of the computation of both the payoff matrices. Given that both the terrorist and the shipper/carrier want to maximize their expected payoffs, the shipper/carrier pursues the optimization described by Eqs. (1)–(3) and the terrorist pursues the optimization described by Eqs. (4)–(6).

max x0 Ay x

0

such that e x ¼ 1 xP0 max x0 By y

0

such that l y ¼ 1 yP0

ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ

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where x is a mx1 vector which gives the frequency with which the shipper/carrier uses each of the m routes, y is a nx1 vector which gives the frequency with which each of the n links are targeted by terrorist, and e and l are mx1 and nx1 vectors of 1s, respectively. A Nash equilibrium point is defined by the pair of strategies x and y where the objectives of both optimizations are satisfied simultaneously (Nash, 1951). Mangasarian and Stone (1964) demonstrated that a necessary and sufficient condition that a point (unique vectors x and y) be a Nash equilibrium of a two-person non-zero sum game for which there is a finite number of pure strategies is that the point be a solution to the following nonlinear optimization. It is important to notice that while there may not be a unique solution to this optimization, the objective value for each of the optimal solutions is 0.

maxx0 ðA þ BÞy  a  b x;y;a;b

ð7Þ

s:t:Ay 6 ae

ð8Þ

B0 x 6 bl e0 x ¼ 1

ð9Þ ð10Þ

0

ly¼1

ð11Þ

xP0 yP0

ð12Þ ð13Þ

where a and b are scalars that represent the expected payoffs for the shipper/carrier and the terrorist, respectively. The government can influence the decisions of the shipper/carrier by prohibiting the use of certain facilities at certain times of the day or for the entire day. Thus the shipper/carrier may not use routes which would violate those prohibitions. Also, terrorists will know of these prohibitions and therefore they will not stage attacks on facilities when these prohibitions are in effect. We assume that the goal of the government is to enact prohibitions which ensure that the expected payoff to the terrorist (b) is less than some threshold value n. Further, assume that for a given set of prohibitions, the shipper/carrier will select the Nash equilibrium that gives them the largest expected payoff (realizing that the payoff calculation includes the impact of an attack if the shipper/carrier uses a route that traverses a link that the terrorist attacks) if multiple Nash equilibrium points exist regardless of the expected payoff to the terrorist (which is constrained by the government restrictions). Let zj be a variable that takes on a value of 0 if a prohibition is enacted on the link j (i.e. the shipper/carrier cannot use the routes that include the link j and consequently this link will not be targeted by the terrorist) and is 1 otherwise. Let z be an nx1 vector of the variables zj. Further, let cij be 1 if link j is included in the route i used by the shipper/carrier and 0 otherwise. Finally, let C be an mxn matrix of the values cij. Let a* be a decision variable which is the expected payoff to the shipper/carrier. The formulation is then as given below:

a ¼ max a x;y;z;a;b

0

s:t:x ðA þ BÞy  a  b ¼ 0 Ay 6 ae

ð14Þ ð15Þ ð16Þ

B0 x 6 bl

ð17Þ

e0 x ¼ 1

ð18Þ

0

ly¼1

ð19Þ

z P C0x

ð20Þ

zPy

ð21Þ

xP0

ð22Þ

yP0 z ¼ f0; 1g a P a for all valid Nash equlibria

ð23Þ ð24Þ ð25Þ

b6n

ð26Þ

Eq. (15) requires that the solution selected for x, y, a and b define a Nash equilibrium for the shipper/carrier and the terrorist given the prohibitions identified by z. Eqs. (16)–(19) (22), and (23)are the remainder of the constraints needed to ensure that the solution is a valid Nash equilibrium. Eq. (20) requires that if a prohibition exists on a link, the probability that the shipper/ carrier uses a route that requires that link is zero. Similarly, Eq. (21) ensures that if a prohibition exists on a link then that link is not targeted by the terrorist. Eq. (24) represents the binary restrictions on the link prohibition variables. Eq. (25) requires that the expected payoff to the shipper/carrier (a) for the selected Nash equilibrium is the maximum of all the valid Nash equilibrium given the prohibitions enacted. Eq. (26) requires that the Nash equilibrium associated with the largest expected payoff to the shipper/carrier also leads to an expected payoff to the terrorist (b) that does not exceed the threshold value n. We assume that, given the prohibitions implemented, the shipper/carrier will consider all the Nash equilibrium strategies available and regardless of the expected payoff to the terrorist, he will select that strategy that yields the highest expected

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payoff to himself. The goal of the government is to identify the prohibitions such that when the shipper/carrier selects the Nash equilibrium with the highest expected payoff to himself, the associated expected payoff to the terrorist (b) is less than or equal to n. Hence the aim of the objective function given in Eq. (14) and the constraint given in Eq. (25) is to identify the Nash equilibrium with the largest expected payoff to the shipper/carrier across all Nash equilibria given the prohibitions. This Nash equilibrium also must have an expected payoff to the terrorist (b) which is less than or equal to n as required by Eq. (26). We develop our model based on the Mangasarian and Stone (1964) formulation because it yields a compact mathematical statement that expresses the goals of all three players. It is also amenable to solution for realistic problems. It assumes that each player knows the payoff matrix of the other two players and uses this information when making their decisions. It effectively treats the game between the shipper/carrier and the terrorist as a leader–follower game (assuming the decision of the government is fixed) because we assume the shipper/carrier will select the strategy which maximizes their payoff given the decision made by the government. Given the choice made by the shipper/carrier, the terrorist selects the strategy that maximizes their payoff. Of course, the shipper/carrier can predict what the choice of the terrorist will be because they are assumed to know the payoff matrix of the terrorist. A parallel argument can be made for the interaction between the government and the combined play of the shipper/carrier and the terrorist (interaction between the outer and the inner game illustrated in Fig. 1). It is useful to note that our formulation expands on the notion of leader–follower as it is typically applied in Stackelberg games/bilevel optimization because it admits randomized policies by the shipper/carrier and the terrorist. This is a critical extension because if the terrorist knows the route to be taken by the shipper/carrier each time they execute a delivery (when p is sufficiently high), the terrorist will be significantly more effective. The shipper/carrier, by selecting a randomized strategy, increases their payoff and by extension reduces the payoff to the terrorist (though this is not directly their goal). Similarly, since a randomized strategy is likely to be associated with a higher payoff to the shipper/carrier, this is likely to be also true for the terrorist. The Mangasarian and Stone (1964) formulation provides an excellent mechanism address this. 3.2. Illustrative example To gain a better understanding of the model, we apply it to the example network illustrated in Fig. 2. The shipper/carrier is interested in undertaking repetitive shipments from origin node 1 to destination node 20 starting at 7 AM. This network has 34 links and for the purpose of this example, we find 100 paths (assuming a departure time of 7 AM) using the K shortest path algorithm developed for stochastic dynamic networks by Dadkar et al. (2008). Each of these 34 links is a pure strategy available to the terrorist and each of the 100 routes is a pure strategy available to the shipper/carrier, respectively. Note that the government can enact prohibitions on any subset of these 34 links. This example identifies pure strategies given the assumption that the shipments will depart the origin at 7 AM. In practice the pure strategies could be developed for each departure time separately and integrated into the payoff matrices for the shipper/carrier and terrorist. This would allow the game to also identify a time of departure for each of the routes selected and prohibitions that vary by time of day. Since this extension to the analysis is straightforward we omit it in the analysis thereby simplifying the discussion and focus on a single departure time of 7 AM. For the purposes of this analysis we assume the following: 1. Link travel time is modeled as a sum of free–flow time plus an exponentially distributed random delay with a mean of 10% of the free flow travel time. Even though the random delay on a link is modeled as a constant plus an exponential random delay we assume that the travel time across a path is approximately normal based on the Central Limit Theorem. 2. Accident probability varies according to a gamma distribution based on the work of Nembhard (1994). The scale parameter is based on whether the link is urban or rural and whether the link is traversed in the day or night. The day–time accident rate is assumed to hold between 8 AM and 8 PM and the night–time accident rate is assumed to hold between 8 PM to 8 AM based on the relative changes in heavy truck accident rates found by Harwood et al. (1993).

2

3

8

13

5

1

14 12

7 10

6 4

19

9

20 15

18 21

16

11 17

22

Fig. 2. The Nash equilibrium with no link restrictions for p = 0.01% and p = 0.1%.

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3. The major hazards are fire or vapor exposition, or both, in the event of an accident or an attack. The at-risk population includes other travelers on the highway near the shipment and the population residing near the road. We assume that the on-link exposure can be described by the normal distribution and varies for urban freeways and rural freeways according to the hourly distributions developed by Tittemore et al. (1972) and New York State Department of Transportation (1982), respectively. The off-link exposure is obtained from the census tracts provided by TransCAD and is assumed to vary normally as well. Further since we are interested in one single measure to determine the value of using one route over the other, we define a composite measure as the equally weighted sum of travel time and consequence measure (a combination of population exposure and accident probability as described in the second section). Increase in this composite measure signifies that the shipper/carrier is worse off. Thus the negative value of the mean of this composite measure is taken as the utility of a route to the shipper/carrier and the shipper/carrier will choose the route with the lowest composite measure mean for higher utility. Note that the shipper/carrier achieves this utility only in case of no attack. For example, consider Route A: 1–2–5– 12–14–20 departing at 7 AM. The various routing attributes as propagated over this route are shown in Table 1. Route A is found to have the lowest composite measure mean (102.58) as defined above and the utility of Route A is 102.58. In case of an attack, the shipper/carrier would like to minimize the population that is exposed to the hazmat shipment and the terrorist would like to maximize this exposure so as to maximize the damage inflicted. Population exposure on the link attacked at the time of the attack is used to represent the repercussions faced by the shipper/carrier in case of an attack as well as the utility of a link to the terrorist in case of an attack. If there is no attack, the terrorist receives no utility. We assume that the population exposure for a link consists of the other travelers on the highway within ½ mile of the shipment when the attack occurs as well as the population on either side of the link when the attack occurs. This assumption for the shipper/carrier can be replaced with one focused on the economic consequences to the shipper/ carrier of an attack with no difficulty. Given our focus on extremely dangerous materials, minimizing exposure in the event of an attack is more appropriate. As mentioned previously, we assume that if the shipper/carrier uses a route that includes the link attacked, the attack occurs when the shipment is on that link and the attack is successful. This assumption can be relaxed by assuming specific probabilities that the shipment is on the link when the attack occurs and that the attack is successful. For the purposes of this case study we assume that these probabilities are 1. A is the 100  34 payoff matrix for the shipper/carrier where as the B is the 100  34 payoff matrix for the terrorist. For example, consider link 5–12 on Route A: 1–2–5–12–14–20 departing at 7 AM. The population exposure on this link at the time that Route A traverses it is estimated to be 16,354. Thus the payoff to the terrorist for this route–link combination for the probability of attack p = 0.01% is 0.01%  16,354 = 1.64. Similarly the payoff to the shipper/carrier for this route–link combination for p = 0.01% is (0.01%  16,354 + 99.99%  102.58) = 104.20. Thus the payoff of Route A to the shipper/carrier varies with p and is different for each link that is present on this route unlike the utility of Route A (102.58) which represents the intrinsic value of the route to the shipper/carrier and is a constant. If we assume that no prohibitions are enacted (i.e. we set zj = 1 " j), the expected payoff to the terrorist (b) is allowed to be arbitrarily high and thus we are only interested in maximizing the expected payoff to the shipper/carrier (a). For a probability of a terrorist attack (p) of 0.01%, the optimal solution implies a Nash equilibrium with an expected payoff to the shipper/carrier (a) of 104.20 and an expected payoff to the terrorist (b) of 1.64. Since there are no restrictions on any of the links in the network and the probability of an attack is very low, the shipper/carrier will always use the route that offers the highest utility (Route A: 1–2–5–12–14–20 as shown by the thicker lines in Fig. 2). Since the shipper/carrier will always use Route A, the terrorist will attack the link on that route with the largest value of population exposure (link 5–12 in this case with an estimated population exposure 16,354). Next suppose the expected payoff to the terrorist (b) is constrained to be no greater than 0.8 which implies that if an attack occurs, the population exposure is restricted to be less than 8000. Fig. 3 represents the Nash equilibrium found which has an expected payoff to the shipper/carrier (a) of 135.78 and an expected payoff to the terrorist (b) of 0.73. The estimated population exposure in case of an attack drops to 7328. The government, in order to constrain (b) to no more than 0.8, prohibits the movement of hazmat on four links: links 2–5, 3–5, 7–9 and 19–20 (each marked by an X in Fig. 3). The routes chosen by the shipper/carrier are marked by thicker lines and the links chosen by the terrorist are marked by the dashed lines in Fig. 3. The route with that offers the highest utility to the shipper/carrier (Route A as shown in Fig. 2) is no longer available because link 2–5 is now restricted from use and similarly other routes, that use the restricted links, are now prohibited. The shipper/carrier is forced to choose the route or routes that that offer the highest utility among the set of routes that do not utilize the restricted links. In this case there are two such routes with almost identical utilities: Route B (1–4–6–7–5– 12–14–20) with utility 134.81 and Route C (1–2–8–13–14–20) with utility 135.38. The model predicts that Route B will Table 1 Probability distributions estimated for Route A. Travel time mean (h)

Accident probability mean (10–6)

Population exposure mean (people)

Consequence measure mean (people)

Composite measure mean

6.77

38.68

51,008

1.97

102.58

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2

13

8

19 3

5

1

14 12

7 6 4

10

9

20 15

18 21

16

11 17

22

Fig. 3. The Nash equilibrium for p = 0.01% and b 6 0.8.

be chosen with a probability of 56% and Route C with a probability of 44%. Since these routes have lower utility values than Route A (whose utility is 102.58), the expected payoff to the shipper/carrier (a) decreases when these restrictions are imposed. All other routes that offer a utility greater than these two routes are not feasible since they utilize one or more of the restricted links. Narrowing the shipper/carrier’s choices also constricts the choices of the terrorist. The terrorist prefers to attack links with the largest values of exposure but now the links on the set of routes chosen by the shipper/carrier have lower exposure thus restricting b. Thus the terrorist targets links 5–12 and 14–20 with a probability of 43% and 57%, respectively. Given that p is a rate of attack, this implies that the terrorist will attack link 5–12 43% of the time and link 14–20 57% of a time. If p is very small it may take a very long time to see multiple attacks. The model aims to restrict the movement of hazmat on a combination of links such that the shipper/carrier is forced to use those routes that utilize links with lower population exposure. For example, in this case, if link 19–20 is not restricted but links 2–5, 3–5 and 7–9 are still restricted, the shipper/carrier will choose to use route 1–2–8–13–19–20 which offers a utility of 125.93, which is the highest utility among the limited routes feasible after the restrictions. The terrorist now mounts attacks on link 2–8 which has a high population exposure leading to an expected payoff of 1.52 to the terrorist. Even though this Nash equilibrium leads to higher expected payoff to the shipper/carrier (127.44) than when the government restricts all 4 links as described above, the expected payoff to the terrorist is only marginally lower then when no restrictions are allowed making this Nash equilibrium unattractive (and infeasible if b is limited to 0.8). A parallel motivation exists for the other three restrictions identified when b is limited to 0.8. 4. Heuristic solution procedure The formulation presented in the previous section cannot be solved directly as a result of constraint (25). Therefore we construct a heuristic procedure which can accommodate this constraint. The solution procedure can be thought of as a twostep procedure which is applied iteratively. In the first step, the government decisions are made and the values for the z variables are set. In the second step the inner game between the shipper/carrier and the terrorist is modified according to these restrictions as per Eqs. (20) and (21). The inner game is defined as Eqs. (15)–(19) (22), and (23)with the objective given in Eq. (14). The links and the routes that violate the restrictions are eliminated temporarily and the matrices A and B are updated to A** and B**, respectively. The modified inner game is solved to find the Nash equilibrium (i.e. x** and y** are identified for the specific modified game and the expected payoffs to the shipper/carrier (a**) and to the terrorist (b**) are found). The z variables can be changed successively, thus changing the x** and y** values and through them a** and b** till a satisfactory solution is reached. We use a Tabu Search algorithm (TS) to update the z variables. The TS algorithm is as follows: 1. Choose a starting solution randomly and calculate the fitness function for it. 2. Set the current solution as the best solution. 3. If the Tabu List is populated, increment the time the solutions have spent in the list by 1 unit. Remove those solutions from the Tabu List that have spent time equal to the Tabu Tenure in the Tabu List thus allowing the algorithm is to choose these solutions again. 4. Find all the neighbors of the current solution and eliminate the ones on the Tabu List. 5. Calculate the fitness functions for all the neighbors. 6. Choose the next solution from the pool of the neighbors and set that as the current solution. 7. Add the change from the previous solution to the current solution to the Tabu List to prevent back tracking to the previous solution. 8. Check the neighbors found this iteration to see if any of them are better than the best solution found thus far. If yes, update the best solution. Note that the current solution is updated at every iteration but the best solution is not necessarily updated. 9. Repeat steps 3–7 for a fixed number of iterations.

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Thus, in order to solve the formulation using the TS algorithm, we need to define the structures associated with the solution string, the starting solution, the fitness function, the neighborhood of each string, and the selection of the next solution. We assume that each solution is an n-long binary–integer string representing the restrictions on the n links such that zj is 0 if there is a restriction on link j and 1 otherwise. The starting solution is chosen randomly such that there are restrictions on a few of the links. The fitness function associated with every solution string can be calculated by solving the optimization represented by Eqs. (14)–(19) (22), and (23)when the value of the z variables are assumed to be determined by the solution string. Depending on the restrictions represented by the solution string, the payoff matrices A and B are updated to A** and B** and the modified non-linear program is solved to find the Nash equilibrium with the maximum expected payoff to the shipper/carrier so as to obtain the values a** (the expected payoff to the shipper/carrier) and b** (the expected payoff to the terrorist) as well as the values for x** and y** (the equilibrium strategies for the shipper/carrier and the terrorist, respectively). After the solution is found, it is penalized if it violates the threshold limit, i.e. by setting the fitness function to be a** if b** 6 n and M*(n  b**) otherwise where M is a large constant. This encourages solutions in the TS algorithm that satisfy the government’s plan of restricting the expected payoff to the terrorist for the Nash equilibrium the shipper/carrier will select. The neighborhood of every solution string is defined by the strings that differ from the current solution string with respect to only one link such that a link restricted in the current solution string is unrestricted in the neighbor string and vice versa. If the neighbor string violates the Tabu List, which contains the solutions that have been visited in the recent past, it is excluded from the neighborhood. At every iteration, fitness function evaluations are performed for each of these neighbors and the next solution string is chosen from them using tournament selection, i.e. they are chosen at random, biased by solution quality as measured by the value of the fitness function associated with every solution string. Changes in government policy (the z vector) are dependent on the threshold value n. We are interested in how changes in n affect the recommended prohibitions on the links as well as the equilibrium strategies for the shipper/carrier and the terrorist (the x and the y vectors). An efficient frontier between the expected payoff to the shipper/carrier (a) and the expected payoff to the terrorist (b) reflects this change and can be generated by running the TS algorithm multiple times for different threshold values n.

5. Case study To illustrate the use of the formulation and the solution procedure, described in the third and the fourth sections, on a realistic problem, we consider routing a hypothetical shipment from Jackson, MS to Tallahassee, FL over the highway network, as shown in Fig. 4. We assume that the shipment departs Jackson at 7 AM. The key question investigated in this case study is how the restrictions imposed by the government should change as the probability of terrorist attack (p) and threshold value (n) change. In order to analyze this situation, we need to identify the pure strategies available to the shipper/carrier, the terrorist and the government as well as the payoff matrices for the shipper/carrier and the terrorist that reflect their key concerns and their behaviors. The shipper/carrier can use any of the different available routes to travel from Jackson, MS to Tallahassee, FL and these routes are thus the pure strategies available to the shipper/carrier. We use the K shortest path algorithm developed for stochastic dynamic networks by Dadkar et al. (2008) to generate 600 paths given the assumption that the shipments will depart the origin at 7 AM. In practice the pure strategies could be developed for each departure time separately and integrated into the payoff matrices for the shipper/carrier and terrorist. This would allow the game to also identify a time of departure for each of the routes selected and prohibitions that vary by time of day. Since this extension to the analysis is straightforward we omit it in the analysis thereby simplifying the discussion and focus on a single departure time of 7 AM.

Fig. 4. The case-study network.

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This case-study network has 1403 bidirectional links but not all 1403 links are a part of the 600 paths that we are focusing on. Thus only the 604 links that are utilized by these 600 paths are of interest to the government. These are the pure strategies available to the government since the government will want to restrict the movement of hazmat on only these links. Similarly these 604 links are also the pure strategies available to the terrorist since the terrorist will want to attack only those links that are utilized by the shipper/carrier. The payoff matrices can be formulated as discussed in the third section. The key question investigated in this case study is how the restrictions imposed by the government change as probability of terrorist attack (p) and threshold value (n) change. The algorithm was run 30 times with an assumed p = 0.01% and p = 0.1% for three different threshold values (n) each. The string used by the algorithm to solve the optimization described in the third section is n = 604 entries long and each entry is either 0 or 1 depending on if there is a restriction on that link or not. The starting solution is selected randomly with 5 links set as restricted. The Tabu Tenure is set to 3. Each trial took about 10 h on average to complete. The remainder of this section focuses on interpreting the efficient frontiers found for the two different assumed probabilities of attack in order to understand how the strategies change for the shipper/carrier, the terrorist and the government. TOMLAB/NPSOL was used to solve the non-linear program through a MATLAB interface and all the experiments were run on a Pentium 4 3.4 GHz PC. Fig. 5 and Table 2 show the efficient frontier estimated from the solutions found for p = 0.01% for different threshold values. The algorithm was run first allowing expected payoff to the terrorist (b) to be arbitrarily high and then restricting it as per the threshold values shown in Table 2. Point P represents the Nash equilibrium found when the maximum benefit to the terrorist is allowed to be arbitrarily high leading to the highest expected payoffs to both the shipper/carrier and the terrorist. As is the case with the example network in the fourth section, this point represents a solution when there is no constraint on the expected payoff to the terrorist (b) and thus there are no restrictions on any of the links in the network. Since all routes are available to the shipper/carrier and the probability of an attack is very low, the shipper/carrier will always use the route that offers the highest utility (Route A which has the lowest value of the composite measure mean). As illustrated in Fig. 6, Route A heads east from Jackson, MS on I-20 to Meridian, MS where it continues eastwards on US80 to Montgomery, AL. Then it turns south and follows US231, US84 and US27 until it reaches Tallahassee, FL. Further, the expected consequences associated with a terrorist attack are very low and are insufficient to dissuade use of this route. The portion of this route using US84 (Link X) has the largest value of population exposure among all the links in this route. Hence the terrorist will attack that link. Fig. 7 represents Point Q on the efficient frontier when the government wishes to limit the expected payoff to the terrorist (b) to below 0.08. This leads to a significant drop in the expected payoff to the terrorist but only a slight decrease in the

0.09

Expected Payoff to the Terrorist (β )

P

0.07

Q

0.05

R

0.03 -60

-59

-58

-57

-56

Expected Payoff to the Shipper/carrier (α ) Fig. 5. Efficient frontier for the case study (p = 0.01%). Table 2 Efficient frontier for the case study (p = 0.01%). Point

P Q R

n (threshold value)

None 0.08 0.06

Number of links restricted by the govt.

– 1 3

a (expected payoff to the shipper/ carrier)

56.02 56.50 57.56

b (expected payoff to the terrorist)

0.0835 0.0639 0.0428

Strategy adopted by the shipper/ carrier at the Nash equilibrium

Strategy adopted by the terrorist at the Nash equilibrium

Route Chosen

Probability with which the route is chosen (%)

Link Chosen

Probability with which the link is chosen (%)

A B C D

100 100 45 55

X Y X Z

100 100 17 83

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Fig. 6. Nash equilibrium for Point P (p = 0.01% and no restrictions).

Fig. 7. Nash equilibrium for Point Q (p = 0.01% and b 6 0.08).

expected payoff to the shipper/carrier. At this point, the government restricts 1 link (as marked by the large X in Fig. 7). Since now there is a restriction on one of the links in the network, some of routes are no longer feasible for the shipper/carrier. However, since the probability of an attack happening is very low, the shipper/carrier will always use the route with the highest utility among the feasible routes, i.e. a route that avoids the link that the government has restricted while still offering highest possible utility to the shipper/carrier (Route B as shown in Fig. 7). Since the shipper/carrier will always use Route B, the terrorist will always attack the link (Link Y) on that route with the largest value of exposure. Route B is significantly different than Route A. It follows US49 south to Collins, MS and then turns east on US84 and north on US331 and continues east on I-10 up to Tallahassee, FL. The link restricted by the government is a part of Route A that is a portion of US231. Note that the US84 portion of Route A has the highest population exposure for that route and not the portion of US231 which is restricted. However, restricting use on this portion of US84 (Link X) does not lead to lower expected payoff to the terrorist because it still allows use of certain routes that have other links with high population exposure. For example, Route K as shown in Fig. 8 has the highest utility to the shipper/carrier among all the feasible routes when Link X is restricted. Thus if Link X is restricted, the shipper/carrier prefers to use Route K but it contains Link Z which has a high population exposure yielding a payoff to the terrorist of 0.0972. Since this payoff exceeds the threshold value n = 0.08 this restriction is not effective.

Fig. 8. Nash equilibrium when p = 0.01% and Link X is restricted.

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Fig. 8 represents Point R, the Nash equilibrium found when the government sets the threshold value (n) to 0.06. This solution offers an expected payoff to the terrorist which is almost half the expected payoff to the terrorist offered by Point P. In this case the government restricts three links. Each link that is restricted by the government is indicated with an X in Fig. 9. The increased restrictions make some of the routes, which offer the highest utility, infeasible. The shipper/carrier now chooses to use two routes (Route C shown by the thicker lines and Route D shown by the lines with the arrowheads) with 45% and 55% probability each leading to lower actual damage in case of an attack and hence a lower value of b. Since the shipper/carrier diversifies the routes taken, the terrorist also diversifies the links targeted and targets Link Z with probability 83% and Link X with probability 17%. Route D is similar to the route with the highest utility (Route A) except for the Link S that is restricted by the government. Thus Route D offers a slightly lower value of utility than Route A offers. Restricting the use of Link T (portion of US49 that travels southeast from Jackson, MS) as well as Link U (portion of I-10) prohibits the shipper/carrier from choosing routes similar to Route B. From Fig. 5 and Table 2, it is clear that in order to achieve a lower expected payoff to the terrorist, the shipper/carrier receives a lower expected payoff as well and there exist solutions along the efficient frontier that achieve this lower expected payoff to the terrorist without diminishing the expected payoff to the shipper/carrier significantly. Fig. 10 and Table 3 show the efficient frontier estimated from the solutions found for p = 0.1% for different threshold values. The algorithm was run first allowing expected payoff to the terrorist (b) to be arbitrarily high and then restricting it as per the threshold values shown in Table 3. As we observed with the example network in the fourth section, for a higher probability of attack, the shipper/carrier chooses to be more conservative and the government does not need to apply as many restrictions. Point P on the efficient frontier represents the solution obtained when there is no limit on the expected payoff to the terrorist (b) and thus there are no restrictions on the network. In this case, the probability of attack is high enough to dissuade the shipper/carrier from using the route with the highest utility all the time (Route A as shown by the thicker lines in Fig. 11). Instead the shipper/carrier chooses to use Route A with 76% probability and another route (Route E) with lower utility as well as lower population exposure with 24% probability. Route E (as shown by the lines with the arrowheads in Fig. 11) is similar to Route A except initially it uses US80 before joining I-20 instead of using just I-20 as Route A does. Also instead of following

Fig. 9. Nash equilibrium for Point R (p = 0.01% and b 6 0.06).

Expected Payoff to the Terrorist (β )

0.9

0.7

P

0.5

Q R

0.3 -60

-59

-58

-57

Expected Payoff to the Shipper/carrier (α ) Fig. 10. Efficient frontier for the case study (p = 0.1%).

-56

279

Y. Dadkar et al. / Transportation Research Part B 44 (2010) 267–281 Table 3 Efficient Frontier for the case study (p = 0.1%). Point

n (Threshold value)

Number of links restricted by the govt.

a (Expected payoff to the shipper/ carrier)

b (Expected payoff to the Terrorist)

Strategy adopted by the shipper/ carrier at the Nash equilibrium

Strategy adopted by the terrorist at the Nash equilibrium

Routes chosen

Probability with which the route is chosen (%)

Links chosen

Probability with which the link is chosen (%)

P

None

–

56.67

0.628

A E

76 24

X V

71 29

Q

0.06

1

56.74

0.449

E F

46 54

X Z

73 27

R

0.04

2

58.83

0.394

G H

39 61

Z W V

59 29 12

Fig. 11. Nash equilibrium for Point P (p = 0.1% and no restrictions).

US84 down to Tallahassee, FL, it continues on US231 to I-10. Thus it avoids the US84 portion (Link X) of Route A that is targeted by the terrorist under solution P and this minimizes the expected payoff to the terrorist. Hence the terrorist diversifies the links targeted and targets Link X with probability 71% and Link V (which is the US80 portion on both the Routes A and E) with probability 29%. Fig. 12 represents Point Q on the efficient frontier when the government wishes to limit the expected payoff to the terrorist (b) to below 0.6 and hence restricts 1 link. The east bound portion of I-20 originating in Jackson, MS is restricted. It cannot be marked in Fig. 12 since it is in close proximity to US80 which is a part of Route E. The shipper/carrier still prefers to use Route E (as shown by the lines with the arrowheads in Fig. 12) but with a higher probability of 46% as opposed to 24% when there are no restrictions on the network. Further, the shipper/carrier no longer uses Route A (the route with the highest utility) since it traverses the link that is restricted. Instead the shipper/carrier prefers to use Route F (as shown by the thicker lines in Fig. 12) with probability 54%. Since the shipper/carrier diversifies the routes taken, the terrorist also diversifies the links targeted and targets Link X with probability 73% and Link Z with probability 27% which are the links with the highest population exposure on the two routes chosen by the shipper/carrier. The Nash equilibrium obtained when the government wishes to limit the expected payoff to the terrorist (b) to below 0.4 and hence restricts two links is represented by Point R on the efficient frontier. Each link that is restricted by the government

Fig. 12. Nash equilibrium for Point Q (p = 0.1% and b 6 0.6).

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Fig. 13. Nash equilibrium for Point R (p = 0.1% and b 6 0.4).

is indicated with an X in Fig. 13. The shipper/carrier chooses Routes G (as shown by the thicker lines) and H (as shown by the lines with the arrowheads) with probabilities of 39% and 61%, respectively. These routes differ significantly geographically from the other routes that we have discussed so far; Route G continues eastward longer than any of the other routes before turning south on US27 whereas Route H joins I-10 at Mobile, AL which much earlier than the other routes. These routes offer lower utility to the shipper/carrier but they traverse links with lower population exposure as well. Hence while the expected payoff to the shipper/carrier (a) associated with this Nash equilibrium is significantly lower, the expected payoff to the terrorist (b) is significantly lower as well. The terrorist maximizes the expected payoff to themselves by targeting links Z, V and W with probabilities 59%, 12% and 29%, respectively. When there is a very low probability of attack, the shipper/carrier’s route preferences focus on the utility offered by the route, effectively ignoring the impacts of an attack. However, as p increases, the payoffs associated with the routes decreases because the effect of having an attack significantly increases. Thus some of the routes that offer high utility values are no longer attractive to the shipper/carrier since they may contain a link or links with higher population exposure which leads to worse payoffs to the shipper/carrier when an attack occurs. Thus the shipper/carrier prefers to control the consequences of a successful attack by avoiding such routes. As p increases, the shipper/carrier gets more conservative with their route choices and thus the government needs to restrict lower number of links to achieve better reduction in the expected payoff to the terrorist (b). 6. Conclusions and opportunities for further research The objective of this paper was to develop a model of the interactions between a shipper/carrier of hazardous material, a terrorist and the government to understand: (1) how governments might set link prohibitions to cope with the threat of attack on a shipment of hazardous materials; (2) how shippers/carriers might make decisions of which routes to use with what frequencies in response to these prohibitions and the underlying threat of terrorism and (3) what links terrorists might then target with what frequencies. In order to achieve this we construct a non-linear integer optimization which is an extension of a two-person, non-zero sum game given in Mangasarian and Stone (1964). In order to effectively solve the resultant optimization for realistic instances a heuristic procedure that integrates Tabu Search and gradient based non-convex optimization was developed. The model was then applied to a realistic case study focused on the repetitive movement of shipments from Jackson, MS to Tallahassee, FL. These analyses illustrated that as the probability of an attack rises the shipper/carrier should select more and more conservative routes which causes a decline in the expected payoff to the shipper/carrier but also controls the damages caused by an attack. This behavior is of particular importance because historically when considering routing decisions for hazardous material shipments, the emphasis has been on the identification of the single ‘‘best” route to use repetitively. This game shows the weakness in that strategy and that it is dominated when the probability of an attack is significant. These analyses also show that as the probability of an attack rises the goals of the government and the shipper/carrier align and this results in the shipper/carrier making decisions that control the consequences of an attack. Therefore the government does not need to enact quite as many prohibitions to control the consequences of an attack. Shippers/carriers now have sufficient incentives to avoid links that are attractive targets to the terrorist. This paper contributes to the literature by developing a non-cooperative non-zero sum game which represents the interactions between a shipper/carrier, terrorist and the government for the movement of hazardous materials. It creates reasonable rules to predict what the shipper/carrier and terrorist are likely to do in response to government prohibitions and their interactions with each other. This allows the identification of a single Nash equilibrium point for the interactions between the shipper/carrier and the terrorist for a given value of p and a maximum allowable expected payoff to the terrorist which can then be used in for decision-making by all three parties. Finally a solution procedure that is effective for realistic problems is also developed. The key area for additional research is the extension of the game to simultaneously consider shipments of a variety of hazardous materials with different origins and destinations. This would substantially enrich the decision-making included

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