In this paper, we investigate toll setting as a policy tool to regulate the use of roads for dangerous goods

Transcription

1 Vol. 43, No. 2, May 2009, pp in ein inform doi /trc INFORMS Toll Policie for Mitigating Hazardou Material Tranport Rik Patrice Marcotte, Anne Mercier Univerité de Montréal, Montréal, Québec, and CIRRELT, Montréal, Québec H3C 3J7, Canada Gille Savard École Polytechnique de Montréal, Montréal, Québec, and GERAD, Montréal, Québec H3T 1J4, Canada, Vedat Verter McGill Univerity, Montréal, Québec H3A 1G5, Canada, and CIRRELT, Montréal, Québec H3C 3J7, Canada, In thi paper, we invetigate toll etting a a policy tool to regulate the ue of road for dangerou good hipment. We propoe a mathematical formulation a well a a olution method for the hazardou material toll problem. Baed on a comparative analyi of propoed mathematical model, we how that toll policie can be more effective than the popular network deign policie that identify road egment to be cloed for vehicle carrying hazardou material. We preent a ummary of computational experiment on a problem intance from Wetern Ontario, Canada. Key word: hazardou material tranportation; toll etting; network deign; bilevel programming Hitory: Received: April 2007; reviion received: December 2007; accepted: March Publihed online in Article in Advance Augut 15, Introduction An increaing amount of hazardou material (hazmat) are hipped by road, rail, waterway, and air. Exploive, gae, flammable liquid, poionou ubtance, infectiou ubtance, and radioactive material are among the hazmat that are tranported in large volume. Thee hipment are indipenable to our modern way of life, although they can be harmful to the people and the environment in the event that they are releaed from their container a a reult of an accident. Due to the inherent tranport rik, the tranportation of hazmat i regulated under the Federal Hazardou Material Tranportation Act (which wa amended by the Patriot Act in 2001) in the United State and under the Federal Tranportation of Dangerou Good Act in Canada. The policie that are available to government agencie for hazmat tranport rik management can be categorized into two main group with repect to their underlying philoophy: proactive and reactive. The latter group of policie aim at confining the undeirable conequence of a hazmat incident after the occurrence of the accident. The development of emergency repone plan which involve the etablihment of a network of reponder team pecializing in different hazmat type i the mot popular example of reactive policie. Clearly, the conequence of an accident can be mitigated through better coordination of reponder team and fater repone time to the incident ite. In contrat, the proactive rik mitigation policie aim at reducing the likelihood and conequence of hazmat incident a priori. The etablihment of inpection center for hazmat truck i a common example in thi category. Our focu i on the proactive policie regulating the ue of road egment by hazmat carrier. In North America and Europe, government agencie do not have the authority to dictate route to hazmat carrier for moving their hipment. Thee agencie mitigate hazmat tranport rik by impoing (permanent or time-baed) curfew on the ue of the road egment under their juridiction. The cloure of certain road egment to hazmat vehicle i a policy that i being ued (or conidered) in many large citie, uch a Wahington D.C., Montréal, and Pari. In their eminal paper, Kara and Verter (2004) formulated the problem of identifying the road egment to be cloed to hazmat hipment a a network deign problem (ND). In their ND formulation, a regulator chooe the road egment to be cloed for hazmat tranportation o a to minimize population expoure (i.e., the total number of people within a threhold ditance from the road egment that are utilized by hazmat vehicle), while taking into account the 228

2 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 229 carrier route choice baed on tranport cot. Thi problem involve two level of deciion, which cannot be olved equentially. There are uually more than one path left available for ome carrier even after the deign deciion are made (all carrier ue the ame road network and the cloed arc are the ame for all hipment of the ame type of hazmat). If the deign deciion were made baed olely on the objective of the regulator (i.e., without keeping in mind the carrier behaviour), then it i probable that the total rik aociated with the carrier route choice made ubequently would be much higher than the rik anticipated by the regulator. One uch example i provided in Erkut, Tjandra, and Verter (2007). Hence, thi problem i conidered a a bilevel problem. For a recent urvey on bilevel programming, ee, e.g., Colon, Marcotte, and Savard (2005) or Dempe (2005). In thi paper, we propoe an alternative policy tool to regulate the ue of road for hazmat tranport, i.e., the ue of toll to deter the hazmat carrier from uing certain road egment, which we refer to a toll-etting policie (TS). Thi policy, alo modeled a a bilevel problem, entail impoing toll on certain road egment o a to channel the hipment on le-populated road. Although TS ha been tudied for regular freight tranportation, to the bet of our knowledge, thi i the firt paper that propoe TS a an effective mean to mitigate hazmat tranport rik. We are alo unaware of the ue of thi policy tool by regulator around the globe. Nonethele, our finding indicate that TS ha ignificant potential a a policy tool becaue it i more flexible and effective than the popular ND policie for mitigating tranport rik. Becaue dangerou good contitute an integral part of indutrialized ocietie, the economic viability of the hazmat tranport ector cannot be ignored while attempting to reduce the public and environmental rik. On the other hand, the carrier mut take into account the rik a well a the cot aociated with their routing deciion to both minimize their inurance cot (Verter and Erkut 1997) and manage their public image. Therefore, in thi paper, we firt extend the work of Kara and Verter (2004) to incorporate the cot and rik (i.e., population expoure) conideration at both the regulator and the carrier level. Baed on thi extended framework, we alo preent ome improvement on the ND olution methodology that permit u to olve much larger intance than thoe reported in Kara and Verter (2004). However, our main contribution i the propoed methodology for implementing the TS policy for hazmat tranportation. To thi end, we preent a mathematical formulation for the bilevel hazmat TS problem and how that thi model can alo be poed a a ingle-level mixed-integer programming (MIP) formulation. Perhap more importantly, we how that not only can TS be more effective than ND in reducing hazmat tranport rik, it can alo be much eaier to olve. A a matter of fact, when the objective of the government i to minimize rik only (i.e., when cot are only included at the carrier level), we how that the toll problem i not truly bilevel in that it reduce to a linear program that can efficiently be olved. Finally, becaue the effectivene of the hazmat tranport policie devied by a government agency depend on the extent of buy-in received from the hazmat carrier during the conultation proce (Verter and Kara 2008), we elaborate the ue of our methodology on a retricted et of road egment conidered for etting toll to produce olution that are more acceptable to the carrier. The remainder of the paper i organized a follow. An overview of the relevant literature that highlight the contribution of thi paper i provided in 2. Section 3 preent the mathematical formulation for the network deign and the toll problem in the context of hazmat tranportation. In thi ection, we alo provide an example howing that thee two model are not equivalent. Section 4 how how the toll problem can be ued by a regulator to obtain minimum rik olution very efficiently. Our olution methodology i outlined in 5, which i followed by a ummary of our computational experiment in 6. Our experiment are baed on the problem intance in Wetern Ontario, Canada, tudied by Kara and Verter (2004). Section 7 conclude the paper. 2. Overview of the Literature In thi ection, we provide a elective overview of the two tream of reearch that are mot relevant for our work, i.e., hazmat ND model and TS application in tranportation. Although hazmat logitic i a mature field of reearch (ee the comprehenive and recent review by Erkut, Tjandra, and Verter 2007), the regulation of the ue of road egment ha attracted the attention of reearcher only fairly recently. Alo, we are not aware of any work on the hazmat TS problem dicued in thi paper. Nonethele, the literature contain numerou application of TS to road pricing and regular freight tranportation, which we will review at the end of thi ection. A mentioned in the previou ection, Kara and Verter (2004) were the firt to propoe a bilevel programming formulation for the hazmat ND problem. The outer-level problem chooe road egment to cloe for hazmat tranportation o that the total number of people expoed to dangerou good i minimized, taking into account that the inner-level problem route all orgin-detination (O-D) hipment

3 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 230 Tranportation Science 43(2), pp , 2009 INFORMS o that the carrier cot are minimized. Uing complementary lackne condition, the author reformulate the problem a a ingle-level MIP that i olved uing an off-the-helf linear programming olver (CPLEX). The problem i modeled a an optimitic bilevel problem; i.e., it i aumed that the carrier would take the lower rik path in cae of equal cot, which i a reaonable aumption. Kara and Verter (2004) preent an application of the propoed methodology in Wetern Ontario, Canada (the cae alo ued for the computational experiment reported in thi paper). Their reult how that ignificant reduction in population expoure can be achieved through government intervention on the ue of road egment by hazmat vehicle. However, their method cannot olve large-cale intance in a reaonable amount of computation time. Erkut and Gzara (2008) alo formulate the network deign problem a a bilevel problem, but intead of olving the complete problem, they propoe a heuritic algorithm that iterate between the outer-level and the inner-level problem (that are both pure network flow problem). A a reult, they improve the computational performance of the olution methodology but obtain uboptimal olution. The author alo generalize the model to a biobjective model by including the traveling cot in the regulator objective function (outer-level). Erkut and Alp (2007) formulate the minimum-rik network deign problem a a Steiner tree election problem. By reducing the poibilitie of the carrier to a ingle path for every O-D hipment, thi methodology reduce the bilevel problem to a ingle-level problem. However, it can reult in increaed population expoure a well a higher travel cot for the carrier. To circumvent the latter weakne, the author propoe a greedy heuritic to add edge to the tree (correponding to hortet path) while keeping the rik increae to a minimum. They alo include traveling cot to the rik in the objective function of the tree election problem. Finally, Verter and Kara (2008) introduced a inglelevel path-baed formulation for the hazmat ND problem where only thoe path that are acceptable to the carrier are included in the model. Thee path are determined a priori for each O-D hipment and are ranked according to the carrier preference. Conequently, the optimal olution of the Verter and Kara (2008) model determine not only the road egment to be cloed to hazmat hipment by the regulator but alo the route that would be ued for each hipment on the reulting network. The propoed methodology i intended to facilitate the conultation between the regulator and the hazmat carrier during the policy deign proce. Some other intereting work on the hazmat global routing problem can be found in the operation reearch literature. Thi planning problem belong to a government agency whoe mandate i to route the hazmat hipment within and through it juridiction. Thi problem i not modeled a a bilevel problem becaue it i ued in a context where the regulator can decide on the route ued by the carrier. The objective of the regulator i to minimize the total rik for the population but alo to enure equity in the patial ditribution of the rik. Recent contribution include Marianov and Revelle (1998); Akgün, Erkut, and Batta (2000); Dell Olmo, Gentili, and Scozzari (2005); and Carotenuto, Giordani, and Ricciardelli (2007). We now turn to an overview of the application of TS in tranportation. The congetion pricing problem uually conider a regulator etting toll o a to minimize the total traveling time for the uer (or maximize the ocial welfare), wherea an optimal olution to the uer problem i an equilibrium where none of the uer i intereted in altering hi path choice. When all road egment are ubject to toll, marginal cot pricing induce the optimal ue of the network (Morrion 1986). In that cae, toll can be een a the difference between the ocial cot (contribution to total traveling time) and the perceived cot for the uer. If there i more than one toll cheme inducing an optimal ue of the road network, then a cheme optimizing a econdary objective uch a minimizing the total toll collected can be utilized (ee, e.g., Bergendorff, Hearn, and Ramana 1997 or Laron and Patrikon 1998). In many ituation, however, only econd-bet olution are implementable, i.e., olution in which not every road egment can be tolled. For example, ituation calling for econd-bet olution occur when pricing i allowed on certain highway only, or in the preence of pay lane or a toll cordon around a city. Thee problem are uually more realitic, but a lower ocial welfare i expected and they are alo more difficult to olve. Intead of maximizing total welfare, owner of private road might wih to maximize the profit related to the toll et on the road egment. Among other, Viton (1995); Liu and McDonald (1999); De Palma and Lindey (2000); and Verhoef (2005) tudied econdbet pricing. In uch problem, the optimal location of the toll point can alo be conidered (ee, e.g., Verhoef 2002). Labbé, Marcotte, and Savard (1998) introduced a general bilevel toll model where a regulator eek to maximize the profit generated by toll put on a ubet of road egment, taking into account that the uer chooe minimum-cot path with repect to the choen toll. Thee author have hown that thi problem, having bilinear objective function at both level, i trongly NP-hard, wherea primal-dual

4 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 231 algorithm aimed at olving large-cale intance of regular freight tranportation problem were derived by Brotcorne et al. (2000, 2001). A heuritic approach to a imilar problem wa propoed by Catelli et al. (2004). Alo in the context of a profit-maximizing firm, Lederer (1993); Bahyam (2000); and Brotcorne et al. (2008) conider the problem of jointly deigning and pricing a network. Conflicting objective between the leader and the follower are not preent in the bilevel hazmat TS where the regulator may even want to minimize, in part, the revenue raied from toll. Finally, we mention the related work of Bouhtou, Erb, and Minoux (2007), who extended the bilevel framework to the analyi of pricing and reource allocation for telecommunication network. 3. Mathematical Formulation Let G = N A be a road network where N i the node et and A i the arc et. Each node i N correpond to an interection in the road network, and each arc i j A correpond to a road egment between two interection. Conider a et H of hazmat type and a et S of O-D hipment to be performed. Ideally, S i compried of all hipment uing one of the road egment of the choen geographical region. For each hipment S, let k be the aociated carrier and n be the number of truck needed to complete the hipment. Let h be the rik on arc i j A when hazmat type h H i carried. If h i the type of hazmat tranported by hipment S, then h i the rik on arc i j A aociated with hipment S (per truck). Alo, let c be the cot of traveling on arc (i j). Note that throughout the paper, we ue the term carrier cot and traveled ditance interchangeably. For each node i N and each hipment S, let ei take the value 1 (repectively, 1) if node i i the origin (repectively, detination) of hipment. Finally, let x and y h be binary variable that take the value 1 if arc i j A i ued for hipment S and if it can be ued for hazmat type h H, repectively, and let t h be the toll on arc i j A for hazmat type h H. Table 1 provide a ummary of the notation ued in the formulation The Network Deign Problem The general network deign problem can be modeled a the following bilevel program: (ND) min y x S n ( h ) + c x (1).t. y h 0 1 i j A h H (2) min n ( c + h ) x (3) x S Table 1 A H N S S h c e i h k M n h t h x y h Mathematical Notation Set of arc Set of hazmat type Set of node Set of O-D hipment Set of O-D hipment of hazmat type h H Parameter converting ditance into population expoure unit Parameter converting population expoure into ditance unit The length of arc i j A Equal 1, 1, or 0depending if node i N i the origin, the detination or a tranhipment node for hipment S Hazmat type carried by hipment S Carrier hipping S Big-M contant Number of truck needed by hipment S Number of people expoed on arc (i j) when hazmat type h H i carried Continuou variable that repreent the toll on arc i j A for hazmat type h H Binary variable that repreent the flow on arc i j A for hipment S Binary variable that indicate if arc i j A i opened for hazmat type h H.t. x j i A x ji = e i i N S (4) x yh i j A S (5) 0 1 i j A S (6) x where it i undertood that in the outer (or leader) problem (1) (2), the vector x mut be an optimal olution of the inner (or follower) problem (3) (6). The parameter and allow the comparion between population expoure and carrier cot. Their value i fixed by the regulator. If = = 0, then the general ND problem reduce to that propoed by Kara and Verter (2004). In ND, the leader (regulator) deign a network that minimize a combination of population expoure and traveling cot, taking into account that carrier optimize their individual utility (it i aumed that all truck aociated to the ame hipment take the ame path). What make the problem hard i the fact that the trade-off value between rik and cot may differ for the leader and the follower; i.e., 1/. Although, for the ake of notational implicity the parameter i identical for all carrier, one could make it dependent on the index k (carrier hipping S) without changing the nature of the problem. Note that ND i eparable by hazmat type. Note alo that if tie between inner-level olution (route) occur, the bilevel formulation implie that carrier adopt the one that minimize the leader objective, i.e., mainly rik Single-Level MIP Reformulation. For fixed deign variable y h, the inner problem i a

5 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 232 Tranportation Science 43(2), pp , 2009 INFORMS network flow problem. The binary requirement on x can thu be replaced by nonnegativity contraint, and the bound x 1, i j A, S, can be dropped becaue it i implied by the contraint y h 1. Hence, the follower linear problem can be replaced by it primal-dual optimality condition. Let i, i N S and, i j A S be the dual variable aociated with contraint (4) (5), repectively. With = = 0, Kara and Verter (2004) reformulated ND a the following ingle-level program: min y x.t. S n ( h ) + c x (7) x j i A x ji = e i i N S (8) x yh i j A S (9) i j n( c + h ) i j A S (10) yh x = 0 i j A S (11) ( x i j n( c + h )) = 0 i j A S (12) x y h 0 i j A S (13) 0 i j A S (14) 0 1 i j A h H (15) Contraint (8), (9), (14), and (15) enure primal feaibility, contraint (10) and (13) enure dual feaibility, wherea contraint (11) and (12) force complementary lackne. Reetting x to be binary, the latter two nonconvex group of logical contraint can be linearized in the uual way. If M are big-m contant, contraint (11), (12), and (14) can be replaced with the following contraint: M ( ( h )) 1 y x i j A S (16) i j n( c + h ) M 1 x i j A S (17) x to yield a MIP. 0 1 i j A S (18) An Alternative MIP Reformulation. Recall that when the deign variable y h are fixed, the follower problem i linear. The ingle-level network deign problem can thu be modeled with contraint that impoe the equality of the objective function value of the follower primal and dual problem intead of complementary lackne condition. In that cae, contraint (11) and (12) can be replaced with the following contraint: n ( c + h ) x = e i i yh i N S (19) One can oberve that the latter contraint are nonconvex. Following the trategy decribed in Labbé, Marcotte, and Savard (1998), the bilinear term can be linearized by introducing the variable = yh in the model. The following linear contraint are added to enure that = 0 when yh = 0 and = when y h = 1: 0 i j A S (20) M yh 0 i j A S (21) 0 i j A S (22) + M yh M i j A S (23) and contraint (19) can be replaced with the following linear contraint: n ( c + h ) x = e i i i N S (24) to yield an alternative MIP formulation olely baed on the integrality of y A Toll Approach An alternative approach to inducing the ue of afe route can be achieved by a toll policy. It mathematical formulation i a follow: (TS) min t x S i j A n ( h + ( c +t h )) x (25).t. t h 0 i j A h H (26) min n ( c +t h + h ) x (27) x.t. S i j A i j A x j i A x ji =e i i N S (28) x 0 1 i j A S (29) In TS, the leader et toll that minimize a combination of population expoure and travel cot, taking into account that the inner problem (27) (29) minimize the carrier utility (with repect to the toll policy). A wa the cae for ND, TS i eparable by hazmat type, and one can ue in the follower objective function (27) parameter S, which are pecific to each carrier. Note again that if tie between inner-level olution (route) occur, the bilevel formulation implie that carrier adopt the one that minimize the leader objective. Actually, with TS a oppoed to ND, tie could be broken through an arbitrarily mall perturbation of the toll.

6 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS Single-Level MIP Reformulation. Following our earlier trategy, the inner program can be replaced by it primal-dual optimality condition. Upon the introduction of dual variable i i N S, thi yield the ingle-level program: min t x.t. S n ( h ) + c x + n t h x (30) x j i A i j n t h ( x t h x i j n t h x ji = e i i N S (31) n ( c + h ) i j A S (32) n ( c + h )) = 0 i j A S (33) 0 i j A h H (34) 0 i j A S (35) Again, after reetting the binary contraint on x, one may linearize the complementarity contraint (33): i j n t h x n ( c + h ) M ( ) 1 x i j A S (36) 0 1 i j A S (37) a well a the bilinear term of the leader objective: 0 i j A S (38) M x 0 i j A S (39) th 0 i j A S (40) th M x M i j A S (41) to yield a MIP formulation. An alternative MIP formulation can be obtained by replacing contraint (36) with contraint impoing the equality of the objective function value of the follower primal and dual problem: n ( c + h ) x + n e i i = 0 i N S (42) where the value of the variable, i j A, S, i already properly et by contraint (38) (41). Both MIP formulation require the ame integer variable. When = 0, becaue the outer-level objective i linear, variable and contraint (38) (41) are redundant in the complementary lackne formulation The Toll Problem I Not Equivalent to the Deign Problem When there i only one O-D hipment, it can be eaily hown that ND i equivalent to TS, in the ene that they yield the ame optimal value and all uboptimal path are made either unattractive (large toll in TS) or unavailable (ND). Becaue the problem i eparable by hazmat type, thi alo hold if there i more than one O-D hipment, provided that each carrie a different type of hazmat. When more than one hipment carrie the ame type of hazmat, it i eay to ee that by etting high-enough toll, TS can alway reduce to ND. However, the revere i not true. We next provide an example that illutrate the added flexibility of TS over ND. Let u conider the example of Figure 1, which involve three O-D hipment, O 1 D 1, O 2 D 2, and O 3 D 3, compried of only one truck each. One can notice that there i only one poible path for each of O 2 D 2 and O 3 D 3, which i to go through interection B, C, and O 1, B, repectively. Suppoe that the government ole objective i to minimize the rik for the population ( = 0), and the carrier ole objective i to minimize their traveling cot ( = 0). Suppoe alo that the rik on arc (B C) i larger than the one on arc (A D 1 ), i.e., r B C >r A D1, and that the rik on all other arc i null (nobody live within the evacuation area). Suppoe, finally, that the traveling cot on all arc i one unit, except for arc (A D 1 ) and (A C), both having a traveling cot of three unit. In that cae, if the regulator doe not interfere, then O 1 D 1 would take the path going through interection B, C (hortet path) and the total rik would be 2r B C (recall that path O 2 D 2 alo ue (B C)). With a network deign policy, the regulator would cloe arc (C D 1 ) becaue it i the only way to prevent the ue of arc (B C) for O 1 D 1.Arc B C and (O 1 B) cannot be cloed becaue they mut be ued by O 2 D 2 and O 3 D 3, repectively. Hence, the total rik would be r B C + r A D1 (path O 2 D 2 ue (B C) and path O 1 D 1 ue (A D 1 )). On the other Figure 1 O 3 O 2 O 1 1 B 1 A D 3 D 2 r (A, D1) 3 r (B, C) 1 C 1 An Example Where Toll Setting Give a Lower Optimal Rik 3 D 1

7 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 234 Tranportation Science 43(2), pp , 2009 INFORMS hand, a olution to the toll problem would et the toll o that path O 1 D 1 ue neither arc (B C) nor arc (A D 1 ). For intance, the regulator could et a toll of two unit on arc (B C) and one unit on arc (A D 1 ). The rik-free path O 1 D 1 going through interection A, C thu become a attractive for the carrier a the other more riky path, and the total rik would be r B C (arc (B C) mut till be ued for O 2 D 2 ). In thi example, if r B C r A D1, then a network deign policy would thu only marginally reduce rik (r B C + r A D1 veru 2r B C ), wherea it would be halved with a toll policy (r B C veru 2r B C ). Both rik-mitigating policie are, thu, clearly not equivalent. The main difference between the network deign and the toll policie i that the latter can actually differentiate between carrier. A toll can be high enough to deter a carrier from uing the correponding arc, wherea another carrier moving the ame type of hazmat might till ue the arc. ND doe not have the ame flexibility becaue the deign deciion have to be the ame for all carrier moving the ame type of hazmat. 4. MinimizingHazmat Tranport Rik via Toll Setting The previou ection demontrated that toll policie are more flexible than network deign policie and can thu induce lower hazmat tranport rik for the population. In thi ection, we will further how that it i alway poible for a regulator to find a toll policy that induce minimum rik and that finding uch a olution i an eay tak. A minimum-rik flow i a olution correponding to the minimum level of rik at which all hipment are delivered, i.e., a regulator ideal olution. The problem of finding a minimum-rik (MR) flow can be tated a follow: (MR) min x.t. S x n h x (43) x j i A x ji = e i i N S (44) 0 i j A S (45) MRi compried of the objective function of TS (25), where the value of i fixed to zero, and the flow conervation contraint. One can oberve that MR i a pure network flow problem (one hortet-path problem per carrier). Let x denote the minimum-rik flow obtained by olving MR. For x to be the optimal olution to TS (i.e., for a toll policy to induce minimum rik for the population), then toll have to be et on the road egment in uch a way that x become the carrier optimal flow a well (an optimal olution to the inner problem (27) (29)). Thi can be achieved by imply etting a toll on every arc with a value equal to the difference between the arc coefficient in the objective function of MR and the one of the follower (27). Thi procedure i akin to marginal cot pricing (Pigou 1920). In the preent cae, for a given arc i j A and a given hipment S, thi marginal cot i 1 h c. When toll are et to thee value, then the objective of the carrier matche that of the leader, and the carrier optimal flow obviouly coincide with the minimum-rik flow. However, nothing prevent a toll calculated in thi fahion from being negative (population expoure can be null on ome road egment). When ubidie are not permitted, ome of the contraint (26) might thu be violated. In addition, all road egment can potentially be tolled in uch a olution, which make it implementation economically and technologically difficult, if not impoible. Alternatively, the problem of finding a et of nonnegative toll that yield the minimum-rik flow can be olved by invere optimization, which conit of inferring the value of ome model parameter (in thi cae the toll can be een a a part of the cot coefficient) given the value of the deciion variable. See Dial (1999) or Ahuja and Orlin (2001) for ome other application of invere optimization (IO). In our context, one might wih to minimize the um of toll raied from the carrier beide enforcing the minimum-rik olution x. Thi i achieved by the following linear mathematical program: (IO( x)) min t S i j A.t. i j n t h n t h x (46) n ( c + h ) i j A S (47) x ( i j n t h n ( c + h )) =0 i j A S (48) t h 0 i j A h H (49) where nonnegative toll are choen o that the complementarity lackne condition of the follower problem (carrier) are atified at x. It i, in fact, the ingle-level model (31) (35), where the variable x are et at x. One can notice that the flow conervation contraint and the nonnegativity contraint on x are not neceary in IO( x) becaue they are trivially atified at x. If one elect to impoe the equality of the objective function value of the follower primal and dual problem intead of complementary lackne condition, contraint (48) can be replaced with the following equivalent linear contraint: n ( c + h ) x + n t h x e i i = 0 i N S (50)

8 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 235 Propoition 1. When all road egment are ubject to toll, there exit a et of nonnegative toll that yield a minimum-rik olution; i.e., IO( x) i alway feaible. Proof. Firt, we note that there alway exit a cycle-free minimum-rik olution x, becaue thi olution i the olution of a linear program and can therefore be aumed to be an extreme point of a flow polyhedron. The proof i baed on an argument of Yang and Huang (2004), initially propoed in the context of forcing the optimal ue of a congeted tranportation network involving cutomer with different valuation of travel time. Let u conider the following auxiliary linear program: (AP) min x.t. S n ( c + h ) x (51) x j i A x ji = e i i N S (52) n x n x i j A h H (53) S h S h 0 i j A S (54) x where S h i the et of O-D hipment carrying hazmat type h H. Let i, i N S, and h, i j A h H, be the dual variable aociated with contraint (52) and (53), repectively. For a feaible olution of AP to be optimal, then it mut alo atify the following primal-dual optimality condition (after contraint (53) are multiplied by 1): i j n h n ( c + h ) i j A (55) )) = 0 ( x i j n h n( c + h i j A S (56) h 0 i j A h H (57) One can notice that the complementary lackne condition tating that either a contraint (53) i active, or the correponding dual variable h i null, are not included in the latter optimality condition. Becaue all coefficient are nonnegative in the objective function of MR(ued to obtain x), contraint (53) are, in fact, alway active in AP. Otherwie, a feaible olution to MRwith a lower rik than x would exit, which i impoible becaue x i an optimal olution to MR. One can alo oberve that the optimal value of AP n c + h x (the total flow (51) i S on every arc for every hazmat type i known becaue all contraint (53) are active in AP for every feaible olution). Hence, x, which atifie contraint (52) (54), i an optimal olution of AP. The optimality condition of AP are thu atified at x, and IO( x) i feaible (IO( x) contraint (47) (49) are equivalent to AP optimality condition (55) (57), where the toll variable correpond to the nonnegative dual variable h ). Therefore, there alway exit a olution to TS where all toll are nonnegative and for which the correponding cot i equal to the minimum rik. Hence, when the regulator only wihe to minimize rik, i.e., when = 0 in the leader objective function (25), TS i not a bilevel problem. However, the more general toll problem, where the regulator rather wihe to minimize a combination of population expoure and traveling cot (including paid toll), cannot be olved by invere optimization and i thu a true bilevel problem. Neverthele, when x i an optimal olution to MR, where the objective function (43) i replaced with min x S n ( h ) + c x (58) IO( x), although not equivalent to TS, can be ued a a proxy. The latter invere optimization problem indeed find a et of minimum toll yielding a olution that itelf minimize a combination of population expoure and ditance traveled. It ha the advantage of being very eay to olve and providing olution with a reduced combination of rik and traveled ditance for the carrier, but may mean higher paid toll compared with directly olving one of the MIP formulation for TS. 5. Solution Methodology A demontrated in the previou ection, the toll problem i efficiently olved by invere optimization when the ole objective of the regulator i to minimize hazmat tranport rik. However, the general toll problem i, like the network deign problem, truly bilevel. The MIP formulation propoed in 3 for TS and ND can be olved directly with a powerful linear programming oftware, but ome enhancement are required to obtain optimal olution in reaonable computing time Boundingthe Big-M Contant In all MIP formulation preented, large contant are ued. It i well known in the integer programming field that the value of uch contant ha an impact on the olution proce, and our formulation are no exception to the general rule MIP Formulation with Equality of the Primal and Dual Objective. Dewez et al. (2006) have propoed tight and valid bound for toll problem where the formulation impoing the equality of the primal and dual objective of the follower problem i ued. Among other valid bound, the author

9 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 236 Tranportation Science 43(2), pp , 2009 INFORMS propoe to calculate, for a given arc i j A and a given hipment S (an O-D pair), the difference between the hortet ditance from the origin of hipment (O ) to it detination (D ) on a toll-free path (a path compried of nontollable arc), and the hortet ditance from O to D uing arc (i j) (when all toll are fixed to 0). The idea i to compute the maximum toll that could be et on every arc for every carrier. A imilar procedure could be applied to network deign problem. From contraint (20) (23) and the binary contraint on y, for a given arc i j A and a given hipment S: (i) if y h = 0, then = 0 and M ; (ii) if y h = 1, then M and =. Hence, the dual variable aociated with contraint (5),, are valid upper bound for M, and, becaue repreent the increae in the carrier cot of hipment when arc (i j) i cloed, they are themelve bounded by the value of the hortet ditance from O to D on a path compried of noncloable arc. In the cae of the hazmat tranportation problem, the exitence of toll-free path (or path compried only of noncloable arc) i not guaranteed. Unlike other toll problem, the leader objective in TS i not to maximize revenue raied from toll but to minimize population expoure (and even minimize a fraction of the paid toll becaue they contribute to the carrier cot). Hence, the problem i bounded without having to uppoe that there exit toll-free path between each origin-detination pair. A valid upper bound on the hortet ditance between an O-D pair, although le tight, can be provided by the longet path between the O-D pair. However, the problem of finding longet path i NP-complete (unle it i on a directed acyclic graph, which i not the cae here becaue arc can repreent two-way road). Neverthele, one can efficiently generate valid bound by olving, for every O-D pair, a maximum-cot flow problem (minimum-cot flow where all cot are multiplied by 1 in the objective function), which i linear and bounded (every arc ha a capacity of one unit of flow). Let B1 be the upper bound on M obtained by calculating the difference between the value of the maximum-cot flow problem from O to D (upper bound on the longet path) and the hortet ditance from O to D uing arc (i j). Although valid, thee bound are obtained from olution that are not necearily path becaue they can contain cycle, both attached to the O-D path or dijoint. For example, Figure 2 how the olution of a maximumcot flow problem where a unit of flow goe from O to D. One can oberve that the cycle (2 3 2) and (4 5 4) are preent in the olution becaue they contribute to increaing the cot of the olution, but they break up the path. However, jut removing both O 4 Figure D A Maximum-Cot Flow Solution That Doe Not Correpond to a Path cycle (O 1 2 D) may yield an invalid bound. On the other hand, one can improve the bound by limiting the flow to one at every node, and thu eliminating the cycle attached to the O-D path (like (2 3 2)). One can oberve that thi can be done without breaking the network tructure, by plitting in two every node, which are afterward linked together with a capacity of one unit. Let B2 B1 be the upper bound on the longet path obtained from thee modified maximum-cot flow problem. When one wihe to further improve the bound through the incorporation of cycle elimination contraint (dijoint cycle like (4 5 4)), then the network tructure collape and the olution proce ha to be embedded within a branch-and-bound procedure. Our numerical reult how that when even the implet uch contraint are added (two-cycle contraint), the improved quality of the bound i offet by the CPU time required for their computation. Nonethele, the linear relaxation of thee contrained problem yield valid upper bound (maximization problem), denoted by B3, which improve on B2. Our computational experiment how that the value of the linear relaxation and the total CPU time required to olve the network deign problem are indeed improved by uing B3 compared with uing B1 and B2, but alo compared with a bet empirical bound obtained by gradually decreaing a unique M appearing in every contraint (21) and (23) until the objective value top being optimal MIP Formulation with Complementary Slackne Condition. From contraint (16) (18) and the binary contraint on y, for a given arc i j A and a given hipment S, we may write: (i) if x = 0 and y h = 1, then M j i + n c + h ; (ii) if x = 0 and yh = 0, then M + j i + n c + h ; (iii) if x = 1, then yh = 1 and M. Hence, the contant M n c + h i bounded by + j i, where i the increae in the carrier cot when arc (i j) i cloed for hipment, and j i i the difference between the hortet ditance from O to j and the hortet ditance from O

10 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 237 to i. When the exitence of a toll-free path (or a path compried of noncloable arc) i not guaranteed for every pair of node, a valid upper bound on j i i given by the longet path between O and D, which i alo a valid upper bound for (ee 5.1.1). can thu be et to 2B3 + n c + h, where M B3 i the upper bound provided by the LP value of a maximum-cot flow problem augmented with the k-cycle contraint (ee 5.1.1). When the formulation including complementary lackne condition i ued, upper bound on M can be more than twice a large a the bound correponding to the equality formulation Warm-Startingwith a Toll Scheme In ome cae, the regulator can be more intereted in network deign olution than in toll olution if it feel that the former are eaier to implement. Neverthele, the toll problem (or it proxy) can be ued to contruct a feaible olution to the network deign problem to accelerate the olution proce. Thi i particularly true when the computing time i limited and a good feaible olution i needed rapidly (e.g., when evaluating different cenario). A feaible olution to ND can be found by olving a minimum-cot flow problem on a reduced network where all arc that are tolled in the optimal olution of TS, but unued by a carrier, are removed. Thi latter olution (x y) can be ued, after the value of the remaining variable ( f ) have been computed, a an upper bound in a branch-and-bound procedure. It i intereting to add that a olution obtained with the toll problem proxy can alo be ued to warm-tart the general bilevel toll problem. The computational experiment found in 6 how that thi enhancement i helpful in reducing computing time. Thi latter tatement i true even when the improved olution proce i compared with a olution proce uing CPLEX 10.0, which include MIP heuritic that have been known to efficiently find integer olution. 6. Computational Experiment In thi ection, we preent computational experiment that were carried out on the data found in Kara and Verter (2004). We firt provide a decription of thee intance, followed by a ummary of our computational experiment The Data Set The tet data are baed on the highway ytem of Wetern Ontario, Canada. Geographical information ytem were ued to obtain a decription of highway egment and information on population expoure within the region of interet. Artificial node were added to the real road network to enure that the denity of the population along any given arc i contant. The 1998 record of Statitic Canada (available by requet from Statitic Canada) provided the lit of hazmat hipment with correponding origin, detination, hazmat type, and the number of truck ued. Four different hazmat type, accounting for 56% of all the hazmat tranported, are conidered: gaoline, fuel oil, alcohol, and petroleum and coal tar. However, becaue the firt three type poe the ame expoure (evacuation of the people within 800 meter, according to Tranport Canada, 1996), they were grouped together. Therefore, the data et i compried of 287 hipment of either one of two hazmat type. The road network i compried of 48 node and 114 arc (57 two-way link) affecting 31 population center. In the tudy of Kara and Verter (2004), only the 53 hipment with an annual volume of 500 truck or more were kept in the data. In the preent paper, our tet are done on the ame ubet of hipment (500 + truck), but alo on all 287 hipment (all hipment). The partial data et i included in our computational experiment for the reader to appreciate the increaed difficulty of olving the complete data et Computational Experiment To evaluate the benefit of olving TS veru ND, we olved both formulation of each problem: the formulation with complementary lackne condition (CS) (ued by Kara and Verter 2004 for ND) and the alternative formulation involving the equality of the primal and dual objective (PD). For TS (with poitive ) and for ND, the ingle-level MIP formulation were olved, uing CPLEX The big-m contant were et to B3 (ee 5.1). TS wa alo olved by invere optimization (IO), and we warm-tarted the olution proce of ND with a feaible olution contructed from the optimal et of toll (IS) obtained by IO. All experiment were performed on an AMD Opteron Proceor 248, 2,191 MHz computer, uing two proceor. When IS i ued (and only in that cae), the heuritic ued by CPLEX 10.0 to generate integer olution became unneceary and were thu deactivated. We firt preent numerical reult for the cae where the leader objective i olely to minimize the population expoure, and then for the more general cae where a fraction of the carrier cot i alo minimized in the leader objective function. Population expoure (PopExp), traveled ditance (Dit), and computational effort (CPU) required to olve both data et with the different approache are compared. We alo indicate, for all approache, the number of cut generated by CPLEX (Cut), the number of node evaluated in the branch-and-bound tree (BBn), and the number of cloed, or tolled, arc (Nc-Nt) out of the 228 poibilitie (114 arc and two hazmat type).

11 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 238 Tranportation Science 43(2), pp , 2009 INFORMS Table 2 Notation (Numerical Experiment) Table 3 Network Deign Problem v. Toll Problem CS Formulation including complementary lackne condition IO Invere optimization proce IS Network deign problem with an initial olution contructed from TS ND Network deign problem PD Formulation where primal and dual objective are equal TS Toll-etting problem % chg Change, in percentage, from a pecified ND model to a TS model CPU Total CPU time (in minute) BBn Total number of node in B&B tree Cut Number of cut generated by CPLEX 10.0 PopExp Total population expoure (in million of peron) Dit Total ditance traveled (in million of kilometer) ObjVal Optimal value of the function combining rik and traveled ditance ObjVal+ Optimal value of the function combining rik, traveled ditance, and paid toll Tpaid Total amount of toll paid by the carrier (in million of dollar) Nc-Nt Number of arc cloed or number of arc tolled For the toll problem, we alo report the total amount of toll paid by the carrier (Tpaid). Finally, we computed the percentage change in population expoure and traveled ditance between ND and TS (% chg). The notation i diplayed in Table MinimizingPopulation Expoure. Even when the government ole objective i to minimize population expoure, it i advantageou to et to a mall poitive value to favor, among minimum-rik olution, one that minimize carrier cot. For both data et teted, = 1 wa uitable. ND and TS were olved for = 0 and = 1. The reult are preented in Table 3. For ND, the alternative PD formulation i fater than the current CS formulation, with the exception of the mallet data et ( = 1), where the optimal olution i found at node 0 + of the branch-and-bound procedure, i.e., exploiting CPLEX heuritic and/or cut at node 0. Recall that, in CS, binary contraint are required on x, wherea they are not in PD (ee 3.1.1). Formulation CS could not even olve the data et involving all hipment within 36 hour of computing time unle warm-tarted (IS). The warm-tart procedure actually improve the running time of both CS and PD, either with = 0or = 1. In Table 3, one oberve that the number of arc cloed i ignificantly reduced under PD. The latter MIP formulation thu yield, in a reduced computing time, more attractive olution for the regulator (le expenive to implement). When the leader objective function i perturbed to allow the minimization of the number of arc cloed, a a econd objective, the problem become much harder to olve, without achieving a ignificant improvement. For intance, PD cloe 14 arc for truck, wherea the minimum poible i 11 arc. For thi reaon, the reult of the latter CPU BBn Cut PopExp Dit Tpaid Nc-Nt truck ND ( = 0) 1. CS CS IS PD PD IS ND ( = 1) 5. CS CS IS PD PD IS TS ( = 0) 9. IO TS ( = 1) 10. IO PD % chg 9 v All hipment ND ( = 0) 1. CS +36h 2. CS IS PD PD IS ND ( = 1) 5. CS CS IS PD PD IS TS ( = 0) 9. IO TS ( = 1) 10. IO PD % chg TS v Note. Unle otherwie pecified, all CPU time are in minute. problem are not reported. If, to gain more control over the carrier, one i intereted in a olution involving a large number of cloed arc, then one imply ha to cloe all arc that carry no flow in the ND PD IS olution. The ame reult can be achieved under TS by etting the toll on all unued arc to arbitrarily large value. When = 0, the ditance traveled by the carrier can vary for the ame level of rik, up to 3.6% higher for ND in all hipment, i.e., million kilometer compared with the minimum of million. Becaue the incluion of a fraction of the carrier traveling cot within the leader objective function actually make ND eaier to olve, PD IS with = 1 (model 8) eem to be the bet choice for ND when the government objective i to minimize population expoure. TS i olved very quickly, and it yield the minimum-rik flow while minimizing the ditance traveled (with = 1) and etting poitive toll on a mall number of arc. With a mall poitive, iti

12 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 239 Table 4 From ND to TS: Some Intereting Data Table 5 Comparion of Bounding Method for M Type % of all hipment ND PD IS ( = 1) LP value CPU 1. Same path no tax Same path taxed in TS New path in TS overall cot decreae New path in TS overall cot increae 0 3 intereting to note that for both data et, the olution given by TS IO provide the ame optimal value a the true bilevel model (TS PD, where the traveling cot include the paid toll on top of the traveling ditance); i.e., there i no olution yielding the minimum rik while reducing a combination of traveled ditance and paid toll compared with the olution provided by the proxy. For thee intance, the proxy i thu equivalent to TS when the traveling cot are only minimized a a econd objective. When comparing TS with the bet ND (model 8), one can oberve that the total population expoure i only reduced by 0.03% for truck, but by a higher percentage (0.64%) when all hipment are conidered. The total traveled ditance can alo be lightly reduced under TS. For truck, the increae in total cot related to the toll actually paid i lightly maller than the decreae in total traveling cot (0.14%), wherea it i lightly higher when all hipment are conidered (0.76%). It i important to note that thi lat tatement applie no matter what the traveling cot per kilometer are. Depending on the ize of the truck and on the annual utilization, the operating cot of a liquid tanker in Ontario lie between $1.40 and $2.30 per kilometer (Tranport Canada 2005). Becaue arc cot are contant in the model, modifying the traveling cot only cale the model, a long a the large contant M are caled proportionally. Algorithmic efficiency i the ame, and the toll vector are only caled. Finally, Table 4 give a ummary of ome data that can be obtained when comparing TS to ND olution. From thi table, one can oberve that for more than 90% of all hipment, the path that i taken from the origin to the detination remain the ame, and only 3.8% of the hipment incur a cot increae (path change or toll increae) under TS Boundingthe Big-M Contant. A decribed in 5.1, the big-m contant ued in the inglelevel MIP formulation can be et to the difference between the hortet ditance from O to D uing arc (i j), and an upper bound on the longet ditance from O to D. Recall that B1 i obtained by olving a maximum-cot flow problem from O to D, wherea B2 i obtained by olving a modified problem where the flow i limited to one at every node and B3 i obtained by olving the linear relaxation of the latter modified maximum-cot flow problem with added 1. B B Bet empirical M B Total arc cot h Note. Unle otherwie tated, CPU time are in minute. contraint to forbid k-cycle. In our numerical reult, the addition of k-cycle contraint did not improve B3 for k>2. Table 5 compare the different bounding method on the bai of the total CPU time required to olve ND PD IS, with = 1, and the linear relaxation value (LP value) they yield. Beide the bound decribed in 5.1 (B1, B2, and B3 ), two other method were teted. Total arc cot ue a common M that i et equal to the value of the um, for all carrier, of all arc cot in the network (it i valid ince every arc ha a capacity of one for every carrier). Thi trivial bound wa then decreaed empirically until the optimal value of the reulting problem topped being optimal. The latter bound (Bet empirical M), for which the optimal value need to be known a priori, erved only a a comparion point. The method are ranked on the bai of their correponding LP relaxation. One oberve in Table 5 that the CPU time decreae ignificantly with every light improvement in the LP value (with = 1, the optimal integer objective function value i ), and that B3 i clearly the bet choice. It i intereting to note that the LP value with the bet empirical M (common contant) can be lower than with other bounding method with a pecific large contant for every arc and every carrier (M ). Our numerical experiment have alo hown that the linear programming (LP) value of the exiting MIP formulation (ND CS IS baed on B3 ) i equal to the one obtained with the bet empirical M (687.20). Even though the upper bound baed on B3 can be more than twice a large under CS a under PD, they are neverthele quite good for CS, and the comparion between formulation CS and PD found previouly remain relevant. The fact that the LP value obtained under PD i better than the one obtained under CS provide numerical evidence that PD i more efficient The General Hazmat Tranportation Problem. For the problem where the leader objective function involve a carrier term, Figure 3 illutrate the compromie between the population expoure and the traveled ditance when the parameter i gradually increaed in TS (all hipment), i.e., the Pareto boundary.

13 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 240 Tranportation Science 43(2), pp , 2009 INFORMS Ditance covered Figure ,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 Population expoure How the Population Expoure Increae a the Carrier Cot Decreae One can notice in Figure 3 that the population expoure lowly increae at firt, when the carrier cot rapidly decreae, and then more rapidly (when the population expoure goe beyond about 850 million). Thi turning point correpond to a value of = 70. A imilar curve i found when i gradually increaed in the maller data et. Table 6 preent a comparion of the population expoure, the traveled ditance, and the computational effort needed to olve ND and TS on both data et for the turning point value. For TS and ND, both ingle-level MIP formulation were olved (PD and CS), but becaue PD wa again clearly the bet choice, only the reult with PD are included in the table. A a comparion to the true bilevel TS, we alo olved a proxy of TS by IO (ee 4). The olution to the latter problem alo erved to warm-tart ND and TS. For thi general problem, we indicate the leader objective function value when it combine rik and traveled ditance (ObjVal), but alo when it combine rik, traveled ditance, and the paid toll (ObjVal+). The abbreviation for all the given olution characteritic are defined in Table 2. The ame concluion a the minimum-rik problem can be drawn for the general network deign problem; i.e., the propoed alternative formulation warm-tarted from an initial olution contructed from a et of toll (model 2, PD IS) i clearly the bet choice for ND. It i olved efficiently, and the olution contain a mall number of cloed arc. For the maller data et, one can oberve that when TS i approximated with IO (model 3), the olution obtained i equivalent to the one given by the true bilevel model (model 4, PD IS). When all hipment are conidered, the decreae in the combination of rik and traveling cot (ObjVal) i higher with the proxy than with the true model ( 0 34% for TS IO and 0 19% for TS PD IS, compared with ND). On the other hand, when the paid toll are taken into account, the proxy actually increae the combination of rik and total carrier cot (ObjVal+) compared with ND, wherea the true model doe not. The bilevel TS i a lot harder to olve than it proxy, but not ignificantly harder than ND. The different methodologie can provide different cenario to be analyzed by the regulator. Finally, Table 7 give a ummary of ome data that can be obtained when comparing TS (proxy) to ND olution. One can notice that the reult found in the table are imilar to the one that were given in Table 4 for the minimum-rik problem. Table 6 General Network Deign Problem v. General Toll Problem = 70 CPU BBn Cut PopExp Dit ObjVal Tpaid ObjVal+ Nc-Nt truck ND 1. PD PD IS TS 3. IO PD IS % chg TS v All hipment ND 1. PD PD IS TS 3. IO % chg 3 v PD IS % chg 4 v Note. All CPU time are in minute.

14 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 241 Table 7 Type From ND to TS: Some Intereting Data General Problem % of all hipment 1. Same path no tax Same path taxed in TS New path in TS overall cot decreae New path in TS overall cot increae Summary of Computational Experiment Contrained Cae In the model preented in the previou ection, it wa aumed that all road egment were ubject to retriction (toll or curfew). In real-world ituation, however, it i poible that ome of them are free of retriction for economical, political, or technical reaon, irrepective of the actual rik. When it i the cae, TS become a combinatorial program that can no longer be olved a a linear program. By contrat, the correponding network deign problem may become eaier to olve becaue the number of feaible combination i reduced. When ome road egment have to tay toll-free, the optimal value of the objective function i likely to deteriorate (econd-bet pricing). A weakne of the toll problem i that toll are not necearily et on the riky arc. For intance, one of the optimal olution to the example hown in Figure 1 et toll on arc (A D 1 ) and (O 1 B) (intead of (A D 1 ) and (B C)). The olution, although equivalent for the regulator, can be viewed a inequitable becaue O 3 D 3 ha to pay a toll although it ue a rik-free path, wherea O 2 D 2 i toll-free, even though it ue a riky arc! Thi weakne alo arie in the network deign problem, becaue nothing prevent rik-free arc to be cloed and thu lengthen a path for a carrier that would not have gone through a populated area but can be partially dealt with by retricting the et of arc ubject to toll or curfew, according to their aociated rik. Some rule might allow an arc to be tolled, or cloed, only if the correponding population expoure exceed a given threhold value, yielding olution that may be more acceptable to the carrier. For the numerical reult preented in thi ection, thi minimum level of rik (Rmin), given by a minimum number of people expoed, wa gradually increaed, and Table 8 compare ND and TS for the different value, when all hipment are conidered and = 1. The inglelevel MIP formulation impoing the equality of the objective function value of the follower primal and dual problem (PD) wa olved (with M = B3 ) for ND and TS. We alo tried to improve the olution proce of ND by warm-tarting it from a feaible olution contructed with an optimal toll cheme; and the total CPU time of the latter olution proce (including the computing time for olving TS) i given in the table (CPU IS). We alo indicate, for every level of rik, the percentage of arc that are ubject to retriction among all arc in the network (Arc%) and the percentage decreae in population expoure ( R%) obtained with TS a oppoed to ND. The abbreviation for all other olution characteritic are defined in Table 2. One can ee, from Table 8, that 55% of the network arc do not involve any population expoure. Once thee arc are taken out of the ubet of arc that are ubject to retriction, i.e., when Rmin = 1, the number of cloable combination i reduced and ND become much eaier to olve. The oppoite phenomenon can be oberved for TS becaue it top being linear when Rmin 1. A Rmin increae, ND continue to get eaier to olve, wherea the CPU time for TS i more table. The fact that TS i harder to olve when Rmin 1 make the ue of it olution a a tarting point for ND le attractive, but one can oberve that ND PD IS i neverthele generally lightly fater to olve than ND PD (recall that CPU TS i already included in CPU IS). One can alo oberve in Table 8 that from Rmin = 0 to 1,500, the total carrier cot (ditance traveled and toll) and the total population expoure do not increae for TS, while the number of tolled arc decreae a Rmin increae. In addition, the et of tolled arc in the olution alo change. Hence, the Table 8 Network Deign Problem v. Toll Problem Contrained Cae Network deign (ND PD) Toll-etting (TS PD) Rmin Arc% PopExp Dit Nc CPU CPU IS PopExp Dit Tpaid Nt CPU R% , , , , , Note. All CPU time are in minute.

15 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik 242 Tranportation Science 43(2), pp , 2009 INFORMS olution with Rmin = i a intereting for the regulator a the one with Rmin = 0, while being perceived a more fair from the carrier. It ha the advantage of not retricting the ue of le-riky arc and thu preventing ituation where a carrier pay a toll even if it doe not go through any populated area (jut to prevent another carrier from uing a riky path). For the network deign problem, the population expoure tart increaing when Rmin = At that minimum-rik level, there i, in fact, a 1% difference in population expoure when the deign problem i olved intead of the toll problem, which i olved about ix time fater than ND. One can finally oberve that when Rmin = , TS become equivalent to ND. When only the mot riky arc are ubject to retriction, no toll are paid by the carrier (they only erve to dicourage the carrier to ue the correponding arc) and the optimal rik for the population and cot for the carrier are the ame in TS and in ND. It i intereting to add that when the optimal value i the minimum rik, i.e., up until Rmin = 3 000, it wa alway poible to olve TN with invere optimization, in a fraction of TS PD CPU, which make warm-tarting ND with TS even more advantageou. In ummary, thee contrained hazmat problem are intereting becaue they produce olution that can be more acceptable to the carrier. For thee problem, toll etting till find better olution than network deign in a reduced computing time. 7. Concluion Thi paper ha introduced toll etting a an efficient policy tool for mitigating the public and environmental rik aociated with dangerou good hipment. We compared the hazmat TS problem, where toll are impoed on road egment in order to channel the hazmat hipment toward le-populated road; with the more popular hazmat ND problem, where certain road egment are cloed to hazmat tranportation. We demontrated that TS, by being able to differentiate between carrier, can achieve higher reduction in the aociated tranport rik while only lightly increaing the carrier cot, and can be ued by a regulator to obtain minimum-rik olution very efficiently. The paper ha alo propoed a more efficient ND formulation requiring a reduced number of integer variable and introduced an improved olution methodology where the toll problem i ued to contruct an initial olution. Finally, thi paper ha propoed valid and eaily calculated bound for the value of the large contant ued in the MIP formulation. The bound that are propoed in the literature for other toll problem alway rely on the exitence of a toll-free path, which i not the cae for thi hazmat tranportation problem. Together, the propoed enhancement have allowed u to olve a much larger intance of the network deign problem in reaonable computing time, wherea the former approach propoed by Kara and Verter (2004) could not. The paper further propoed to limit the et of road egment ubject to retriction to improve the buy-in received from the hazmat carrier. Future development of our approach will conider both rik equity among the different population center and cot equity among the carrier. Acknowledgment Thi reearch ha been upported in part by a team grant from FQRNT and by NSERC dicovery grant (firt, third, and fourth author). The author are member of the Center for Reearch on Tranportation (CRT) in Montréal, which provided the infratructure for reearch. The generoity of Bahar Kara of Bilkent Univerity in haring the Wetern Ontario data et with the author i greatly appreciated. The author are alo grateful to three anonymou referee for their valuable comment. Reference Ahuja, R. K., J. B. Orlin Invere optimization. Oper. Re. 49(5) Akgün, V., E. Erkut, R. Batta On finding diimilar path. Eur. J. Oper. Re Bahyam, T Service deign and price competition in buine information ervice. Oper. Re. 48(3) Bergendorff, P., D. W. Hearn, M. V. Ramana Congetion toll pricing of traffic network. P. M. Pardalo, D. W. Hearn, W. W. Hager, ed. Network Optimization. Springer-Verlag, New York, Bouhtou, M., G. Erb, M. Minoux Joint optimization of pricing and reource allocation in competitive telecommunication network. Network 50(1) Brotcorne, L., M. Labbé, P. Marcotte, G. Savard A bilevel model and olution algorithm for a freight tariff-etting problem. Tranportation Sci. 34(3) Brotcorne, L., M. Labbé, P. Marcotte, G. Savard A bilevel model for toll optimization on a multicommodity tranportation network. Tranportation Sci. 35(4) Brotcorne, L., M. Labbé, P. Marcotte, G. Savard Joint deign and pricing on a network. Oper. Re. 56(5) Carotenuto, P., S. Giordani, S. Ricciardelli Finding minimum and equitable rik route for hazmat hipment. Comput. Oper. Re Catelli, L., G. Longo, R. Peenti, W. Ukovich Two-player noncooperative game over a freight tranportation network. Tranportation Sci. 38(2) Colon, B., P. Marcotte, G. Savard Bilevel programming: A urvey. 4OR Dell Olmo, P., M. Gentili, A. Scozzari On finding diimilar pareto-optimal path. Eur. J. Oper. Re Dempe, S Bilevel programming. C. Audet, P. Hanen, G. Savard, ed. Eay and Survey in Global Optimization. Springer-Verlag, New York, De Palma, A., R. Lindey Private road: Competition under variou ownerhip regime. Ann. Regional Sci. 34(1) Dewez, S., M. Labbé, P. Marcotte, G. Savard New formulation and valid inequalitie for a bilevel pricing problem. Technical report, Univerité Libre de Bruxelle, Bruxelle, Belgium.

16 Marcotte et al.: Toll Policie for Mitigating Hazardou Material Tranport Rik Tranportation Science 43(2), pp , 2009 INFORMS 243 Dial, R. B Network-optimized road pricing: Part I: A parable and a model. Oper. Re. 47(1) Erkut, E., O. Alp Deigning a road network for hazardou material hipment. Comput. Oper. Re Erkut, E., F. Gzara Solving the hazmat tranport network deign problem. Comput. Oper. Re. 35(7) Erkut, E., S. Tjandra, V. Verter Hazardou material tranportation. C. Barnhart, G. Laporte, ed. Handbook in Operation Reearch and Management Science, Tranportation. North- Holland, Amterdam, Kara, B. Y., V. Verter Deigning a road network for hazardou material tranportation. Tranportation Sci. 38(2) Labbé, M., P. Marcotte, G. Savard A bilevel model of taxation and it application to optimal highway pricing. Management Sci Laron, T., M. Patrikon Traffic management through link toll An approach utilizing ide contrained traffic equilibrium model. P. Marcotte, S. Nguyen, ed. Equilibrium and Advanced Tranportation Modelling. Kluwer, Dordrecht, The Netherland, Lederer, P. J A competitive network deign problem with pricing. Tranportation Sci. 27(1) Liu, N. L., J. F. McDonald Economic efficiency of econdbet congetion pricing cheme in urban highway ytem. Tranportation Re. Part B 33(3) Marianov, V., C. Revelle Linear, non-approximated model for optimal routing in hazardou environment. J. Oper. Re. Soc Morrion, S. A A urvey of road pricing. Tranportation Re. Part A 20(1) Pigou, A. C., ed Wealth and Welfare. Macmillan, London. Tranport Canada Operating Cot of Truck in Canada Tranport Canada, OperatingCot2005/2005-e.htm. Verhoef, E. T Second-bet congetion pricing in general network. Heuritic algorithm for finding econd-bet optimal toll level and toll point. Tranportation Re. Part B 36(8) Verhoef, E. T Second-bet congetion pricing cheme in the monocentric city. J. Urban Econom. 58(3) Verter, V., E. Erkut Incorporating inurance cot in hazardou material routing model. Tranportation Sci. 31(3) Verter, V., B. Y. Kara A path-baed approach for hazmat tranport network deign. Management Sci. 54(1) Viton, P. A Private road. J. Urban Econom Yang, H., H.-J. Huang The multi-cla, multi-criteria traffic network equilibrium and ytem optimum problem. Tranportation Re. Part B 38(1) 1 15.