Implications of Cost Equity Consideration in Hazmat Network Design

1 The hazmat network design problem (HNDP) aims to reduce the risk of transporting hazmat in the 2 network by enforcing regulation policies. The goal of reducing risk can increase cost for different 3 hazmat carriers. Since HNDP involves multiple parties, it is essential to take the cost increase of all 4 carriers into consideration for the implementation of the regulation policy. While we can consider 5 cost by placing upper bounds on the total increase, the actual cost increase for various OD pairs 6 can differ, which results in unfairness among carriers. Thus we propose to consider the cost equity 7 issue as well in HNDP. Additionally, due to the existence of multiple solutions in current HNDP 8 models and the possibility of unnecessarily closing road segments, we introduce a new objective 9 considering the length of all the closed links. Our computational experience is based on a real 10 network and we show results under different cost consideration cases. 11 Sun, Karwan, and Kwon 2 INTRODUCTION 1 For hazmat transportation, the number of accidents is small compared to the number of shipments. 2 However, the consequence is very severe in terms of fatalities, injuries, large-scale evacuations and 3 environmental damage. Hence hazmat transportation usually remains part of a government’s man4 date. Government authority regulates hazmat transportation on the network under its jurisdiction 5 by the following methods: banning or putting tolls on certain road segments, curfews (banning 6 certain road segments for certain durations) and enforcing carriers to go through a set of chosen 7 checkpoints. 8 Here we consider the hazmat network design problem (HNDP) with the regulation method 9 of banning certain road segments. Kara and Verter (1) define the problem as follows: (1) given an 10 existing road network, the hazmat network design problem involves selecting the road segments 11 that should be closed so as to minimize total risk given that, (2) the carriers will then choose 12 the minimum cost routes on the resulting network. Hence the government should consider the 13 behaviours of the carriers when designing the road network. 14 Kara and Verter (1) formulate HNDP as a bilevel model with the government as a leader 15 (upper level) and the carriers as followers (lower level). They transform the bilevel model into a 16 single mixed integer problem by substituting the lower level problem with its KKT conditions and 17 solve the single model with a standard optimization solver (CPLEX). Erkut and Alp (2) consider 18 HNDP as a tree selection problem. In this way, the carriers have no alternative routes. They solve 19 the problem using a commercial solver and develop a simple construction heuristic to expand the 20 solution by adding road segments. This allows authorities to trade off risk and cost. Erkut and 21 Gzara (3) generalize the problem considered by Kara and Verter (1) to the undirected case and 22 propose a heuristic solution method. They also formulate the problem as a bi-objective bilevel 23 model to include trade-offs between risk and cost. Alternatively, in consideration of a compromise 24 between cost and risk, Verter and Kara (4) present a path-based formulation to identify paths 25 that are mutually acceptable to the government and the carriers. Amaldi et al. (5) provide an 26 exact formulation with fewer binary variables for HNDP. Gzara (6) proposes a family of valid cuts 27 and incorporates them within an exact cutting plane algorithm to solve the HNDP. Xin et al. (7) 28 consider a robust HNDP with risk interval data. Sun et al. (8) consider HNDP with risk uncertainty 29 using robust optimization with a cardinality uncertainty set to allow for flexible decision making. 30 Taslimi et al. (9) study HNDP by incorporating location of hazmat response teams and risk equity. 31 Fan et al. (10) consider the regulation method of closing road segments for certain durations and 32 present a path based model to mitigate risk. 33 Besides banning certain road segments, government can also set tolls to regulate hazmat 34 transportation. Marcotte et al. (11) first propose the use of tolls in mitigating hazardous materials 35 transport risk. Wang et al. (12) extend the approach to a dual toll pricing method to simultaneously 36 control both regular and hazmat vehicles to reduce risk. Esfandeh et al. (13) enhance the dual toll 37 pricing model by considering nonlinear delay time to more accurately measure the risk and model 38 equilibrium. Bianco et al. (14) consider toll policies to regulate hazardous material transportation 39 considering both total risk and risk spreading. Esfandeh et al. (15) propose and analyze a dual-toll 40 setting policy for both hazmat and regular carriers to minimize total risk on the network while 41 considering stochastic driver preferences in route selection. Bruglieri et al. (16) propose another 42 risk mitigation regulation to select a set of gateways in the network and enforce carriers go through 43 these checkpoints for their chosen routes. 44 Risk equity is also a major issue in hazmat transportation. In hazmat routing, models 45 Sun, Karwan, and Kwon 3 have been proposed for determining paths of minimum total risk while guaranteeing equitable risk 1 spreading (17). Gopalan et al. (18) study a single hazmat trip and limit the risk difference between 2 each pair of partitioned zones. Gopalan et al. (19) further develop the model into multiple O-D 3 pairs of hazmat transportation. Carotenuto et al. (20) consider the risk equity issue by placing an 4 upper limit on the total hazmat transportation risk over populated links. For HNDP, Bianco et al. 5 (21) consider risk equity by assuming the regional authority aims to minimize the total transport 6 risk induced over the entire region in which the transportation network is embedded, while local 7 authorities want the risk over their local jurisdictions to be as low as possible. Bianco et al. (14) 8 consider toll policies to regulate hazardous material transportation to not only minimize the total 9 risk but also to spread the risk in an equitable way. Taslimi et al. (9) minimize the maximum risk 10 among territory zones to address risk equity. 11 In HNDP, because government authority regulates different carriers likely leading to higher 12 costs for the carriers, cost should be a consideration of the HNDP as well. Erkut and Gzara (3) 13 extend the link based bilevel model to account for the cost/risk trade-off by including cost in the 14 first-level objective weighting both total risk and cost. The same model is considered by Gzara 15 (6) in analyzing a proposed cutting plane algorithm. Verter and Kara (4) consider a path based 16 formulation with cost/risk trade-offs for government and carriers. Specifically, they consider a 17 K-shortest path algorithm to generate all the paths. Alternatively, the paths with lengths that are 18 within a certain percentage of the length of the shortest path can also be used. Cappanera and 19 Nonato (22) study how to obtain the nondominated solutions considering risk and cost for gateway 20 location risk mitigation strategies. 21 Cost, however, has not been fully systemically studied in the literature. Moreover, cost 22 equity among different carriers is not considered in any of the current models. Closing certain 23 road segments can result in higher cost for carriers. But the cost increase for carriers could be 24 significantly different, resulting in unfairness of the regulation policy. In some extreme cases, for 25 example, one carrier’s cost could remain the same but another carrier could have its cost doubled. 26 Thus we propose to consider cost equity in HNDP. 27 In this paper, we study different HNDP models with various cost considerations, particu28 larly the cost equity issue, while addressing the existence of multiple optimal solutions. The re29 mainder of the paper is organized as follows. The next section introduces the HNDP models in the 30 literature. Then we provide different HNDP models with multiple cost consideration. Computa31 tional results are shown in the numerical experiments section. Finally, conclusions and suggestions 32 are given. 33 HNDP DESCRIPTION 34 In this section, we first describe the leader-follower bilevel model for the HNDP. Due to the uni35 modularity of the lower level problem, it can be linearized and the bilevel model can be transformed 36 into a single level model. We will then discuss the linearization methods. 37 Problem Description and Formulation 38 We consider the HNDP in which the government determines the available road segments to mini39 mize total risk and carriers choose routes on the resulting network to minimize cost. Suppose we 40 have a transportation network that is defined by a graph G = (N,A), where N denotes the set of 41 nodes (road intersections) and A denotes the set of arcs (road segments). HNDP involves trans42 porting S shipments between different origins and destinations. For each shipment s ∈ S, ns is the 43 Sun, Karwan, and Kwon 4 corresponding number of shipments, ri js and ci js are the risk and cost associated with arc (i, j)∈ A. 1 For simplicity, we assume the cost is independent of each shipment, resulting in ci js = ci j for any 2 shipment s∈ S. Let xi js = 1 if arc (i, j) is used to transport shipment s and yi j = 1 if arc (i, j) is open 3 to hazmat traffic. Then the problem can be formulated using a bilevel integer linear programming 4 model (1) as 5 min yi j∈{0,1} ∑ (i, j)∈A ∑ s∈S nsri jsxi js, (1) 6 where xi js is obtained by 7 min xi js ∑ (i, j)∈A ∑ s∈S ci jsxi js, (2) 8