Locating alternative-fuel refueling stations on a multi-class vehicle transportation network

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Production, Manufacturing and Logistics

Locating alternative-fuel refueling stations on a multi-class vehicle transportation network Seong Wook Hwang, Sang Jin Kweon, Jose A. Ventura∗ The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802, USA

a r t i c l e

i n f o

Article history: Received 12 January 2016 Accepted 21 February 2017 Available online xxx Keywords: Location Refueling infrastructure Transportation networks Multi-class alternative fuel vehicles 0-1 linear programming model

a b s t r a c t The existing literature regarding the location of alternative fuel (AF) refueling stations in transportation networks generally assumes that all vehicles are capable of traveling the same driving range and have similar levels of fuel in their tanks at the moment they enter the network and when they exit it. In this article, we relax these assumptions and introduce a multi-class vehicle transportation network in which vehicles have different driving ranges and fuel tank levels at their origins and destinations. A 0-1 linear programming model is proposed for locating a given number of refueling stations that maximize the total traffic flow covered (in round trips per time unit) by the stations on the network. Through numerical experiments with the 2011 medium- and heavy-duty truck traffic data in the Pennsylvania Turnpike, we identify the optimal sets of refueling stations for AF trucks on a multi-class vehicle transportation network. © 2017 Published by Elsevier B.V.

1. Introduction Compressed natural gas (CNG) and liquefied natural gas (LNG) have recently been considered as candidate next generation fuels to improve fuel economy and produce lower greenhouse gas emissions than traditional fossil fuels, such as gasoline and diesel. In particular, LNG is the top alternative fuel (AF) for diesel-powered trucks, which move approximately 90% of the freight tonnage in the US, because LNG has higher energy density than other AFs (Whyatt, 2010). LNG engines are suited for mid- and heavy-duty trucks, which are classified by the Federal Highway Administration (FHWA) into classes 6–10 according to the number of axles and trailer units (Randall, 2012, Appendix A), as shown in Fig. 1. LNGpowered trucks can reduce up to 16% of greenhouse gas emissions and 73% of volatile organic compounds emissions compared to diesel-powered trucks (Tiax LLC, 2007). Furthermore, the low fuel price of LNG compensates for the high cost of purchasing LNGpowered trucks (Garthwaite, 2013). Myers et al. (2013, Appendix C) state that trucking companies can reduce operational costs if the LNG price is at least $0.52 per diesel gallon equivalent less than diesel price, given that trucks travel at least 120,0 0 0 miles annually over 6 years. A proper AF refueling availability is necessary to encourage the use of AF vehicles, including LNG-powered trucks. There exists a variety of approaches to develop an AF infrastructure on a ∗

Corresponding author. E-mail address: [email protected] (J.A. Ventura).

transportation network. Vehicles may need multiple refuelings for long-distance trips because fuel tank capacities are limited; the driving range of AF vehicles is, in fact, even shorter than that of traditional fuel counterparts. A set of refueling stations must be located along a path to cover the corresponding traffic flow when the path distance is longer than the driving range. Kuby and Lim (2005) develop the flow refueling location model (FRLM) to locate AF refueling stations on a transportation network with the objective of maximizing the total traffic flow covered by the stations. The FRLM requires a pre-generation stage to establish the combinations of refueling stations that can cover vehicles on each path for a given driving range. Evaluating the station combinations on all paths of a given network requires a large computational effort. To resolve the computational burden of the FRLM, Lim and Kuby (2010) suggest the use of three heuristics, namely greedy, greedy substitution, and genetic algorithms, and Kuby et al. (2009) integrate these heuristic algorithms to a geographic information system to analyze scenarios and evaluate the tradeoffs for the development of a hydrogen refueling infrastructure in Florida. As an alternative approach to reduce the complexity of the FRLM, Capar and Kuby (2012) propose an efficient formulation of the FRLM. Since their model does not need the pre-generation stage of feasible refueling station combinations, it is capable to efficiently find exact solutions for large-scale network problems. While the FRLM mainly finds the locations of refueling stations that maximize the origin–destination (OD) flow refueled, a set-covering formulation of the problem can also be applied to locate the stations with the objective of minimizing the total

http://dx.doi.org/10.1016/j.ejor.2017.02.036 0377-2217/© 2017 Published by Elsevier B.V.

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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(6) Three Axles Single Unit

(7) Four or More Axles Single Unit

(8) Three or four Axles Single Trailer

(9) Five Ax A Axles les Single Trailer (10) Six or More Ax A Axles les Single Trailer

Fig. 1. FHWA vehicle classification scheme for mid and heavy duty trucks (Randall, 2012, Appendix A).

cost of building the refueling stations to cover all traffic flow on a given network. In general, the set-covering approach uses the OD distance matrix, instead of the OD flow volume matrix required by the FRLM. OD distances are easily collected, while OD flow volumes are usually estimated, as it is difficult to obtain their exact values or probability distributions. We note that the set-covering approach also requires the use of the OD flow volume matrix when the set-covering approach defines path coverage with respect to station capacity. Considering the advantage of the set-covering approach on collecting the OD distance matrix, Wang and Lin (2009) extend the basic concept of the set-covering problem to formulate a mixed-integer programming model that determines locations of AF refueling stations with the objective of minimizing the total building cost of the stations. The model evaluates whether a vehicle at a given site can arrive to the next site with the fuel remaining on the tank. This evaluation procedure for each path is, however, very computationally costly. To reduce the computational burden of the set-covering approach, MirHassani and Ebrazi (2012) provide a new reformulation of the set-covering model for AF refueling stations, which is able to solve large-scale set-covering problems much faster. Also, the new reformulation can be simply changed to a flow-base maximum coverage model. In addition to the solution approaches for the AF refueling station location problem discussed above, there exist several extensions of the FRLM that consider additional situations. First, some drivers may be willing to detour from their pre-planned paths for refueling. According to a survey that compares spatial refueling behaviors between CNG and gasoline vehicle drivers, CNG vehicle drivers seem to detour more than gasoline vehicle drivers do for refueling services (Kuby, Kelley, & Schoenemann, 2013). In order to account for driver deviation behavior, Kim and Kuby (2012) propose the deviation version of the FRLM with the objective of maximizing the total traffic flow covered by the stations on deviation paths. Kim and Kuby (2013) then develop heuristic algorithms for the deviation version of the FRLM to solve large network problems. Second, external restrictions on setting up refueling stations can be considered in real-world networks. Upchurch, Kuby, and Lim (2009) consider capacity constraints that limit traffic flow volumes at the refueling stations, and Capar, Kuby, Leon, and Tsai (2013) consider a budget constraint to analyze the effects of different land values. Third, it may be possible to cover more traffic flow volume when refueling stations are able to be located along the arcs. Kuby and Lim (2007) propose an approach for adding a single site on the middle of paths and suggest two dispersions methods, one minimizes the maximum length of subdivisions of the original arcs and the other maximizes the minimum length of subdivisions of the original arcs. The FRLM with original and additional candidate sites can provide better optimal locations for AF refueling stations to maximize the traffic flow volume covered by stations. Ventura, Hwang, and Kweon (2015) introduce

the continuous version of the refueling station location problem where a single refueling station can be located anywhere on a tree transportation network. Lastly, there are other station location network design studies considering different initial states of charge of electric vehicles at origins and integrated decision-making models for marketing, engineering, and operational decisions (Kang, Feinberg, & Papalambros, 2015; Lee, Kim, Kho, & Lee, 2014). In order to invigorate the use of AF vehicles in intercity freight transportation, Hwang, Kweon, and Ventura (2015) propose a new mathematical model for developing AF infrastructures on directed (symmetric) transportation networks when vehicles traveling the network have the same driving range and similar fuel levels at ODs. A directed transportation network consists of two divided-pathways, which are separated by a traffic barrier, and is only accessible from entrance and exit ramps, so that vehicles can drive at high speed safely for long-distance travel without any interruption such as traffic signals and intersections. Such a transportation network is called a dual carriageway or a divided highway, and many countries apply this road system to motorways, freeways, expressways, and toll roads. In general, a directed transportation network has built-in service facilities that provide service such as travel information, restrooms, food, and fuel for drivers’ convenience, so that drivers use these facilities on the network without deviating from their preplanned trips. Built-in service facilities are classified according to accessibility; a single-access service facility can provide refueling service only to vehicles in one driving direction, while a dual-access service facility can offer its service to vehicles in both driving directions. This paper proposes a new mathematical model for a refueling station location problem on a directed transportation network where AF vehicles have different driving ranges and fuel tank levels at the entrances and exits. In general, some vehicles have higher fuel efficiency or carry a larger fuel tank than others, so that vehicles have different driving ranges depending on vehicle classes. The information on traffic flow distribution of the vehicle classes in a road system is easily obtained from federal agencies or related corporations. For example, in case of Pennsylvania, the information on the traffic flow distribution of vehicle classes is available from Pennsylvania Spatial Data Access (Pennsylvania Spatial Data Access, 2016). Korea Expressway Corporation (2014) also annually publishes a summary of the traffic flow statistics in South Korea. Next, we consider that a vehicle has different remaining fuel tank levels at the entrances and exits of a transportation network. A vehicle would depart from its home location, such a transportation company, travel to a road network, go through a particular entrance and exit pair of the network, and then exit the network to reach a customer location. After that, the vehicle would return to the home location. During this round trip, the vehicle may be refueled outside the road network in or near its home location and customer locations, so that it is reasonable to consider

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different entry and exit fuel levels for vehicles on the transportation network. Note that, for convenience, we regard entrances and exits of a transportation network as origins and destinations (ODs) of traffic flow, respectively. We suggest three methods to estimate fuel tank levels at OD pairs. First, we can survey drivers at each entrance and exit tollbooth in toll roads or at refueling stations in regular transportation networks asking how much fuel is left in their tank. Sperling and Kitamura (1986) and Kuby et al. (2013) surveyed drivers at refueling stations in the San Francisco Bay Area and Los Angeles County asking for their refueling patterns and preferences. Secondly, using a global positioning system tracker for drivers can be another way to estimate fuel remaining in the tank at OD pairs. Note that, for mobile device users, Google Map currently provides the Timeline service that tracks users’ itineraries (Google, 2016). By investigating where drivers have refueled before entering or after exiting a transportation network, fuel tank levels at OD pairs can be estimated. Lastly, fuel consumption data can be collected from an on-board diagnostics (OBD) system, which monitors emissions, speed, mileage, and other useful data on vehicles (Alessandrini, Filippi, & Ortenzi, 2012). By US Federal Law, all light-, medium-, and heavy-duty vehicles in the US produced after 1996 support an OBD system primarily for emissions inspections (Environmental Protection Agency, 2015). The statistics of fuel tank levels can be obtained and analyzed through an OBD acquisition tool. In this paper, a directed transportation network with multiple vehicle classes and different fuel levels at ODs is defined as a multi-class vehicle transportation network. The objective of the problem is to determine the locations of the AF refueling stations that maximize the total traffic flow covered on the multi-class vehicle transportation network under consideration. The remainder of the paper is organized as follows. In Section 2, we first describe the general problem settings and assumptions. Next, we group OD pairs into two types depending on the OD distance. Then, covering conditions for round trips are established for each type of OD pair. Considering the specific covering conditions, a 0-1 linear programming model is formulated for the refueling station location problem on a multi-class vehicle transportation network. Section 3 describes the Pennsylvania (PA) Turnpike and discusses an approach to reduce the size of the network. Then, the proposed model is applied to the simplified PA Turnpike under different settings in terms of vehicle classes and available fuel tank levels at ODs. We also consider random fuel tank levels to test the robustness of the model. Lastly, Section 4 provides conclusions and topics for future research. 2. Model development Let G(V, E ) be a directed (symmetric) transportation network with set of vertices V for the origins and destinations (ODs) and set of edges E = {(vi , v j )|for some vi , v j ∈ V }, where |V | = nV and |E | = nE . The distance between origin vi and destination v j is denoted by d (vi , v j ), and it is considered to be the length of the shortest path between vi and v j . Note that, since network G is symmetric, (vi , v j ) ∈ E if and only if (v j , vi ) ∈ E; in addition, d (vi , v j ) = d (v j , vi ). Instead of distance, the length of a path can also be measured in terms of travel time or fuel consumption. The set of OD pairs is designated as Q = {q(vi , v j )|vi , v j ∈ V, i < j}. All vehicles traveling on a particular OD pair, q(vi , v j ) ∈ Q, are considered to make a round trip using the corresponding shortest path from vi to v j , denoted as P (vi , v j ), and from v j to vi , denoted as P (v j , vi ). We call P (vi , v j ) and P (v j , vi ) the original and return paths, respectively. If an OD pair has multiple shortest paths, then one of these paths is arbitrarily selected in the proposed model. Let θ (vi , v j ) denote the average flow volume (in round trips per time unit) on both the original and return paths. In order to locate AF refueling

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stations within this network, we consider two different types of candidate sites; single- and dual-access candidate sites. We define K = {kt |t = 1, . . . , nK } as the set of candidate sites. Since the network is divided into two driving directions, from vi to v j and from v j to vi , each path in a particular OD pair has two sets of candidate sites for the AF refueling stations: CS(vi , v j ) = {kt ∈ K |kt ∈ P (vi , v j )} and CS(v j , vi ) = {kt ∈ K |kt ∈ P (v j , vi )}. Note that dual-access candidate sites belong to both sets. The distances between two candidate sites, an origin and a candidate site, and a candidate site and a destination for each OD pair are denoted as d (kt , kt  ), d (vi , kt ), and d (kt , v j ) for kt , kt  ∈ K, vi , v j ∈ V , respectively. Network G is a multi-class vehicle transportation network, where vehicles are distinguished by two factors: vehicle classes and fuel tank levels at ODs when they enter or exit the network. First, let C = {1, 2, . . . , mC } be the set of vehicle classes. A vehicle of class cu ∈ C can travel up to a distance Rcu under free flow conditions, where Rcu is defined as the limited driving range of class cu vehicles. The fuel tank level is measured in terms of its driving range equivalent; that is, a vehicle of class cu with a half-full tank and Rcu = 100 miles means that it is capable of traveling at most 50 miles without refueling. The average traffic flow of class cu vehicles in OD pair q(vi , v j ) ∈ Q is denoted as θcu (vi , v j ) = wcu (vi , v j ) × θ (vi , v j ), where wcu (vi , v j ) is the portion of class cu vehicles in the OD pair. Additionally, vehicles can have different fuel tank levels at ODs because the entrances and exits of a directed transportation network such as toll roads, expressways, and highways are not usually vehicles’ home and customer locations. Some vehicles may be refueled outside the network before they enter or after they leave the network. Let P be the finite set of possible fuel tank levels, where |P | = mP , when a vehicle enters or leaves network G. Since each vehicle is assumed to make a round trip between its origin vi and its destination v j , the fuel tank level of a vehicle is characterized by four parameters, pi , p j , pi , p j ∈ P : pi : fuel tank level at vi when the vehicle enters the network in the original path, P (vi , v j ), p j : fuel tank level at v j when the vehicle exits the network in the original path, P (vi , v j ), p j : fuel tank level at v j when the vehicle reenters the network in the return path, P (v j , vi ), pi : fuel tank level at vi when the vehicle exits the network in the return path, P (v j , vi ). For a particular round trip between OD pair q(vi , v j ), vehicles of class cu may have different fuel tank levels at vi and v j in the original path P (vi , v j ) and return path P (v j , vi ). Let fr = ( pi , p j , p j , pi ) be a possible combination of four fuel tank levels for the round trip and F = { fr |r = 1, . . . , mF } be the set of combinations of four fuel tank levels. As shown in Fig. 2, for OD pair q(vi , v j ), a vehicle of class cu with fuel tank combination fr = ( pi , p j , p j , pi ) can travel up to a distance pi Rcu from its origin before refueling for the first time when it enters network G. Then, it can travel a distance Rcu between consecutive refueling. After that, it should be refueled at a station within a distance (1 − p j )Rcu from its destination in order to reach its customer location. Similarly, the refueling requirements for the return trip on P (v j , vi ) have to be held at both interchanges. When this vehicle goes back to its new origin, which is prior destination v j , after visiting its last customer location, it may use a route to be able to refuel outside the network. Thus, the vehicle’s fuel level at v j may be different when it exists the network in the original trip than when it re-enters it for the return trip. The average traffic flow of class cu vehicles with tank combination fr in OD pair q(vi , v j ) ∈ Q is denoted as θcu , fr (vi , v j ) = w fr (vi , v j ) × θcu (vi , v j ), where w fr (vi , v j ) is the portion of vehicles with tank combination fr in this OD pair.

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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Fig. 2. Representation of a class cu vehicle with fuel tank combination f r = ( pi , p j , p j , pi ).

Fig. 3. Illustration of a round trip on a multi-class vehicle transportation network.

Fig. 3 illustrates a round trip example between two interchanges vi and v j . Vehicles are grouped into two classes C = {c1 , c2 } with Rc1 = 100 and Rc2 = 300, and they have two fuel tank combinations F = { f1 = (1/4, 3/4, 3/4, 1/4 ), f2 = (1/4, 3/4, 1/2, 1/2)}. All candidate sites in CS(vi , v j ) = {k1 , k2 , k3 , k4 } and CS(v j , vi ) = {k5 , k6 } are single-access sites. Class c1 vehicles with fuel tank combination f2 = (1/4, 3/4, 1/2, 1/2) should be refueled at a station in candidate site k1 in path P (vi , v j ) because they can drive at most a distance Rc1 /4 = 25 when they enter the network via vi . Then, vehicles with a full tank can reach candidate sites k2 and k3 . After filling up at a station in one of these two candidate sites, vehicles have to be refueled at a station in candidate site k4 to keep a three-quarter fuel tank at v j . Similarly, vehicles can complete the trip in the return path after refueling at two stations located in candidate sites k5 and k6 . Thus, the feasible sets of candidate sites for a round trip of class c1 vehicles with fuel tank combination f2 are {k1 , k2 , k4 , k5 , k6 }, {k1 , k3 , k4 , k5 , k6 }, and {k1 , k2 , k3 , k4 , k5 , k6 }. On the other hand, class c2 vehicles with fuel tank combination f1 = (1/4, 3/4, 3/4, 1/4) can complete their round trip between vi and v j with only two refueling stops. The two proportions of the fuel tank that the vehicles have in path P (vi , v j ) are the same as those of class c1 vehicles with tank combination f2 , but class c2 vehicles can travel from vi to v j with only two refueling stops at candidate sites {k1 , k3 }, {k1 , k4 }, {k2 , k3 }, and {k2 , k4 }, because Rc2 > Rc1 . Furthermore, when class c2 vehicles with fuel tank combination f1 reenter the network, vehicles can travel back to vi without any refueling stop because d (vi , v j ) = 140, which is less than Rc2 /2, and still have more than a quarter-fuel tank level when they exit the network at vi . In the above example, class c2 vehicles with fuel tank combination f1 do not require to be refueled when they travel from v j back to vi because they have enough fuel at v j to reach vi , i.e., d (vi , v j ) ≤ Rc2 /2. From this observation, we derive Property 1. Let F S(vi , v j ; cu , fr ) = {F Kl : l = 1, . . . , mL } be the family of sets of feasible refueling stations that can cover round trips in OD pair q(vi , v j ) for class cu vehicles with tank combination fr = ( pi , p j , p j , pi ). Note that all stations defining the feasible sets in F S(vi , v j ; cu , fr ) must be located either in the original path P (vi , v j ), the return path P (v j , vi ), or in both, and the locations must be in set CS(vi , v j ) ∪ CS(v j , vi ). Property 1. Suppose that class cu vehicles with tank combination fr = ( pi , p j , p j , pi ) make round trips in OD pair q(vi , v j ). If d (vi , v j ) ≤ ( pi − p j )Rcu and d (vi , v j ) ≤ ( p j − pi )Rcu , then vehicles can complete the round trips without refueling, i.e., ∅ ∈ F S ( vi , v j ; cu , f r ).

Proof. When class cu vehicles with tank combination fr = ( pi , p j , p j , pi ) in OD pair q(vi , v j ) enter the network, they are capable of traveling a distance of pi Rcu on the original path P (vi , v j ). After traveling over a distance d (vi , v j ), their fuel tank level at v j is pi Rcu − d (vi , v j ). Since pi Rcu − d (vi , v j ) ≥ p j Rcu , no refueling is required on the original path. A similar argument can be provided to reach the same conclusion on the return path P (v j , vi ). Thus, if d (vi , v j ) ≤ ( pi − p j )Rcu and d (vi , v j ) ≤ ( p j − pi )Rcu , no refueling stops are necessary during the round trip, i.e., ∅ ∈ F S(vi , v j ; cu , fr ).  As shown in the previous example, if five stations are located at the candidate sites in set {k1 , k3 , k4 , k5 , k6 }, then both class c1 and class c2 vehicles with fuel tank combination f2 can make round trips between vi and v j . However, if only two stations are located at the candidate sites in set {k2 , k3 }, then only class c2 vehicles with fuel tank combination f1 can be covered. In this respect, for a given OD pair, Property 2 establishes conditions under which the family of sets of feasible refueling stations corresponding to certain vehicle class and fuel level combination includes another family of sets of feasible refueling stations associated to a different vehicle class and fuel level combination. Property 2. Let cu , cs ∈ C be two vehicle classes and fr , fr ∈ F be two possible fuel tank combinations in OD pair q(vi , v j ), where fr = ( pi , p j , p j , pi ) and fr = (ei , e j , e j , ei ). If Rcs ≤ Rcu , ei Rcs ≤ pi Rcu , e j Rcs ≥ p j Rcu , e j Rcs ≤ p j Rcu , and ei Rcs ≥ pi Rcu , then F S ( vi , v j ; c s , f r  ) ⊆ F S ( vi , v j ; c u , f r ). Proof. We need to show that if F Kl ∈ F S(vi , v j ; cs , fr ), then F Kl ∈ F S(vi , v j ; cu , fr ). First, let F Ka = {k1 , . . . , kg } be a subset of F Kl that covers class cs vehicles with fuel tank combination fr on the original path P (vi , v j ). Without loss of generality, suppose that the candidate sites in F Ka are sequentially arranged on the original path. Then, d (vi , k1 ) ≤ ei Rcs , d (kg , v j ) ≤ (1 − e j )Rcs , and d (kh , kh+1 ) ≤ Rcs , for h = 1, . . . , g − 1. Since Rcs ≤ Rcu , ei Rcs ≤ pi Rcu , and e j Rcs ≥ p j Rcu , class cu vehicles with fuel tank combination fr on P (vi , v j ) can also refuel at stations in the candidate sites in F Ka when they travel on the original path. Second, let F Kb be a subset of F Kl that covers class cs vehicles with fuel tank combination fr on the return path P (v j , vi ). Similarly, it can be shown that stations located in the candidate sites in F Kb can cover class cu vehicles with fuel tank combination fr on the return path. Thus, F Kl ∈ F S(vi , v j ; cu , fr ).  2.1. Classification of candidate refueling station locations In order to establish the covering conditions for a round trip in OD pair q(vi , v j ) ∈ Q for vehicles of class cu ∈ C with fuel tank

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Fig. 4. Sets of candidate refueling station locations for a round trip in OD pair q(vi , v j ).

combination fr = ( pi , p j , p j , pi ), we define six sets of candidates sites, three sets for the original path P (vi , v j ) and three sets for the return path P (v j , vi ). For the original path, first, the set of initial candidate station locations, denoted as SScu (vi , v j ; pi ), enumerates all the candidate sites within a distance of pi Rcu from vi . Second, the set of terminal candidate station locations, denoted as E Scu (vi , v j ; p j ), identifies all the candidate sites within a distance of (1 − p j )Rcu from v j . Lastly, the set of intermediate candidate station locations, denoted as IScu (vi , v j ; pi , p j ), lists all the candidate sites which belong neither to SScu (vi , v j ; pi ) nor to E Scu (vi , v j ; p j ). Three sets of candidate sites are similarly identified for the return path. Then, the following six sets of candidate sites can be defined for vehicles of class cu with fuel tank combination fr = ( pi , p j , p j , pi ) when they perform a round trip between OD pair q(vi , v j ):

SScu (vi , v j ; pi ) = {kt ∈ CS(vi , v j )|d (vi , kt ) ≤ pi Rcu },





E Scu (vi , v j ; p j ) = kt ∈ CS(vi , v j )|d (kt , v j ) ≤ (1 − p j )Rcu , I Scu ( vi , v j ; pi , p j )



= kt ∈ CS(vi , v j )|kt ∈ / SScu (vi , v j ; pi ) ∪ EScu





 vi , v j ; p j , 

SScu (v j , vi ; p j ) = kt ∈ CS(v j , vi )|d (kt , v j ) ≤ p j Rcu ,





E Scu (v j , vi ; pi ) = kt ∈ CS(v j , vi )|d (vi , kt ) ≤ (1 − pi )Rcu , I Scu ( v j , vi ; p j  , pi )





= kt ∈ CS(v j , vi )|kt ∈ / SScu (v j , vi ; p j ) ∪ EScu (v j , vi ; pi ) . Fig. 4 illustrates the above six sets of refueling station locations for class cu vehicles with fuel tank combination fr = ( pi , p j , p j  , pi ). For the round trip illustration in Fig. 3, the candidate sites in set CS(vi , v j ) = {k1 , k2 , k3 , k4 } and CS(v j , vi ) = {k5 , k6 } for vehicles of class c1 with limited driving range Rc1 = 100 and fuel tank combination f2 = (1/4, 3/4, 1/2, 1/2) are grouped into SSc1 (vi , v j ; 1/4) = {k1 }, E Sc1 (vi , v j ; 3/4) = {k4 }, I Sc1 ( vi , v j ; 1/4, 3/4 ) = {k2 , k3 }, S Sc1 ( v j , vi ; 1/2 ) = {k5 }, E Sc1 (v j , vi ; 1/2) = {k6 }, and ISc1 (v j , vi ; 1/2, 1/2) = ∅. 2.2. Classification of origin–destination pairs We classify OD pairs into two types for each vehicle class according to their given limited driving range. For the first type of OD pairs, refueling stations do not need to be placed in intermediate candidate sites to cover round trips, while in the second type of OD pairs, some refueling stations may need to be placed in intermediate candidate sites to cover round trips depending on the fuel tank combinations. Thus, OD pairs are grouped as follows:

Type 1: Qc(u1 ) = {q(vi , v j ) ∈ Q |0 < d (vi , v j ) ≤ Rcu }, Type 2: Qc(u2 ) = {q(vi , v j ) ∈ Q |Rcu < d (vi , v j )}.

Since the maximum path length in Type 1 OD pairs is Rcu , vehicles of class cu can travel between origin vi and destination v j by only using stations located in initial and terminal candidate sites. Since the path length in Type 2 OD pairs is greater than Rcu , vehicles first need to refuel at stations placed in initial and terminal candidate sites, and depending on the distance between these stations, they may need to refuel one or multiple times at stations located in intermediate candidate sites. 2.2.1. Coverage for Type 1 OD pairs On the original path P (vi , v j ) of a Type 1 round trip, vehicles of class cu with fuel tank combination fr = ( pi , p j , p j , pi ) do not need to refuel from vi to v j if d (vi , v j ) ≤ ( pi − p j )Rcu . Otherwise, vehicles need to refuel either at a single station located on a candidate site in SScu (vi , v j ; pi ) ∩ EScu (vi , v j ; p j ) or at two stations, one located on a candidate site in SScu (vi , v j ; pi )\E Scu (vi , v j ; p j ) and the other located on a candidate site in E Scu (vi , v j ; p j )\SScu (vi , v j ; pi ). Vehicles have similar refueling requirements on the return path P (v j , vi ). In order to describe refueling conditions for round trips in Type 1 OD pairs, we introduce indicators αcu (vi , v j ; pi , p j ) and αcu (v j , vi ; p j , pi ) that designate the necessity of refueling on each path:

αcu (vi , v j ; pi , p j )  1, i f d (vi , v j ) > ( pi − p j )Rcu ; one or two re f uelings are needed on P (vi , v j ), = 0,

otherwise; no re f ueling is needed,

αcu (v j , vi ; p j , pi )  1, i f d (vi , v j ) > ( p j − pi )Rcu ; one or two re f uelings are needed on P (v j , vi ), = 0,

otherwise; no re f ueling is needed.

According to the two indicators, the set of Type 1 OD pairs is divided into the following four categories for round trips of class ci vehicles with fuel tank combination fr = ( pi , p j , p j , pi ): Type 1 (a): αcu (vi , v j ; pi , p j ) = 0, αcu (v j , vi ; p j , pi ) = 0. Round trips do not require any refueling stops. Type 1 (b): αcu (vi , v j ; pi , p j ) = 1, αcu (v j , vi ; p j , pi ) = 0. One or two refueling stops are required on the original path P (vi , v j ). Candidate station locations need to be selected from SScu (vi , v j ; pi ) and E Scu (vi , v j ; p j ). Type 1 (c): αcu (vi , v j ; pi , p j ) = 0, αcu (v j , vi ; p j , pi ) = 1. One or two refueling stops are required on the return path P (v j , vi ). Candidate station locations need to be selected from SScu (v j , vi ; p j ) and E Scu (v j , vi ; pi ). Type 1 (d): αcu (vi , v j ; pi , p j ) = 1, αcu (v j , vi ; p j , pi ) = 1.

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Type 1 (a): No refueling stops are required.

Type 1 (b): Two refueling stops are required on . original path

Type 1 (c): Two refueling stops are required on return path .

Type 1 (d): Two refueling stops are required on each path.

Fig. 5. Four cases of round trip covering requirements for a Type 1 OD pair.

One or two refueling stops are required on both the original path P (vi , v j ) and the return path P (v j , vi ). Candidate station locations need to be selected from SScu (vi , v j ; pi ), E Scu (vi , v j ; p j ), SScu (v j , vi ; p j ), and E Scu (v j , vi ; pi ). Fig. 5 depicts round trips of Type 1 OD pairs covered by refueling stations for class cu vehicles with fuel tank combination f r = ( pi , p j , p j  , pi ). 2.2.2. Coverage for Type 2 OD pairs On a Type 2 round trip, in addition to refueling at the initial and terminal stations on the original path P (vi , v j ), some stations in intermediate candidate sites may be necessary to be able to cover the round trip for vehicles of class cu with fuel tank combination fr = ( pi , p j , p j , pi ). Let kt ∈ SScu (vi , v j ; pi ) and kt  ∈ E Scu (vi , v j ; p j ) be the locations of the initial and terminal stations. Then, if d (kt , kt  ) > Rcu , trips on subpath P (kt , kt  ) need to be covered by intermediate stations located at some candidate sites in IScu (vi , v j ; pi , p j ) so that the distance between any pair of consecutive stations on path P (vi , v j ) is at most Rcu . For the return path, similar placements of refueling stations are required. In order to establish the above refueling conditions for Type 2 OD pairs, we need to define binary identification coefficients for each vehicle class to designate whether or not the distance between two candidate sites on each path is less than or equal to Rcu . Identification coefficients βcu ,kt ,k  (vi , v j ; pi , p j ) can be defined in t matrix form on the original path for vehicles of class cu with fuel tank combination fr . Rows correspond to all candidate sites kt in USc(u1 ) (vi , v j ; pi , p j ) and columns represent non-initial candidate sites kt  in USc(u2 ) (vi , v j ; pi , p j ), where

USc(u1) (vi , v j ; pi , p j ) = {kt |kt ∈ SScu (vi , v j ; pi ) ∪ EScu (vi , v j ; p j ) ∪ IScu (vi , v j ; pi , p j )}, USc(u2) (vi , v j ; pi , p j ) = {kt  |kt  ∈ USc(u1) (vi , v j ; pi , p j )\SScu (vi , v j ; pi )}.

βcu ,kt ,kt  (vi , v j ; pi , p j ) for kt ∈ USc(u1) (vi , v j ; pi , p j ) and kt  ∈ USc(u2) (vi , v j ; pi , p j ) on path P (vi , v j ) are Then,

coefficients

set to 0 or 1 as follows:

βcu ,kt ,kt  (vi , v j ; pi , p j )  1, i f d (vi , kt ) < d (vi , kt  ) and 0 < d (kt , kt  ) ≤ Rcu , =

0,

otherwise.

Each identification coefficient indicates whether or not vehicles of class cu with fuel tank combination fr are able to reach the next station kt  after refueling at the current station kt . Similarly, we can define the identification coefficients βcu ,kt ,k  (v j , vi ; p j , pi ) t

for kt ∈ USc(u1 ) (v j , vi ; p j , pi ) and kt  ∈ USc(u2 ) (v j , vi ; p j , pi ) on the return path P (v j , vi ), where

USc(u1) (v j , vi ; p j , pi ) = {kt |kt ∈ SScu (v j , vi ; p j ) ∪ E Scu (v j , vi ; pi ) ∪ IScu (v j , vi ; p j , pi )}, USc(u2) (v j , vi ; p j , pi ) = {kt  |kt  ∈ USc(u1) (v j , vi ; p j , pi )\SScu (v j , vi ; p j )}. Then, the binary coefficients are defined as follows:

βcu ,kt ,kt  (v j , vi ; p j , pi )  1, i f d (kt , v j ) < d (kt  , v j ) and0 < d (kt , kt  ) ≤ Rcu , =

0,

otherwise.

Fig. 6 shows the values of identification coefficients in an example where class cu vehicles with fuel tank combination fr = ( pi , p j , p j , pi ) travel on path P (vi , v j ). Part (a) of the figure shows the locations of the eight candidate sites on the path. In this example, the sets of initial, intermediate, and terminal candidate sites for the vehicles are SScu (vi , v j ; pi ) = {k1 , k2 }, IScu (vi , v j ; pi , p j ) = {k3 , k4 , k5 }, and E Scu (vi , v j ; p j ) = {k6 , k7 , k8 }, respectively. Part (b)

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(a) Original path

,

7

with eight candidate sites

(b) Three sets of feasible station locations Fig. 6. Illustration of feasible station locations on a path for a Type 2 OD pair.

of the figure depicts three valid sets of station locations, which enable vehicles to travel from vi to v j without running out of fuel. In each set, one station is located in SScu (vi , v j ; pi ), another is located in E Scu (vi , v j ; p j ), and, if necessary, additional stations are located in IScu (vi , v j ; pi , p j ). The three sets are feasible because distances between consecutive stations, kt and kt  , are less than or equal to Rcu , i.e., βcu ,kt ,k  (vi , v j ; pi , p j ) = 1.



The sets of candidate sites for AF refueling stations and the covering conditions for class cu ∈ C vehicles with fuel tank combination fr ∈ F traveling between OD pair q(vi , v j ) ∈ Q on network G have been defined in Sections 2.1 and 2.2, respectively. To find the locations of a given number s of refueling stations on the network, a 0-1 linear programming (LP) model is proposed with the objective of maximizing the total traffic flow covered by the stations. Recall that the average flow volume of vehicles of class cu with fuel tank combination fr in OD pair q(vi , v j ) is denoted as θcu , fr (vi , v j ). The following decision variables are used in the proposed 0-1 LP model:

i f a re f ueling station is located at candidat e sit e kt ∈ K, 0, otherwise,

ycu , f r ( vi , v j )



=

t

2.3. 0-1 linear programming model

1,

xkt =

1, 0,

i f tra f f ic f low θcu , fr (vi , v j ) is covered by a set o f re f ueling stations, otherwise,

z = total flow volume covered by s refueling stations. Then, the model is formulated as follows:

(M1 ) maximize z =





{θcu , fr (vi , v j ) ycu , fr (vi , v j )},

(1)

q(vi ,v j )∈Q cu ∈C fr ∈F

subject to



xkt ≥ ycu , fr (vi , v j ),

kt ∈SScu (vi ,v j ;pi )

if

αcu (vi , v j ; pi , p j ) = 1, ∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u1) , ∀cu ∈ C, (2)

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xkt ≥ ycu , fr (vi , v j ),

kt ∈E Scu (vi ,v j ;p j )

if

αcu (vi , v j ; pi , p j ) = 1, fr ∈ F , ∀q(vi , v j ) ∈ Qc(u1) , ∀cu ∈ C, 

(3)

xkt ≥ ycu , fr (vi , v j ),

αcu (v j , vi ; p j , pi ) = 1, fr ∈ F , ∀q(vi , v j ) ∈ Qc(u1) , ∀cu ∈ C, (4) 

xkt ≥ ycu , fr (vi , v j ),

kt ∈E Scu (v j ,vi ;pi )

if

αcu (v j , vi ; p j , pi ) = 1, ∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u1) , ∀cu ∈ C, (5) 

xkt ≥ ycu , fr (vi , v j ),

∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C,

kt ∈SScu (vi ,v j ;pi )

(6) 

xkt ≥ ycu , fr (vi , v j ), ∀ fr ∈ F ,

∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C,

kt ∈E Scu (vi ,v j ;p j )

(7) 

xkt ≥ ycu , fr (vi , v j ), ∀ fr ∈ F ,

∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C,

kt ∈SScu (v j ,vi ;p j )

(8) 

xkt ≥ ycu , fr (vi , v j ),

by constraint set (10). For each station kt  ∈ USc(u2 ) (vi , v j ; pi , p j ), we formulate covering constraints that determine whether or not vehicles can travel from station kt ∈ USc(u1 ) (vi , v j ; pi , p j ) to the next station kt  after refueling at the current stations kt . Note that, if station kt  follows station kt and the distance between two stations is less than or equal to Rcu , βcu ,kt ,k  (vi , v j ; pi , p j ) = 1; otherwise, t

kt ∈SScu (v j ,vi ;p j )

if

[m5G;March 16, 2017;23:15]

∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C,

kt ∈E Scu (v j ,vi ;pi )

βcu ,kt ,kt  (vi , v j ; pi , p j ) = 0 in constraint set (10). Similar refueling

logic is employed in constraint set (11) to verify the coverage of the return path using coefficient βcu ,kt ,k  (v j , vi ; p j , pi ). The t number of refueling stations on network G is predetermined in constraint (12). The last sets of constraints (13) and (14) define the domains of the decision variables. We can reduce the size of the problem because round trips in some Type 1 OD pairs do not require any refueling stops during the journey. When αcu (vi , v j ; pi , p j ) = αcu (v j , vi ; p j , pi ) = 0, the corresponding ycu , fr (vi , v j ) is unrestricted and set to 1 in the optimal solution, which means that class cu vehicles with fuel tank combination fr = ( pi , p j , p j , pi ) can make round trips without refueling in OD pair q(vi , v j ). 3. Computational experiments The PA Turnpike road network is chosen as the case study network to find the locations of AF refueling stations for LNG trucks. The proposed model is applied to a simplified version of the turnpike network with a reduced number of (aggregated) interchanges considering multiple truck classes using the 2011 truck volume data (Myers et al., 2013). Three truck classes with different limited driving ranges and four different fuel tank combinations are considered, so as to analyze the effect of fuel levels on the coverage of truck traffic flow.

(9) 

3.1. The Pennsylvania Turnpike

βcu ,kt ,kt  (vi , v j ; pi , p j ) xkt ≥ ycu , fr (vi , v j ),

kt ∈USc(u1) (vi ,v j ;pi , p j )

∀kt  ∈ USc(u2) (vi , v j ; pi , p j ), ∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C, (10) 

βcu ,kt ,kt  (v j , vi ; p j , pi ) xkt ≥ ycu , fr (vi , v j ),

kt ∈USc(u1) (v j ,vi ;p j , pi )

∀kt  ∈ USc(u2) (v j , vi ; p j , pi ), ∀ fr ∈ F , ∀q(vi , v j ) ∈ Qc(u2) , ∀cu ∈ C, (11) 

xkt = s,

(12)

kt ∈K

xkt ∈ {0, 1}, ∀kt ∈ K, ycu , fr (vi , v j ) ∈ {0, 1},

(13)

∀q(vi , v j ) ∈ Q, ∀ fr ∈ F , ∀cu ∈ C.

(14)

The objective function (1) maximizes the total traffic flow covered by a given number s of stations on network G. Constraint sets (2) and (3) impose refueling requirements that a candidate site in SScu (vi , v j ; pi ) and another one in E Scu (vi , v j ; p j ) need to be selected to cover the original path P (vi , v j ) of a trip for a Type 1 OD pair if αcu (vi , v j ; pi , p j ) = 1. Similar conditions are provided by constraint sets (4) and (5) to cover the return path P (v j , vi ) of a trip for a Type 1 OD pair. Next, round trips in Type 2 OD pairs need covering conditions similar to those of Type 1 OD pairs as shown in constraint sets (6)–(9). Furthermore, additional refueling stations located in IScu (vi , v j ; pi , p j ) may be necessary to cover round trips in Type 2 OD pairs. To cover the original path P (vi , v j ) of a trip, distance requirements for consecutive stations are ensured

The PA Turnpike road network comprises two main segments, the East–West mainline between Pittsburgh and Philadelphia (I-70, I-76, and I-276) and the Northeast extension between Philadelphia and Scranton (I-476). This article refers to these two segments as the PA Turnpike, which forms a tree network with 47 active interchanges/toll plazas, as shown in Fig. 7. Note that the interchanges of the PA Turnpike represent entries and exits on the network. The distance between Pittsburgh and Scranton is approximately 400 miles, which is the longest distance between any two points in the turnpike. Currently, there are 17 open service plazas, and some additional service plazas are temporarily or permanently closed (Myers et al., 2013). The PA Turnpike has two types of service plazas: single-access plazas provide refueling service only to vehicles driving along the same side of the roadway where the plaza is located, and dual-access plazas provide refueling service to vehicles in both directions of the roadway. In this case study, we consider 19 candidate sites for the LNG refueling stations, including the 17 open service plazas and 2 temporarily closed service plazas. Fig. 7 and Table 1 provide information about the locations of active interchanges and service plazas. Note that 3 service plazas are dual-access and the other 16 service plazas are single-access. The PA Turnpike Commission (PTC) provided the OD traffic flow data along the PA Turnpike from January 1 to December 31, 2011 (Myers et al., 2013). The data was sorted by interchanges and PTC vehicle classification. The PTC vehicle classification scheme is different than the one developed by the FHWA. The PTC classifies vehicles into nine categories depending on the number of axles and vehicle weight, while the FHWA has thirteen categories based on the number of axles and units, after distinguishing between passenger and commercial vehicles. The original OD traffic flow matrix based on the PTC vehicle classification was transformed to

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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9

Fig. 7. PA Turnpike with 47 interchanges and 19 service plazas (Myers et al., 2013).

Table 1 List of potential LNG refueling station (service plaza) locations (Myers et al., 2013). Code

Service plaza

Milepost

ZEN OMP NST SSS NSS SMW NMW SLH BMT CLV HSP LAW BMV PJC VFG KPR NNM ALT HKR

Zelienople (closed) Oakmont Plum New Stanton South Somerset North Somerset South Midway North Midway Sideling Hill Blue Mountain Cumberland Valley Highspire Lawn Bowmansvill Peter J. Camiel Valley Forge King of Prussia North Neshaminy (closed) Allentown Hickory Run

T21.70 EB T49.30 EB T77.60 WB T112.30 EB 112.37 WB T147.31 EB T147.32 WB T172.27 EB/WB T202.63 WB T219.12 EB T249.70 EB T258.77 WB T289.87 EB T304.84 WB T324.55 EB T328.40 WB T351.90 WB A55.90 NB/SB A86.14 NB/SB

the OD truck flow matrix used in our case study with three categories of trucks: single unit trucks (FHWA classes 6 and 7), single trailer trucks with 3–5 axles (FHWA classes 8 and 9), and single trailer trucks with 6 or more axles (FHWA class 10). Table 2 shows an aggregated version of the conversion factors (in percentages) provided by the PTC. As an example, note that 13% of the PTC class 2 vehicles traveling along the PA Turnpike correspond trucks in FHWA classes 6 and 7. Service plazas within a road segment in the PA Turnpike offer convenient refueling services for vehicles; thus, they do not have to exit to secondary roads for refueling. However, when there is no service plaza available between a pair of interchanges, vehicles

Table 2 Conversion factors from PTC to FHWA vehicle classification (Myers et al., 2013). Conversion factor (%)

Aggregated FHWA Class

PTC class

1–5 6–7 8–9 10 11–13

1

2

3

4

5

6

7

8

9

100 0 0 0 0

62 13 25 0 0

27 31 12 30 0

3 7 8 81 1

1 1 2 88 8

0 0 1 86 13

0 1 1 91 7

0 1 1 90 8

0 50 0 30 20

traveling along this OD pair have no choice but to exit the turnpike to be able to refuel, and the trip is considered not covered. Based on this observation, we decided to reduce the computational effort of our model by developing simplified version of the PA Turnpike, where sets of consecutive interchanges without any service plaza between them were consolidated into a single (aggregated) interchange. Note that a set of consecutive interchanges can only be consolidated into a single interchange when the road segment containing these interchanges does not include any intersection point. The aggregation process works in the following manner. Let CV = {v1 , . . . , vl } be a set of interchanges that are sequentially located between a pair of consecutive candidate sites (service plazas), namely kt and kt  , where kt and kt  are located right next to v1 and vl , respectively. In some cases, however, if the sequence of interchanges is located in the last segment of the road network, which end may be directly connected to another network, then there exists no service plaza either next to v1 or next to vl . The location of the aggregated interchange used to replace CV , denoted as vCV , is calculated as the weighted average of the original interchange locations, where the weight of an interchange, denoted as (vg ), for vg ∈ CV , is set to be the sum of its annual inflow and outflow volumes. Then, the weighted distance between kt and vCV

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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Fig. 8. The simplified PA Turnpike with 19 (aggregated) interchanges.

Table 3 List of the 19 (aggregated) interchanges on the PA Turnpike. Code

Interchange

Code

Interchange

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

T2, T10, T13 T28, T39, T48 T57, T67, T75 T91, T110 T146 T161 T180, T189, T201 T226, T236, T242, T247 T266, T286 T298

v11 v12 v13 v14 v15 v16 v17 v18 v19

T312 T326 T333 A20 T339, T340, T343, T351 T352, T358, T359 A31, A44 A56, A74 A95, A105, A115, A122, A131

is determined by the following equation:



d (kt , vCV ) =

(vg )d (kt , vg ) . vg ∈CV (vg )

vg ∈CV



The aggregated traffic flow for class cu trucks with tank combination fr between OD pair q(vCV , v j ) can be computed by the following equation:



θcu , fr (vCV , v j ) =

θc u , f r ( v g , v j )

vg ∈CV ; g< j

+



θcu , fr (v j , vg ), ∀vg ∈ V \CV.

vg ∈CV ; j
Note that similar equations can be derived to compute the weighted distance between vCV and kt  , and the aggregated traffic flow for OD pair q(vCV , v j ). By applying this aggregation process, the 47 original interchanges on the PA Turnpike were consolidated into 19 (aggregated) interchanges. Fig. 8 shows the simplified PA Turnpike with 19 service plazas and 19 (aggregated) interchanges. Table 3 summarizes the PA Turnpike information with the aggregated interchanges. Note that the notations ‘T’ and ‘A’ represent

the classification of toll roads on the PA Turnpike: ‘T’ denotes the East–West mainline (I-70, I-76, and I-276), and ‘A’ denotes the Northeast extension (I-476). Throughout this consolidation process, we noted that the truck flow volume within the sets of interchanges that were aggregated into single interchanges accounted for 23.46% of the total OD truck flow volume on the turnpike. This lack of coverage is due to the absence of service plazas on the original and return paths for these OD pairs. In other words, only 5,776,0 0 0 round trips, or equivalently 76.54% of all round trips, could be covered by the 19 candidate sites. The term ‘effective coverage’ is used in the case study to indicate the proportion of trips that can be refueled by a particular set of station locations with respect to the 5,776,0 0 0 refuelable trips. Hwang (2016) provides detailed information about the aggregated OD truck flow matrix for refuelable trips, the distance matrix between pairs of (aggregated) interchanges, and the distance matrix between (aggregated) interchanges and service plazas on the simplified PA Turnpike. 3.2. Case study According to the American Trucking Association, an LNG truck with two 75 diesel gallon equivalent (DGE) fuel tanks is capable of driving up to 700 miles per refueling (American Trucking Association, 2011). However, a practical driving range generally varies because fuel efficiency changes depending on the carrying weight and road conditions. Given that these factors drop the mileage per gallon of LNG trucks, we assume a conservative limited driving range of 300 miles per 75 DGE fuel tank (Myers et al., 2013). We also consider that trucks in FHWA classes 8 and 9 are mounted with a single 75 DGE fuel tank on one side while trucks in FHWA class 10 are usually mounted with two 75 DGE fuel tanks on their left and right sides; thus, the limited driving range of FHWA truck classes 8 and 9 is set to be 300 miles and the limited driving

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ranges for FHWA truck class 10 is set to be 600 miles (Myers et al., 2013). Moreover, FHWA classes 6 and 7 consist of single unit trucks with three or more axles, are usually mounted with a 50 DGE fuel tank on one side, and can drive up to 250 miles per refueling (Autocar ACMD, 2014; Lyford-Pike, 2003). Thus, a moderate limited driving range for FHWA classes 6 and 7 vehicles can be set to be 200 miles. Considering the various limited driving range for LNG trucks in FHWA classes 6–10, in this case study we assume that the mid- and heavy-duty trucks traveling along the PA Turnpike can be grouped into the following three truck classes: Class c1 : FHWA truck classes 6 and 7 with Rc1 = 200 miles, Class c2 : FHWA truck classes 8 and 9 with Rc2 = 300 miles, and Class c3 : FHWA truck class 10 with Rc3 = 600 miles. In 2011, 8.39% of the total OD truck trips on the PA Turnpike were made by class c1 trucks, 11.28% by class c2 trucks, and 80.33% by class c3 trucks. Assuming that each OD pair has the same distribution of truck classes, the coefficients of truck class portions are set to wc1 (vi , v j ) = 0.0839, wc2 (vi , v j ) = 0.1128, and wc3 (vi , v j ) = 0.8033 for all OD pairs in the turnpike. The prevailing literature regarding the location of alternative fuel (AF) refueling stations in transportation networks generally assumes that vehicles have at least a half-full tank each time they enter and exit the network, and are capable of covering the same driving range. This half-full tank assumption enforces drivers to refuel their vehicles at least twice to make a round trip between ODs. In our model, if a vehicle has enough fuel at the origin to reach its destination along a path from origin to destination, the vehicle does not need to be refueled in the corresponding path. However, if a vehicle does not have enough fuel at its destination to reach the origin in its return trip, then it has to be refueled. In this respect, we relax the half-full tank assumption by considering four different fuel tank combinations. Let F = { f1 , f2 , f3 , f4 } be the set of fuel tank combinations that LNG trucks can have at the OD pairs when they enter or exit the PA Turnpike, where f1 = (1/2, 1/4, 1/2, 1/4), f2 = (1/2, 1/4, 1/4, 1/2), f3 = (1/4, 1/2, 1/2, 1/4), and f4 = (1/4, 1/2, 1/4, 1/2). Fuel tank combination f1 is the least restrictive fuel tank combination in our case study. Some drivers may not need to refuel their vehicles depending on their OD travel distance, vehicle driving range, and fuel tank combination f1 . By Property 1, class cu trucks with fuel tank combination f1 can travel between OD pair q(vi , v j ) without any refueling if d (vi , v j ) ≤ ( pi − p j )Rcu and d (vi , v j ) ≤ ( p j − pi )Rcu . On the other hand, fuel tank combination f4 represents the most restrictive fuel tank combination in our case study, i.e., any truck with fuel tank combination f4 has the most restrictive refueling conditions to complete a round trip between any OD pair because at least one refueling stop is necessary on both the original and return paths. Lastly, the two other fuel tank combinations, f 2 and f3 , are generated by mixing the least and most restrictive fuel tank combinations. Since p j < pi in fuel tank combination f2 and pi < p j in fuel tank combination f3 , at least one refueling stop is required in the corresponding path. Recall that the portion of vehicles with tank combination fr for each OD pair is denoted as w fr (vi , v j ). In this case study, the values of w fr (vi , v j ) are considered to be the same for all OD pairs in the PA Turnpike. 3.2.1. Trucks with identical fuel tank combination In order to identify the effect of each fuel tank combination on the coverage of truck traffic flow, we consider the first scenario where all trucks have an identical fuel tank combination at the OD pairs when they enter and exit the network, i.e., w fr (vi , v j ) = 1 for trucks with fuel tank combination fr . Let K ∗f (s ) be the optimal r set of s refueling station locations when trucks have a fuel tank combination fr . Fig. 9 shows the effective coverage of truck traffic

Fig. 9. Tradeoff between number of refueling stations and effective coverage for four fuel tank combinations.

flow for each fuel tank combination as the number of refueling stations increases from 0 to 19. The table shows that, for fuel tank combination f1 , which is the least restrictive fuel tank combination, 80.30% of the trucks can make round trips without refueling and 91.80% of the round trips can be covered by locating just one refueling station. However, for trucks with fuel tank combination f4 , the table shows the lowest overall effective coverage because the trucks with fuel tank combination f4 have the most restrictive refueling requirements to travel between any OD pair. Note that only 69.09% of trucks with fuel tank combination f4 are able to complete their round trips even though all 19 candidate sites are selected; that is, 30.91% of the round trips cannot be covered. In contrast, for fuel tank combinations f2 and f3 , the coverages are close to each other because the fuel tank portions are symmetric, i.e., the fuel tank portions for the original trip in one combination are identical to the fuel tank portions for the return trip in the other combination. Furthermore, as shown in Table 4, the effective coverage of traffic flow with an identical fuel tank combination is broken down by truck class and type of OD pair. Note that all OD pairs for class c3 trucks are Type 1 because the maximum distance between OD pairs is approximately 400 miles in the PA Turnpike and their limited driving range is 600 miles. Regardless of how much fuel is in the tank, the coverage of traffic flow for Type 1 OD pairs by K ∗f (s ) is the highest for a small number of stations r because these trips require fewer refueling stops than the trips for Type 2 OD pairs. When the coverage by truck class is considered, since class c3 trucks have the longest driving range in this problem and 80.33% of the truck trips on the PA Turnpike are made by class c3 trucks, the most coverage of truck traffic flow for any number of stations in the PA Turnpike comes from class c3 trucks. 3.2.2. Trucks with four mixed fuel tank combinations In reality, trucks with different fuel tank levels at their ODs use the PA Turnpike. It would require considerable time and cost to determine the precise distribution of fuel tank levels for trucks. In this respect, the second scenario considers that, if trucks have an equally distributed fuel tank combination among f1 , f2 , f3 , and f4 , i.e., w fr (vi , v j ) = 0.25, for r = 1, . . . , 4, then the truck traffic flow covered by the optimal set of station locations, K ∗f (s ), can be deterr mined by solving Model (M1 ) for the first scenario. Thus, we first compute the optimal set of station locations and effective coverage for a given number of stations s, s = 0, . . . , 19, for trucks with four equally mixed fuel tank combinations. We define K(∗25,25,25,25) (s ) as

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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Table 4 Effective coverage (C) of each fuel tank combination by three truck classes and two types OD pairs. Number Optimal coverage (%) of

K ∗f (s )

stations

C (1 )

80.30 91.80 94.31 96.25 97.09 98.64 99.19 99.47 99.55 99.62 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65

K ∗f (s )

2

c2

c1

(s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

K ∗f (s )

1

(2 )

(1 )

c3 (2 )

Qc1

Qc1

Qc2

Qc2

3.66 3.91 4.93 6.26 6.59 7.11 7.46 7.61 7.66 7.72 7.76 7.76 7.76 7.76 7.76 7.76 7.76 7.76 7.76 7.76

0.00 0.00 0.00 0.00 0.00 0.15 0.17 0.25 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28

7.26 8.55 9.36 9.96 10.18 11.02 11.21 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25 11.25

0.00 0.00 0.00 0.00 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03

C

c1

(1 )

(1 )

Qc3

69.38 79.34 80.02 80.03 80.30 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33 80.33

0.00 17.38 31.64 46.50 54.77 61.90 67.86 73.01 75.51 77.97 80.42 82.64 83.45 83.82 84.14 84.21 84.21 84.21 84.21 84.21

K ∗f (s )

3

c2 (2 )

(1 )

c3 (2 )

Qc1

Qc1

Qc2

Qc2

0.00 0.98 1.42 1.49 1.65 2.25 2.75 3.65 3.84 5.06 5.24 6.12 6.40 6.53 6.72 6.78 6.78 6.78 6.78 6.78

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.16 0.19 0.19 0.28 0.28 0.28 0.28 0.28 0.28

0.00 2.02 3.03 4.69 4.95 5.76 6.43 6.95 7.28 8.03 8.35 9.20 9.40 9.43 9.47 9.47 9.47 9.47 9.47 9.47

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03

C

(1 )

(1 )

Qc3

0.00 14.37 27.19 40.32 48.17 53.89 58.68 62.40 64.38 64.88 66.83 67.13 67.44 67.65 67.65 67.65 67.65 67.65 67.65 67.65

0.00 16.90 30.29 43.20 54.28 61.33 66.18 70.81 73.89 76.66 79.18 81.29 82.73 83.71 84.42 84.50 84.54 84.54 84.54 84.54

4

c1

c2 (2 )

(1 )

c3 (2 )

C (1 )

Qc1

Qc1

Qc2

Qc2

0.00 0.51 0.94 1.19 1.62 2.37 2.78 3.16 3.34 4.10 5.26 6.14 6.26 6.34 6.41 6.45 6.49 6.49 6.49 6.49

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.08 0.25 0.25 0.25 0.25 0.28 0.28 0.28 0.28 0.28

0.00 2.02 3.01 3.43 4.65 5.68 6.23 6.75 7.03 8.00 8.19 9.19 9.35 9.46 9.54 9.55 9.55 9.55 9.55 9.55

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03

Qc3

0.00 14.37 26.35 38.59 48.01 53.28 57.18 60.89 63.52 64.47 65.65 65.67 66.83 67.62 68.19 68.19 68.19 68.19 68.19 68.19

c1 (1 )

0.00 16.71 27.69 33.83 42.01 48.23 53.05 57.67 59.72 62.26 63.36 64.25 64.97 65.91 66.92 67.60 68.21 68.70 69.09 69.09

c2 (2 )

(1 )

c3 (2 )

Qc1

Qc1

Qc2

Qc2

Qc(31)

0.00 0.32 0.37 0.50 1.04 1.15 1.73 2.20 2.25 3.00 3.99 4.89 4.96 4.93 5.10 5.16 5.21 5.36 5.51 5.51

0.00 0.00 0.00 0.00 0.01 0.02 0.02 0.14 0.15 0.18 0.18 0.18 0.18 0.18 0.28 0.28 0.28 0.28 0.28 0.28

0.00 2.02 2.42 2.26 3.34 3.15 3.67 5.90 6.75 7.02 7.03 7.03 7.11 7.23 7.55 7.63 7.70 7.74 7.77 7.77

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.03 0.03 0.03 0.03 0.03

0.00 14.37 24.90 31.07 37.61 43.92 47.64 49.43 50.56 52.05 52.15 52.15 52.71 53.57 53.95 54.50 54.99 55.29 55.50 55.50

Table 5 Effective coverage (C) and relative effective coverage (RC) for equally distributed trucks with four fuel tank combinations. K(∗25,25,25,25) (s )

Number of stations (s)

C (%)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20.08 33.31 45.36 50.53 57.34 62.70 66.35 69.76 72.30 74.84 76.62 78.34 79.66 80.87 82.01 82.73 83.44 83.84 84.18 84.37

K ∗f (s )

K ∗f (s )

1

coverage of K ∗fr (s )

K ∗f (s )

3

4

C (%)

RC (%)

C (%)

RC (%)

C (%)

RC (%)

C (%)

RC (%)

20.08 32.12 37.29 50.53 53.87 62.47 65.69 68.12 68.88 71.57 72.66 76.31 78.85 80.63 81.85 82.54 82.93 83.62 84.01 84.37

10 0.0 0 96.43 82.21 10 0.0 0 93.95 99.63 99.01 97.65 95.27 95.63 94.83 97.41 98.98 99.70 99.80 99.77 99.39 99.74 99.80 10 0.0 0

20.08 33.31 36.87 49.92 51.99 53.77 56.04 59.77 60.40 64.93 65.55 73.42 75.69 75.87 78.80 81.68 82.90 83.62 84.01 84.37

10 0.0 0 10 0.0 0 81.28 98.79 90.67 85.76 84.46 85.68 83.54 86.76 85.55 93.72 95.02 93.82 96.09 98.73 99.35 99.74 99.80 10 0.0 0

20.08 33.31 36.66 48.71 51.75 53.91 55.12 58.77 59.54 65.26 68.16 77.65 78.01 78.40 78.58 79.91 80.77 82.56 84.18 84.37

10 0.0 0 10 0.0 0 80.82 96.40 90.25 85.98 83.07 84.25 82.35 87.20 88.96 99.12 97.93 96.95 95.82 96.59 96.80 98.47 10 0.0 0 10 0.0 0

20.08 33.31 45.36 50.53 57.34 62.70 66.35 69.76 71.04 72.78 75.32 75.96 76.32 77.06 80.44 82.22 82.62 82.96 83.16 84.37

10 0.0 0 10 0.0 0 10 0.0 0 10 0.0 0 10 0.0 0 10 0.0 0 10 0.0 0 10 0.0 0 98.26 97.25 98.30 96.96 95.81 95.29 98.09 99.38 99.02 98.95 98.79 10 0.0 0

the optimal set of s station locations when the distribution of fuel tank combinations is uniform among f1 , f2 , f3 , and f4 . Then, the optimal sets of station locations, K ∗f (s ), are used to determine the r effective coverage for trucks with an equally distributed fuel tank combination among f1 , f2 , f3 , and f4 throughout the network. From the optimal coverage of K(∗25,25,25,25) (s ) and the coverage of K ∗f (s ), for r = 1, . . . , 4, we can compute the relative coverage r that shows the percentage of coverage of K ∗f (s ) with respect to r the optimal coverage. For s = 0, . . . , 19, the relative coverage is calculated using the following formula:

RC (s ) =

K ∗f (s )

2

coverage of K(∗25,25,25,25) (s )

× 100(% ).

Table 5 provides a comparison of the optimal coverage of K(∗25,25,25,25) (s ) versus the coverage and relative coverage of K ∗f (s ), r for r = 1, . . . , 4, and s = 0, . . . , 19. For example, for s = 8, stations

in K(∗25,25,25,25) (8 ) cover 72.30% of the equally distributed trucks, while stations in K ∗f (8 ) only cover 59.54% of the equally dis3

tributed trucks, i.e., stations in K ∗f (8 ) cover 82.35% of the trucks 3

that are covered by stations in K(∗25,25,25,25) (8 ). The five curves in Fig. 10 are constructed with different sets of station locations when trucks have an equally distributed fuel tank combination among f1 , f2 , f3 , and f4 . The first curve with black circle symbols shows the optimal effective coverage provided by the stations in K(∗25,25,25,25) (s ) for s = 0, . . . , 19. The other four curves with triangle, asterisk, diamond, and square symbols show the coverages provided by the sets of station locations in K ∗f (s ), r for r = 1, . . . , 4, respectively. As shown in Fig. 10, the curves with fuel tank combinations f1 and f4 show coverages similar to those provided by the optimal curve for different ranges of s. In particular, for s = 1, . . . , 7, the optimal sets of station locations for trucks

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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Fig. 10. Comparison of effective coverages between stations in K(∗25,25,25,25) (s ) and K ∗fr (s ), for r = 1, . . . , 4.

with fuel tank combination f4 can also provide maximum coverage for equally distributed trucks. A possible explanation is that stations in K ∗f (s ) can also cover the other trucks with fuel tank com4

binations f1 , f2 , and f3 , by Property 2 for small values of s. Before the effective coverage reaches 70%, stations in K ∗f (s ) are able to 4

cover the equally distributed trucks to optimality, for s = 1, . . . , 7. When the number of stations reaches the range 8 to 10, the curve with K ∗f (s ) does not provide optimal coverage for equally dis4

tributed trucks, but it shows a better performance than the other curves with K ∗f (s ), K ∗f (s ), and K ∗f (s ). When s reaches large values, 1

2

3

i.e., s = 12, . . . , 17, stations in K ∗f (s ) cover more round trips of 1

equally distributed trucks than those covered by stations in K ∗f (s ), 2

K ∗f (s ), and K ∗f (s ). Note that, at s = 11, 18, stations in K ∗f (s ) cover 3

4

3

slightly more round trips of equally distributed trucks than those covered by stations in K ∗f (s ) because some truck trips belong 1

to round trips in Type 1 (c) OD pairs, which require one or two refueling stops on return paths. The coverages highlighted in bold strokes in Table 5 are the highest value among the coverages by stations in K ∗f (s ), for r = 1, . . . , 4, for a given number of stations. r

3.2.3. Trucks with least and most restrictive fuel tank combinations Previously, in Section 3.2.1, the coverages of K ∗f (s ) and K ∗f (s ) 1

4

were found to provide upper and lower bounds for the effective coverage, as shown in Fig. 9. From this observation, in this subsection, we suggest a method that uses only the least and most restrictive fuel tank combinations f1 and f4 to estimate the coverage of LNG trucks having fuel tank combinations distributed between f1 and f4 . First, in order to obtain the correct optimal solutions before estimating the values, the proposed model is run for a varying percentage of the trucks with fuel tank combination f 1 from 0% to 100%, in increments of 25%, i.e., w f1 (vi , v j ) = γ1 and w f4 (vi , v j ) = γ4 , where γ1 and γ4 are the portions of trucks with fuel tank combinations f1 and f4 , respectively. Let K(∗γ ,γ ) (s ) be the optimal set 1

4

of s station locations when 100 × γ1 % of trucks have fuel tank combination f1 and 100 × γ4 % of trucks have fuel tank combination f4 . Fig. 11 and Table 6 show effective coverage (C) as a function of the number of stations for each distribution of fuel tank combinations. Recall that, if all trucks have an identical fuel tank combination at ODs, then 80.30% of trucks with fuel tank combination f 1 can travel between OD pairs without refueling, while 30.91% of trucks with fuel tank combination f4 cannot be covered by all with sta-

13

Fig. 11. Upper and lower bounds, and gaps for the effective coverage for different percentages of trucks with two fuel tank combinations.

tions in all 19 candidate sites. Let U Bs and LBs be upper and lower bounds, respectively, on the effective coverage for a given number s of stations. In Fig. 11, these upper and lower bounds can be identified by the two curves where all trucks have either fuel tank combination f1 or fuel tank combination f4 . As shown by the only decreasing curve (dash curve with star symbols) in the figure, the coverage gap between the upper and lower bounds decreases when the number of stations increases from 0 to 19. Now, by applying linear interpolation between bounds U Bs and LBs , we can estimate the effective coverage, denoted as Cˆ(γ1 ,γ4 ) (s ), when a certain percentage (100 × γ1 %) of trucks have fuel tank combination f1 and the remaining percentage (100 × γ4 %) of trucks have fuel tank combination f4 without running the proposed model. That is, Cˆ(γ1 ,γ4 ) (s ) = (100 × γ1 % ) × U Bs + (100 × γ4 % ) × LBs . The error between the optimal effective coverage of K(∗γ ,γ ) (s ) 1

4

and its estimate, denoted as E(γ1 ,γ4 ) (s ), can then be calculated as follows:

E ( γ1 , γ4 ) ( s ) =

|Cˆ(γ1 ,γ4 ) (s ) − coverage of K(∗γ1 ,γ4 ) (s )| × 100(% ). coverage of K(∗γ ,γ ) (s ) 1 4

For example, when s = 10, the optimal effective coverage of trucks consisting of 50% of fuel tank combination f1 and 50% of fuel tank combination f4 is 81.35%. This value can also be estimated by using a linear interpolation of the bounds U B10 and LB10 ; that is, Cˆ(50,50) (10 ) = 50%(U B10 + LB10 ) = 81.51%, with an error of 0.20%. Table 6 shows every error E(γ1 ,γ4 ) (s ) using the linear interpolation and their averages. The average errors in Table 6 are less than 1%, which implies that this interpolation method is efficient and reliable for estimating the optimal solutions. 3.2.4. Trucks with random fuel tank combinations In order to test the robustness of the proposed model, the original problem is rerun with random variations of the four fuel tank levels, ( pi , p j , p j , pi ), considering two scenarios: the least and most restrictive fuel tank combinations, i.e., f1 = (1/2, 1/4, 1/2, 1/4) and f4 = (1/4, 1/2, 1/4, 1/2). We assume that the t-th random fuel tank combination fr,t consists of random four tank levels ( pit , p jt , p j t , pi t ), where each fuel tank level follows a uniform distribution on the ±15% interval of the corresponding value of a given fuel tank combination fr = ( pi , p j , p j , pi ). For each fuel tank combination f1 and f4 , 10 random fuel tank combinations are generated, and then we determine the optimal set of station locations and effective coverage

Please cite this article as: S.W. Hwang et al., Locating alternative-fuel refueling stations on a multi-class vehicle transportation network, European Journal of Operational Research (2017), http://dx.doi.org/10.1016/j.ejor.2017.02.036

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S.W. Hwang et al. / European Journal of Operational Research 000 (2017) 1–17 Table 6 Effective coverage (C) for different distributions of two fuel tank combinations, coverage error (E) by the linear interpolation, and coverage gap between upper and lower bounds. K(∗100,0) (s )

Number of stations (s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Average error

K(∗75,25) (s )

K(∗50,50) (s )

C (%)

E (%)

C (%)

E (%)

C (%)

E (%)

C (%)

Coverage gap (%)

80.30 91.80 94.31 96.25 97.09 98.64 99.19 99.47 99.55 99.62 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65 99.65

60.23 71.59 77.22 80.65 82.36 85.94 87.53 88.74 89.37 90.07 90.34 90.62 90.89 91.19 91.47 91.64 91.79 91.91 92.01 92.01

0.00 2.01 0.57 0.00 1.17 0.12 0.15 0.32 0.25 0.23 0.27 0.20 0.10 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.27

40.15 51.38 60.71 65.04 68.80 73.34 75.97 78.38 79.49 80.80 81.35 81.79 82.15 82.73 83.28 83.62 83.93 84.18 84.37 84.37

0.00 5.61 0.48 0.00 1.09 0.14 0.20 0.24 0.19 0.17 0.20 0.20 0.19 0.06 0.01 0.01 0.00 0.00 0.00 0.00 0.44

20.08 33.09 44.20 49.44 55.40 60.78 64.41 68.03 69.61 71.53 72.35 73.02 73.56 74.28 75.10 75.61 76.07 76.44 76.73 76.73

0.00 7.22 0.34 0.00 0.69 0.08 0.28 0.13 0.10 0.10 0.11 0.11 0.11 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.47

0.00 16.71 27.69 33.83 42.01 48.23 53.05 57.67 59.72 62.26 63.36 64.25 64.97 65.91 66.92 67.60 68.21 68.70 69.09 69.09

80.30 75.09 66.63 62.42 55.08 50.41 46.15 41.80 39.82 37.36 36.30 35.40 34.69 33.74 32.73 32.05 31.44 30.95 30.56 30.56

r,t

optimal set of s refueling station locations for trucks corresponding to the t-th random fuel tank combination fr,t . Recall from Section 3.2.1 that the optimal set of s refueling station locations for trucks with a fuel tank combination fr is denoted as K ∗f (s ). r As shown in the boxplots of Fig. 12(a), for a small number of stations, s = 0, . . . , 4, the variations between the optimal effective coverages of K ∗f (s ) for t = 1, . . . , 10 are noticeable because the 1,t

traffic flow of trucks that do not require refueling to cover their round trips decreases. However, as the number of stations increases, s = 5, . . . , 19, the coverage variation gradually diminishes, but there still exist some outliers. On the other hand, effective coverages of K ∗f (s ) for t = 1, . . . , 10 have small variations, as shown 4,t

in Fig. 12(b), and the average coverages of K ∗f (s ) are very close 4,t

to the optimal coverages of K ∗f (s ), as shown in Fig. 12(c), for any 4

number of stations s, s = 0, . . . , 19. In Fig. 12(b), note that there is only one outlier in all the coverages of K ∗f (2 ). These results show 4,t

that, when trucks have random fuel tank combinations, the most robust solutions provided by our model occur when restrictive fuel tank combinations are restrictive. Furthermore, in order to measure the changes of station locations by randomly generated fuel tank combinations f r,t for a given number of stations s, we introduce the relative difference (RD) of station locations, denoted RDL ft (s ). It can be calculated as follows:

RDL ft (s ) =

t=1

|K ∗fr (s )\K ∗fr,t (s )| n×s

K(∗0,100) (s )

C (%)

for a given number of stations s, s = 0, . . . , 19. Let K ∗f (s ) be the

n

K(∗25,75) (s )

× 100(% ),

4

that more conservative fuel tank level combinations reduce the variability of the solutions. 3.3. Computational performance All scenarios were run on an Intel i5 3.5 gigahertz Quad-Core 8 gigabytes RAM PC through the following three stages. The first stage partitions all OD pairs into two types according to the limited driving range and OD distance for each truck class. Then, the sets of initial, intermediate, and terminal candidate station locations are generated based on the four fuel tank combinations in set F for each OD pair. Additionally, for Type 1 OD pairs, indicators are designated depending on whether or not refueling stops are required on the original and return paths. For Type 2 OD pairs, identification coefficients are expressed in matrix form. The second stage generates Model (M1 ) with the corresponding candidate sites, indicators, and identification coefficients in CPLEX LP file format. The first two stages are built with MATLAB R2013a. In the third stage, the coded 0-1 LP model is solved in CPLEX version 12.4 for s = 0, 1, . . . , 19. Table 8 provides the CPU time (in seconds) for each stage and total CPU time for different fuel tank combinations. The CPU times for K(∗γ ,γ ) (s ) are the average of the five different distributions, 1

where n refers to the total number of random fuel tank combinations generated, and s is the total number of station locations selected by our model. Note that |K ∗f (s )\K ∗f (s )| indicates the r

of stations, and the average value of RDL f1 (s ) is quite noticeable (25.74%). On the other hand, the average value of RDL f4 (s ) is relatively small, i.e., only 9.77% of the selected station locations for trucks with random fuel tank combinations are different from the stations in K ∗f (s ). These results also lead to the same conclusion

r,t

number of different station locations for trucks to be selected between a fixed fuel tank combination fr and the t-th random fuel tank combination fr,t . Table 7 summarizes the average effective coverages for trucks with random fuel tank combinations f r,t , RD of station locations, and average number of different locations, for s = 0, . . . , 19 and r = 1, 4. Similar to the results in the effective coverages for randomly generated fuel tank combinations, the values of RDL f1 (s ) are unstable, especially for a small number

4

K(∗100,0) (s ), K(∗75,25) (s ), K(∗50,50) (s ), K(∗25,75) (s ), and K(∗0,100) (s ), where γ1 and γ4 are the portions of trucks with fuel tank combinations f1 and f4 , respectively. Similarly, the CPU time of K ∗f (s ) is calr,t

culated by taking the average CPU time of randomly generated fuel tank combinations fr,t for r = 1, 4 and t = 1, . . . , 10. There is no notable difference in the CPU times of the first and second stages for fuel tank combinations f1 to f4 . Note that the CPU time of the first stage in K(∗25,25,25,25) (s ) is the sum of the four CPU times of the first stage for each individual fuel tank combination. The CPU time of the first stage in K(∗γ ,γ ) (s ) is calculated in a 1

4

similar manner. However, in the second stage of K(∗25,25,25,25) (s ) and K(∗γ ,γ ) (s ), both CPU times increase because Model (M1 ) 1

4

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S.W. Hwang et al. / European Journal of Operational Research 000 (2017) 1–17

(a) Boxplots for random fuel tank combinations

(b) Boxplots for random fuel tank combinations

∗(

(c) Comparisons of coverages of

∗

∗

15

,

( ) for = 1, … ,10

,

( ) for = 1, … ,10

) and average coverages of

∗

,

( ) for

= 1,4

Fig. 12. Tradeoff between number of refueling stations and effective coverage for randomly generated fuel tank combinations.

is formulated with additional candidate sites, indicators, and identification coefficients in CPLEX LP file format in MATLAB. The CPU times of the third stage vary depending on the size and complexity of the model. In particular, the CPU time of K ∗f (s ) is 4

larger than the CPU times of K ∗f (s ), K ∗f (s ), and K ∗f (s ) because 1

2

3

the model for K ∗f (s ) is solved with the most restrictive fuel tank 4

combination f4 . On the other hand, the CPU times of the third stage for K(∗25,25,25,25) (s ) and K(∗γ ,γ ) (s ) are directly proportional 1 4 to the number of decision variables and constraints. Lastly, since a random fuel tank combination is generated with a uniform distribution on the interval of ±15% of a fixed fuel tank combi-

nation fr , the CPU time of K ∗f (s ) is slightly larger than that of K ∗f (s ).

r,t

r

4. Conclusions In this article, we have introduced a multi-class vehicle transportation network, where vehicles have different driving ranges and fuel tank levels at ODs on a directed-transportation network with single-access and dual-access candidate sites for locating AF refueling stations. Vehicles are first grouped into multiple classes depending on their limited driving ranges. Since all vehicles are assumed to travel between ODs in round trips, a set of four different fuel levels remaining in the tank is defined as a fuel

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S.W. Hwang et al. / European Journal of Operational Research 000 (2017) 1–17 Table 7 Average effective coverage (C¯) for randomly generated fuel tank combinations, RD of station locations (RDL ft (s )), and average number of different locations. Number of stations (s)

Fuel tank combination f 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Average

RDL f1 (s ) (%)

Avg. no. of diff. locations

C¯ (%)

RDL f4 (s ) (%)

Avg. no. of diff. locations

58.20 71.29 79.14 83.41 87.10 90.24 92.37 93.83 94.86 95.62 95.95 96.22 96.45 96.55 96.58 96.62 96.65 96.65 96.65 96.65

– 30.00 90.00 53.33 52.50 24.00 33.33 20.00 23.75 22.22 29.00 26.36 23.33 20.00 13.57 8.67 6.88 8.24 3.89 0.00 25.74

– 0.3 1.8 1.6 2.1 1.2 2 1.4 1.9 2 2.9 2.9 2.8 2.6 1.9 1.3 1.1 1.4 0.7 0 1.68

0.00 15.40 26.70 32.71 39.94 45.71 50.63 53.93 56.25 58.39 60.33 61.56 62.65 63.38 64.45 65.31 65.82 66.32 66.68 66.86

– 0.00 0.00 20.00 12.50 14.00 15.00 14.29 18.75 11.11 3.00 7.27 13.33 20.77 17.86 9.33 5.63 1.18 1.67 0.00 9.77

– 0 0 0.6 0.5 0.7 0.9 1 1.5 1 0.3 0.8 1.6 2.7 2.5 1.4 0.9 0.2 0.3 0 0.89

Table 8 CPU time (in seconds) for s = 0,

( ( ( (

) ) ) )

K ∗f s 1 K ∗f s 2 K ∗f s 3 ∗ Kf s 4 K(∗25,25,25,25) K(∗γ ,γ ) s 1 4 K ∗f s 1,t K ∗f s 4,t

( ) ( )

( )

(s )

Fuel tank combination f 4

C¯ (%)

1,

...,

19.

First stage

Second stage

Third stage

Sum

3.64 3.61 3.63 3.64 14.52 7.28 4.10

0.38 0.41 0.43 0.41 1.27 0.58 0.41

1.03 1.13 1.00 1.63 10.93 1.85 1.09

5.05 5.15 5.06 5.68 26.72 9.52 5.60

4.28

0.36

1.59

6.23

tank combination, which indicates how much fuel is estimated to remain in the tank at ODs when they enter and leave the road network in each direction. Regarding vehicle refueling on a multi-class vehicle transportation network, two properties have been derived. The first property identifies conditions under which vehicles are capable of performing round trips between some ODs without refueling. In the second property, dominance relationships between two vehicle classes and two fuel tank combinations are provided for coverage of round trips between an OD pair with a given set of valid refueling station locations. In this research work, we have also proposed a new mathematical model to locate a given number of AF refueling stations at preselected candidate sites to maximize the coverage of traffic flow (in round trips per time unit) on a multi-class vehicle transportation network. In order to define covering conditions for round trips in OD pairs, OD pairs are partitioned into two types for each vehicle class. Then, sets of constraints are generated for each type of OD pairs to ensure that vehicles can be refueled successfully in their round trip according to their fuel tank level combinations. The proposed model has been applied to a simplified version of the PA Turnpike using the 2011 annual traffic data for FHWA truck classes 6–10. Given the traffic flow distribution in the PA Turnpike, we have considered four different scenarios to locate LNG refueling stations at existing service plazas under different conditions defined for four fuel tank combinations. In the first scenario, we have solved the station location problem for the four fuel tank combinations and have noticed that they significantly affect the effective coverage. In the second scenario, we have used the optimal sets of station locations generated in the first scenario

and have applied them to the case where trucks have different fuel tank combinations, which are equally distributed among the four fuel tank combinations used in the first scenario. Interestingly, for a small number of stations, the solutions where station locations are based on the fuel tank combination with low fuel levels at the entrance points and high levels at the exit points show the best performance. Conversely, for a large number of stations, the equally distributed trucks with the four fuel tank combinations can be covered more efficiently by station locations obtained for the fuel tank combination with high fuel levels at the entrance points and low levels at the exit points. Also, in the third scenario, we have shown that the least restrictive fuel tank combination among four fuel tank combinations provides robust upper bounds on the effective coverage for intermediate fuel tank combinations. Lower bounds can also be generated by the most restrictive fuel tank combination. By using the upper and lower bounds, we have shown experimentally that linear interpolation can be used to find approximate effective coverage values for trucks with two fuel tank combinations. The results of the four scenarios show the need of considering different distributions of fuel tank levels in AF refueling station location problems to generate robust solutions. Lastly, in order to evaluate the robustness of the solutions found by the proposed model, we have rerun our model with randomly generated fuel tank combinations using a uniform distribution. Interestingly, we have concluded that the most robust solutions in terms of effective coverage and set of station locations are obtained under the most restrictive random fuel tank level combinations. The proposed model can be expanded to various transportation network location problems. One possible extension of the model is to locate additional stations on a network when there already exists an initial AF refueling infrastructure. By considering the initial set of station locations, the proposed model can also be extended to determine new locations that avoid retail competition and maximize the coverage of additional traffic flow. We can also consider locating AF refueling stations on a transportation network when taking into account station capacities, and AF handling and storage costs for various AF types. For example, the perishable nature of LNG may affect the location and capacity of AF refueling stations. In addition, year-to-year traffic flow, including the transportation demand for AFs on a transportation network, may fluctuate. In this case, considering scenarios corresponding to different ranges of traffic flow, we can apply a tractable robust optimization approach

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