Research Article | Open Access
An Alternative Fuel Refueling Station Location Model considering Detour Traffic Flows on a Highway Road System
With the development of alternative fuel (AF) vehicle technologies, studies on finding the potential location of AF refueling stations in transportation networks have received considerable attention. Due to the strong limited driving range, AF vehicles for long-distance intercity trips may require multiple refueling stops at different locations on the way to their destination, which makes the AF refueling station location problem more challenging. In this paper, we consider that AF vehicles requiring multiple refueling stops at different locations during their long-distance intercity trips are capable of making detours from their preplanned paths and selecting return paths that may be different from original paths for their round trips whenever AF refueling stations are not available along the preplanned paths. These options mostly need to be considered when an AF refueling infrastructure is not fully developed on a highway system. To this end, we first propose an algorithm to generate alternative paths that may provide the multiple AF refueling stops between all origin/destination (OD) vertices. Then, a new mixed-integer programming model is proposed to locate AF refueling stations within a preselected set of candidate sites on a directed transportation network by maximizing the coverage of traffic flows along multiple paths. We first test our mathematical model with the proposed algorithm on a classical 25-vertex network with 25 candidate sites through various scenarios that consider a different number of paths for each OD pair, deviation factors, and limited driving ranges of vehicles. Then, we apply our proposed model to locate liquefied natural gas refueling stations in the state of Pennsylvania considering the construction budget. Our results show that the number of alternative paths and deviation distance available significantly affect the coverage of traffic flows at the stations as well as computational time.
Reducing greenhouse gas (GHG) emissions in the transportation sector is one of the most vital steps in fighting against global warming in the United States (U.S.). According to the U.S. Environmental Protection Agency , the transportation sector generates the largest share of GHG emissions. In order to cut down tail-pipe emissions in the transportation sector, vehicles using alternative fuel (AF), such as biodiesel, hydrogen, electrical energy, and natural gas, have received significant attention in recent years because they emit less well-to-wheel GHG than that of vehicles using traditional fossil fuels, such as diesel and gasoline. Currently, 367 light-duty and 216 medium- and heavy-duty vehicles for the 2018 and 2019 model years are available in the U.S. AF vehicle market .
While the public interest in using AF vehicles instead of conventional vehicles has increased, the number of public refueling stations for AF vehicles is still insufficient, especially for intercity trips between urban and rural counties. Table 1 shows the 2010 census total population, number of electric charging stations, and the number of electric charging stations per 100,000 residents in the U.S. based on the Census Bureau’s urban-rural classification [3, 4]. In this table, counties are categorized into three groups according to their population density: mostly urban, mostly rural, and completely rural. It indicates that urban areas have more than twice as many electric charging stations than rural areas with a similar population size. One of the barriers to the invigoration and development of AF infrastructures in highway systems, which play a major role in intercity trips among urban and rural counties, is high construction cost. For example, the construction cost of one natural gas refueling station is at least $1.8 million in the Pennsylvania Turnpike .
Due to the sparse distribution of AF refueling stations between urban areas in highway systems, AF vehicles with a short driving range that travel long-distance intercity trips may need to use longer paths with refueling availability, including multiple refueling stops at different locations on the way to their destination, rather than their shortest paths without available refueling stations, so they can safely complete their trips. Thus, AF vehicle drivers may need to make detours to be able to refuel their vehicles.
Generally, road structures in highway systems are different than those in other transportation networks. First, highway roads are divided into two pathways separated by either a raised barrier or an unpaved median. In order to offer uninterrupted traffic flow, highway roads do not have any traffic intersections, and vehicles are only able to enter/leave highway systems through entrance and exit ramps. Next, highway systems have built-in service facilities where drivers are able to take a rest and refuel their vehicles. Since highway roads are physically partitioned by a barrier or an unpaved median, some built-in service facilities, called single-access stations, can only be accessed from one side of the road, while the rest, called dual-access stations, can be accessed from both sides of the road. Hwang et al. [6, 7] and Ventura et al.  model this type of transportation systems as directed transportation networks.
In general, concerning the design of an AF refueling infrastructure along transportation networks, a number of studies allowing repeated trips between origin/destination (OD) vertices assume that vehicles make symmetric round trips traveling along preplanned (shortest) paths between the corresponding OD vertices. It also assumes that the set of candidate station locations is a subset of vertices in the network, and therefore, all the candidate locations are dual-access sites. These assumptions imply that, if a vehicle travels from an origin to a destination on a single path by filling up at some stations, it also stops by the same stations on the return path back to the origin. These assumptions, however, have made the refueling station location problem for AF vehicles less practical since they do not reflect the characteristics of AF vehicles well at its early stage of market.
By relaxing the assumptions listed above, this paper aims at determining more reliable locations of AF refueling stations in real-world applications based on the distinct features of AF vehicles traveling intercity trips on directed transportation networks. We first consider that some candidate sites for AF refueling stations are single-access and drivers may choose different return paths from original paths to be able to refuel their vehicles in both directions. Also, since several paths are available between ODs, AF refueling availability is considered as one of the AF vehicle drivers’ top priorities when they select paths to travel in highway systems. Thus, we allow AF vehicles to make nonsymmetric round trips between their ODs. We first generate multiple paths between all OD pairs through a revised -th shortest path algorithm considering a maximum deviation distance. Then we formulate a new mixed-integer programming model that considers a set of predetermined paths for each OD pair and a preliminary set of candidate station location sites, including single-access and dual-access sites, on a directed transportation network in which AF vehicles are able to use any of the corresponding OD paths depending on the availability of refueling service. The proposed model determines the optimal set of station locations for a given number of stations and the selected round trips for all ODs that maximize the total traffic flow covered (in round trips per time unit) by the stations. For computational experiments, we first apply the proposed model to a classical 25-vertex network with 25 candidate sites through various scenarios. We also validate our proposed model with a budget constraint to construct AF refueling stations in the state of Pennsylvania.
Our model proposed in this study is applicable to various types of AF vehicles, especially to liquefied natural gas (LNG) vehicles, with their refueling station location problems. LNG vehicles are similar to the existing long-haul vehicles powered by diesel in terms of powertrain and refueling, but LNG vehicles are known to provide economic and environmental benefits; thus, LNG vehicles are well-suited for replacing the current long-haul vehicles powered by diesel on a highway road system. For example, UPS has been working to shift their high carbon-fueled vehicles to new generation of LNG vehicles since the late 20th century and has been increasing their number . In the U.S., UPS has deployed their LNG vehicles mainly in Indianapolis, Chicago, Earth City, and Nashville, and plans to use these LNG vehicles in larger areas . In addition to LNG vehicles, battery-electric vehicles and hydrogen fuel cell vehicles will also be applied to our proposed model once they are fully available for long haul logistics on a highway system.
The remaining of this paper is organized as follows. Section 2 reviews the related literature and shortly discusses the main distinctions of our research work over the existing studies as well. In Section 3, we first introduce the AF refueling station location problem with detour traffic flows on a highway road system and provide properties of feasible paths. Next, an algorithm that generates a set of multiple feasible paths for which the properties are satisfied on a given network is proposed. In Section 4, we provide covering conditions with candidate sites to cover round trips for each OD pair. Then, we propose a mixed-integer programming model to locate a given number of AF refueling stations on a directed transportation network with the objective of maximizing the total traffic flow covered, considering multiple paths for each OD pair. In Section 5, the proposed model is tested on a well-known 25-vertex network to evaluate the effects of the number of multiple paths, maximum deviation distance, and vehicle driving range on the coverage of traffic flows. The proposed model is then validated in Section 6 with an application to the state of Pennsylvania to demonstrate its performance on a large-size problem. Lastly, we provide conclusions and discuss the future work of AF refueling station location problems in Section 7.
2. Literature Review
In this section, we take a look at the literature relevant to this paper. First, in Subsection 2.1, we review the literature addressing -shortest path problems, and in Subsection 2.2, we examine the literature related to AF refueling station location problems. Then, we organize the main distinctions of our research work by comparing with the relevant literature in Subsection 2.3.
2.1. -shortest Path Problems
The -shortest path problem is to find shortest paths between two vertices in a given network in a nondecreasing order of length, where refers to the number of shortest paths to find. The -shortest path problem can be classified into two types according to whether paths are allowed to have cycles or required to be simple with no cycles.
The first type of -shortest path problem, proposed by Hoffman and Pavley , allows repeated vertices along any path. Fox  develops an algorithm to apply this type of problem to probabilistic networks. Eppstein  uses the concept of binary heap data structure in a nondecreasing order of additional path length due to deviation to find the shortest paths in polynomial time. Since the experimental results of Eppstein’s algorithm still take considerable time to find the shortest paths, Jiménez and Marzal  present a modified version of Eppstein’s algorithm to improve its practical performance. Aljazzar and Leue  propose a directed search algorithm to search for the shortest paths between two given vertices of a network. Their algorithm provides the same asymptotic worst-case complexity but uses less memory than Eppstein’s algorithm. Liu et al.  propose a novel -shortest path algorithm for neural networks. Given a set of battery exchange stations, Adler et al.  suggest a polynomial time algorithm to solve the electric vehicle shortest-walk problem with battery exchanges considering vehicle’s limited driving range. This algorithm shows the chance of extension to the -shortest path problem by transforming the original traffic network into the so called refueling shortest path network.
The second type of -shortest path problem does not allow any repeated vertex along a path, and is thus called the -shortest simple (or loopless) path problem. Since this type of problem needs an additional constraint to allow only loopless paths, it turns out to be more challenging than the first type of problem . Pollack  introduces the concept of the -shortest simple path problem and solves it by modifying Hoffman and Pavley’s method , so as to avoid paths containing repeated vertices. Clarke et al.  present a branch-and-bound procedure to find the -shortest simple paths, but this method requires a significant computational effort and storage requirements in the main memory. Sakarovitch  first identifies several shortest paths that may contain repeated vertices by using the efficient version of Hoffman and Pavley’s method , and then picks up -shortest simple paths among them. By applying a procedure that partitions a path into two subpaths, Yen  develops an efficient algorithm to find the -shortest simple path. Lawler  generalizes a procedure that can reduce the amount of storage required in solving the -shortest simple path problem. Katoh et al.  present an improved version of Yen’s algorithm that solves the problem efficiently in an undirected network. The practical performance of Yen’s algorithm is comparatively analyzed with other -shortest path algorithms [24, 25]. An implementation of Yen’s algorithm is also studied to improve its practical performance [17, 26]. Hershberger and Suri [27, 28] suggest the efficient replacement path algorithm for finding shortest simple paths in a directed network. Zeng et al.  present a heuristic -shortest path algorithm that is based on Yen’s algorithm when determining an eco-friendly path that results in minimum carbon dioxide emissions from light-duty vehicles.
2.2. Alternative Fuel Refueling Station Location Problems
The maximal covering location and the set-covering location approaches are two main streams of research for addressing AF refueling station location problems. Kuby and Lim  are one of the first applying a maximal covering location model to solve an AF refueling station location problem. They introduce the flow-refueling location model (FRLM) to find the optimal location of refueling stations for AF vehicles by considering their limited driving range per refueling with the objective of maximizing the total traffic flow covered. Upchurch and Kuby  show that the FRLM identifies more stable locations for AF refueling stations than the -median model, especially in a statewide case study. In general, the FRLM requires a significant computational effort to pregenerate all feasible location combinations of refueling stations that allow vehicles to make round trips between ODs. Lim and Kuby  propose three heuristic versions for the FRLM. Kuby et al.  apply two of them to locate hydrogen refueling stations in Florida. Capar and Kuby  present a new formulation for the FRLM that skips the pregeneration of all feasible combinations on every path. Capar et al.  suggest an arc-cover-path-cover model that focuses on the arcs comprising each path, so as to solve the problem efficiently without the pregeneration of all feasible location combinations of stations on each path for the FRLM. Jochem et al.  apply the Capar et al.  model to allocate charging stations in the German autobahn. While most AF refueling station location problems assume that the AF refueling stations can only be located at the vertices, Kuby and Lim  and Ventura et al.  consider additional candidate sites along arcs. Kweon et al.  extend the approach suggested in Ventura et al.  to locate a refueling station anywhere along a tree network to the case where a portion of drivers are willing to deviate to receive refueling service. He et al.  propose a bilevel model to solve the optimal locations of electric charging stations, through taking the driving range limitation of an electric vehicle, the battery charging time required, and the situation in which some electric vehicle drivers possibly charge at home into account.
While Kuby and Lim  and the following studies solve the AF refueling station location problem using a maximal covering location problem, Wang and Lin  solve this problem using a set-covering model with the objective of minimizing the total cost of locating stations to cover all the traffic flow on a given transportation network. Wang and Wang  integrate Wang and Lin’s model  into the classic set-covering model considering vertex-based and flow-based demands for the AF refueling station location problem. Since Wang and Lin’s model  requires a significant computational effort to evaluate the effect of the limited vehicle driving range on the number of charging stations needed for achieving multiple origin-destination intercity travel with electric vehicles on Taiwan, MirHassani and Ebrazi  propose a novel approach by using the conservation of flow law, which is able to solve large-scale problems. Chung and Kwon  extend MirHassani and Ebrazi’s model  to a multiperiod planning problem for allocating charging stations in the Korea Expressway. Hosseini and MirHassani  use the idea of MirHassani and Ebrazi’s model  to propose a two-stage stochastic mixed integer programming model for the refueling station location problem, where the traffic flow of AF vehicles is uncertain and portable AF refueling stations are considered. Kang and Recker  use the idea of the set-covering problem to locate hydrogen refueling stations with the assumption that at most one refueling stop per trip is required in a city. Assuming that vehicles only require one refueling stop per trip, two refueling station location models, including the versions that consider limited capacity of refueling stations , and refueling demand uncertainty and driver route choice behavior , are developed to minimize the total cost imposed on a planner and drivers over multiple time periods. Using the Adaptive Large Neighborhood Search algorithm  and the Adaptive Variable Neighborhood Search algorithm , locations for battery swap stations and electric vehicle routes are determined to provide services with the objective of minimizing the sum of the station construction cost and routing cost. To minimize the total cost to locate electric vehicle charging stations in road networks, Gagarin and Corcoran  suggest a novel approach that searches for the dominating set of locations among the candidate locations whose distance is below a certain threshold from a given driver. Using a parallel computing strategy, Tran et al.  propose an efficient heuristic algorithm for location of AF refueling stations based on the solution of a sequence of subproblems.
In general, drivers often deviate from their original paths of shortest travel time or distance to be able to refuel their vehicles . Both the maximal covering location and the set-covering location approaches have been extended to consider driver’s deviation options under a variety of situations. Kim and Kuby  address the deviation version of the FRLM (DFRLM) where drivers are able to deviate from their paths of shortest length between ODs, and Kim and Kuby  propose a heuristic algorithm for the DFRLM to solve large-scale problems. Huang et al.  develop a new model, called the multipath refueling location model (MPRLM), by considering multiple deviation paths between ODs. For intracity trips that require at most one refueling stop, Miralinaghi et al. [47, 48] suggest deviation versions of the set-covering location problem to find potential locations of AF refueling stations. For intercity trips of large-scale problems, Yıldız et al.  use a branch and price algorithm, which does not need pregeneration of path generation, for the AF refueling station location problem adding the routing aspect of drivers. Most recently, Göpfert and Bock  and Arslan et al.  suggest novel projection and branch and cut methods in dealing with the deviation version of the refueling infrastructure planning, so as to extend the computational efficiency even further to solve large-size problem instances with less computational effort.
2.3. Main Distinctions of Our Research Work
Comparing with the literature listed above, we shortly provide the main distinctions of our research work over the existing studies as follows:(i) Our problem is an extension of Hwang et al.  problem to consider potential deviation paths on directed transportation networks, such as highway network systems. This leads to consider (1) the mixed set of single-access and dual-access candidate sites to locate AF refueling stations, and (2) nonsymmetric round trips between ODs, where return paths are allowed to be different from original paths for refueling services in both directions.(ii) Our study is well-suited for the AF refueling station location problem, specially with LNG vehicles traveling long-distance intercity round trips. Some number of recent studies, including Miralinaghi et al. [47, 48], limits this type of problem only for AF vehicles with intracity trips, which needs at most one refueling stop per trip. On the other hand, we apply covering condition procedures depending on LNG vehicle driving range, so as for LNG vehicles traveling intercity trips to allow multiple refueling stops at different locations.(iii) In our research work, paths are not fixed for every OD pair. Instead, paths are flexible to consider detour traffic flows. While Kim and Kuby [54, 55] and Huang et al.  apply Hoffman and Pavley’s algorithm  and Yen’s algorithm  to take deviation paths into account, we develop a new algorithm based on Eppstein’s algorithm  to rigorously and efficiently find shortest paths allowing repeated vertices along paths within the tolerance (i.e., maximum deviation distance) of the driver. This helps reduce the computational effort in solving our proposed mixed-integer programming model.(iv) We conduct computational experiments on a classical 25-vertex network (with 300 OD pairs) and a large network at Pennsylvania state (with 2,211 OD pairs) (1) to evaluate the performance of our research work; and (2) to analyze the effects of number of alternative paths including deviation paths, maximum deviation distance, and vehicle driving range on the optimal location as well as the corresponding total traffic flow covered.
The major differences between the proposed model and the existing studies that are directly relevant to ours are summarized in Table 2.
3. The AF Refueling Station Location Problem with Detour Traffic on a Highway Road System
In this section, we first introduce the AF refueling station location problem with detour traffic flows on a highway road system, where AF vehicles are able to make detours for refueling and to select different paths between original and return trips. Next, we provide three small instances to describe concepts of feasible paths on a directed simple network in this problem. Also, based on the examples, we derive four properties of feasible paths. Then, an algorithm is presented to determine multiple paths for which the four properties are satisfied for all ODs. This paper aims to locate a given number of AF refueling stations on a directed transportation network so as to maximize the coverage of detour traffic flows by considering multiple paths between ODs. To this end, a mixed-integer programming model will be presented in Section 4.
3.1. Problem Statement
We define the problem on a simple directed network , where is the set of vertices for ODs, is the set of arcs having nonnegative lengths, , and . The road network in this problem has neither loops nor multiple parallel road segments in the same direction between any pair of vertices. Let be the set of OD pairs. For any , vehicles perform the same round trip, which can be divided into the original path from origin to destination , and the return path from to . For convenience, and are defined as the sets of original and return paths, respectively. Let be the set of constituent vertices in path ; then, each path can be represented as a sequence of vertices such that , where , , is the -th vertex in and is the number of vertices in . Also, the length of is calculated as , where is the arc length from the -th vertex to the next vertex in . Similarly, is defined as the length of the return path. If and are the shortest distances from to and from to , respectively, then the corresponding original and return paths are denoted as and . Also, we define and as the -th shortest feasible original and return paths for OD pair . In Subsections 3.2 and 3.3, we will discuss in detail the properties of feasible paths.
Next, the locations of candidate sites for AF refueling stations, denoted as , are assumed to be predetermined, where some candidate sites can only be accessed from one side of the road (i.e., single-access) and the rest can be accessed from both sides of the road (i.e., dual-access). We define as the sequence of candidate sites in , where , such that . In order to represent the distances between , , and site , such that , denotes the distance from to in . If is the arc that passes through in , such that , then is calculated as , where is the vertex adjacent to and is the distance from to . Similarly, , which refers to the distance from to , is calculated as , where indicates the distance from to . Furthermore, the distance between two candidate sites and , where , is denoted as . When we select the two arcs, and , which include and , respectively, such that , . Figure 1 shows a representation of vertex and candidate site sequences, as well as distances between vertices and candidate sites.
We consider that vehicles make a complete round trip between their ODs. They have a limited driving range under free flow conditions, denoted as , which refers to the maximum travel distance with a single refueling. Since vehicles’ home locations or final destinations are generally far away from highway interchanges, we assume they have at least a half-full tank when they enter and exit the road network. From now on, we call this assumption the half-full tank assumption. The half-full tank assumption was first introduced by Kuby and Lim , considering that data about the actual fuel tank level of vehicles when entering and exiting a highway road system are hard to obtain or likely to be inaccurately estimated. This assumption has been then followed by the existing literature. The half-full tank assumption originally aims at ensuring that vehicles can repeat round trips several times without running out of fuel during their round trip. That is, a vehicle accessing the last refueling station and reaching the origin/destination with at least a half-full tank is able to make the original/return trip from the origin/destination with a half-full tank and access the same station without running out of fuel. The half-full tank assumption in our study similarly makes vehicles refueled at stations positioned within a distance of from their origin interchanges and again at stations placed within the same distance from their destination interchanges in both the original and return trips. In this respect, we define , , and as the following three sets of candidate sites in , which are categorized depending on their distances from and :
Note that we consider only for such that . For return paths, we similarly define , , , and . When a feasible set of AF refueling stations is located in , , , , , and , the corresponding paths can be covered by the refueling stations. In Section 4, we will discuss in detail the covering conditions that depend on path lengths. The half-full tank assumption can be relaxed using Hwang et al.’s  model to consider different fuel tank levels of vehicles at their origins and destinations when detours are available on directed transportation networks, but it is left for future research to mainly focus on addressing the refueling station location problem with deviation options on a highway road system in this study.
In general, drivers deviate from their preferred paths, e.g., the least time or shortest distance paths, in as short a distance as possible if the preferred paths do not offer refueling availability. Also, a GPS navigation system provides a certain number of routes to travelers who detour from their familiar paths. In this respect, we first consider that vehicles can deviate from their shortest path up to a maximum deviation distance, which is calculated by multiplying the length of the shortest path by a positive deviation factor, denoted as . Then, a predetermined maximum number of paths, denoted as , whose length does not exceed the maximum deviation distance is nominated for each driving direction. This implies that, if a shortest distance path does not provide refueling service, then drivers can select up to alternative paths for the same OD pair depending on their refueling availability and the limited additional deviation travel distance. Table 3 summarizes the relevant notation and parameters. Note that data about the actual values of deviation factor and maximum number of paths for each OD pair are difficult to obtain or likely to be inaccurately predicted. Besides, different drivers would have different standards for the proper values of and , as well as different vehicle driving ranges . Thus, this study does not fix their values. Instead, in Section 5 we change their values for a given number of refueling stations to demonstrate the coupled effects of these parameters on the coverage of OD traffic flows.
3.2. Three Small Instances
The concept of feasible path on a directed transportation network is illustrated with three small disconnected networks with seventeen vertices , two dual-access candidate sites , and five single-access candidate sites in Figure 2. Suppose we are trying to determine whether vehicles with can perform round trips without running out of fuel. First, since vehicles are considered to have at least a half-full tank at theirs ODs, for , shortest paths and are feasible because the corresponding trips can be covered by placing a single refueling station at site . However, for , since candidate sites are available in neither nor , drivers need to deviate from these paths to receive refueling service in their round trips. If a refueling station is located at site , then vertex sequence for the original path and opposite sequence for the return path are feasible paths for . Second, suppose that there are two AF refueling stations available at sites and . Then, the shortest original path of does not offer AF refueling availability because vehicles cannot reach with at least a half-full tank when they exit the network. For the shortest return path, vehicles that reenter the network with a half-full tank at cannot reach the AF refueling station at site . Thus, vehicles in would need to make detours using vertex sequence for the original path and opposite sequence for the return path. These two deviation paths are feasible paths for . Lastly, suppose that sites and are selected as AF refueling stations. Vehicles in cannot be refueled by these two AF refueling stations if they use the shortest original and return paths. However, if vehicles make multiple cycles at vertex , then vertex sequence becomes a feasible original path and vertex sequence turn out to be a feasible return path for . Note that another vertex sequence also provides AF refueling availability for the return path, but drivers would not choose this vertex sequence because this path is longer than the path defined by the previous vertex sequence . From these observations, we can state four feasible path properties in the next subsection.
3.3. Four Properties of Feasible Paths
We derive four properties of feasible paths on a directed transportation network. Based on these four properties, an existing -th shortest path algorithm will be modified in the next subsection to generate multiple feasible paths for all OD pairs. Since we consider round trips, the four properties are applicable to both feasible original and return paths. The first property represents that feasible paths have a proper sequence of candidate sites to be selected for AF refueling stations which cover trips in feasible paths.
Property 1. Let be the -th shortest feasible original path from to that contains feasible candidate sites for AF refueling stations to be able to cover trips in , and denote the sequence of candidate sites in , where . According to the cardinality of , denoted as , one of the following two cases must be satisfied:(a) If , then . (b) If , then , , and for any two adjacent candidate sites .
Proof. Since is a feasible path, the half-full tank assumption implies that . In case (a), there is a single candidate site available for a refueling station on , and by the half-full tank assumption, this site is located within a distance of from and to ensure that vehicles on make a successful trip from to . Then, by definition of and , this site should be located in the common segment in . This implies that .
In case (b), there are at least two candidate sites available on . If either or , then cannot offer any AF refueling service. Thus, and . For any two adjacent candidate sites , if , then vehicles at site cannot reach site , i.e., does not have AF refueling availability. It also contradicts the definition of . Thus, for any two adjacent candidate sites .
In practice, vehicles do not need to visit the same refueling station located at a single-access site several times. For example, let us consider two return path alternatives of in Figure 2. While a vertex sequence goes over site twice, vehicles can also use a vertex sequence to travel from to with a single visit to site . That is, a shorter feasible path is always preferred to a longer feasible path that goes through a single-access candidate multiple times. On the other hand, it is unavoidable for vehicles to revisit the same refueling station when the station is placed at a dual-access candidate site. From one of the original paths of in Figure 2, a feasible vertex sequence passes through a refueling station at dual-access site twice, but in opposite directions. In this respect, prior to checking whether paths go over candidate sites multiple times, we treat a dual-access candidate site as two distinct single-access sites. From this observation, we derive the second property for candidate sites of feasible paths.
Property 2. For , suppose that is an original path that satisfies Property 1 and passes through a single-access candidate site twice:
Then, there must exist an original path, denoted as , that satisfies Property 1 and goes over site only one time, such that :
Proof. Let and be the -th and -th candidate sites in which are located at the same site such that , where . By eliminating all consecutive candidate sites from to and the corresponding arcs in , we can construct path with . Then, from Property 1, since , vehicles at can reach in . This implies that vehicles at can reach in because and are placed at the same site . Thus, is a feasible sequence of candidate sites for and .
When drivers decide to make detours from their preplanned paths for refueling, they would be reluctant to travel along unnecessary paths to reach available refueling stations. For instance, let us take an original path of from the previous network in Figure 2. Recall that a vertex sequence is one of the feasible original paths for . Now, let us consider a different vertex sequence defined by . The candidate site sequence for this second path is , which enables successful trips from to without running out of fuel. However, drivers would not travel further along the subpath of the second path, because they can reach without refueling at site . They would return at after visiting the refueling station at site , i.e., they would use the first feasible path, , instead of the second one for the original path of . Similarly, drivers can consider two possible paths for the return path of . While vehicles using a vertex sequence require each refueling stop at both sites and , vehicles in a vertex sequence need only one refueling stop at site . Since the length of the latter path is shorter, vehicles in the return path of would use the second vertex sequence instead of the first one. Based on these observations, we conclude that drivers would not make detours along a vertex sequence which requires unnecessary travel to reach available refueling stations. From now on, we call a site in unnecessary subpaths an irrelevant candidate site, so that feasible paths can be represented by vertex sequences without irrelevant candidate sites. Property 3 provides conditions under which there exist irrelevant candidate sites in paths.(a)If , then (1) , and (2) either or .(b)If , then (1) , (2) , (3) , and (4) , where .
Proof. In case (a), suppose that . By Property 1, , which implies that . Thus, any candidate site in is irrelevant because vehicles covered by a single refueling station at site can complete trips from to . Similarly, it can be shown when .
In case (b), by Condition (4), such that , , and . This implies that any additional refueling stop is unnecessary between and . Then, by Conditions (1), (2), and (3), , which implies that includes irrelevant candidate sites.
Given two feasible paths, and such that , for the same OD pair , if any subset of candidate sites that covers the trips in is also able to cover the trips in , then we say that is dominated by . This implies that is unnecessary to be considered in our proposed model. In practice, drivers would use only because . For example, let us take the previous example of in Figure 2. Recall that . Then, we can find another feasible original path, which is for vehicles with . They have the same sequence of candidate sites, but vehicles may prefer to because . Furthermore, in order to clarify the concept of dominated feasible paths, let us see another example of in Figure 2. For the return path, we have two feasible return paths, and , such that and . For , we have five possible sets of candidate sites to cover the trip, i.e., , , , , and . In case of , is the only feasible set. That is, . It means that is dominated by because . In this respect, Property 4 shows that the shortest feasible paths can dominate other feasible paths when the candidate sites of the shortest feasible paths include the candidate sites of the other paths.
Property 4. Given , let be the shortest feasible original path with . Suppose that there exists another feasible original path, denoted as , such that and . Then, is dominated by .
Proof. By definition of dominated paths, we need to show that any subset of candidate sites to cover the trips in is also able to cover the trips in . Suppose that is a set of candidate sites that covers trips in . This implies that , and therefore, because . If , then it is trivial to show that can cover trips in because . If , then we select the first and last sequences of candidate sites in , denoted as and , respectively. When we decompose into , , and , we have that . If , then we can construct a shorter feasible path whose length is . It contradicts that is the shortest feasible path. Thus, . Similarly, it can be shown that . Furthermore, if such that , then it also contradicts that