1. Introduction
The increasing rate of urbanization has added pressure on the already constrained transportation networks in our communities. For example, a recent estimate indicates that approximately
people move to the Nashville metropolitan area per day (Bureau, 2016). Commuters predominantly prefer using their personal vehicle rather than transit options; e.g., according to US Census Bureau’s 2013 survey (McKenzie, 2015; Bureau, 2014), 86% of all workers commuted to work by private vehicle, either driving alone or carpooling. This leads to increased congestion, especially during peak morning and evening travel times. Figure 1 shows the average loss of traffic speed, defined as the relative ratio of decreased speed compared with free flow speed and the free flow speed, on the road during a particular time interval. The fact that the majority of personal vehicles used by commuters are single occupancy only serves to exacerbate the issue (Hu and Reuscher, 2004).Technological innovations have enabled shared mobility options which are increasingly being used by commuters often in lieu of a personal vehicle. To support the demand for such options, companies are increasingly investing in making shared mobility services readily available to the user ondemand. For example, ridesharing apps such as Uber and Lyft are becoming more common place, particularly in urban environments. While such services may be deemed more convenient than riding public transit due to their ondemand nature, reports show that they do not necessarily decrease the congestion in major cities (NACTO, 2016).
To fix these problems many cities in the United States have started implementing Transportation Demand Management programs (TDM). For instance, the city of Nashville has the Nashville Complete Trips Transportation Demand Management program (Tennessee Department of Transportation, 2017) whose goal is to increase the efficiency of the transportation system and improve the air quality of the region by reducing single occupancy vehicle travel. While these programs are focusing primarily on adhoc incentives and introducing new modes of travel (e.g., bike and carshare options), there is an intrinsic problem that we believe is not being handled. This problem relates to how individuals make a decision of when to travel, which mode of travel to use, and what routes to use and the fact that decisions are collectively executed on the same constrained, public resource—i.e. the transportation network. Our hypothesis, which is explored in this paper, is that the current transportation decision support systems that are commercially available largely focus on shortterm planning for individuals (e.g., each user receives information or routing suggestions determined based on their stated preferences) rather than analyzing the shortterm societal or systemlevel impacts of routing decisions across users and then using that analysis in distributing the mobility demand across the dimensions of space, time, and modes of transport.
1.1. Commercial Mobility Decision Support Solutions
To help us illustrate our point, consider that a large number of U.S. adults use smartphone apps as decision support systems for planning their mobility choices; e.g., Google Maps, Waze, Apple Maps are some of the major trip planners employed by users. In fact, the number of commuters utilizing these services has gone up from 74% in 2013 to 90% in 2015 according to one study (Anderson, 2016).
Users can specify their preferences for certain routes, departure time, and desired means of transport; the mobility apps give a set of itineraries from which the user can select an option that is most closely aligned with their preferences. They also provide users with a traffic heat map showing estimated realtime congestion levels, either predicted via historical data or obtained via realtime samples from other users using the service.
A prevalent solution with current trip planners is that, at peak hours when routes become more congested, new routes are offered to users based on estimated realtime traffic conditions. While current planners have longterm prediction models that give individual users routes determined to be optimal for that user given estimated current conditions, such a solution is reactive in the sense that users change or adjust their travel plans after congestion occurs.
As an example of this phenomena, we used Google Maps(Alphabet, 2016) to analyze the routes given to users during during peak hours. Specifically, we conducted an experiment to see what happens when 1000 users make a request to go from the same general origin to same general destination in an interval of 5 minutes (requests are distributed uniformly across the 5 minute period). Information on routing algorithms and the congestion model that Google Maps employs is not publicly available. However, by querying their system, we are able to analyze the routes provided via their platform. As shown in Figure 2, one route was provided to all users and, assuming all users take this route, it becomes heavily congested. We hypothesize that routes produced by Google Maps’ are done so in a reactive manner.
Efficient multimodal routing has been studied extensively. While routing algorithms are practical and are used by current trip planners, there is a large gap between what is socially optimal (i.e. at the systemlevel) and what users choose to do as selfish individuals, as has been studied extensively in the routing game literature. Researchers largely agree that distribution of mobility demand is a potentially viable approach to addressing this gap. Yet, there are some practical challenges: (1) we do not know how many people are participating, (2) there are no simulation tools to study and analyze the policy assumptions (which can be used to design incentives and then improve the solution applications), (3) the dynamic routing problem across multiple modes is a difficult problem. We discuss these problems further in the related work section 2.
1.2. Contributions
This paper has following contributions:

We propose a User Optimal MultiModal Router (UOMMR) in which the best possible multimodal paths are given to the user that maximizes their multiple objectives. It improves upon stateoftheart multimodal, multiobjective routing algorithms by using switch conditions to provide a feasible multimodal path in the network and gives results that are useroptimal in the sense that the user objectives are maximized.

Building on the UOMMR extension, we propose a Social Optimal MultiModal Router (SOMMR) which takes a proactive approach to avoid congestion by suggesting route choices that are more socially optimal in the sense that they are considerate of the choices made by other users and work towards improving systemlevel performance.

We present a simulation analysis of the effects of UOMMR and SOMMR on the traffic congestion. To this end, we leverage the MATSim simulation tool—an opensource framework for implementing largescale agentbased transport simulations (Horni et al., 2016). We assess the effect of the ratio of SOMMR users to UOMMR users in the population on system level performance as measured by average traveltime across users^{1}^{1}1Of course, other metrics can be used and our framework is agnostic to the particular metric..
Our results indicate that as the number of users using SOMMR generated routes increases, the average travel time of all agents in the system decreases and so does the variation in travel times. Our results also indicate that to decrease congestion, SOMMR increasingly offers transit routes to the users. This reconfirms that multimodal routing is one of the many important strategies that, if appropriately leveraged, can decrease congestion while also decreasing the traveltime of a significant proportion of users in the system. Not surprisingly, as there is no free lunch, our results also show that some users do experience longer travel times under the SOMMR generated routes as compared to UOMMR routes; however, the increase in travel time is not significant in the case study we explore. This motivates a future line of research into the fairness of such strategies as well as incentives to encourage participation and route adoption.
1.3. Paper outline
In Section 2, we provide a brief overview of the current literature on routing algorithms and routing games as it relates to the work in this paper. Subsequently, in Section 3, we describe the setup of our multimodal graphbased approach and, in Section 4, we describe the proposed algorithms for multimodal routers. We detail the simulation framework, execution procedure, and key results findings in Section 5. We provide a discussion of the results and future directions in Section 6 and make concluding remarks in Section 7.
2. Related work
There are two bodies of work that we draw on to construct our approach to more socially optimal transportation resource consumption. The first is the literature on routing algorithms from which we build on the idea of multiobjective routing in networks with constraints. The second is the literature on routing games. In this latter body of work, there is a long history of studying inefficiencies arising from selfish routing.
2.1. Routing Algorithms
Routing is a widely researched topic in large part due to its practical relevance in realworld applications. Much of the work on route planning focuses on finding the shortest path on a directed weighted graph. Dijkstra (Dijkstra, 1959) and Bellman and Ford (Ford Jr, 1956; Bellman, 1958) proposed some of the first algorithms to solve this problem. Although these algorithms are quite old, they are still fundamental to many route planning algorithms that exist today.
While these algorithms compute optimal shortest paths, they are too slow to process realworld data sets such as those deriving from largescale road networks. To address this issue, there are many techniques aimed at speeding up these algorithms. Such techniques often are based on clever heuristics that accelerate the basic shortest paths algorithms by reducing their search space. Bidirectional search
(Dantzig, 2016; Goldberg and Harrelson, 2005), e.g., not only computes the shortest path from the source to the target , but simultaneously computes the shortest path from to on the backward graph. Goaldirected search such as A (Hart et al., 1968) uses other heuristics to guide the search. Goldberg et al. proposed the ALT approach in which they enhance A by introducing landmarks to compute feasible potential functions using the triangle inequality (Goldberg and Harrelson, 2005; Goldberg and Werneck, 2005). In other work, contraction techniques are used to speedup the shortest path computation; e.g., highway hierarchies (Sanders and Schultes, 2005, 2006) exploits the hierarchical in road networks, while contraction hierarchies (Geisberger et al., 2008) is based on contracting the graph.Routing in public transportation is more difficult than in road networks due to scheduling which has to be done using largely the same network edges as are used by personal vehicles, thereby introducing uncertainties in the resulting timing plan. There are two approaches to cope with the inherent timedependency of a public transportation schedule: the timeexpanded and the timedependent approach. The timeexpanded model is intuitive and allows more flexibility in constraints while the timedependent approach has a smaller graph size and, thus, faster query times (Pajor, 2009).
However, in much of this existing work as identified above, only single criteria is used for finding paths. In public transportation networks, multicriteria optimization is important due to the many potentially conflicting objectives that exist including minimizing traveltime, costs, transfers, etc. This gives rise to a multiobjective search problem, which is an extension of the shortest path problem where link costs are expanded to vectors containing several objectives. This problem has been studied extensively
(MullerHannemann and Weihe, 2001; MüllerHannemann and Schnee, 2007; Disser et al., 2008).The multiobjective A search algorithm finds all of the Paretooptimal paths (nondominated paths) in a multicriteria network (Tung and Chew, 1992; Mandow et al., 2005). It has been shown that at worst, multiobjective A search requires an exponential size graph in space and time (Mandow and De La Cruz, 2010). Some variants such as relaxed Pareto dominance (dominance) are used in practice (Perny and Spanjaard, 2008).
We, on the other hand, are interested in the multimodal setting in which users can choose amongst a combination of modes of transport. Our approach of developing a multimodal graph (detailed in Section sec:graphsetup) is similar to the work in (Pajor, 2009). To avoid arbitrary modes of transportation at arbitrary points of the network in a multimodal network, the label constrained shortest path problem (Barrett et al., 2008; Barrett et al., 2000), where regular languages model reasonable path restrictions, can be used. However, this approach does not take into account the realtime properties of the network and user context. Hence, we propose to use switch conditions (Liu, 2010) to construct a feasible multimodal path in a network. Our approach improves upon multiobjective A (Tung and Chew, 1992) to include congestion information and route users based on the plans of other users.
2.2. User Choice versus Social Optimum
In a related body of work referred to as static routing games the notion of inefficiency is wellstudied (see, e.g., (Van Huyck et al., 1990; Roughgarden and Tardos, 2002; Roughgarden, 2005; Roughgarden and Tardos, 2004)). Static routing games are ones in which there is a graph representing a network with a number of commodities (i.e. sourcedestination pairs). Populations of users must allocate themselves amongst a finite set of routes (paths connecting sources to destinations) associated with their commodity. Paths are made of up network edges, each of which has an associated congestionrelated cost or latency that the users experience if the edge is selected. Applications range from wireless networks to transportation (see, e.g., (Vickrey, 1969; Boyce et al., 2005; Youn et al., 2008; Sheffi, 1985; Correa et al., 2008)). There is a large body of work in this area that focuses on characterizing the gap between the selfishly selected user equilibrium (Wardrop equilibrium) (Wardrop, 1952) and the social optimal solution. This gap is commonly referred to as the price of anarchy.
One avenue of research focuses on how mechanisms such as tolling, uncertainty, and altruism affect equilibrium quality and can serve to reduce the price of anarchy. Indeed, the approach of tolling places additional costs on edges in order to redistribute or alleviate congestion leading to a narrowing of the gap between the social optimum and userselected equilibrium. For instance, there are a number of solutions for obtaining the optimal edge pricing (Cole et al., 2003a; Fleischer et al., 2004; Cole et al., 2003b; Jelinek et al., 2014). Similarly, a number of works have examined how users’ uncertainty level regarding edge costs or travel information (see, e.g., (Liu et al., 2016; Wu et al., 2017; Thai et al., 2016; Sekar et al., 2017)) impacts the price of anarchy. The price of anarchy under stochastic selfish routing game with riskaverse players has also been studied (Nikolova and StierMoses, 2011). It has been shown that, counterintuitively, under some conditions on structure of edgelevel congestion, equilibrium quality under uncertainty is better compared to the full information case (Sekar et al., 2017). The study of altruism is analogous to uncertainty in that it is a property associated with the user. In this case, it is assumed users are (at least partially) willing to suffer (take on additional cost) for the good of society. For example, the impact of the degree of cooperation on equilibrium quality has been explored in (Azad et al., 2010) and, in the case when users are partially altruistic, the price of anarchy has been studied in (Chen and Kempe, 2008; Çolak et al., 2016).
In large part, the key idea in each of these avenues of work is that users perceive or experience different edge costs than they would under normal operating conditions. Yet, the static approach adopted in much of the routing game literature has some limitations as far as the realism with which it represents the actual process that gives rise to congestion and increased travel time (Chiu et al., 2011). The socalled Dynamic Traffic Assignment (DTA) problem, as its name suggests, seeks to address the routing problem in a dynamic setting by determining routes for users over a period of time. There are largely two directions of research into the DTA problem: noniterative or iterative. The latter has the benefit of repeatedly solving the problem in order to achieve a more efficient routing solution, however, it cannot be solved in a realtime setting where users are arriving to a system and routes need to be recommended. It is our aim to address the routing problem in this context.
While there are some approaches aimed at closing the gap between the social (or systemlevel) optimum and the user select behavior (Samaranayake et al., 2015; Qian et al., 2012). In line with this body of work, our approach leverages an agentbased simulation environment to assign routing choices that are more socially optimal in the sense that they are considerate of the choices made by other users and work towards improving system level performance which can be metrized, e.g., by average travel time across all users. In addition, we consider multiple modes of travel (e.g., bike, personal or shared vehicle, transit, etc.). Specifically, users are offered route choices aligned with their specified preferences dynamically. Moreover, we assess the level of adoption of socially considerate routes on performance. Such analysis will be the foundation for understanding the level of incentives needed to have an appreciable effect on system level performance.
3. Problem Setup
3.1. Preliminaries
Let be a directed graph, where is the set of vertices, the set of edges. We say there is an edge from to if and only if . Note that we use the terms graph and network interchangeably. Each edge has a set of edge labels that denotes the different modes of transportation allowed on , where .
A graph is said to be weighted when a numerical label (i.e. weight) is assigned to each of its edges. For instance, there might be a cost involved in traveling from a vertex to one of its neighbors, in which case the weight assigned to the corresponding edge can represent such a cost. All edges in our graphs are weighted by periodic timedependent travel time functions (Pajor, 2009) where depicts a set of time points or timeperiod (seconds, minutes or hours of a day). If is constant over , we call as timeindependent (Pajor, 2009). In timedependent graphs, the shortest path depends on the departure time of the source node. This might result in shortest paths of different length for different departure times or even a completely different route. A timequery has as input and a departure time . It computes a shortest path tree to every node when departing at at time .
3.2. UniModal Graph
Our multimodal graph is composed of different unimodal graphs for each mode of transportation. We briefly discuss different unimodal graphs relevant in our work. In road networks, nodes model intersections or point of interests (POI) and edges depict street segments. A street segment is often shared by different modes such as . In this case, the traveltime for each mode is dependent on other modes with shared street segments.
In some street segments, the paths for each mode are clearly divided and hence, traveltime for one mode is independent of other modes. In the case of segments that represent roads for vehicles, the segments are shared amongst a variety of vehicle types including cars, trucks, and other heavy duty vehicles. Travel time on a road segment depends on the congestion. The effect of congestion in road transport, is primarily that the travel time of a congested road segment increases as the traffic load approaches the traffic capacity of the road segment. The travel time increase can be modeled with the BPR (Bureau of Public Roads) (Manual, 1964) formula:
where

: time at which the user departs node along edge connecting to

: function that provides expected volume of vehicles on edge at time

: capacity of edge (max number of vehicles that could fit on that edge)

: freeflow travel time of edge (time it would take to traverse edge if vehicles were moving at the freeflow velocity—i.e. the speed limit—on that edge)

: a constant in the BPR function (usually taken to be )

: travel time on edge as a function of the volume

: a constant in the BPR function (usually taken to be )
As can be seen from the formula the travel time will stay at the free flow travel time until the flow is very close to the capacity. The travel time then increases rapidly as the expected volume of vehicles approach the capacity of the edge.
3.3. MultiModal Graph
For building a multimodal graph , we merge the node and edge sets of each individual unimodal graph. Combining the graph requires us to identify neighbors of each node which has different modes, where the user can switch from one mode to another. Such a node is called switch node. These switch nodes are candidate nodes where we merge and link the unimodal graphs. The act of merging unites two different unimodal graphs and the act of linking inserts link edges to connect the nodes having different modes. We only link nodes that are no more than distance apart, a parameter determined by each switch node. Associated with each switch edge are list of preconditions, that needs to be satisfied to traverse that edge. Such conditions are called the switch condition module. The switch condition module depends on the properties of the switch nodes, switch edge connecting the switch nodes, user profile and time. Specifically, define , where is the switch edge with and as the switch nodes, is the profile of user and is the departure time at node . Some of the conditions that are used in the switch condition module include the following:

Does the user have a mode with it (e.g., car or bike)?

If yes, does the user need to park its mode?

If not, can the new mode store the current mode (e.g., bike rack on bus)?

Does the user have mode rental options (e.g., car or bikesharing)?

What is the associated cost for switching modes?

Is the cost less than a fixed amount (e.g., bound on the user’s available capital)?

How much time does it take to switch modes?

Are their physical restrictions such as turn movements?

Are their additional cost such as toll payments or parking?
To determine the nearest neighbors, we use dimensional trees (d trees) (Bentley, 1975), a data structure specifically designed for geometric search algorithms. The idea is to generalize a binary search tree to dimensions. Queries of dimensional points can be answered in average logarithmic time.
4. MultiModal Routing
In this section, we describe our approach to UOMMR, an improvement of existing multimodal routing algorithms, that determines which paths should be given to users provided multiple objectives. In addition, we describe our SOMMR which builds on UOMMR to offer more socially considerate route suggestions to users aimed at reducing congestion in the overall network, thereby improving system performance.
4.1. User Optimal MultiModal Router (UOMMR)
In UOMMR, the best possible multimodal paths that maximizes multiple objectives are given to the user. UOMMR is based on the MultiObjective A algorithm. We use UOMMR for routing on our multimodal graph described in Section 3.3. To find feasible and meaningful multimodal paths in a network, we are using Algorithm 1 to get outgoing edges which satisfy the set of conditions in the switch condition module.
4.2. Social Optimal MultiModal Router (SOMMR)
Much like UOMMR, SOMMR returns the set of best possible multimodal paths that maximize multiple objectives to the user. Extending beyond UOMMR, SOMMR takes a proactive approach to avoiding congestion by suggesting route choices that are more socially optimal in the sense that they are considerate of the choices made by other users and work towards improving system level performance. Unlike UOMMR, SOMMR takes into account the route plan given to the user and updates the congestion model of all links present in the route plan.
To add a user’s plan, we take all the links given in the route plan and the departure time present in nodes connecting each link and add congestion to each link as shown in Algorithm 2. The amount of congestion we add depends on the mode the user is using to traverse a link. For a link, if a user is in a car, we increment the congestion level in the time interval the car traverses the link, if it does at all. We note that buses have a physical capacity limiting the number of users that can be on a bus at a given time. In addition, for a link, we update the congestion according the the bus schedule and delays incurred in the simulation—that is, in the time interval that a bus traverses a link we update the congestion on that link accordingly. The difference between cars and a bus in terms of the congestion model is that cars can be associated with a user’s plan whereas buses have a schedule independent of any particular user. In our graph, we have assumed that the schedule of the bus is fixed. For , we assume that adding a user will have negligible effect on congestion on road and hence, we do not update the congestion.
As shown in the Algorithm 3, to get the predicted time dependent congestion level of a link at the departure time of user , we use interval trees (Cormen et al., 1990) to get overlapping intervals and look for intervals whose minimum value is greater than the given time . The predicted congestion level varies from zero to one, where zero means that there is no congestion and one implies that the link is fully congested.
Finally, we update the Algorithm 1 to also take into account congestion of links and the capacity of buses (or any transit vehicle). The updated algorithm is shown in Algorithm 4 in which we introduce the social ratio defined to be the ratio of the total population using SOMMR generated routes—that is, if is the total population size, then the population of SOMMR users is with cardinality and the population of UOMMR users is with cardinality . As indicated in the Algorithm 1, congestion should be less than .
5. Simulation
5.1. MATSim Implementation
We use MATSim (Horni et al., 2016) for our simulation. MATSim supports implementing largescale agentbased transport simulations and is based on iterative dynamic traffic assignment. That is, every agent repeatedly optimizes its daily activity schedule while in competition for spacetime slots with all other agents on the transportation infrastructure. Every agent possesses a memory containing a fixed number of day plans, where each plan is composed of a daily activity chain and an associated score. The score can be interpreted as an econometric utility.
While MATSim is capable of iteratively finding best routes for users, we are interested in creating a simulation that mirrors the realworld in the sense that given a set of routes (generated via UOMMR or SOMMR), the users execute their routes and the effect is observed. In this sense, we use MATSim as a oneshot simulator to test the viability and performance of the different UOMMR and SOMMR generated routes.
5.2. Simulation setup
Let user be defined by a tuple , where is its origindestination pair and is the departure time of the user from its origin . There are two types of population in our simulation: (i) population which contains users that have the UOMMR generated paths and (ii) population which contains users that have the SOMMR generated paths. As previously mentioned, the social ratio along with the total population size determines the size of the two populations (i.e. and . In our simulations, we vary the social between zero and one (i.e. ) with increments of . This allows us to assess the penetration level of our platform and the routes it suggests (assuming participants accept the suggested routes) required to have an appreciable impact on performance.
Prior to the start of the simulation, users do not have routes assigned to them. Instead, as each user is slotted to enter the network, say at time , they are given UOMRR or SOMMR routes, depending on their population assignment (i.e. for UOMMR and for SOMMR), generated using the current state of the simulator. In the case of the SOMMR, generated routes are a function of historical route assignment (from the beginning of the simulation) and current network conditions.
We define the state to be the volume of users for each edge . For a given time , if users enter the network, then edgespecific costs are determined by the following BPR (Manual, 1964) model:
where and is the number of additional users assigned to edge .
5.3. Simulation Results
We have used Vanderbilt employees trip distribution data for modeling agents used in our simulation. and Traffic analysis zone (TAZ) dataset from US Census (Bureau, 2016) to model demand distribution of Vanderbilt employees in each TAZ. TAZ is a special area delineated by state and/or local transportation officials for tabulating trafficrelated data, especially journeytowork and placeofwork statistics. Based on the demand distribution in each TAZ, we have sampled random points in each TAZ based on likelihood. So, if one TAZ ha 100 people and another TAZ has 1000 people, then the likelihood of picking people from 1000 people TAZ will be 10 times compared to TAZ of 100 people. Figure 3 shows the demand rate of Vanderbilt employees in each TAZ.
In our Vanderbilt dataset, we have job type for each employees. Time for each employee (agent in simulation) is randomly assigned based on their job types as shown in Table 1. So, simulation is done for peak hour traffic at 7.3010am for onward journey and 4pm7pm for return journey.
Job type  Onward Journey  Return Journey 

Faculty  7.30am9am  5pm7pm 
Students  8am10am  4pm6pm 
Staff  7am8am  5pm6pm 
We have collected historical traffic data for Nashville from January 1 to January 31 and have used Random Forest regression
(Natingga, 2017) to build a traffic speed prediction model for a typical weekday (Ignoring weekend data). This model takes total capacity, free flow speed, number of lanes of the link and hour of the day to predict the traffic speed.The simulation is executed as described in Section 5.2. We use average travel time (from their origins to destinations) across all users in both and , considered together, as a performance metric for the system (of course other performance metrics can and should be considered as appropriate). Since the best traveltimes across roads are different, we normalize the results:
Figure 4 shows the change in congestion heatmaps of the links used by agents between the simulation having no agents following SOMMR suggestions (i.e. ) and the simulation where all the agents follow SOMMR suggestions (i.e. ). This heatmap contains congestion both from MATSim agents and background traffic. As the heatmap shows, agents in social ratio 1.0 prefer alternative routes and modes than agents for social ratio 0.0. Since the heatmap contain background traffic too, there are links which are heavily congested in both the heatmaps and our SOMMR router plays no role in decreasing congestion in those links.
Figure 5
shows the normalized travel times of all the agents in simulation. The figure shows that as the number of users using SOMMR increases, the average travel time of all agents in the system decreases as does the variation in traveltimes. This implies that as more agents follow SOMMR suggestions, congestion in the system decreases as measured by the average traveltime in the network. This decrease in travel times is because agents are provided alternate routes with car, bus and walk. After social ratio of 0.7, the mean remains almost same, while the variance continues to decrease.
Figure 6 shows the mode distribution for different ratios of agents using SOMMR. The results imply that as ratio of users using SOMMR increases, the number of agents using transit increases, while at the same time number of agents using their personal car decreases. That is, as increases, SOMMR is routing more agents through transit and walk to decrease congestion. It should be noted that in this plot, agents are using multiple modes to reach their destination. Large increase in transit ridership is due to the fact that agents increasingly use bus only in some legs of their entire trip. Having rental data and parking location data for cars might give a better result.
Figure 7 shows the change in traveltime each agent experiences between the simulation having no agents following SOMMR suggestions (i.e. ) and the simulation where all the agents follow SOMMR suggestions (i.e. ). Total decrease in travel time for some agents outweighs the total increase in travel time of rest of the agents in population. There is a huge variation in this figure, with many agents observe drastic increase in travel time (). Such variations are due to increased congestion on roads and limited bus availability (frequency) for many trips. It also depends on the transit network. In Nashville transit network has hubspoke pattern, where most of the agents have to go to downtown, wait for bus and then go to Vanderbilt on another bus. Having more frequency of buses in some routes and more pointtopoint connections can give better results.
6. Discussion
Our proposed routing algorithm improves on multiobjective A (Tung and Chew, 1992) by including congestion information and by suggesting routes to users that are constructed given the plans of other users and are nearoptimal routes that can decrease congestion in system. Specifically, even at a low penetration (i.e. small values of the social ratio ), we are able to achieve an improvement in systemlevel performance. To see a significant improvement, we are developing techniques that actively increase the social ratio by incentivizing users to participate in our SOMMRbased platform as well as to adopt the routes it suggests.
Our simulation results are based on the assumption that users are taking the route proposed to them by SOMMR. This assumption will however, not hold true in real life situations where users may not be inclined to take a socially optimal route. Moreover, as our results indicate, there are some users who are worse off when using SOMMR as compared to UOMMR. Hence we need our routing model to consider the uncertainty of a user actually taking the route, consider fairness in terms of route assignment (as some users may be more likely to get slower routes as a function of other socioeconomic characteristics such as income level that are not accounted for in our model), and also propose mechanisms to incentivize users to take a socially optimal route. We are developing a probabilistic model in which we update a prior for the routes different user types adopt. This probabilistic model will serve as a starting point for designing incentives to increase SOMMR route adoption.
While we were able obtain simulation results that support our arguments, we are also aware of that this is a closed simulation and we did not have any input from the real world. Furthermore, the simulation was conducted with only a few fixed itineraries and we have only considered travel time as a metric for providing useroptimal routes to the agents in simulation. To show more realistic simulation, however, we need to consider common travel demands of users and include fare model for transit, shared vehicles, parking, and other costly services in our simulation.
7. Conclusions
We have demonstrated that by taking a proactive approach and leveraging multiple modes in routing decisions, congestion can be decreased. As part of our future work, we will extend our current approach to include (a) incentives for users to take a socially optimal route, (b) modeling uncertainty of users actually following the route proposed by router and (c) realistic simulation that can help us better understand urban vehicular dynamics in a city which potentially can be a tool for city planners to understand the interventions they need to make for encouraging the transition towards a socially optimal multimodal routing platform.
Acknowledgments
This work is sponsored in part by the National Science Foundation under the award number CNS1646912 (UW), CNS1647015 (Vanderbilt) and CNS1528799 (Vanderbilt) and in part by the Vanderbilt Initiative in Smart City Operations and Research, a transinstitutional initiative funded by the Vanderbilt University.
References
 (1)
 Alphabet (2016) Alphabet. 2016. Google Maps Directions. https://www.google.com/maps. (2016). Accessed: 20171203.
 Anderson (2016) Monica Anderson. 2016. More Americans using smartphones for getting directions, streaming TV. http://www.pewresearch.org/facttank/2016/01/29/ussmartphoneuse/. (2016). Accessed: 20171203.
 Azad et al. (2010) Amar Prakash Azad, Eitan Altman, and Rachid ElAzouzi. 2010. Routing games: From egoism to altruism. In Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), 2010 Proceedings of the 8th International Symposium on. IEEE, 528–537.
 Barrett et al. (2008) Chris Barrett, Keith Bisset, Martin Holzer, Goran Konjevod, Madhav Marathe, and Dorothea Wagner. 2008. Engineering labelconstrained shortestpath algorithms. Algorithmic aspects in information and management (2008), 27–37.
 Barrett et al. (2000) Chris Barrett, Riko Jacob, and Madhav Marathe. 2000. Formallanguageconstrained path problems. SIAM J. Comput. 30, 3 (2000), 809–837.
 Bellman (1958) Richard Bellman. 1958. On a routing problem. Quarterly of applied mathematics 16, 1 (1958), 87–90.
 Bentley (1975) Jon Louis Bentley. 1975. Multidimensional binary search trees used for associative searching. Commun. ACM 18, 9 (1975), 509–517.
 Boyce et al. (2005) David E Boyce, Hani S Mahmassani, and Anna Nagurney. 2005. A retrospective on Beckmann, McGuire and Winsten’s Studies in the Economics of Transportation. Papers in regional science 84, 1 (2005), 85–103.
 Bureau (2014) U.S. Census Bureau. 2014. Biking to Work Increases 60 Percent Over Last Decade, Census Bureau Reports. https://www.census.gov/newsroom/pressreleases/2014/cb1486.html. (2014). Accessed: 20171203.
 Bureau (2016) U.S. Census Bureau. 2016. QuickFacts, Davidson County, Tennessee; UNITED STATES. https://www.census.gov/quickfacts/fact/table/davidsoncountytennessee,US/PST045216. (2016). Accessed: 20171202.
 Chen and Kempe (2008) PoAn Chen and David Kempe. 2008. Altruism, selfishness, and spite in traffic routing. In Proceedings of the 9th ACM conference on Electronic commerce. ACM, 140–149.
 Chiu et al. (2011) YiChang Chiu, Jon Bottom, Michael Mahut, Alex Paz, Ramachandran Balakrishna, Travis Waller, and Jim Hicks. 2011. Dynamic traffic assignment: A primer. Transportation Research ECircular EC153 (2011).
 Çolak et al. (2016) Serdar Çolak, Antonio Lima, and Marta C González. 2016. Understanding congested travel in urban areas. Nature communications 7 (2016).
 Cole et al. (2003a) Richard Cole, Yevgeniy Dodis, and Tim Roughgarden. 2003a. How much can taxes help selfish routing?. In Proceedings of the 4th ACM conference on Electronic commerce. ACM, 98–107.
 Cole et al. (2003b) Richard Cole, Yevgeniy Dodis, and Tim Roughgarden. 2003b. Pricing network edges for heterogeneous selfish users. In Proceedings of the thirtyfifth annual ACM symposium on Theory of computing. ACM, 521–530.
 Cormen et al. (1990) Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. 1990. Introduction to algorithms mit press. Cambridge MA (1990).
 Correa et al. (2008) José R Correa, Andreas S Schulz, and Nicolás E StierMoses. 2008. A geometric approach to the price of anarchy in nonatomic congestion games. Games and Economic Behavior 64, 2 (2008), 457–469.
 Dantzig (2016) George Dantzig. 2016. Linear programming and extensions. Princeton university press.
 Dijkstra (1959) Edsger W Dijkstra. 1959. A note on two problems in connexion with graphs. Numerische mathematik 1, 1 (1959), 269–271.
 Disser et al. (2008) Yann Disser, Matthias MullerHannemann, and Mathias Schnee. 2008. Multicriteria shortest paths in timedependent train networks. Lecture Notes in Computer Science 5038 (2008), 347–362.
 Fleischer et al. (2004) Lisa Fleischer, Kamal Jain, and Mohammad Mahdian. 2004. Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on. IEEE, 277–285.
 Ford Jr (1956) Lester R Ford Jr. 1956. Network flow theory. Technical Report. RAND CORP SANTA MONICA CA.
 Geisberger et al. (2008) Robert Geisberger, Peter Sanders, Dominik Schultes, and Daniel Delling. 2008. Contraction hierarchies: Faster and simpler hierarchical routing in road networks. Experimental Algorithms (2008), 319–333.
 Goldberg and Harrelson (2005) Andrew V Goldberg and Chris Harrelson. 2005. Computing the shortest path: A search meets graph theory. In Proceedings of the sixteenth annual ACMSIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, 156–165.
 Goldberg and Werneck (2005) Andrew V Goldberg and Renato Fonseca F Werneck. 2005. Computing PointtoPoint Shortest Paths from External Memory.. In ALENEX/ANALCO. 26–40.
 Hart et al. (1968) Peter E Hart, Nils J Nilsson, and Bertram Raphael. 1968. A formal basis for the heuristic determination of minimum cost paths. IEEE transactions on Systems Science and Cybernetics 4, 2 (1968), 100–107.
 Horni et al. (2016) Andreas Horni, Kai Nagel, and Kay W Axhausen. 2016. The multiagent transport simulation MATSim. Ubiquity Press London.
 Hu and Reuscher (2004) Patricia S Hu and Timothy R Reuscher. 2004. Summary of travel trends: 2001 national household travel survey. (2004).
 Jelinek et al. (2014) Tomas Jelinek, Marcus Klaas, and Guido Schäfer. 2014. Computing optimal tolls with arc restrictions and heterogeneous players. In LIPIcsLeibniz International Proceedings in Informatics, Vol. 25. Schloss DagstuhlLeibnizZentrum fuer Informatik.
 Liu et al. (2016) Jeffrey Liu, Saurabh Amin, and Galina Schwartz. 2016. Effects of Information Heterogeneity in Bayesian Congestion Games. arXiv preprint arXiv:1603.08853 (2016).
 Liu (2010) Lu Liu. 2010. Data model and algorithms for multimodal route planning with transportation networks. Ph.D. Dissertation. München, Techn. Univ., Diss., 2011.
 Mandow and De La Cruz (2010) Lawrence Mandow and José Luis Pérez De La Cruz. 2010. Multiobjective A* search with consistent heuristics. Journal of the ACM (JACM) 57, 5 (2010), 27.
 Mandow et al. (2005) Lawrence Mandow, JL Pérez De la Cruz, et al. 2005. A New Approach to Multiobjective A* Search.. In IJCAI, Vol. 8.
 Manual (1964) Traffic Assignment Manual. 1964. Bureau of public roads. US Department of Commerce (1964).
 McKenzie (2015) Brian McKenzie. 2015. Who drives to work? Commuting by automobile in the United States: 2013. American Community Survey Reports (2015).
 MüllerHannemann and Schnee (2007) Matthias MüllerHannemann and Mathias Schnee. 2007. Finding all attractive train connections by multicriteria pareto search. In Algorithmic Methods for Railway Optimization. Springer, 246–263.
 MullerHannemann and Weihe (2001) Matthias MullerHannemann and Karsten Weihe. 2001. Pareto shortest paths is often feasible in practice. Lecture notes in computer science 2141 (2001), 185–198.
 NACTO (2016) NACTO. 2016. RideHailing Services: Opportunities and Challenges for Cities. https://nacto.org/policy2016/ridehailingservicesopportunitiesandchallenges/. (2016). Accessed: 20171203.
 Natingga (2017) David Natingga. 2017. Data Science Algorithms in a Week: Top 7 algorithms for computing, data analysis, and machine learning. (2017).

Nikolova and
StierMoses (2011)
Evdokia Nikolova and
Nicolas E StierMoses. 2011.
Stochastic selfish routing. In
International Symposium on Algorithmic Game Theory
. Springer, 314–325.  Pajor (2009) Thomas Pajor. 2009. Multimodal route planning. Universität Karlsruhe (2009).
 Perny and Spanjaard (2008) Patrice Perny and Olivier Spanjaard. 2008. Near Admissible Algorithms for Multiobjective Search.. In ECAI. 490–494.
 Qian et al. (2012) Zhen Sean Qian, Wei Shen, and HM Zhang. 2012. Systemoptimal dynamic traffic assignment with and without queue spillback: Its pathbased formulation and solution via approximate path marginal cost. Transportation research part B: methodological 46, 7 (2012), 874–893.
 Roughgarden (2005) Tim Roughgarden. 2005. Selfish routing and the price of anarchy. Vol. 174. MIT press Cambridge.
 Roughgarden and Tardos (2002) Tim Roughgarden and Éva Tardos. 2002. How bad is selfish routing? Journal of the ACM (JACM) 49, 2 (2002), 236–259.
 Roughgarden and Tardos (2004) Tim Roughgarden and Éva Tardos. 2004. Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior 47, 2 (2004), 389–403.
 Samaranayake et al. (2015) Samitha Samaranayake, Jack Reilly, Walid Krichene, JeanBaptiste Lespiau, Maria Laura Delle Monache, Paola Goatin, and Alexandre Bayen. 2015. Discretetime system optimal dynamic traffic assignment (SODTA) with partial control for horizontal queuing networks. In American Control Conference (ACC), 2015. IEEE, 663–670.
 Sanders and Schultes (2005) Peter Sanders and Dominik Schultes. 2005. Highway hierarchies hasten exact shortest path queries. In European Symposium on Algorithms. Springer, 568–579.
 Sanders and Schultes (2006) Peter Sanders and Dominik Schultes. 2006. Engineering highway hierarchies. In ESA, Vol. 6. Springer, 804–816.
 Sekar et al. (2017) Shreyas Sekar, Liyuan Zheng, Lillian Ratliff, and Baosen Zhang. 2017. Uncertainty in MultiCommodity Routing Networks: When does it help? arXiv preprint arXiv:1709.08441 (2017).
 Sheffi (1985) Yosef Sheffi. 1985. Urban transportation network. Pretince Hall 4 (1985).
 Tennessee Department of Transportation (2017) Tennessee Department of Transportation. 2017. Transportation Demand Management Program. https://www.tn.gov/tdot/article/transportationdemandmanagementtdm. (2017).
 Thai et al. (2016) Jérôme Thai, Nicolas LaurentBrouty, and Alexandre M Bayen. 2016. Negative externalities of GPSenabled routing applications: A game theoretical approach. In Intelligent Transportation Systems (ITSC), 2016 IEEE 19th International Conference on. IEEE, 595–601.
 Tung and Chew (1992) Chi Tung Tung and Kim Lin Chew. 1992. A multicriteria Paretooptimal path algorithm. European Journal of Operational Research 62, 2 (1992), 203–209.
 Van Huyck et al. (1990) John B Van Huyck, Raymond C Battalio, and Richard O Beil. 1990. Tacit coordination games, strategic uncertainty, and coordination failure. The American Economic Review 80, 1 (1990), 234–248.
 Vickrey (1969) William S Vickrey. 1969. Congestion theory and transport investment. The American Economic Review 59, 2 (1969), 251–260.
 Wardrop (1952) John Glen Wardrop. 1952. ROAD PAPER. SOME THEORETICAL ASPECTS OF ROAD TRAFFIC RESEARCH. Proceedings of the institution of civil engineers 1, 3 (1952), 325–362.
 Wu et al. (2017) Manxi Wu, Jeffrey Liu, and Saurabh Amin. 2017. Informational aspects in a class of bayesian congestion games. In American Control Conference (ACC), 2017. IEEE, 3650–3657.
 Youn et al. (2008) Hyejin Youn, Michael T Gastner, and Hawoong Jeong. 2008. Price of anarchy in transportation networks: efficiency and optimality control. Physical review letters 101, 12 (2008), 128701.
Comments
There are no comments yet.