Advances in the construction of a time dependent multimodal hypergraph with real time information
David López Flores
Instituto de Ingeniería, Universidad Nacional Autónoma de México dlopezfl@iingen.unam.mx
Angélica Lozano
Instituto de Ingeniería, Universidad Nacional Autónoma de México alozanoc@iingen.unam.mx
Abstract
In many cities of the world public transport is frequency based, however some modes are starting to use real time information on arrivals. This paper describes a first approach to model a multimodal transport system with these characteristics, the model is based on the works of Ziliaskopoulos & Wardell (2000), Lozano & Storchi (2002) and Artigues, et al., (2013). This kind of model has not been yet reported in literature and is a part of a work on progress for the development of an Advanced Traveler Information Systems (ATIS).
Keywords: hypergraph; real time; time dependent; multimodal.
1 Introduction
The aim of this paper is to model a multimodal transport network where some modes of the public transport system have real time information on the arrivals and other modes are frequency based, the model presented will be use, in future works, to find shortest paths in multimodal transport systems. A model of these characteristics has never been addressed in literature.
Spatio-temporal graphs are used to model public transport systems with real time information (Zhang, Li, et al., 2011, Li et al., 2010, Jariyasunant et al., 2011), however as time intervals are added, the network grows exponentially, so is necessary to reduce the size of the network through heuristics (Hedi Ayed et al.
2008) or to pre-calculate some results in order to reduce loading work of the algorithm (Jariyasunant et al.
2011). Other approach for model transport systems is using time dependent networks where the nodes only represent stops or intersections of streets and the weight of the arcs is a time dependent function (Ziliaskopoulos & Wardell 2000, Häme & Hakula, 2013 y Noh et al., 2012). In Schulz (2005) the performance of different labeling algorithms for finding shortest paths with minimum transfers were tested, and the author proved that the algorithms that solve the problem with time dependent networks are between 1.9 and 2 times faster than the algorithms that solve the problem using time expanded networks.
Hypergraphs are used to model public transport systems that are frequency based (Lozano & Storchi 2002) because it is possible to reproduce the action of waiting a line in a stop. Unlike arcs in digraphs which represent a cost (distance, time, convenience, etc.), hyperarcs in hypergraphs represent a probability distribution, which in case of public transport systems defines the lines frequency.
In this paper the main concepts of hypergraphs are defined in Section 2. In Section 3 there is a brief description of time dependant graphs. In Section 4 the time dependent multimodal hypergraph model is
R. Z. Ríos-Mercado et al. (Eds.): Recent Advances in Theory, Methods, and Practice of Operations Research, pp. 160-167, UANL - Casa Universitaria del Libro, Monterrey, Mexico, October 2014.
defined, where the concepts of states, viability, nodes, arcs and paths are fully described. Finally in Section 5 conclusion and future work are presented.
2 Hypergraphs
A directed hypergraph or h-graph is a couple H = (V,E), where V = (v1,v2,... vn) is the set of nodes and E = (e1,e2,... em) is the set of hyperarcs. A hyperarc or h-arc e E is defined as the couple e = (t(e),h(e)) where t(e) ⊂ V is the set of tail nodes and h(e) ⊂ V is the set of head nodes, i.e., the h-arc connects subsets of V (Vološin, V. 2009).
The forward start of u V(H) is the set of arcs coming out of u and is denoted by FS(u) = {(u,y) E such that y V}. The backward star of u V(H) is the set of arcs coming in u and is denoted BS(u) = {(y,u) E such that y V} (Ausiello, G. et. al 2001).
Let H be a hypergraph. A path, rod, that connects an origin o and a destination d is a sequence of nodes and hyperarcs, rod = (o = t(e1),e1,t(e2),e2,... em,d), where t(ei+1) h(ei) for i = 1,2... m-1 y d h(em). A hyperpath rod is the minimum set of acyclic paths rod, such that the destination d is connected to any node that belongs to rod (Lozano & Storchi 2002).
3 Time dependent graphs
Time dependent graphs are directed graphs where each stop (of a mode belonging to public transport) is represented by only one node; on the contrary in spatio-temporal graphs each stop is represented by a set of nodes standing for the arrivals and departures of the bus at the stop in each interval of time. The arcs connecting the stops have associated a function f whose domain is in the defined time interval T (StøltingBrodal & Jacob 2004). This function f holds f(t) ≥ t, such that t is in the time intervals. The length of an arc (u,v) is defined as l = f(t) - t where t is time at the node u, so l is positive. In addition f(t) = t’ for some t’ ≥ t (Schulz 2005).
4 Time dependent multimodal hypergraph
Let G be a hypergraph that model a public transport system where some modes are frequency based and other modes have real time information about arrivals at stops. As G represents public transport systems that are frequency based, hence G need to be a hypergraph in order to model the waiting times at stops. So G = (V,H,T,M) is a multimodal time dependent hypergraph where V is the set of nodes, H is the set of hyperarcs, T = {t0, t0 + Δt, t0 + 2Δt,... (|T| - 1)Δt} is the discretized time intervals, where Δt is the minimum time interval, and M is the set of modes.
4.1 Modes and states
The presented modes are an adaptation of the classification proposed by Lozano & Storchi (2002). Let M
= {M1, M2, M3, M4, M5} such that:
M1:= Pedestrian mode.
M2:= Public transport mode with real time information (bus, tram, subway, etc.).
M3:= Public transport mode frequency based (bus, tram, subway, etc.).
M4:= Private mode (automobile, motorcycle or bicycle).
M5:= Transfer mode.
The user behavior is modeled by means constraints on the used modes. A constraint is boarding the private mode M4 no more than once in a trip, because it is impossible leaving a mode M4 and then boarding it again in a different part of the trip. Constraints on the sequence of modes used modes in a trip are also included, for example it may be illogical to start the trip by bicycle and to continue the trip using the bicycle share system.
Hence, two types of constraints can be defined: logical constraints and preference constraints. Logical constraints represent physical restrictions (for example, boarding private car no more than once in the trip). Preference constraints are represent subjective user preferences, like will to pay versus time saved or travel comfort (transfers could be uncomfortable due to walking or using stairs).
The viability of a path was defined by Lozano & Storchi (2002) in order to know which combination of modes are admissible on paths, according to logical and/or preferences constraints. A multimodal path is viable if the paths composing the path do not use a restricted mode more than once. Note that according to preference constraints, a path may be viable in a network but unviable in others. All the paths have a code associated, which identifies the current combination of modes of a path and let to identify what modes could be used for continuing the path. Lozano & Storchi (2001) define this code as the state of the path.
So in order to know which sequence of states is viable, a non-deterministic state automaton (NSA) is defined (Artigues, el al., 2013). An NSA is defined by A = (S, , s0, F), where S = {1,2,… |S|} is the set of states, s0 is the initial state, F is the final set of states and : M×M×S→s, is a transition function such that
(m,m’,s) gives the resulting state from the state s when a modal transfer is made from the mode m to the mode m’. For the modes M1, M2,… M5 the NSA graph is shown in Figure 1. For example, according to Figure 1, (M4, M5, 3) = 9.
Figure 1 NSA graph
The NSA graph shown in Figure 1 only takes into account the logical constraint related to boarding a private mode M4 no more than once in a trip. The case where a traveler has two cars parked in different locations are not considered in this graph.
In Figure 1, a node represents the state associated with a viable path, and a Mi-arc represents a journey via a mode of the set Mi, i {1,2,3,4,5}. A trip starts at the origin without an associated state (node 0).
For example, if a trip starts via a mode M4 (private vehicle), a path of state 3 is obtained, as long as the path continues via this mode M4, the path remains state 3. If a modal transfer is performed (M5-arc), the path can only continue via a mode M1, M2, M3 or M5.
The state of a hyperpath p indicates the combination of the modes used in p. Let e = (i,h(e)) be a hyperarc with h(e) = {j1,j2} where sx and sy are the states of hyperpaths pj1d and pj2d, respectively. If e is concatenated to the hyperpath pj1d or and hyperpath pj2d, the resulting hyperpath must reflect the modes used in sx and sy, so the resulting state will be s, where s points the sequence of modes used in pj1d and pj2d.
For each pair of states s and s’, such that the modes accepted in s are also accepted in s’, it is possible to establish an order that determines what state is preferable to another. This relationship defines an order between any pair of paths, letting us to know what path has a better chance to grow. Let call this relationship the dominance between states. The state of a path p let us to known the restricted modes used in p and then what modes can be concatenated to p. In this paper, it is proposed an extension of the dominance rules by Artigues et. al., (2013).
Let s and s’ be two states such that all modes accepted in s are also accepted in s’. Define s « s’, s dominates s’, if for any pair of modes (m, m’) M, such that m is a feasible mode for s, one of the following conditions holds:
(m, m’, s’) =
(m, m’, s’) = (m, m’, s)
(m, m’, s) = s and (m, m’, s’) = s’
trans_s < trans_s’,
where trans_s is the minimum number of transfers (modal changes) necessary to reach s.
4.2 Nodes
Each node v V, has associated a modal change delay ξxyikj(t), i.e., the time required to traverse the arc (i, k) with mode x to the arc (k, j) with mode y at time t. Note that if x = y, then ξxyikj(t) = 0 (this model does not takes into account turning delays). The modal change delay for public transport consists of two actions: the time required to board or leave the bus in a stop plus the waiting time at the stop for the bus.
A set of virtual nodes called twin nodes i’ are added to hypergraph, these nodes join almost every node i through virtual arcs. Each arc (i’,i) has a null travel time and ξxyii’j(t) 0. These nodes represent the time for entering the hypergraph, for example, leaving a parking lot or waiting for a bus. For every hyperarc e such that |h(e)| > 1 all the nodes i h(e) have no twin nodes associated because is not possible to enter the hypergraph directly by boarding a bus, first it is needed waiting for it at a stop. Figure 2 shows an example of the twin nodes (gray color) in a hypergraph.
Figure 2 Twin nodes Nodes are stored in a matrix i, which contains:
All the optimum hyperpaths from node i to the destination, for all time intervals and modes.
The number of transfers of the hyperpath.
The state of the hyperpath.
i is defined similarly as in Ziliaskopoulos & Wardell (2000). The matrix i of Ziliaskopoulos & Wardell (2000) only works for networks whose arcs have one and only one head node. In this paper, modes which are frequency based are included, then hyperarcs are needed.. Then, in order to store the information about all the optimum hyperpaths at nodes, a matrix (i = ’i ×’’i) is defined where the x axis contains the forward star of node i, the y axis has all the time intervals multiplied by the number of modes and the z axis includes the set of head nodes of the hyperarc entering to node i.
So each node i has associated a matrix ’i with labels (xik(t), tr, s), where xik(t) is the optimum travel time of the hyperpath from de node i to the destination when entering node i by the arc (k,i) of mode x; tr is the number of transfers of the path and s is the current state of the hyperpath. Therefore, ’i is defined as in equation (1):
[
) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | | ) ) ) | | ) ) ) | | ) ) ) | | ) ) )]
(1) Where xm M; i, k V, and tu T, for all m |M| and u |T|. Also, trl is a natural number and sg {1,2,…7}.
Additionally, as some of the arcs that enter node i are hyperarcs, it is needed to relate all the head nodes of the hyperarc where i belongs. So for every node ip h(e), a matrix ’’i is defined, whose labels are
) ) where ) is the optimum travel time of the hyperpath from node ip to the destination, when entering node ip by the hyperarc e of mode x; tr is the number of transfers of the hyperpath and s is the current state of the hyperpath. Then, ’’i indicates the optimum hyperpaths from the destination to all the head nodes of the hyperarc when entering node i trough e.
[
) ) ) ) ) )]
(2)
’’i is shown in equation (2), where xm M, ip,k V, ip h(e) and k t(e), tu T, for all m |M| y j
|T|. Also, tr is a natural number and s {3, 4, 5, 6, 7}. Note that tr and s do not vary, that is because the hyperpath from i to the destination when entering by e must have same state and number of transfers.
Also s cannot be 1 or 2 because these states are for modes that are not modeled by means hyperarcs.
4.3 Hyperarcs
The hyperarcs are divided into two groups Ht y Hf, where Ht Hf = H. Ht is the set of arcs associated to all the modes except the modes that are frequency based. Let τxij(t) be the travel time of arc (i,j) Ht of mode x at the time t. For the modes with real time information, if there is not an available vehicle in node i at time t, then τxij(t) = , otherwise τxij(t) = c, c ℝ For the modes M1 and M4, τxij(t) = c, c ℝ for all t T, because this modes are always available.
Hf is the set of hyperarcs which represent the frequency based modes. Let φxj(t) the frequency of a line of transport j using the mode x at the time t, such that j h(e). Now, let Li be the set of lines that stop at i, so given an origin and destination, it is possible to assume that some of the lines stopping at i can transport the user near to the destination. So define L’iLi the attractive set as the subset of lines that stop at i that the user could board to reach the destination (Lozano & Storchi 2002). The attractive set defines the concept of boarding hyperarc h’ = (i,h(e’)) where h(e’) h(e). A boarding hyperarc contains the lines j in the attractive set where j h(e).
If φxj(t) is the frequency of the line j using the mode x at time t, it is define the next concepts for L’i:
e’(t) =xj(t) such that j h(e’).
e’(t) = 1 / e’(t).
e’,j(t) = j(t) / e’(t),
where e’(t) is the combined frequency of the attractive set at the time t; e’(t) is the average waiting time at the stop i of the attractive set at the time t; and e’,j(t) is the probability of boarding line j in the attractive set at time t.
4.4 Real time information
The model supposes that the modes of transport with real time information have schedules and these are updated if new information is available. Suppose that the travel time of the arc (i,j) at time tc is τxij(tc) = de, where de > 0 and x is a scheduled mode. Now if the bus is ahead two time intervals, the new travel times for the arc (i,j) are τxij(tc) = and τxij(tb) = de, where tb = tc + 2Δt. This update is possible only if real time information is available. In general, predictions for arrivals in public transport only reach 20 minutes to the future (Jariyasunant et al., 2010), so only the part of the path that are within this period of time will have real time information. However, as the predictions models for the arrivals improve, the model presented in this paper will have accurate predictions for longer periods.
4.5 Paths and expected travel time
Given an origin o, a mode m, a time interval t, and destination d, a hyperpath p is defined as a sequence of hyperarcs, modes and time intervals. Let pod(t) = {(h0,m0,t0),(h1,m1,t1),... (hk,mk,tk)} a hyperpath from o to d, where hi H, mj M, tk T, for all i |H|, j |M|, k |T|.
The hyperarcs in public transport represent frequencies and the action of boarding lines, then it is illogical that two hyperarcs are consecutive in a hyperpath pod(t), since this implies two consecutive boarding actions of different lines. So if e and e’ are hyperarcs of H, then h(e) t(e’) = Also for all e H, |t(e)|
= 1, because there is only one stop for a subset of lines.
The expected travel time of a hyperpath is defined as an adaptation of the Ziliaskopoulos & Wardell (2000) and Lozano & Storchi (2002) models. Let xik(t) the expected travel time of a path p from the node i to the destination when entering to p by the arc (k,i) of mode x. So xik(t) is calculated as in equation (3).
) {
) ( )) ( ) ( ))) | )|
) ∑ ) ) ) ) ⌈ )⌉ (3) When arcs are concatenated, i.e., when |h(e)| = 1 the time of the hyperpath is calculated based on:
The time associated to the change of mode ( )).
The travel time of the arc (i,j), for the current time interval plus the delay associated to modal change, ), i.e. ( )),.
The expected travel time till node j when entering the node through arc (i,j) for the current time interval, plus the delay ), plus the travel time of arc (i,j), ( )), i.e. (
) ( ))).
Note that expected travel time in an arc concatenation, is defined similarly to the Ziliaskopoulos &
Wardell (2000) model.
When a boarding hyperarc is concatenated in the hyperpath, the expected travel time is calculated based on:
The average waiting time of the attractive set, e’(t), at the current interval of time.
The combined probability of the time required to take any of the lines of the attractive set, ∑ ) ) ).
The expected travel time for concatenation of hyperarcs is calculated like in Lozano & Storchi (2002) model, but making their formula time dependent, which requires to avoid discontinuities in time. So the expected travel time till the node j when enter the node by arc (i,j), ), has to be evaluated for the current time interval t plus the average waiting time for the lines of the attractive set. Hence, ) is evaluated in t’ equal to the floor function of t + ωe’(t). Observe that the time associated to the change of mode ) does not account in the hyperarc concatenation because the consecutive arc of an hyperarcs is always of the same mode.
5 Conclusions
This is just the beginning of a research whose objective is to develop an Advanced Traveler Information Systems (ATIS) that merges in a multimodal transport network the use real time information on bus arrivals and frequency based public transport. Some improvements need to be made in the presented model. One of them is reducing the size of the matrix i, by limiting the number of modes considered in
the matrix. This reduction will be made with the aid of the NSA graph which allows eliminate the sequence of modes that are not viable.
New concepts were defined (to our knowledge are not reported in the literature) like the classification of constraints on modes, which will be useful in the future for personalizing the ATIS. Also, an extension of the dominance rule of Artigues et al., (2013) is presented; this extension is useful for compare pair of states that are no comparable in the dominance of Artigues et al., (2013). Finally, the obtaining of travel time in time dependent hyperpaths where some modes are frequency based was proposed. Moreover, the model reported in this paper has not been addressed in the literature and will be useful for cities where public transport is frequency based but there are some modes that have real time information on bus arrivals.
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