Enhanced Shortest Path Computation for Multiagent-based Intermodal

Transport Planning in Dynamic Environments

Christoph Greulich

, Stefan Edelkamp

, Max Gath

, Tobias Warden

, Malte Humann

Otthein Herzog

and T. G. Sitharam

Center for Computing and Communication Technologies, University of Bremen, Bremen, Germany

Center for Infrastructure, Sustainable Transportation and Urban Planning, Indian Institute of Science, Bangalore, India

Keywords:

Multiagent-based Simulation.

Abstract:

This paper addresses improved urban mobility using multiagent simulation. We provide a description of the

agent model and the routing infrastructure as a step towards a rich model of the interactions that happen

in intermodal transport planning tasks. The multiagent model is generic in the sense that different public and

individual transport agents and transportation agencies can be added and parameterized on-the-ﬂy. It integrates

planning with execution. We show that a sequence of calls to Dijkstra’s single-source shortest-paths algorithm

is crucial for planning and provide an efﬁcient memory-less implementation with radix heaps in order to make

this application feasible with respect to scalability. As a case study, we implement a scenario for Bangalore

(India), starting on a higher level of abstraction and drilling down to a running program.

1 INTRODUCTION

Under the umbrella term of smart mobility, the devel-

opment of forward-looking trafﬁc concepts for fast-

growing metropolitan areas has attracted considerable

interest of regional authorities and transport planners.

In this context, optimized utilization of existing and,

potentially, planned trafﬁc infrastructure, using new

mobility concepts has come in sharp focus.

Applying new trafﬁc concepts to the real world

can be very expensive, especially when the given in-

frastructure has to be altered or new ressources have

to be acquired. Therefore, multiagent-based simula-

tion (MABS) can be used to procure well-founded as-

sessments of the impact of potential changes before

actual deployment. Agent-based trafﬁc simulation

is an active research area (Chen and Cheng, 2010).

Various simulators are available, e.g., MATSIM and

SUMO.

Similar to Kl

ugl and Rinsf

user (Kl

ugl and

Rindsf

user, 2011), we do not concentrate on traf-

ﬁc ﬂow but on complex multimodal transportation

tasks. However, they provide a rather abstract model

that only contains dedicated route selection agents.

Our model is also related to the one introduced by

Meignan, Simonin, and Koukam (Meignan et al.,

2006) but is far more dependent on inter-agent com-

munication. We assume that information about possi-

ble travelling options can not be obtained directly but

must be requested from the respective transportation

service. Our main contributions are:

• push-button infrastructure map import from the

OpenStreetMap database yielding simulation re-

sults based on real-world transportation data.

• dynamic creation of agents in a multimodal route

planning environment and scalable integration in

a modern MABS system.

• a generic model for interleaved travel planning

and plan execution which includes replanning on

failure. While planning, the agents consider both

public and individual transportation options.

• enhanced shortest path planning in optimal linear

time with off-line space overhead of two pointers

per node and no additional space allocated during

the search.

Section 2 introduces MABS. Section 3 discusses

the proposed simulation model with a focus on the

implemented multiagent system. Section 4 provides

a time and space optimal implementation of Dijksta’s

algorithm. Section 5 speciﬁes the agent conﬁguration

used in our scenario and report on the feasibility as-

sessment of our implementation with results on sim-

ulation efﬁciency on the trafﬁc infrastructure of Ban-

galore.

324

Greulich C., Edelkamp S., Gath M., Warden T., Humann M., Herzog O. and Sitharam T..

Enhanced Shortest Path Computation for Multiagent-based Intermodal Transport Planning in Dynamic Environments.

DOI: 10.5220/0004262103240329

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 324-329

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

2 AGENT-BASED SIMULATION

Any trafﬁc simulation model can be classiﬁed by

its level of detail. While macroscopic models only

consider aggregated information, microscopic models

are based on detailed modeling of individual entities

(Hoogendoorn and Bovy, 2001).

Multiagent systems provide the capability to im-

plement microscopic models. Even though the deﬁni-

tion of an agent is widely discussed, the most impor-

tant characteristic of an agent is its autonomy (Macal

and North, 2010): Every agent is a unique entity and

gains information by sensing its environment or by

social interaction with other agents. In addition, an

agent acts and reacts upon its own decisions which

are based on the individual goals of the agent and its

knowledge of the environment.

A MABS system can combine distributed

discrete-event or time-stepped simulation with

decision-making encapsulated in agents as separate

and concurrent logical processes. In classical simula-

tion systems, the involved logical processes as well

as interaction links have to be known in advance and

must not change during simulation. This is not the

case in MABS systems, as each agent may interact

with all other agents. Agents may join or leave

simulation during execution, e.g., depending on a

stochastic simulation model.

We use our event-driven MABS system which

has been designed to solve and evaluate scenarios

in the logistics domain. The system is based on

the Java Agent DEvelopment framework (Bellifem-

ine et al., 2007) and adds the functionality of a dis-

crete time simulation as well as conservative syn-

chronization with time model adequacy, causality and

reproducibility (Gehrke et al., 2008). In general,

our system architecture has the following compo-

nents: World model, physical objects, infrastructure

and agents including behaviour deﬁnitions.

The (physical) simulation world model of a sce-

nario can be modeled as a graph so that the infrastruc-

ture can be mapped accordingly. Graph nodes rep-

resent, e.g., trafﬁc junctions. Graph edges represent

roads, rails, waterways, etc. In order to model real

transport processes, we extended our system to sim-

ulate scenarios within real trafﬁc infrastructures that

BEH: Init

BEH: HandleInformationRequests

BEH: HandleBoardingQueries

BEH: HandleTransportRequests

Agent: ScheduledVehicleAgent

Agent: Person

Simple Communication

FIPA IP Communication

State Transition

condition

Legend

Figure 1: Scheduled Transportation Company Agent.

are imported from OpenStreetMap databases. Agents

act either as artiﬁcial autonomous decision makers on

behalf of their associated entity within this model or

as background services, e.g., for creating additional

agents or providing information.

3 SIMULATION MODEL

We distinguish between simple behaviors (like Init or

HandleInformationRequests or Driving) and complex

behaviors (like TransportPassengers) that itself con-

sist of an arrangement of behavioral states (see Fig. 1

- 3). Complex behaviors are implemented in terms of

Finite-State-Machine (FSM) behaviors. FSM behav-

iors are labeled transition systems with a starting state

(indicated with an incoming arc), one or more ter-

minal states (encircled node), and several conditional

state transitions. The behaviors also include commu-

nication arcs (dashed) that show whether or not a be-

havior communicates with another agent. Addition-

ally, we show which agent poses shortest path queries

(SPS).

The Scheduled Transportation Company Agent is

an implementation of a transportation company that

provides an information infrastructure for answer-

ing initial transport queries (HandleInformationRe-

quests), sets-up and maintains a list of persons to

be picked up at stops for a speciﬁc vehicle (Handle-

BoardingQueries), and receives and answers trans-

port requests of persons at stops (HandleTransportRe-

quest).

The corresponding agent model is shown in Fig. 1.

After registration with the system wide directory fa-

cilitator and acquisition of time-table information

from a conﬁguration ﬁle, the agent is responsible for

invoking time-dependent initialization of Scheduled

Vehicle Agent instances that represent the means of

transport. The sub-behaviors are spawned, and run in

parallel.

On its initialization, a Scheduled Vehicle Agent re-

ceives a schedule from the agency that it has to follow

on a daily basis. The time table information contains

the arrival and departing time of each stop in the tour.

If a vehicle is running late it usually tries to catch up

with its schedule, reducing waiting times. The Sched-

uled Vehicle Agent’s model is shown in Fig. 2. The

FSM behavior for scheduled driving mainly imple-

ments a cycle of boarding, debarking and moving.

Furthermore, it uses an interaction protocol to com-

municate with the transportation company agent to re-

ceive new instructions.

In addition, we implemented a Autonomous

Transportation Vehicle Agent which basically com-

EnhancedShortestPathComputationforMultiagent-basedIntermodalTransportPlanninginDynamicEnvironments

325

BEH: Init

BEH: ScheduledDriving

FSMBehaviour

Unloading Done

Received List

Loading Done

Arrived at next Stop

BEH: Deboarding

BEH: RequestLoadInstructions

BEH: Boarding

BEH: MoveToNextStop

Agent: PersonAgent

Perf: INFORM

Agent: PersonAgent

Perf: INFORM

Agent: ScheduledTransportationCompanyAgent

BEH: Takedown

No next Stop left

Figure 2: Scheduled Vehicle Agent.

bines the functionality of a scheduled transportation

agent and a vehicle. Instead of running on schedule,

the autonomous vehicle agent picks up a person on

demand, just like a taxi would. The answer of the au-

tonomous transportation vehicle agent to a transport

query (a, b) is a single transport option which covers

the whole route from a to b since the autonomous ve-

hicle is not bound to scheduled routes.

The Person Agent is a complex agent which com-

municates with transportation agencies and vehicles.

It plans and executes travel routes, either by using

public transport passively or by walking actively. The

implemented model is shown in Fig. 3. We see a hi-

erarchy of complex FSM behaviors. It shows that a

person operates in a loop of planning and executing

plans.

For the generation of plans (Planning behavior)

shortest paths have to be generated for walking, as

well as transportation agencies have to be contacted

for options. While the start and goal locations are

assumed to be on an arbitrary node in the map, not

all transportation requests can be satisﬁed by bus, so

that planning includes to pad vehicle usage with walks

to or from the stops of the vehicles. Once the plan

is ﬁxed, it goes to execution (ExecutePlan behavior),

where each step of the plan will be executed sequen-

tially, either as an active (walking) or passive trans-

port. We see that passive transport has to be moni-

tored and can fail (e.g., by a timeout on the waiting

time), so that the planning behavior has to be rein-

voked on termination.

The Birth-Giver Agent initializes person agents

with several parameters like the start location, a cer-

tain budget in time and cost, as well as a target loca-

tion. All values are random numbers, drawn accord-

ing to a given probability distribution. Optionally,

the start and end location can be speciﬁed manually

by determining a ﬁxed node. The realistic modeling

of the random process is crucial for the applicabil-

ity of the simulation outcome. The better the model

BEH: Init

Init done

Plan Generated

Plan Executed

Failure

Arrived at

Destination

BEH: Planning

FSMBehaviour

Acquire Options

Options Acquired

Plan

Generated

BEH: ExecutePlan

FSMBehaviour

Agent: All„Transport“Agents

Active

Passive

Success

Plan Executed

Success

Failure

BEH: PassiveTransport

FSMBehaviour

Success

Failure

Agent: „Transport“ServiceAgent

Failure

Agent: Vehicle Agent

Perf: INFORM

SPS

BEH: PersonFSM

BEH: Planning

BEH: ExecutePlan BEH: Takedown

BEH: GeneratePlan

BEH: AcquireOptions

BEH: FSMEndPoint

BEH: ExecuteNextStep

BEH: ActiveTransport

BEH: PassiveTransport

BEH: SealTransportContract

BEH: FSMEndpoint

BEH: TransportMonitoring

Figure 3: Person Agent.

the better its prediction. These data might be indi-

rectly inferred by information on where people live

and where they work, or by monitoring their current

use of vehicles. In reality, the choice of a transport

mode depends on several details, such as spatial dis-

tance or social-economic variables (Buehler, 2011).

In the end, a rather complex probability distribu-

tion for transport requests, humans and their queries

should be derived.

4 SHORTEST PATH SEARCH

Single-source shortest paths search with Dijkstra’s or

the A* algorithm (Dijkstra, 1959) was already present

in our MABS system, but showed performance prob-

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

326

lems and slowed down the overall MABS process

considerably. Therefore we engineered the imple-

mentation by using a memory-efﬁcient joint repre-

sentation of graph and radix heap nodes. A joint

node representation includes a state label (unlabeled,

labeled or scanned), a linked list of edges, the ele-

ment for storing the distances, the radix bucket a node

is stored in as well as two pointers (pred, succ) for

linking the elements in the radix heap. An edge is

a pair of a successor node ID and according weight

(cost/distance). We observed that the joint node rep-

resentation is more crucial to the performance of the

search than the proper choice of the data structure.

One reason is to avoid memory allocation, another is

that efforts for maintaining handles to the nodes can

be avoided.

We also employed a key-based priority queue, ex-

ploiting the maximum weight C of all edges. A

radix heap (Ahuja et al., 1990) maintains an array

of dlg(C + 1)e + 1 buckets of sizes 1, 1, 2, 4, 8, 16,

etc. Elements in the buckets are doubly-linked. More

precisely, we maintain buckets b[0..B] and bounds

u[0..B + 1] with B = dlog(C + 1)e + 1 and u[B + 1] =

∞. The invariants of the algorithms are: 1) all keys in

b[i] are in [u[i], u[i + 1]], 2) u[1] = u[0]+ 1, and 3) for

all i ∈ {1, . . . , B − 1} we have 0 ≤ u[i + 1] − u[i] ≤

i−1

The main difference to one-level buckets (Dial,

1969) is to use buckets of exponentially increasing

sizes. Therefore, only O(lgC) buckets are needed. If

edge weights are integers or ﬂoating point numbers

O(lgC) can be interpreted as a constant independent

from the number of nodes n and edges m.

For shortest path search, in the presence of a lower

bound heuristic function h, A* (Hart et al., 1968) can

be applied. Without reopening, A* resorts to a variant

of Dijkstra’s algorithm with f (s) = h(s) for start node

s and new weight w

(u, v) = w(u, v) − h(u) + h(v) for

all edges (u, v).

Theorem 1 (Time Optimality Shortest Paths Explo-

ration). Given that the edge weights are computer

words (64-bit integer or ﬂoating-point numbers) and

provided a matching number representation for stor-

ing the accumulated distances at each node, our im-

plementation of the Single-Source Shortest Paths Al-

gorithm of Dijkstra (or A* with consistent heuristic)

has optimal linear time complexity.

Proof. The radix heap assumes that all edge costs in

the graph are integers bounded above by C. The re-

sult is that Dijkstra’s algorithm can be implemented

with a time complexity of O(m + n lgC), where n is

the number of nodes and m is the number of edges.

Given that the logarithm of a 64-bit integer is bounded

by a constant lgC = 64, the running time on a mod-

ern computer is linear O(m + n). If edge weights are

doubles, lgC = lg(1.79769 · 10

308

) = O(1)

Furthermore, the radix heap is signiﬁcantly sim-

pler to implement compared to the Fibonacci heap

and similar data structures. The radix heap achieves

this time complexity by taking advantage of the prop-

erty that shortest path distances fall into a ﬁnite range

during the computation shortest paths by Dijkstra’s

algorithm.

The practical savings were considerable. Erst-

while unanalyzable systems turned out to be quickly

analyzable, and even a full Dijkstra exploration was

much faster than the original implementation of A*.

In smaller graphs (200,000 nodes, 2,000,000 edges),

generating the graph turned out to be more com-

plex (4,443 ms) than actually searching it (1,668 ms).

But even larger graphs (1,000,000 nodes, 10,000,000

edges) could be generated (30,574 ms) and searched

(34,631 ms) in adequate time.

Recall, that shortest path queries are frequently

posed by different agents such as persons and vehi-

cles; buses in a dynamic world may have to recom-

pute shortest paths from every bus station to the next

(based on dynamic changes to the road network due

to trafﬁc jams). Frequent shortest path queries are

also needed for preprocessing the graph in order to

solve so-called vehicle routing problems. The deci-

sion making then relies on efﬁcient solutions to the

Traveling Salesman Problem (TSP), a touchstone for

many general approaches in combinatorial optimiza-

tion: Genetic algorithms, simulated annealing, tabu

search, ant system, just to name a few. The prob-

lem is strong NP-hard and difﬁcult to approximate

unless the triangular inequality holds ((Christoﬁdes,

1976); (Arora, 1998)). In an application of a forward-

ing agency, the TSPs are generated via shortest paths

reductions of route networks. Each order to be served

corresponds to one city in the TSP.

On the ﬁrst glance, in case a better shortest path

search performance is needed, A* is an obvious alter-

native to Dijkstra’s algorithm. For a consistent heuris-

tic evaluation function, it is optimal efﬁcient and (up

to tie-breaking) A* will expand the minimum number

of nodes. For a heuristic that is strictly more informed

than the trivial 0-heuristic applied in Dijkstra’s algo-

rithms, it is guaranteed to expand less nodes. How-

ever, at this level of speed per node, the number of

expanded nodes is not the only key performance mea-

sure. We experienced that the time needed for com-

puting the lower bound heuristic during shortest path

search can negatively inﬂuence the overall perfor-

mance. The computation of shortest paths of several

hundred TSP matrices based on the distance calcu-

lations according to the Haversine formula resulted in

EnhancedShortestPathComputationforMultiagent-basedIntermodalTransportPlanninginDynamicEnvironments

327

A* requiring 330s, while Dijkstra’s algorithm only re-

quired 274s. The net computing time for computing

all heuristic values took 207s, so that despite the sav-

ings in the number of node in A*, the heuristic itself

is too complex. Hence, computationally more sim-

pler heuristics are needed to beat Dijkstra’s algorithm.

Devising an efﬁciently computable consistent lower

bound heuristic is non-trivial, as approximations of

the heuristic are often inconsistent and thus sacriﬁes

optimality, or call more involved data structures.

As an alternative improvment to the search, bidi-

rectional shortest paths search can be executed. More-

over, there are many speed-up techniques that prepro-

cess the graph for a much better search time (Bast

et al., 2007). However, such off-line improvements

hardly translate to a dynamic scenario.

5 EXPERIMENTAL SETUP

In the context of a study on opportunities for novel

last-mile connectivity, in a preliminary case study we

looked at a MABS model for the Bangalore urban re-

gion, based on a user needs study (Akhilesh et al.,

2012) and the 2010 Trafﬁc Management Plans for

Major Towns in Bangalore Metropolitan Region for

the Bangalore Metropolitan Region Development Au-

thority conducted by WilburSmith Associates.

To perform the simulation we extracted route data

from the OpenStreetMap. As Bangalore was not pre-

deﬁned as a coherent district, we deﬁned a bounding

box and included also streets that cross the bound-

aries instead of clipping them exactly at the border of

the bounding box. The road infrastructure contains

49,399 nodes and 134,222 edges and includes the In-

ternational Airport of Bangalore (BIAL).

The bus routes and associated information on

travel fares as well as ﬂeet sizes is supplied by

the Bangalore Metropolitan Transport Corporation

(BMTC) authorities. They refer to real-life data. We

decided to focus on the last-mile connectivity of pas-

sengers at BIAL. The distribution of passenger trans-

ports measured for a particular week shows that there

are about 6,000 persons who use the bus lines on a

day. In comparison there are about 10,000 persons

taking a taxi (trips per day). Information on real-life

bus schedules as well as bus stops is added manually.

The simulated scenario models 8 bus lines, whose ﬁrst

or last stop is the BIAL.

The implementation of agents for the initial study

is fully operational. In our scenario, a ﬁrst birth-giver

agent is responsible for creating person agents that act

for persons located in the inner city district. The de-

sire of this persons is to arrive the airport as soon as

Figure 4: Running simulation with busses, persons, rick-

shaws and bikes.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

100

1.000

10.000

100.000

Boarding Debarking

Number of Bus Stop (in Alphabetic Order)

Number of Passengers

Figure 5: Simulation results. The open bars show the num-

ber of boarding and the closed bars the number of debarking

passengers at each bus stop within an whole week. The bus

stops are numbered consecutively to their ofﬁcial name in

alphabetic order.

possible. Person agents that are created by a second

birth-giver agent represent individuals whose desire it

is to get from the airport to the inner city district. Fi-

nally, a third birth-giver agent generates a ﬁxed num-

ber of rickshaws which start at the airport and also

drive to the inner city district.

To generate the start and end location randomly,

we implemented a random walk strategy starting at

a bus stop to generate requests at certain nodes with

an arbitrary distance. Therefore we ensure that trans-

port requests are not far from the encoded bus sta-

tions. Simulating dynamically changing trafﬁc con-

ditions is the subject of further research. As a result,

edge following is mainly determined by the speed of

the human or the vehicle as well as the type of the

road.

Moreover, we added bikes, so that persons have

the ﬂexibility to decide how to get to the bus stop or

home: if the distance is close enough (smaller than

some predeﬁned threshold) they walk, otherwise they

ride by bike (see Fig. 4). The maximum walking

speed of a person is limited to 5 km/h while the max-

imum speed of a bike is limited to 30 km/h. We sim-

ulated a time span of a whole week.

For our sample scenario we are interested in the

number of passengers at each bus stop (see Fig. 5).

No. 2 is the BIAL and, therefore, either the place of

arrival or departure of each person. Low trafﬁc is pri-

marily caused by a high density of bus stops within

a certain district. In total we simulated more than

45,000 agents with up to 1,100 agents running con-

currently.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

328

Performance indicators recorded in different sim-

ulations can easily be combined (e.g., averaged val-

ues). Besides the number of passengers that board or

debark a bus, there are a lot of other interesting per-

formance indicators for the simulation like the time

spend by persons to reach their destination, the bud-

get needed for the travel, or a combination of both.

Experiments have shown that the bottleneck of the

computations is computing several shortest paths on

the routing graph by each person agent. Buses do

not change their routes if there is no occurrence of

unexpected events. As a consequence, the computa-

tion time of the buses are minimized by the use of an

cache.

6 CONCLUSIONS

AND OUTLOOK

This paper reported on the current status of a ﬁne-

grained MABS model for urban mobility in Banga-

lore. The physical simulation world model builds

upon detailed OpenStreetMap data for the basic traf-

ﬁc infrastructure. It allows the mapping of the exten-

sive network of bus lines operated by the BMTC. The

generation of individuals that utilize modeled trans-

port modalities can be conﬁgured according to actual

distributions. Besides agent models for transport op-

erators such as the BMTC and independent operators

(using, for instance, taxis and rickshaws), a particu-

lar focus was put on the rational modeling of trans-

port customers. These are equipped with capabilities

for interleaved planning and execution of intermodal

transport schedules. Experiments have shown that our

system handles a hand-modeled excerpt of BMTC bus

lines connecting airport and city center with a real-

istic number of more than 6,000 passengers per day

in a time frame which make extensive experimenta-

tion practical in spite of complex planning and route-

ﬁnding calculations performed by the simulation ac-

tors. The challenges to be addressed in ongoing multi-

disciplinary research on smart mobility in metropoli-

tan areas are manifold. The results of this research

(e.g. optimized utilization of existing trafﬁc systems

or new emergency strategies) can be compared and

evaluated by using a MABS model like the one we

introduced in this paper.

ACKNOWLEDGEMENTS

The presented research was partially funded by

the BMBF within the acatech project GRIP IT

orderkennzeichen 01/S09048) and the German Re-

search Foundation (DFG) within the Collaborative

Research Centre 637 ”Autonomous Cooperating Lo-

gistic Processes: A Paradigm Shift and its Limita-

tions” (SFB 637) at the University of Bremen, Ger-

many.

REFERENCES

Ahuja, R. K., Mehlhorn, K., Orlin, J. B., and Tarjan, R. E.

(1990). Faster Algorithms for the Shortest Path Prob-

lem. Journal of the ACM, 37(2):213–223.

Akhilesh, K. B., Sitharam, T. G., Goswami, M., and Man-

jula, D. (2012). User Needs Study: Living Lab on

Bangalore Mobility and ICT Research for Smart City

Solutions. Technical report, CiSTUP.

Arora, S. (1998). Polynomial time approximation schemes

for euclidian traveling salesman and other geometric

problems. Journal of the ACM, 45(5):753–782.

Bast, H., Funke, S., Sanders, P., and Schultes, D. (2007).

Fast Routing in Road Networks with Transit Nodes.

Science, 316(5824):566–566.

Bellifemine, F., Caire, G., and Greenwood, D. (2007). De-

veloping Multi-Agent Systems with JADE, volume 5.

Wiley.

Buehler, R. (2011). Determinants of transport mode choice:

a comparison of germany and the usa. Journal of

Transport Geography, 19(4):644 – 657.

Chen, B. and Cheng, H. H. (2010). A review of the appli-

cations of agent technology in trafﬁc and transporta-

tion systems. IEEE Transactions On Intelligent Trans-

portation Systems, 11(2):485 –497.

Christoﬁdes, N. (1976). Worst-case analysis of a new

heuristic for the travelling salesman problem. Tech-

nical Report 388, Graduate School of Industrial Ad-

ministration, Carnegie-Mellon University.

Dial, R. B. (1969). Shortest-path forest with topological

ordering. Comm. of the ACM, 12(11):632–633.

Dijkstra, E. W. (1959). A note on two problems in connex-

ion with graphs. Numerische Mathematik, 1(1):269–

271.

Gehrke, J. D., Schuldt, A., and Werner, S. (2008). Quality

Criteria for Multiagent-Based Simulations with Con-

servative Synchronisation. In 13th ASIM Dedicated

Conference on Simulation in Production and Logistics

(ASIM), pages 177–186.

Hart, P. E., Nilsson, N. J., and Raphael, B. (1968). A formal

basis for the heuristic determination of minimum cost

paths. IEEE Transactions on Systems, Science, and

Cybernetics, SSC-4(2).

Hoogendoorn, S. P. and Bovy, P. H. L. (2001). State-of-the-

art of vehicular trafﬁc ﬂow modelling. Institution of

Mechanical Engineers, Part I: Journal of Systems and

Control Engineering, 215(4):283–303.

ugl, F. and Rindsf

user, G. (2011). Agent-based route (and

mode) choice simulation in real-world networks. In

IAT, pages 22–29.

Macal, C. M. and North, M. J. (2010). Tutorial on agent-

based modelling and simulation. Operational Re-

search Society, 4:151 – 152.

Meignan, D., Simonin, O., and Koukam, A. (2006). Multi-

agent approach for simulation and evaluation of urban

bus networks. In AAMAS’06, pages 50–56.

EnhancedShortestPathComputationforMultiagent-basedIntermodalTransportPlanninginDynamicEnvironments

329