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An application of ant colony systems for DUE and SUE
assignment in congested transportation networks
Matteo Matteucci 1, Lorenzo Mussone 2*
1
Department of Electronics and Information,
Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milan, Italy
tel: +39 02 2399 3470 e-mail: matteucci@elet.polimi.it
2
Dep. of Building Environment Science and Technology,
Politecnico di Milano, Via Bonardi 9, 20133 Milan, Italy
tel: +39 02 2399 5182 e-mail: mussone@polimi.it
*corresponding author
ID: 634
Topic area: D6 Scenario Development & Analysis
1
An application of ant colony systems for DUE and SUE
assignment in congested transportation networks
Matteo Matteucci 1, Lorenzo Mussone2*
1
Department of Electronics and Information, Politecnico di Milano, Italy
Piazza Leonardo da Vinci 32, 20133 Milan
tel: +39 02 2399 3470 e-mail: matteucci@elet.polimi.it
2
Dep. of Building Environment Science and Technology,
Politecnico di Milano, Italy
Via Bonardi 9, 20133 Milan
tel: +39 02 2399 5182 e-mail:vmussone@polimi.it
*corresponding author
Abstract
The paper deals with deterministic and stochastic user equilibrium (DUE and SUE
respectively) well known problems in the transportation field. In order to solve
these problems a modified version of the ant colony system is proposed. The ant
colony heuristic is adapted in order to take into account all aspects characterizing
the transportation problem: multiple ODs (Origin-Destination), link congestion,
non-separable cost link functions, elasticity of demand, multi classes in demand
and different user cost models including stochastic cost perception. Applications to
four different networks are finally reported.
Keywords: Transportation networks, User equilibrium assignment, DUE, SUE,
Ant Colony Optimization, Ant Colony System
1. Introduction
An analysis of transportation networks requires mathematical tools capable of
managing all information related to the interaction between demand and supply
and solving the complex problems developed by this interaction. The features that
must be taken into account concern both the supply and the demand
characteristics. Supply first is made up by four elements: the graph, the flow
propagation model, the route choice model, and the congestion model. Transport
demand characterizes mobility with respect to different choices of travel (e.g.
frequency and reasons of trips, origins and destinations, modal split, paths). Except
the very particular case of non-congested networks, route choice depends on
congestion giving rise to the interaction between demand and supply. If only route
choice depends on congestion a rigid demand assignment model is considered. If
other characteristics of demand depend on congestion an elastic demand
assignment model must be considered.
The traffic assignment problem can be formulated and solved by following two
main approaches depending on the prevailing traffic patterns (steady-state or
dynamics) of the transportation system to be simulated or on the aim of the
analysis. In turns, dynamics can refer to within-day-dynamic day-to-day-dynamic
or both. In this paper the steady state both within-day and day-to-day traffic
assignment will be referred to. The best known approaches to this analysis are
deterministic user equilibrium (DUE) and the stochastic user equilibrium (SUE).
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DUE was formulated by Wardrop (1952) as a criterion to find the distribution
on the routes. This criterion states that as the journey time on all the routes actually
used are equal and less than those which would be experienced by a single vehicle
on any unused route. This is based on strong assumptions that the network travel
times are deterministic for a given flow pattern and that all travelers are perfectly
aware of the travel times on the network and always capable of identifying the
shortest travel time route. To overcome the limits of the deterministic model, some
researchers have proposed different SUE models to relax the assumption of perfect
knowledge of network travel times, allowing travelers to select routes based on
their perceived travel times.
The ant colony system (ACS) is a particular implementation of the Ant Colony
Optimization meta-heuristic (Dorigo and Stützle, 2004; Dorigo and Blum, 2005).
They have both been successfully applied to many discrete optimization problems
also in the transportation field. Typical applications of ant systems are Travelling
Salesman Problem (Dorigo et al., 1996; Dorigo and Gambardella, 1997b; Li and
Gong, 2003), network routing (Di Caro and Dorigo, 1998), the Quadratic
Assignment Problem, QAP, (Gambardella et al., 1999). More recently, ant
colonies have been applied to road traffic management (Bertelle et al., 2003) and
resource allocation in transportation (Doener et al., 2001). D’Acierno et al. (2006)
proposed an MSA (Method of Successive Averages) algorithm for a SUE
simulation based on an ACO paradigm. Other applications of ACO technique in
the transportation field can be found in (Mussone et al., 2005; Matteucci and
Mussone, 2006)
The aims of this paper are manifolds and concern the application of ACS to
cope with several aspects of DUE/SUE traffic assignment. We developed an
algorithm, based on an ACS, able to search the optimal (or quasi-optimal if
stopped in advance) solution of the assignment problem especially when
uniqueness of solution is not guaranteed or when no a-priori hypotheses can be
done about the objective function. Experimentally we have validated this
algorithm also on realistic network scenarios showing that it achieves good
solutions, especially for the SUE assignment, with a reduced computing time with
respect to traditional algorithms. Moreover the algorithm we propose does not
require any assumption on the link cost function or the user choice model so it is
suitable, as we tested in the experiments, also for link cost functions with nonseparable costs (that is the case of links ending with a T-intersection or a
roundabout) and user cost perception such as Gaussian and Lognormal.
In Section two the transportation problem is analytically described in a greater
detail; Section three explains the ACO (Ant Colony Optimization) meta heuristic
and the particular version used, Ant System; in Section four we propose a
modified version of Ant System in order to solve the above mentioned
transportation problems . Section five shows the results obtained by the application
of the modified ant system on some transportation networks characterized by
different size and link cost functions; Section six deals with conclusions and future
research.
2. The Traffic Assignment Problem
2.1 Problem definition
A transportation system may have different admissible states during time. The
evolution mechanism is represented by the circular dependence between traffic
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demand, flows and costs (Figure 1). In some cases the circular dependence evolves
to steady state conditions where demand flow and costs are mutually consistent. In
the following paragraphs some basic notations and formalization for the steadystate (equilibrium) case are introduced and discussed.
Given a directed graph G(V,E), with E being a set of N nodes and V a set of l arcs,
we can identify a subset of n  nodes in E and call them centroids. Let d be a
vector with components representing the average number of trips going from
centroid origin o to centroid destination d within a give time period. Each origindestination flow generates on the network path flows Fi, with iIod, where Iod is
the subset of all admissible paths connecting the pair of centroids o and d. For a
given link/edge eE, the sum of all path flows crossing this link is called the link
load:
f i   k aik Fk
(1)
where aik is 1 if the link i is crossed by the path k and 0 otherwise. In matrix form
f=AF
(2)
where F, f are the vectors of path and link flows, respectively, and A is the linkpath incidence matrix.
A model of a transportation system describes the behavior of traffic demand d
and its relationship with link flows. By introducing a cost ce(f) for traveling on a
certain link e, depending on the observed traffic f, one can express traffic demand
and the way it is distributed on links in function of the vector of costs c, in
particular the relationship between F, f and d becomes
F=P(c(f))d(c(f)) or F*=P(ATc (ATF*))d(ATc (ATF*))
(3)
f=AP(c(f))d(c(f)) or f*=AP(ATc (f*))d(ATc (f*))
(4)
where P is the path choice probability matrix, known also as path choice map,
whose every element expresses the probability that traffic demand di (of the i-th
od-couple) is routed on path k; C=ATc represents the vector of path choices costs.
Equations 2 and 3 or equation 4 describe the circular dependencies, the
consistency of which is at the base of the equilibrium problem. As Figure 1 shows,
neither the way the system evolves nor how it reaches equilibrium is studied. It is
assumed that when the system reaches equilibrium it becomes stationary, because
demand is constant so its link performance only can modify the balance between
demand and costs.
Figure 1: Equilibrium relationship between traffic demand, flows and costs.
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Equilibrium can be analyzed with a further condition on OD trip matrix
elasticity. The OD trip matrix can be rigid, in the sense that the cost variation
caused by congestion affects only the choice of path. This means that vector d is
assumed invariant to link costs. Vector f* or F* are then defined by the equations:
F*=P(C(F*))d where C(F*)=ATc(AF*)
(5)
f*=AP(C(f*))d where C(f*)=ATc(f*)
(6)
Otherwise demand can be elastic; this means that it depends on congestion
costs as well as on system attributes.
2.1.1 Rigid demand: deterministic case
The problem of finding vectors f* and F*, under the assumption of a deterministic
path choice model (DUE) in order to avoid some mathematical difficulties due to
the fact that in the deterministic case equations 3 and 4 (even in the case of rigid
demand) are multi-valued maps, is studied by means of formulations based on the
variational inequalities:
C(F*)T(F-F*)≥0 F  SF
(7)
where SF is the set of admissible path flow vectors. Equivalent variational
inequality models are based on link flow leading to
c(f*)T(f-f*)≥0 f  Sf
(8)
where Sf is the set of admissible link flows.
The calculation of equilibrium link flow with rigid demand and symmetric
Jacobian of cost functions is based on algorithms that solve the previous
variational inequalities by an optimization model. One of the most famous is the
Frank-Wolfe algorithm (FW) (Frank and Wolfe, 1956) from which many variants
were derived. When link costs are non-separable, the Jacobian may not be
symmetric and other algorithms must be used, for example the diagonalization
algorithm, which uses an approximation of the variational inequality by using
separable cost functions. It is worth noting that this not ensures that the underlying
equilibrium has a unique solution.
2.1.2 Rigid demand: stochastic case
A stochastic equilibrium assignment is obtained by applying the equilibrium
approach to the assignment of congested networks on the hypothesis of a
probabilistic behavior of choice. SUE was first introduced by Daganzo and Sheffi
(1977) and developed later in a limited form by Sheffi and Powell (1981); a first
model for optimization was proposed by Daganzo (1983) for symmetric SUE.
In Daganzo and Sheffi (1977) the authors applied the method of fixed point in
order to solve deterministic or stochastic equilibrium. A plain complete review of
SUE models is presented by Cascetta (1988 and 2001).
More recently Ceylan and Bell (2005) presented a bi-level technique oriented to
signalized network design which involves an assignment model to calculate SUE:
genetic algorithms are used to solve the upper-level problem for the signalized
network and a SUE traffic assignment is applied at the lower-level.
A solution to SUE exists under very mild assumptions but uniqueness is
assured in the case of separable cost functions. In other cases more than one
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equilibrium solution may exist and the solution found may not be the one that
interests the transportation analyst (i.e. it may not be that of total minimum cost).
New methods to analyze these transportation problems are therefore of general
interest.
3. ACO: Ant Colony Optimization
3.1 The meta-heuristic technique
Ant Colony Optimization (ACO) (see for example Dorigo et al., 1999) is a metaheuristic technique (in Figure 2 the flow chart of ACO technique process is
described) that implements a colony of artificial ants cooperating in order to find a
solution to a difficult combinatorial optimization problem (Di Caro and
Gambardella, 1999; Dorigo and Stützle, 2004; Gutjahr, 2002; Zlochin et al., 2004).
From a general point of view, ants in ACO can be seen as simple agents
cooperating to build a complex solution by communicating indirectly through the
modification of the environment: this process is called stigmergy (Di Caro and
Dorigo, 1998). Actually each ant builds independently a possible poor solution to
the problem (or part of it) and the optimal solution is obtained as the emerging
behavior of the colony due to stigmergy. Information collected by each ant while
searching for the solution is, in fact, shared with other ants in the colony by
leaving a clue to the solution in the form of a pheromone trail.
Pheromone knowledge built-up by ants is a shared local long-term memory
that influences their decisions. When and how much pheromone should be
released on the “environment” depends on the problem and on the implementation
of the specific solution. It can be released while ants build the solution (on-line
step-by-step), or after a solution has been built moving back to all the visited states
(on-line delayed), or both. While ant internal states can be used to build feasible
solutions using knowledge about effects of actions that can be performed in that
state, pheromone trails and problem-specific heuristic information guide the ant
decisions as a long-term memory playing auto-catalysis on the whole optimization
process. This is clearly explained by thinking that the more ants choose an action,
the more this action is favoured by adding pheromone to its trail and the more
interesting it becomes for the following ants.
Locally available pheromone and heuristics form ant decision tables, i.e.,
probabilistic tables used by each ant decision policy to direct the solution search
towards the most interesting regions of the search space. The stochasticity of ant
action choice and a pheromone evaporation mechanism avoid a rapid drift of the
colony towards the same part of search space. The level of stochasticity in policy
selection and the strength of the updates in pheromone trails determine the balance
between the exploration of new points in state space and the exploitation of
accumulated knowledge. If necessary, ant decisions can be enriched with problemspecific components like backtracking procedures or looking ahead (Russel and
Norvig, 1995). Finally, some extra components which use global information
called daemon actions can be allowed to observe the ants’ behavior, and collect
useful global information to deposit additional pheromone information, biasing, in
this way, the ant search process from a non-local perspective.
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Figure 2: The general Ant Colony Optimization technique.
3.2 Ant System
To better understand the functioning of our algorithm for traffic assignment, we
introduce the classical example of ACO used to solve the Travelling Salesman
Problem (TSP) (Dorigo and Gambardella, 1997a and 1997b; Li and Gong, 2003;
Dorigo et al., 2002): the Ant System.
At each iteration m ants compute m possible tours performing a stochastic local
search. When in node i each ant chooses the node j to move, and the link (i,j) is
added to its tour and this is repeated until the ant has completed its tour. Once the
tour is completed, each ant deposits the pheromone on pheromone trail variables
associated with the visited links in order to make these links more “desirable”. In
AS the pheromone evaporation procedure, which happens just before ants start to
deposit pheromone, is interleaved with the ants’ activity.
The internal state of each ant contains cities already visited and it is called tabu
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list. It is used to define, for each ant k, the set of cities it still has to visit.
Pheromone information is changed during problem solution to reflect the
experience acquired by ants during the optimization process. The amount of
pheromone deposited is usually proportional to the quality of the solutions they
produced: the shorter the tour generated by an ant, the greater the amount of
pheromone it deposits on the links belonging to the tour. Pheromone evaporation
is then interspersed to the ants’ search to avoid “stagnation”, i.e., the situation in
which all ants end up making the same tour.
In TSP, the ant-decision table Ai  [a ij t ] Ni of node i is obtained by the
composition of the local pheromone trail values with local heuristic values as
follows:
aij (t ) 

[ ij (t )] [ ij ] 
lN i
{[ il (t )] [ il ]  }
, j  N i
(9)
where ij(t) is the amount of pheromone trail on link (i,j) at time t, ij=1/dij is
the heuristic value used in this problem for the movement from node i to node j, Ni
is the set of neighbors of node i, and  and  are parameters controlling the
relative weight of pheromone trail and heuristic value, respectively. The
probability an ant k chooses to go from city i to city j Nk when constructing its
journey at the t-th algorithm iteration, is given by:
pijk (t ) 
aij (t )
a
il
(t )
(10)
lN ik
where N ik  N k is the set of nodes connected to node i that ant k has not visited
yet (nodes in N ik are selected from those in Ni by using ant memory Mk).
After all the ants have completed their tour, pheromone evaporation on all links
is triggered, and, after that, each ant k deposits a quantity of pheromone  ijk (t ) on
each used link:
1 / Lk (t ) if
 (t )  
if
 0
k
ij
arc(i, j )  T k (t )
arc(i, j )  T k (t )
(11)
where Tk(t) is the tour made by ant k at iteration t, and Lk(t) is the tour length. It is
clear from (11) that the value  ijk (t ) depends on ant performance: the shorter the
tour made, the greater the amount of pheromone deposited. In practice, the
addition of new pheromone by ants and pheromone evaporation are implemented
by the following rule applied to all the links:
 ij (t )  (1   ) ij (t )   ij (t )
(12)
where  ij (t )  k 1  ijk (t ) , m is the number of ants at each iteration (maintained
m
constant), and (0,1] is the pheromone trail decay coefficient. The initial
amount of pheromone  ij (0) is set to the same small positive constant value0 on
all links in order to randomly explore the network at the first iteration of the
algorithm.
8
4. Ant Colony System for traffic assignment problems
In this paper a modified version of Ant System algorithm, the Ant Colony
System (ACS), to mimic user behavior in a transportation network, is proposed, in
order to solve the different versions of the transportation problems described in the
previous sections (i.e. deterministic and stochastic user equilibrium with separable
and non-separable cost functions). In practice, ACS is a variant of ACO algorithms
that uses IB-update (Iteration Best) or BS-update (Best-So-far) rules instead of an
AS-update (Ant System) rule and additionally includes mechanisms to avoid
premature convergence (Dorigo and Blum, 2005). These characteristics allows to
achieve better results than algorithms using AS-update.
In this approach, ants are used like agents routing traffic in a transportation
network and each OD is represented by one colony because equations 3 and 5 (or 4
and 6) require a different choice probability for each path. The final traffic
assignment flow will be the result of the cooperation of ants belonging to the same
colony that explores the space of feasible flows in order to find the best
equilibrium flow. For this reason the solution cannot be a system optimum but it
is an equilibrium solution because cooperation does not involve ants of different
colonies and therefore the final result cannot take into account the performance of
other colonies.
An ant decides to travel on a path using the information left before by other
ants, then it distributes a new quantity of pheromone in function of the “goodness”
of the path. In order to effectively explore the space of solutions there must be a
relationship between pheromone and flow, because ants deposit pheromone not
flow. For each link, for instance, a quantity of flow proportional to the pheromone
present on it can be associated. Then, after pheromone distribution, the new
proportional flow assignment implies a variation of costs that leads the ants
following to have a different evaluation of paths. This implies a positive or
negative feedback that takes the system to an equilibrium state, for which a
variation in flow assignment implies a variation in pheromone distribution that reestablishes the equilibrium. All this can be viewed in the same manner as the
circular relationships of equilibrium between traffic demand, flows and costs
explained previously (Figure 1), where pheromone substitutes traffic demand
(Figure 3).
Figure 3: Equilibrium relationships between pheromone distribution, flows and costs.
9
At every iteration step each ant deposits a quantity of pheromone  ijk (t ) on
each link equal to:
1 / C k (t ) if
 (t )  
if
 0
k
ij
arc(i, j )  T k (t )
arc(i, j )  T k (t )
(13)
where Tk(t) is the tour made by ant k at iteration t, and Ck(t) is its cost. For every
od-couple there is an ant colony, with its own nest (centroid O) and a food source
(centroid D). Every ant of the same colony distributes pheromone of the same
type, so that the ants in that colony can recognize and follow only paths that lead
to the same food source.
Every ant colony is independent and its ants have to route a quantity of flow
equal to the corresponding flow demand from an origin O to a destination D. This
leads to have on link(i,j) at time t a quantity of flow equal to:
N OD
f ij (t )   d c
c 1
 ijc (t )
c
 FS
(t )
(14)
where dc is the flow demand of colony c, NOD is the number of colonies,  ijc (t )
c
(t ) is the sum of
is the quantity of pheromone of colony c on link(i,j) and  FS
pheromone quantities present on the links of the forwarding star of node i of
colony c.
Convergence for ACS is discussed in (Dorigo and Blum, 2005). Usually
convergence criterium is based on the following test:
y
max
i
t
i

 yit 1

yit 1
(15)
where y it 1 is the value of variable on link i at iteration (t-1), y it is the value of
variable on link i at iteration t and  is a threshold value. The variable considered
for the test is the link flow but it can be useful to verify convergence on link costs
and total cost as well.
4.1 User decision model for DUE and SUE
As we introduced previously an ant decides to travel on a path using the
information left before by other ants. In particular, an ant must follow paths
leading to the food source (i.e., its destination) belonging to its colony, and so it
has to pay attention only to the information (pheromone) left by the other ants of
the colony. According to this vision we have a colony for each origin destination
pair.
The ant-decision table Aic  [aijc t ] Ni of node i and colony c is obtained by
the composition of the local pheromone trail values with a heuristic weight of the
minimum path as follows:
a (t ) 
c
ij
[ ijc (t )  wij (t )]
[
lN i
c
il
(t )  wil (t )]
, j  N i
(16)
where  ijc (t ) is the amount of pheromone trail of colony c on link (i,j) at time t
and wcij(t) is a weight value of link (i,j), set as follows:
10
w if
w c ij (t )  
0
arc (i, j ) belong to the shortest
otherwise
path
(17)
As described in the previous section, the pheromone trail for the specific colony is
updated by  ijc (t ) according to the inverse cost of the path found.
The use of w introduces an heuristic improvement of the algorithm to speed up
the convergence process through the calculation of the minimum path for each
colony by Dijkstra algorithm. To take into account the cost circular relationship
between cost and flow, w needs to be iteratively updated on the value of 1/C(t) of
the shortest path; this implies that finding the optimal values of the weights wcij(t)
requires the computation of the minimum path for every od-couple. As a side
effect, this heuristic assures that an ant can decide to choose links that belong to
the minimum path, even if there is no more pheromone on them. It can be seen as a
probability biasing which always gives an ant a chance of finding a good solution.
To take into account the deterministic or stochastic nature of user decision we
introduced a further improvement in the ant decision table. The value of  ijc (t )
used in equation 16 may be the exact amount of pheromone released by ants or the
perceived one extracted according to a certain. This turns out to model a
deterministic user model in the case the exact pheromone value is used (i.e., the
user choice is the one that deterministically select the minimum cost), as well as a
stochastic one when the pheromone trail is treated as a random variable.
It is worth stressing that the algorithm does not make any assumption on the
distribution of this random variable so we can apply a normal distribution to
represent a symmetric uncertainty on the actual estimate of link cost, or a
lognormal distribution modeling to represent an asymmetric optimistic
uncertainty.
5. Experimental Validation
Tests have been conducted to verify the actual properties of ACS in different
scenarios with both deterministic and stochastic user equilibrium. Four networks
with different structures and OD demand are tested. In Table 1 the main features,
number of links, number of nodes and OD centroids of the networks are reported.
Link type and number of parameters refer to the cost function and parameters used
for that specific network: specifically cost function type 2 is equation 2.3.4 in
(Cascetta, 1998) and type 3 is the BPR function (Bureau of Public Roads).
Two networks are simulated in laboratory:
• the trial network (Figure 4) is used as a preliminary test bench for the
research and as in the case of the Naples network (described in the next
paragraphs), a specific project produced and studied the demand for each
day in the course of a year and we used the demand of a typical weekday
and that of 8 a.m.;
• the non separable cost trial network (Figure 5) is created in order to verify
the capacity of ACS to solve the problem of non-separable links, therefore
links and demand are created with this purpose in mind.
The other two networks are real networks so as the demand used for assignment:
• the network of Milan represents the so called “area Maggi” (Figure 6) which
is quite a large area in the southern part of the city and the demand is that of
the morning peak hour; it is characterized by a limited number of paths and
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has a high number of ODs (1283) ;
• the network of Naples (Figure 7) represents the very large area of the extraurban network of Naples (1363 links); it was the object of the PRIN2002
project (Bifulco, 2005) which produced and studied the demand for each day
in the course of a year; also in these experiments we used the demand of
demand of a typical weekday and that of 8 a.m.
Trial (Rete 1)
Non Separable Costs
Maggi (Milan)
Extra-urban (Naples)
Link
Type
2
3
3
2
No.of Param.
7
8
8
7
Links
12
28
373
1363
Nodes
6
12
189
994
OD
4
8
1283
1483
Table 1: The characteristics of the networks used in experimental validation.
Figure 4: Layout of the trial network.
Figure 5: Layout of the trial network with
non-separable costs link functions.
Figure 6: Layout of the "area Maggi" (Milan) Figure 7: Layout of the Naples extra-urban
network.
network.
The role of pheromone and its evaporation is investigated by varying the
pheromone trail decay coefficient, (eq. 12), in the range (0.005, 0.5), that is from
a very low to a very high level of decay. The max number of iterations is 500 for
all experiments though this is excessively high and useless in many cases.
Comparisons of ACS-DUE results (except for the non-separable costs trial
network) with those obtained by a Frank-Wolfe (FW) algorithm are proposed to
verify performance. Ten simulations for each SUE scenario are carried on and in
the following tables the average solutions are reported.
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SUE are evaluated varying standard deviation of the distribution from 0.01 to
100 on the pheromone trail corresponding respectively to a standard deviation
from 100 to 0.01 [s.] on the perceived path cost; the subscript N indicates that a
normal distribution is applied while the subscript L indicates a lognormal
distribution.
Table 2 and Table 3 report DUE and SUE assignment results, respectively.
They are obtained assuming that a simulation is convergent when the 90% of cases
is convergent with a threshold of 1% (according to eq. 15) for Area Maggi and
Naples networks and with at threshold of 5% for the trial and non separable cost
networks (marked by *). The acronym N.C. stays for “not calculated” and means
that more iterations than 500 are needed for convergence. These tables report
results according to the value of and the number of iterations for the three
variables, link costs, flow and total costs. For each variable two values are given:
the “first” column gives the iteration at which convergence is reached for the first
time; the “last” column gives the last iteration at which convergence does not
occur. Therefore the “first” column gives an idea of speed of convergence while
the “last” one gives the number of iterations minus one needed for convergence.
Simulations for DUE assignments are shown in Table 2. Comparisons with the
FW solution (except for the non separable cost network to which we cannot apply
this algorithm) are quite satisfactory: final total costs are very close (exactly the
same for Naples network) so as flows and costs on links. Computing time is
indeed much higher than for FW algorithm though it is very short (Table 4).
 = 0.005
 = 0.01
 = 0.05
Trial *
link costs
flow
Total costs
first last first last first last
6
14
8
17
3
2
6
14
8
22
3
2
6 421
11 492
3
2
 = 0.005
 = 0.01
 = 0.05
Non Separable costs*
link costs
flow
Total costs
first last first last first last
20
77
16
28
5
4
21 496
18
30
5
4
0 N.C.
86 498
5
4
 = 0.005
 = 0.01
 = 0.05
link costs
first last
30
43
37
79
0 N.C.
area Maggi
flow
Total costs
first last first last
14
20
2
3
21
26
2
3
52 N.C.
5
4
 = 0.005
 = 0.01
 = 0.05
Naples
link costs
flow
Total costs
first last first last first last
2
1
8
7
2
1
2
1
8
7
2
1
2
1
6
5
3
2
DUE simulations
Table 2: Results for DUE assignments.
13
SUE simulations
N
L
N
L
N
L
0.01
0.1
1
10
100
 = 0.005
0.01
0.1
1
10
100
0.01
0.1
1
10
100
 = 0.01
0.01
0.1
1
10
100
0.01
0.1
1
10
100
 = 0.05
0.01
0.1
1
10
100
Trial *
link costs
flow
Total costs
first last first last first last
9
20
9
12
2
3
8
13
11
16
3
2
6
13
9
12
2
1
6
19
6
19
2
1
9
17
17
29
2
4
9
20
9
12
2
3
9
24
6
14
6
5
7
13
7
9
3
2
9
15
11
10
3
2
10
30
3
32
3
2
9
20
9
12
2
3
8
13
11
16
3
2
6
13
9
12
2
1
6
11
9
24
2
1
11
16
19
25
2
4
9
20
9
12
2
3
9
24
6
14
6
5
7
9
7
12
3
4
9
15
11
14
3
2
10
30
3
32
3
2
10 411
11 299
2
3
11 391
12 391
3
2
4 437
4 477
2
1
6 479
9 434
2
1
9 257
17 N.C.
2
4
10 442
10 320
2
4
7 493
8 383
5
4
7 491
7 N.C.
3
4
16 492
13 352
3
2
10 475
3 493
3
4
Non Separable costs*
link costs
flow
Total costs
first last first last first last
19
67
18
21
5
4
19
72
17
21
5
4
20
73
17
25
5
4
20
67
17
35
5
4
18
29
14
29
5
4
19
67
18
21
5
4
19
63
16
21
5
4
20
74
15
17
5
4
18
34
10
19
4
3
13
12
9
11
2
1
19 N.C.
18
33
5
4
21 490
17
30
5
4
22 492
19
37
5
4
13 471
13
36
5
4
15 179
14
30
5
4
19 486
18
33
5
4
19 483
18
33
5
4
19 496
16
30
5
4
20 411
18
17
4
3
11
20
9
17
2
1
0 N.C.
78 N.C.
5
4
0 N.C.
64 N.C.
5
4
0 N.C.
60 N.C.
5
4
0 N.C.
42 N.C.
5
4
66 N.C.
17 N.C.
6
5
0 N.C.
84 N.C.
5
4
0 N.C.
78 N.C.
5
4
0 N.C.
73 N.C.
5
4
20 N.C.
20 N.C.
4
3
15 498
9 496
2
1
area Maggi
link costs
flow
Total costs
first last first last first last
32
53
19
24
5
4
29
77
15
26
2
3
34
51
21
20
4
3
19
62
14
22
6
12
23 493
23 369
10 369
34
51
11
20
2
3
34
57
15
30
4
3
31
51
21
20
4
3
25 103
20
65
10
51
33 479
23
87
13 104
39
84
21
26
4
3
31
60
18
43
4
3
32
81
19
26
4
3
33 113
17
27
6
16
19 472
19 411
14 243
33
70
15
22
2
3
29
70
21
25
4
3
36
57
20
25
4
3
27 312
18 101
8
37
23 493
23 478
12 196
0 N.C.
48 N.C.
4
3
0 N.C.
39 495
4
3
0 N.C.
41 498
5
4
71 N.C.
23 N.C.
7
8
29 497
29 490
9 443
0 N.C.
74 N.C.
2
3
0 N.C.
47 495
4
3
0 N.C.
49 497
4
3
47 497
27 436
9
18
54 N.C.
40 N.C.
12 273
 = 0.005
 = 0.05
 = 0.5
Naples
link costs
flow
Total costs
first last first last first last
2
1
9
8
2
1
2
1
8
7
2
1
2
1
5
4
2
1
2
1
4
3
3
2
2
1
12
26
10
13
2
1
8
9
2
1
2
1
8
7
2
1
2
1
8
7
2
1
4
3
34
40
4
8
6
5
49
67
4
9
2
1
8
9
2
1
2
1
8
7
2
1
2
1
5
4
2
1
2
1
4
3
3
2
2
1
12
36
6
36
2
1
9
8
2
1
2
1
9
8
3
2
2
1
8
7
2
1
3
2
79 N.C.
3
2
6
5
0 N.C.
7
6
2
1
6
5
2
1
2
1
6
5
2
1
2
1
4
3
2
1
2
1
5 423
3
2
2
1
14 495
10 308
2
1
6
5
3
2
2
1
6
5
3
2
2
1
7
6
3
2
4
3
0 N.C.
7 244
20 N.C.
0 N.C.
8 498
Table 3: Results for SUE assignments.
14
Many oscillations are observed both on small networks (trial and non-separable
costs) and “Area Maggi” network. A very small value of  is necessary to
achieve better performance. This is probably due to the fact that flow is near
capacity and some similar paths for the same OD exist: this creates conditions that
a small assignment of flow on these paths can change significantly the cost on
links starting oscillations. The “Area Maggi” network is sensitive to though to a
lesser extent: the many ODs constrained on a limited number of links contribute to
slow down convergence.
Naples network is instead almost insensitive to  since results are very similar
each other. It is due to the particular structure of the network where comparable
paths for each OD are not very numerous due to the considerable length of links.
Simulations for SUE assignments are shown in Table 3. There is no relevant
difference among ten simulations and the final number of iterations has a very low
variance. The role of  is very similar to that of DUE case. The best results are
obtained with a very low value of pheromone, 
(trial and non-separable costs) and “Area Maggi” network. Lognormal distribution
shows a little better results than the Normal one for the same value of . The
“best” standard deviation is the highest in the non-separable cost network probably
due to the necessity of exploring much more different solutions because of the
indirect relationship between link cost and link flow.
Naples network is still almost insensitive to ; for these simulations a little
better results are achieved by using a Normal distribution and generally low values
of standard deviation performs better. This can be explained by considering that,
due to the particular structure of the network, it is not necessary to investigate
many other paths than the present one.
In Table 4 computing time performance for the minimum number of iterations
according to the variable flow is reported for each scenario. Convergence time is
very low, below the second, for the trial and non-separable cost network; only
“Area Maggi” network requires one dozen of seconds for convergence.
Minimum number of iterations
DUE
 = 0.005
 = 0.005
 = 0.005
 = 0.05
Trial *
Non Separable costs*
area Maggi
Naples
SUE
Trial *
Non Separable costs*
area Maggi
Naples
L=10
L=100
L=0.01
N=1
 = 0.005
 = 0.005
 = 0.005
 = 0.5
link cost
flow
Total cost
14
78
44
2
18
29
21
6
3
5
4
3
link cost
flow
Total cost
16
13
52
2
11
12
21
4
3
2
4
2
Time to
convergence [s]
<10-3
0.01
9.18
1.71
-3
<10
0.01
11.25
1.18
Table 4: Computing time for the minimum number of iterations.
15
6. Conclusions
ACS is proposed in order to solve the DUE and SUE assignment problems. The
modified version of this heuristic is suitable for application in almost all real cases
due to its versatility without assuming simplifying hypotheses. The solution found
by ACS does not depend on the shape of the objective function and therefore also
the particular cases of non-separable cost link functions or multi-class demand can
be tackled easy and successfully.
Applications to real networks show a computation time that is short enough
(though not comparable to the speed of the Frank-Wolfe algorithm in the case of
DUE) also in complex networks and it can be improved through a parallel
programming that is easy enough to apply thanks to the similar nature of ACS,
intrinsically parallel. The impact of pheromone decay is analyzed and we can
suggest that its effects depend strongly on network structure and cost functions.
Generally we expect that by increasing the value of  oscillations increase but this
holds true only if the feasible set of paths is wide. Usually a few iterations are
sufficient for the algorithm to converge (also for complex networks) and probably
a better tuning of parameters could reduce it even more. The impact of standard
deviation in cost perception distribution is investigated as well and it shows how
the stochastic nature of ACS can solve the SUE faster than DUE problems.
More research is necessary to investigate convergence mechanisms especially
when the existence and uniqueness of convergence cannot be theoretically
demonstrated and when decreases with the number of iterations as suggested by
some authors (e.g. Li and Gong, 2003). Moreover, other models of cost perception,
for example with a variance distribution function of its mean, could be tested.
Acknowledgements
The MIUR PRIN02 Project “Transportation systems in the information society”
partially supported this work. Thanks are due to Prof. Bifulco for his profitable
suggestions. Thanks are also due to the City Municipality of Milan for having
provided data for the “Area Maggi” network.
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