INTRODUCTION
Generally, a multimodal assignment problem has to be adopted when it is necessary to estimate user flows on different transportation systems under two assumptions: network costs affect not only path choice but also modal split and at least two transportation modes are congested (i.e., their performance depends on network flows). In particular, this problem can be considered an equilibrium problem with elastic demand and can therefore be formulated as a fixedpoint problem.
Elastic assignment problems can usually be classified into deterministic or
stochastic equilibrium models according to the assumption on the random residual
of the path choice model. In both cases, Cantarella (1997)
and Cascetta (2001) proved that, with suitable assumptions
on cost functions, demand functions and path choice models, it is possible to
state the existence and uniqueness of the equilibrium solution. Moreover, Cantarella
(1997) proposed an extension of Blum’s theorem (Blum,
1954) where he showed that under suitable assumptions on cost functions,
demand functions and path choice models, it is possible to state the convergence
of solution algorithms for solving the fixedpoint problem.
The aim of this study is to propose an assignment model where the elasticity of demand is associated to a mode choice model, considering the phases of trip generation and distribution as rigid. Moreover, in order to apply assumptions suitable in urban contexts, we hypothesise that travel times of transit systems depend on flows of private cars and freight vehicles in the case of shared lanes and are constant (with respect to the flows) in the case of exclusive lanes.
The proposed model introduces some theoretical complications because it does
not satisfy some assumptions proposed by Sheffi (1985),
Oppenheim (1995) and Cantarella (1997)
for stating the uniqueness of the equilibrium solution in the case of elastic
demand. This problem was analysed and solved by D'Acierno
et al. (2002), extending cost function assumptions in the case of
a bimodal (road and transit) context. Hence, in this paper, we extend the previous
model to the case in which there is a third congested system, that is the freight
distribution system.
The proposed model is solved by adopting the three different approaches (external, internal and hypernetwork) used to solve elastic demand assignment problems and analysing effects of freight distribution on (road and transit) user choices.
The study is structured as follows: the proposed fixedpoint problem is analysed
in the next section; then extensions of solution algorithms proposed by Cascetta
(2001) and D'Acierno et al. (2002) for managing
elastic demand problems are described. Study describes the application of the
algorithms in the case of a real dimension network and compares them in terms
of calculation times and reliability of solutions. Finally, appendix A proposes
in detail the proof of theoretical properties of the proposed model.
PROPOSED MODEL FORMULATION
The assignment problem proposed in this paper for simulating user flows on an urban network where three transportation systems are considered (i.e., road, transit and freight systems) is based on the following assumptions:
• 
Demand is elastic at mode choice level for the road and transit
systems (that is users can choose whether to travel by road or on the transit
system) 
• 
Demand is rigid for the freight system (that is network performances
affect only path choice and not travel demand or freight quantities) 
• 
The path choice behaviour of the transit user is pretrip/en
route (hyperpath approach proposed by Nguyen et
al. (1998). Indeed, users of the transit systems are unable to determine
exactly which is the first (transit) vehicle belonging to their choice set
that will arrive at a certain stop/station. Therefore, under this hypothesis
it is assumed that the users do not choose a predetermined path but rather
a travel strategy called hyperpath 
• 
Road, transit and freight systems are affected reciprocally
in terms of travel times in the case of shared lanes 
• 
Transit travel times are constant (with respect to road and
freight vehicle flows) in the case of exclusive lanes. Indeed, in this case,
travel times depend only on transit vehicle flows and dwelling times at
stops/stations 
• 
The assignment model is multiclass with undifferentiated
congestion (that is the cost function of each user category can be expressed
as a linear transformation of a cost function common to all the classes
and dependent on total link flows) 
• 
The average occupancy index for the road system is constant
for all user categories 
• 
Capacity constraints are neglected in all transportation systems 
With the above hypotheses, the user flows on the network can be obtained by
means of the interaction between three kinds of models: a supply model, that
simulates transport performances depending on user flows, a demand model, that
imitates user choice influenced by performances of the three considered transportation
systems and a flow propagation model, that describes how path flows load links
generating link flows.
The supply model can be formulated as:
where: C^{i}_{m} is the vector of path generalised costs for mode m associated to user category i; m is the generic transportation system, where m = r for the road system, m = t for the transit system and m = f for the freight system; β^{i}_{m} is a nonnegative coefficient for mode m associated to user category i, which allows the multiclass approach to be considered; A^{i}_{m} is the linkpath incidence matrix for mode m associated to user category i; c_{m} is the vector of link generalised costs for mode m. f_{r} is the link flow vector of the road system (m = r); φ_{t} is the frequency vector of the transit system, which is equal to the vehicle flow vector associated to the transit system (m = t); f_{f} is the link flow vector of the freight system (m = f); C_{m}^{i,NA} is the vector of nonadditive (NA) path costs for mode m associated to user category i, which expresses costs that depend only on paths of mode m (such as road tolls at motorway entrance/exit points).
Likewise, the demand model, based on the assumption that users are rational decisionmakers maximising the utility associated to their choices, can be formulated as:
where: F^{i}_{m} is the path flow vector for mode m associated to user category i; P^{i}_{k,m} is the path choice matrix for mode m associated to user category i; d^{i}_{m} is the travel demand vector for mode m associated to user category i.
Moreover, the flow propagation model is based on the assumption of intraperiod (within day) and interperiod (day to day) stationarity (i.e., it is assumed that demand and supply remain constant over a period of time long enough to allow the system to reach a stationary or steadystate condition). This hypothesis on temporal dimensions allows the flow propagation model to be formulated as an instantaneous propagation model (i.e., path flows are instantaneously propagated on links), that is:
where: f^{i}_{m} is the link flow vector for mode m associated to user category i; B^{i}_{m} is the linkpath incidence matrix for mode m associated to user category i, which is equal to matrix A^{i}_{m} only in the case of road and freight system.
Indeed, the assumption of the hyperpath approach requires the use of different linkpath matrices in Eq. 1 and 3 for the transit system.
Finally, it is worth noting that the link flow vector for mode m, indicated as f_{m}, can be expressed as:
where: w^{i}_{m} is the homogenisation coefficient for mode m associated to user category i, which takes into account the influence of user categories on link performances.
Combining Eq. 1, 2 and 3
with Eq. 4, the demandsupply interaction model, known as
the assignment model, is obtained as a fixedpoint model (Cantarella,
1997) and formulated generically as:
In particular, taking into account the assumptions of the proposed model, the fixed model can be characterised for each mode by means of the following relations:
In particular, the proposed assignment problem can be formulated via three subproblems: a fixedpoint problem with elastic demand for the road system, whose solution is vector f_{r}^{*}; a stochastic network loading for the transit system (due to the fact that transit user flows do not affect network performances), whose solution is vector f_{t}^ and a fixedpoint problem with rigid demand for the freight system (due to the fact that freight demand is not affected by network performances), whose solution is vector f_{f}*.
However, the multimodal aspects are expressed by:
• 
The dependency of link cost vectors (c_{m}) on equilibrium
vehicle flow vectors of all transportation systems (f_{r}*, φ_{t}
and f_{f}*) 
• 
The dependency of each demand vector of congested systems
(road and transit systems), indicated respectively as d^{i}_{r}
and d^{i}_{t}, on all path cost vectors of these systems
(C^{i}_{r} and C^{i}_{t}) and therefore
on equilibrium vehicle flow vectors of all transportation systems (f_{r}^{*},
φ_{t} and f_{f}^{*}) 
These considerations imply that, in order to solve Eq. 68,
it is necessary to solve Eq. 6 and 8 jointly
and then to implement (Eq. 4).
Extending the analysis developed by D'Acierno et al.
(2002), as shown in appendix A, we may state that the fixedpoint problem
described by Eq. 68 has at least
one solution if:
• 
Path choice functions of the road and freight systems (i.e.,
matrices P^{i}_{k, r} and P^{i}_{k, f})
are continuous 
• 
Travel demand functions of the road and the transit systems
(i.e., vector d^{i}_{r} and d^{i}_{t}) are
continuous and upper bounded 
• 
Cost functions of all transportation systems (i.e., vectors
c_{r}, c_{t} and c_{f}) are continuous 
• 
Each OD pair is connected 
The first three assumptions are generally satisfied by almost all functions proposed in the literature, whereas the last hypothesis is related to the network framework analysed and is generally respected.
Likewise, it is possible to extend the considerations of D'Acierno
et al. (2002) in order to state the uniqueness of the equilibrium
solution. In particular, as shown in appendix A, we can show that the fixedpoint
problem described by Eq. 68 has
at most one solution if:
• 
Path choice functions of all transportation systems (i.e.,
matrices P^{i}_{k, r}, P^{i}_{k, t} and
P^{i}_{k, f}) are additive, nonincreasing with respect
to generalised path costs (i.e., vectors C^{i}_{r}, C^{i}_{r}
and C^{i}_{f}), continuous with continuous first partial
derivatives 
• 
Demand functions of the road system (i.e., vector d^{i}_{r})
are nonnegative, upper bounded and nonincreasing with respect to generalised
path costs (i.e., vectors C^{i}_{r}, C^{i}_{t}
and C^{i}_{f}) 
• 
Cost functions of the road and freight systems (i.e., vectors
c_{r} and c_{f}) are increasing with respect to link flows
(i.e., f_{r} and f_{f}) 
The first and second assumptions are satisfied by almost all functions proposed
in the literature, whereas the second hypothesis is generally not satisfied.
However, D'Acierno et al. (2002) showed that
if one condition on a cost function is verified, also the third condition is
verified. In particular, the extension of the cost function condition in the
case of the proposed model can be formulated as:
where: c^{l}_{r} is the lth component of vector c_{r},; c^{l}_{t} is the lth component of vector c_{t}; c^{l}_{f} is the lth component of vector c_{f}; f^{l}_{r} and f^{h}_{r}are respectively the lth and hth component of vector f_{r}, with l ≠ h; φ^{l}_{t} and φ^{h}_{t} are respectively the lth and hth component of vector φ_{t}, with l ≠ h; f^{l}_{f} and f^{h}_{f} are respectively the lth and hth component of vector f_{f}, with l ≠ h; S_{fr} is the feasibility set of vector f_{r}; S_{φt} is the feasibility set of vector φ_{t}; S_{ff} is the feasibility set of vector f_{f}.
The cost function condition described in Eq. 9 can be expressed by means of the following Fig. 1 and 2.
In particular, in Fig. 1 when road flows (or equivalently bus or freight vehicle flows) are low, bus speeds are lower than freight vehicle speeds (bus travel times are higher than those of freight vehicles) and freight vehicle speeds are lower than car speeds (freight travel times are higher than those of cars). Moreover, when road flows (or equivalently bus or freight vehicle flows) increase, the speeds of the three systems tend to converge.
Likewise, Fig. 2 represents the case when the bus travel time on a link is equal to the freight travel time or road travel time plus a constant term that expresses the dwelling time at bus stops.

Fig. 1: 
First case of cost functions 

Fig. 2: 
Second case of cost functions 
However, in urban contexts it is possible to have the first diagram, the second or a linear combination of both. Since derivative operators are linear operators, that is:
the proposed condition, described by Eq. 9, could be always satisfied.
SOLUTION ALGORITHMS
The effectiveness of a model is related to the possibility of developing algorithms
that provide solutions in reasonable computing times. Therefore we developed
four solution algorithms for solving the proposed multimodal assignment problem.
In particular, we provided an extension of algorithms proposed by D'Acierno
et al. (2002) and based on the three algorithmic approaches (external,
internal and hypernetwork) for the solution of elastic demand assignment problems
(as shown by Cascetta, 2001).
Based on the extension of
Blum (1954) theorem, convergence
of internal approach algorithms was stated under some quite general assumptions
(
Cantarella, 1997;
Cascetta, 2001).
Likewise,
Sheffi (1985) and
Cascetta (2001)
showed that hypernetwork approaches are theoretically equivalent to internal
ones and hence the general assumptions of
Cantarella (1997)
and
Cascetta (2001) allow their convergence to be stated.
By contrast, the convergence of external approach algorithms was not proved in
the literature. In any event, if it is possible to state that the solution exists
and is unique and the external approach algorithm converges, it converges just
to the equilibrium solution.
Solution algorithms for all proposed approaches are based on an MSAFA framework
(Cantarella, 1997); inside this framework the stochastic
network loading algorithm adopted is that proposed by Dial
(1971) for road and freight systems and the modified version by Nguyen
et al. (1998) for the transit system. These algorithms generate implicitly
paths (a necessary requirement on real dimension networks) and the immediate
calculation of the EMPU (Expected Maximum Perceived Utility) variables, which
allow us to calculate the demand component in the case of elastic demand (i.e.,
in the case of the road and transit systems).
The algorithm proposed by Dial (1971), as well as its
modified version proposed by Nguyen et al. (1998),
consists of three phases:
• 
Phase 1: Dialefficient path generation 
• 
Phase 2: Weight calculation (by which also the EMPU
variables and hence travel demand vectors may be calculated) 
• 
Phase 3: Network loading 
In all three algorithmic approaches adopted, phase 1 is performed only once, since the assumption of additive path (hyperpath) choice models means that the choice set is independent of the values of generalised costs. Moreover, we applied the property of independence of network costs with regard to transit user flows. Therefore, the loading of the transit network can be performed only once, after the equilibrium flows on road and freight systems have been calculated.
Once a starting modal split has been fixed, the external approach algorithm
calculates the EMPU variables of the road system by an MSAFA algorithm (phase
2 and 3 cycles) applied on road and freight systems and the EMPU variables of
the transit system by means of only one implementation of phase 2 applied on
the transit system. These EMPU variables are used to calculate a new modal split,
thereby recalculating the costs on the networks and the new values of the EMPU
variables. When a stopping criterion on modal split is satisfied, the algorithm
terminates and the corresponding flows of the road and the freight systems are
the multimodal equilibrium ones, that is the solution of Eq. 6
and 8, while a last transit network loading (phase 3) has
to be performed to obtain the transit flows, that is the solution of Eq.
7.
The first version of the externalapproach algorithm differs from the second in the starting flow values adopted in the MSAFA algorithms to calculate the rigid demand flows on road and freight networks. In the first case the starting flows are equal to zero; in the second, instead, they are equal to the flows obtained in the previous assignments.
Once a null starting flow vector has been fixed, the internal approach algorithm
calculates the EMPU variables of both road and transit systems, applying phase
2 to all three (road, transit and freight) systems. The algorithm calculates
the stochastic network loading flows on the road and the freight networks (phase
3) after the calculation of the modal split. With the obtained flows, it recalculates
the EMPU variables until convergence on road and freight flows is reached, that
is the solution of Eq. 6 and 8 are obtained.
Also in this approach, it is necessary to perform a last transit network stochastic
loading (phase 3) in order to obtain the (equilibrium) transit flows, that is
the solution of Eq. 7.
The hypernetwork approach algorithm executes a rigid demand assignment model
on a multimodal supply network model (hypernetwork) that allows the simulation
of the mode choice on the network by using specific dummy links (modal diversion
links). Therefore this algorithm consists in the use of a rigid demand MSAFA
algorithm (phase 2 and phase 3 cycles). This particular network framework is an
extension of the hypernetwork model proposed by
D'Acierno
et al. (2006). A scheme of the adopted hypernetwork supply model is
reported in
Fig. 3.
SOME APPLICATIONS ON A REAL DIMENSION NETWORK
The proposed algorithms were tested on a real dimension network, whose features are reported in Table 1. Comparing the transit system and the road networks, it may be noted that the link ratio is about 7:1. This explains the huge difference (about 36 times) between the computing times (reported in Table 2) of network loading for the road and freight system (phases 2 and 3 in the case of these transportation systems) and demand model performance, which also incorporates the weight calculation of the transit system (phase 2).

Fig. 3: 
The hypernetwork supply model 
Table 1: 
Test network features 

A comparison among the algorithms is reported in Table 3. An important result is that the three algorithms converge to the same solution. Indeed, the maximum percentage error never exceeds 5.00% which represents the threshold used in the stopping criterion of the algorithms. However, though it was not possible to demonstrate the convergence of all solution algorithms, we numerically proved the convergence of the proposed algorithms.
Comparison between the two versions of the external algorithm shows that the second version obviously requires less stochastic loading on road and freight systems. In terms of computing times, the internal approach algorithm requires much lower times than the external ones (about 3 times as long), which can be explained by the different number of iterations required to converge (i.e., the number of required network loadings indicated in Table 3 is lower in the case of the internal approach).
Moreover, the internal and hypernetwork approaches coincide both in terms
of iterations and results, but in the hypernetwork approach the computation
times are slightly higher (about 1.20 times), since the dummy links (i.e., modal
links shown in Fig. 3 increase the number of total links in
the network to be examined.
Table 2: 
Computing times (Intel Core 2 Quad Q6600  2.40 GHz) 

Table 3: 
Comparison among algorithm approaches 

These dummy links, as described in the previous section, allow the implicit
simulation of the mode choice model and therefore the demand model application
is not performed in the hypernetwork approach (as shown in Table
2 and 3). Indeed, as described in the previous section,
the hypernetwork approach simulates implicitly the mode choice model (i.e.,
the demand model) by means of dummy links.
Finally, in order to verify the usefulness of a multimodal approach in urban
freight distribution two experiments were performed. The first aimed to analyse
the presence of freight vehicles on trip choices of conventional (i.e. road
and transit) users. In particular, the simulation obtained considering the proposed
threesystem assignment model was compared with those obtained by neglecting
freight vehicles. Our results showed that the maximum difference in user flows
is 60.00% in the case of the road system and 4.44% in the case of the transit
system; likewise, the average difference in user flows is 6.42% in the case
of the road system and 0.28% in the case of the transit system. These results
show the usefulness of simulating freight and road systems jointly to obtain
a better estimation of road flows; the effects on transit user flows are, instead,
less important.
The second trial was based on the analysis of the influence of road and transit system vehicles on freight vehicle route choices. The simulation obtained by considering the proposed threesystem assignment model was compared with a monomodal assignment model with rigid demand applied only on the freight system. In this case, results showed that the difference in freight vehicle flows is 169.39% in the maximum case and 4.94% in the average case.
Thus the two experiments showed that neglecting freight vehicle presence can substantially modify link flows. Likewise, neglecting the multimodal aspects in the urban context, that is the presence of a road and transit system, can considerably change the minimum cost path used by freight vehicles.
CONCLUSIONS AND RESEARCH PROSPECTS
The proposed multimodal assignment problem allows transportation systems in the urban context to be simulated by considering the effects of private cars and freight vehicles on congestion. Although, it is not possible to state the convergence theoretically, the developed algorithms converge to the same solution in reasonable computation times. Moreover, the comparison among the different approaches showed that, at least in the case of the analysed network, the internalcycle algorithms are more efficient in terms of computing times.
An important result is that neglecting freight vehicles generates average differences in road and transit user flows lower than 5.00%, which is the threshold used in the stopping criterion of the algorithms, although in terms of maximum difference these values can be considerable only in the case of the road system (which may exceed 50.00%). By contrast, simulating the freight system without taking account of congestion generated by road and transit systems (i.e. considering only the freight system and neglecting any other transportation system) can generate maximum differences in freight vehicle flows higher than 169%, even if in terms of averages these values are lower than the stopping criterion threshold (set at 5.00%).
Future research should address the use of the proposed model as a simulation tool in network design problems in order to account for all transportation components in urban contexts.
APPENDIX A
In this appendix, we propose an extension of the proofs proposed by D'Acierno
et al. (2002) for stating the existence and the uniqueness of the
fixedpoint solution described by Eq. 68.
The existence of the fixedpoint solution: The fixedpoint problem (described
by Eq. 68) has at least one solution
if:
• 
Path choice functions of the road and freight systems (i.e.,
matrices P^{i}_{k,r} and P^{i}_{k,f}) are
continuous 
• 
Travel demand functions of the road and the transit systems
(i.e., vector d^{i}_{r} and d^{i}_{t}) are
continuous and upper bounded 
• 
Cost functions of all transportation systems (i.e., vectors
c_{r}, c_{t} and c_{f}) are continuous 
• 
Each OD pair is connected 
Proof: In fact, vectors f_{r}*, f_{t}^ and f_{f}*
are the fixed points of the compound function described by the abovementioned
equations, which under the above assumptions are continuous functions defined
over a nonempty, compact and convex set (if each OD pair is connected). Moreover,
these equations assume values only in the definition set; thus, all assumptions
of Brouwer’s theorem on the existence of fixed points are satisfied (Cantarella,
1997; Cascetta, 2001).
The uniqueness of the fixedpoint solution: The fixed point problem
(described by Eq. 68) has at most
one solution if:
• 
Path choice functions of all transportation systems (i.e.
matrices P^{i}_{k,r}, P^{i}_{k,t} and P^{i}_{k,f})
are additive, nonincreasing with respect to generalised path costs (i.e.
vectors C^{i}_{r}, C^{i}_{t} and C^{i}_{f}),
continuous with continuous first partial derivatives 
• 
Demand functions of the road system (i.e. vector d^{i}_{r})
are nonnegative, upper bounded and nonincreasing with respect to generalised
path costs (i.e., vectors C^{i}_{r}, C^{i}_{t}
and C^{i}_{f}) 
• 
Cost functions of the road and freight systems (i.e., vectors
c_{r} and c_{f}) are increasing with respect to link flows
(i.e. f_{r} and f_{f}) 
Proof: If path choice models are defined by nonincreasing functions
with respect to generalised path costs and demand functions are nonnegative,
upper bounded and nonincreasing with respect to generalised path costs, the
elastic demand stochastic uncongested network assignment function (obtained
by combining Eq. 24) is nonincreasing
monotone with respect to link costs, that is:
with:
where S_{c} is the feasibility set of vector c.
The proof is then completed by reductio ad absurdum. If two different equilibrium
link flow vectors existed, f_{1}* ≠f_{2}*εS_{f}
(with S_{f} = [Sf_{r}, S_{φt}, Sf_{f}]),
assuming c_{1}* = c(f_{1}*) and c_{2}* = c(f_{1}*),
the equilibrium definition f_{1}* = f(c_{1}*) and f_{2}*
= f(c_{2}*) and the monotonicity of the stochastic uncongested network
assignment function with c' = c_{1}* and c" = c_{2}* yields:
Since, vector φ_{t} is fixed, from the monotonicity of the road
and freight cost functions (i.e., vectors c_{r} and c_{f}),
with f' = f_{1}*≠f" = f_{2}*, it follows that:
Thus, there is a contradiction between the monotonicity of the cost functions and that of the stochastic uncongested network assignment function.