Where,
And: 
T_{ij} 
= 
Total number of trips from origin zone i to destination zone
j 
t_{ij} 
= 
Travel time from zone i to zone j 
O_{i} 
= 
Fixed and known total number of trips originating at zone i 
D_{j} 
= 
Fixed and known number of trips destined for zone j and 
q 
= 
Model parameter to be estimated 
The trip distribution model in Eq. 1 is a standard doubly constrained gravity model that can be used for captive trips.
Trip assignment: The trip assignment model used in this study adopts
Wardrop’s UE principle^{[12]}. This principle states that, at equilibrium,
the average travel cost on all used paths connecting any given ij pair will
be equal and the average travel cost will be less than or equal to the average
travel cost on any of the unused paths^{[6]}. This study assumes that
the UE conditions would hold over the network. Thus, the mathematical expression
equivalent to the UE conditions can be stated as follows:
Where, 

= 
Total person trips from i to j using path p 

= 
Average travel cost from origin i to destination j using path p and 

= 
Minimum (or equilibrium) travel cost from i to j 
Equation 4 and 5 state that if the (person)
trip flow from i to j by path p be positive, then the travel cost for that path
equals travel costs of all other path combinations chosen from i to j and that
if the trip flow on a path combination from i to j is zero, its travel cost
is no less than the cost on any chosen path combination. For simplicity of presentation,
it is assumed that the auto occupancy is 1.
When Eq. 4 and 5 are combined with the
flow conservation conditions, one can show that:
And the flow non negativity constraints:
These equations constitute a quantitative statement of Wardrop’s UE principle.
The equilibrium conditions can be interpreted as the KKT conditions for an equivalent
minimization problem, which is:
Subject to Constraints 6 and 7 and a definitional constraint:
Where, 
f_{a} 
= 
Flow of person trips on link a 
t_{a} 
= 
Travel time function on link a at person flow and 

= 
Equal to 1 if path p from i to j includes link a and 0 otherwise 
CTDAM formulation: Previous discussion has treated each model component
as a separate entity. Thus, the trip distribution model would involve fixed
zonetozone travel costs, whereas the trip assignment model would consider
a fixed distribution of trips. In the former case travel costs are not affected
by congestion resulting from increased demand for traveling to particular destinations,
whereas in the latter case, since the demand is constant, travelers do not alter
their choice of destination even when travel to that destination entails additional
costs. This counterintuitive location and travel behavior leads to the consideration
of the CTDAM with which the problems of travel choice are solved jointly^{[6]}.
The proposed CTDAM is specified as follows:
Where,
Equation 1017 constitute a quantitative
statement of UE conditions for the CTDAM. The equilibrium conditions state that
at equilibrium, a set of OD trip flows and path flows must satisfy the following
requirements:
• 
The OD trip flows satisfy a distribution model of Eq.
10 
• 
The flows are distributed in accordance with the UE criterion
(Eq. 11 and 12). 
• 
The number of trips on all paths connecting a given OD pair
equal the total trips distributed from i to j (Eq. 13) 
• 
Each path flow is nonnegative nature (Eq. 14) 
• 
The number of trips from a given origin i to all possible
destinations j is equal to the total trips generated from i (resulting from
summation over j on both sides of Eq. 10) 
• 
The number of trips from all possible origin i to a given
destinations j is equal to the total trips destined for j (resulting from
summation over i on both sides of the Eq. 10) 
• 
The definitional relationship between path and link flows
is satisfied (Eq. 15) 
Equivalent minimization problem: The main idea behind the equivalent optimization problem approach is to construct an intermediate model built around a convenient objective function and the original constraints (or a subset of them) that would permit to recover the model equations from the conditions of optimality of the minimization or maximization problem^{[13]}.
To solve the CTDAM model for equilibrium, the approach involves showing that
an EMP exists whose solutions satisfy the equilibrium conditions (Eq.
1017). In other words, consider the following minimization
problem
Subject to
In this formulation, the objective function (Eq. 18) comprises
two components. The first component can be represented by:
and the second component can be written as:
The function F(f) has as many terms as the number of links in a transportation network. Each term is a function of the traffic flows over all possible paths that share a given link a, which implied by the linkpath incidence relationships (Eq. 15). The second term, G (T), has as many terms as the number of OD pairs in the transportation network. The function G (T), corresponds to the Wilson’s (1967) entropy maximizing doublyconstrained spatial interaction model^{[14]}. The parameter of θ in the objective function is assumed to be determined exogenously.
Equations 1921 are the flow conservation
constraints where the Eq. 22 is the flow non negativity constraint
that is required to ensure that the solution of the program is physically meaningful.
The importance of the EMP is that even with very mild assumptions imposed upon the demand and link cost functions, it is a convex program and has a unique solution that is equivalent to the CTDAM. The objective function (Eq. 18) is strictly convex, since both terms are strictly convex functions. Therefore, there is a unique equilibrium solution. The theorem of equivalence can be proved based on the Lagrangian equation and the KKT optimality conditions for the equivalent minimization problem^{[6]}. Solution algorithm: Implementation of the CTDAM requires an algorithm for obtaining solutions for the EMP. Due to the fact that the EMP is a convex programming problem with linear constraints, it can be solved efficiently by either Evans or FrankWolfe algorithm. The Evans algorithm is much superior to solving the problem and is preferred. It is because; this algorithm requires less iteration than the FrankWolfe algorithm in order to obtain suitable solutions. Moreover, each iteration of the Evans algorithm computes an exact solution for the equilibrium conditions, while in the FrankWolfe algorithm; none of the equilibrium conditions are met until the final convergence^{[15]}. This has an important implication in the largescale network applications because it is often unlikely that either the Evans or the FrankWolfe algorithm will be run to exact convergence due to the high computational costs involved. The Evans algorithm applied to the EMP can be summarized as follows^{[16]}:
Step 0: Initialization: Find an initial feasible solution {}.
Set n: = 0.
Step 1: Travel cost update: Set ,
n: = n+1 and compute minimum cost paths {}
on the basis of updated link costs, for every OD pair.
Step 2: Direction finding: • 
Solve a doubly constrained gravity model as a function of
the shortest path costs, ,
applying the twodimensional balancing method 
• 
Perform an allornothing assignment of demand
to the shortest paths computed with the updated link costs .
This yields .The
and
represent
the auxiliary flow, variables corresponding to ,
respectively 
Step 3: Convergence check: Compute the Relative Gap and test for convergence:
If the Relative Gap is less than predetermined tolerance level ε, the procedure stops, otherwise continues.
Step 4: Stepsize determination: Find a_{n} that solves:
Step 5: Flow update: Revise trip flows as following:
Step 6: Convergence check: Retest the updated value of the objective function for convergence. If the Relative Gap is acceptable, convergence is achieved; otherwise go to Step 1. RESULTS AND DISCUSSION This section presents a behavioral comparative analysis between the application results of the simultaneous approach represented by CTDAM and the sequential approach for work and educational purpose trips. Tehran transportation network: The research area in Tehran comprehensive traffic and transportation studies (TCTTS) consists of 22 municipal districts and 560 traffic zones. The number of external traffic zones is 15. This network is composed of 8363 directed links, representing streets and 5523 nodes, which generally represent intersections. Each link is described by its beginning node, ending node, length, mode, link type (i.e., freeway, expressway, principal arterial, etc and the facility type, i.e., oneway, twoway undivided, twoway divided, etc.) and finally number of lanes and volume delay function^{[17]}.
Volume delay function: In order to formulate the traffic assignment
problem as an optimization problem the Jacobian matrix of the cost function
must be symmetric. To ensure uniqueness of the equilibrium link flows, it is
assumed that the link costs are separable and the cost functions are monotonically
increasing ^{[6]}. These assumptions are satisfied by the most commonly
applied BPRtype functions with:
Where: 

: 
The free flow travel time and 
k_{a} 
: 
The link capacity, as well as by many variants of the BPR function 
A total of 19 different calibrated and adjusted volume delay functions provided
by TCTTS were used as a link performance functions model ^{[17]}.
Solution procedure: All computations are limited to the morning peak period; the total flow for captive trips is approximately 631,500 person trips h^{1}. Total flow is divided into two trip purposes of HomeWork and HomeEducation. In order to compare the models they are solved in the following order; • 
Assigned (UE assignment) OD matrices of trips with work and
educational purposes on Tehran network in TransCAD software. The UE assignment
in the Sequential Model (SM) obtained after 76 iterations (ε = 0.01).
The outputs of the assignment model were link volumes and travel times.
Figure 1 shows traffic volumes that are obtained from
the peak hour SM of Tehran network in 2007 
• 
If observed OD flows are available, θ and d should be
calibrated using real data. Otherwise, these parameters can be set close
to the observed systemwide average travel length in min^{[18]}.
We used d = 26 min that was obtained from OD matrices 
• 
The next step involved solving the Tehran network with CTDAM
model, using d = 26 min and applicable trip production and attraction of
traffic analysis zones 
 Fig. 1: 
Traffic volumes obtained from the sequential model during
a peak hour for Tehran network in 2007 
 Fig. 2: 
Relative gap and optimal step size for CTDAM 
The convergence of the solution to the CTDAM is shown in Fig. 2. As depicted in this figure, after 59 iterations, the Evans algorithm reached relative gap and optimal step size of 0.009924 and 0.044185 respectively. Furthermore, two other convergence criteria were considered, one involving the trip table and another for the link flow array.
For the trip table, a simple criterion is considered in which the Total Misplaced
Flow (TMF) is estimated. This is the sum of the absolute differences of zoneto
zone OD flows in the main problem and sub problem solutions. If these two measures
are equal, then one can conclude that the algorithm has converged with regard
to the trip table^{[19]}. TMF estimates for CTDAM model are presented
in Fig. 3 using the log scale to facilitate the comparison.
 Fig. 3: 
Total misplaced OD flow and maximum link flow change for CTDAM 
 Fig. 4: 
Comparison of CTDAM and SM with observed link flow 
It is shown that the TMF measure reached a minimum value of 2 persons/hour
in the 16th iteration and then remained unchanged.
For the link flow portion of the problem, another convergence criterion that was considered deals with the maximum overall link flow change that is the absolute deviations between the current solution and the solution from the previous iteration^{[18]}. This criterion is also shown in Fig. 3. Comparative results: Our comparative analysis were based on comparing the predicted daily link flows output from each approach to the observed morning peak hourly link flows where we have 48 links with observed traffic counts in 2006 data from TCTTS^{[17]}. With considering the observed link volumes as a base (variable parameters) and link volumes of SM and CTDAM as a function, we can compare the two models with constructing linear regression models. Figure 4 shows the results of calibrating these models. The results of the comparison point out that there is enough correlation between link flows from the CTDAM and observed link flows. The better performance of proposed model (CTDAM) compare to SM grows up of comparison study. The intercepts of both models are not large, however regression coefficient (slope) from CTDAM is statistically accepted and closer to one compare to SM. CONCLUSION The sequential approach used in practice to predict shortrun transportation equilibrium has several inherent weaknesses and is internally inconsistent in its models structure. To overcome these deficiencies, this study applied a combined model for the simultaneous prediction of trip distribution and trip assignment to Tehran Metropolitan Area with the goal to perform a comparison between the sequential and simultaneous approaches. The comparison between the solutions of the sequential procedure and combined models remain as a valid research problem that needs to be examined in many urban areas. In this study the CTDAM is used for predicting captive trips on transportation network of Tehran. It was shown that the proposed combined model can notably satisfy several convergence criterions. Furthermore, different evaluation measures are utilized to compare the results of the combined model with a SM. The modeling results presented in this research suggest that: • 
The UE assignment in the CTDAM can be obtained relatively
faster than the SM. 
• 
There is enough correlation between link flows from the CTDAM
and observed link flows 
• 
When we compare proposed CTDAM and SM with sample observed
link flows, CTDAM results (R^{2}, intercept and regression coefficient)
are statistically more significant rather than SM 
Finally, several avenues for future research have emerged from this study. It appears to be productive to reformulate and apply the model so that it also consists of the modal split step. Furthermore, due to various travelrelated constraints (imposed by personal and household characteristics as well as transportation system), the use of the models capable of distinguishing between captive and noncaptive trips and considering both trips in a single modeling structure should be explored in future research. ACKNOWLEDGEMENTS Authors are thankful to Professor David Boyce for his valuable comments on the first draft of this research. Data and network information used in this study are provided by Tehran Comprehensive Transportation and Traffic Studies Company (TCTTS). " target="_blank">View Fulltext
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