Traﬃc Assignment: Methods and Simulations for an Alternative Formulation of the Fixed Demand Problem

Motorists often face the dilemma of choosing the route enabling them to realise the fastest (i


Introduction
A dilemma often facing transport planners is to choose whether to leave motorists free to make their own route choices where they aim to minimise their own travel times, or to try to actively manage the traffic flows in order to minimise the total journey times for all motorists travelling between origin and destination, i.e., whether to plan or not to plan?
Assuming that journey time is the only criteria for route choice, car travellers may be seen to act selfishly as self optimisers insofar as they usually want to minimise their own journey times. As a consequence of this policy, in the absence of any effective traffic control measures, route switching by the travellers to what they perceive to be the fastest route will act to produce a steady state where all (used) routes have an approximately equal travel time. The resultant total travel time at this equilibrium flow will be greater than that obtained for the optimal flow, achieved in the presence of a perfect traffic control system. These two states of the system, as defined by Wardrop [1], are generally referred to as user equilibrium (UE) and system optimal (SO). This difference between UE and SO travel times can lead to the decidedly counter-intuitive result that additions to road capacity, typically through more road construction, resulting in increased rather than the expected slower journey times.
This class of problems, known as the Traffic Assignment Problem (TAP), was first formulated by Dafermos and Sparrow [2] and has a number of known mathematical programs for solving variations of the fixed demand problem (where the number of cars being transported from an origin to destination is fixed) [3]. We present a closely related formulation of SO and UE using a simplified parallel link model. Several discrete and continuous versions of this model are presented, together with a comparison between various solution methodologies. Various minimisation problems with separable goal function and simple constraints have been treated by numerous authors. The interested reader may consult the survey paper of Patriksson [4], or the more recent work focused on networks involving parallel routes by Krylatov [5].
Concerning features of more realistic traffic models, we just mention the network equilibrium problems under demand uncertainty and capacity constraints studied via scalarization approaches by Cao et al. [6], or the seasonal heteroscedasticity in vehicular traffic flow investigated by Huang et. al [7].
A fundamental feature of road transportation is that car travel time is dependent on the number of cars accessing the route. If there are m ≥ 2 routes between the origin and destination points, the time t i for a car accessing route i (i = 1, . . . , m) is a monotonic increasing polynomial function of the traffic flow x i as measured in"units of vehicle" per "unit of tim" accessing route i, namely proposed by Youn et al. [8] and referred to as the BPR Formula (Bureau of Public Roads [9]).
The term x i /c i is effectively the (traffic) flow to capacity ratio of the road. Road capacity may be conceptualised in different ways (see Minderhoud et al 1997) [10] but here it is taken to mean the specific design capacity of the road. Since many, if not most, roads operate at traffic flows well above their design capacity this allows for the situation where x > c if not x c. The assumptions are that travel time along two roads having the same speed limit and length should be equal when the traffic levels meet the design capacity. It may be noted that as Denoting the cost of transporting x i vehicles along route i (i = 1, . . . , m) by the total travel time T (x) for n vehicles distributed on m routes is given by the formula where x = (x 1 , . . . , x m ) ∈ N m and x 1 + · · · + x m = n. 2

Mathematical Programs
In this section we present some mathematical programs having discrete or continuous state spaces, which are related to optimal flow, equilibrium flow, and optimal equilibrium flow.

Optimal Flow Programs
First, we define two mathematical programs related to optimal flow, whereby a fixed number of cars is assigned to each route in such a way that the total travelling time is minimised. We consider a discrete program having non-negative integer solutions, and a continuous counterpart which has the solution in the set of non-negative real numbers, denoted by subject to

Continuous Optimal Flow [COF]
subject to

Equilibrium Flow Programs
Second, we define mathematical programs for the equilibrium flow assignment, whereby individual travel times are as similar as possible across all routes. The steady state traffic flow along each route could be seen as a solution of the following system of equations: However, this system is often inconsistent, hence one may seek to minimise the variance of travel times.
The following mathematical programs can be formulated as alternatives to the equilibrium system (6): subject to Continuous Equilibrium Flow [CEF] subject to In contrast to the UE defined by Wardrop's first principle [1], "The journey times on all routes are equal, and less than those which would be experienced by a single vehicle on any unused route", the above model considers the cost differences between all roads, whether used or not. If the demand is sufficiently high, then the solution to the above model and Wardrop's first principle will be one and the same. The difference between the two can be especially noticed at low demand.

Formulation of the programs [DOF] (4) and [COF] (5) as equilibrium problems
It was shown in [11] that the solution of [COF] (5) corresponds to that of the equilibrium system where As an alternative of the equilibrium system (9) we can consider the following mathematical programs: subject to

Continuous Optimal Equilibrium Flow [COEF]
subject to

Existence of a Solution for the Continuous Equilibrium Systems
The existence of a solution for all defined equilibrium systems can be shown to depend on the demand, which in turn must be greater than a certain value. Here we briefly discuss a necessary condition which involves the parameters of the model.
Proof. Without loss of generality, functions can be relabeled such that If the system (6) has a solution (x 1 , . . . , x m ), then From the equality f i (x i ) = f m (x m ) written for i = 1, . . . , m, one recovers the unique value Adding for i = 1, . . . , m we obtain D > D 0 .
Conversely, if D > D 0 we prove that the system (6) has a solution. Indeed, equation has a solution x m > 0, hence by (15) we also obtain x i , i = 1, . . . , m − 1 which satisfy (14).
Example. For the functions f i (x) defined by (1), we have the identity 5

Methodology
In this section we discuss some methods used in the analysis of the optimal flow and equilibrium flow problems, providing some details especially for the dynamic programming and tabu search approaches. We also discuss the complexity of these methods, in relation to computations detailed in [12], where we have also analyzed exhaustive search and numerical methods based on steepest descent.

Solution of the mathematical program [DOF] (4) by dynamic programming
The cost function T of the mathematical program [DOF] (4) has separable variables, being a sum of terms containing independent variables (3). Since the feasible set S ∩ N m is finite, problem (4) can be solved using Bellman's algorithm of dynamic programming (see Bellman [13] and Bazaraa et al. [14]).
Defining recursively the Bellman functions Then, the optimal value of problem (4) is given by An optimal solution x 0 = (x 0 1 , . . . , x 0 m ) of problem (4) can be deduced by the backward recursive procedure: Let c := n and choose x 0 m ∈ argmin A full explanation of this method and examples are given in [12].

Numerical optimisation methods
The polynomial functions f 1 , . . . , f m defined by (1) are convex (0, ∞) (as all coefficients are non-negative), hence the exact solution can also be approximated by various numerical methods.
These numerical methods will be particularly relevant in the study of the equilibrium problem for numerous links m and large number of vehicles n. In that case the exhaustive search becomes ineffective, while Bellman's algorithm is not applicable due to the cost function having non-separable variables. [DOEF] (11) and [COEF] (12). The basic 'Tabu Search Step' is detailed below and the implementation of the adaptive step is shown in Figure 1. For the equilibrium flow, the road travel time function (1) is used, whereas the road travel time for the optimal flow is c i (

Tabu Search Step
Step 1. Initialization a) Make an initial allocation of n vehicles to m routes such that the solution satisfies Step 3. Reassign the traffic with the updated load values Step 4. Update the Tabu list. Reduce the Tabu value for all routes, but those for M , N , so that: x M cannot be increased until TabuTime has elapsed; x N cannot be decreased until TabuTime has elapsed. Step 5. Update the value of the objective function OptSol = min{σ 2 (R 1 , . . . , R m ), OptSol}.
Note. The starting step size is initialised as an integer power of 4, ensuring an appropriate started step size, and that the method can be used for both continuous and discrete based solutions. h = 4 log 10 (n) .
In the discrete case the minimum stepsize will result in a value of 1. 8

Complexity calculations
Here we give the computational complexity for exhaustive (presented in more detail in [12]), dynamic programming and tabu search, as a function of the number of links m and the number of vehicles n.

Exhaustive Search
Let N m (n) denote the number of possible configurations (x 1 , . . . , x m ) such that x 1 , . . . , x m ≥ 0 and x 1 + · · · + x m = n (i.e., the size of the feasible space for (4)). Clearly, N 1 (n) = 1 and recursively we have Therefore the complexity for an exhaustive search of n vehicles on m roads is,

Dynamic Programming
The complexity of dynamic programming depends on the main recursive operations given in Section 3.1.
For fixed k ∈ {1, m} and c ∈ [0, n], the number of operations required to compute G k (c) is c + 1 by (17).
Evaluating for m roads, the total number of operations required will be m (n + 1)(n + 2) 2 , thus resulting in a complexity of O mn 2 .

Tabu Search
Step 2 requires 2m operations per iteration. Given the different termination criteria used for the continuous problem we can only state that the maximum number of iterations is given by ijk (set by the user). For the discrete problem we can provide a strict upper bound by utilizing the fact that the step sizes considered are integers and for each iteration i, h is reduced by a factor of 4. An upper bound for the maximum number of iterations is log 10 (n) · j · k, thus giving a complexity O(m · log 10 (n)).

Results
In the classic Beckmann formulation for UE (19), defined for our parallel-links model as subject to The relationship between the optimal and equilibrium states of the system for the variance-based and traditional formulations is illustrated through the price of anarchy [15], defined as P A = Total cost at equilibrium flow Total cost at optimal flow (20) More details on selfish routing and the price of anarchy can be found in the book of Roughgarden [16].

Part A
Travel time functions (1) for a 3 road example are given in Table 1. In this case, moving x 1 cars along . Therefore, the total travel time of n = x 1 + x 2 + x 3 cars along these routes is T (x) = g 1 (x 1 ) + g 2 (x 2 ) + g 3 (x 3 ), according to (3).

Road Parameters and Road Travel Time per Vehicle
Road No.

Comparison of Computation Efficiency of Methods for Discrete Solutions
The execution time required to find the solution for the mathematical program [DOF] (4) by Dynamic Programming and Tabu Search are depicted in Figure 2. The results confirm the analysis presented in the methodology section. For high demand, Dynamic Programming method requires much more time than Tabu Search method. Tabu Search is preferred when handling larger networks and higher demands, however Dynamic Programming serves well as a means to validate and tune the parameters used by the Tabu Search.

Comparison of Discrete Optimal [DOF] and Discrete Equilibrium [DEF] Solutions
Tabu Search solutions for optimal flow [DOF] and equilibrium flow [DEF] programs are plotted in Figure   3 for demands from 1000 up to 50000, in increments of 1000. Vehicles initially prefer road 2, while as demand increases road 1 quickly becomes dominant and the percentage of vehicles on the road settles down. The long-term dominance of a particular road i is determined by a combination of its capacity c i and power p i . Demand Method  For low demands, some roads may be empty. As the demand increases, new roads are brought into use, which reflects in spikes of the Price of Anarchy. The two spikes in Figure 3 (c) correspond to the introduction of new roads, as shown by Figure 3 (a). It appears that the optimal solution responds to such changes faster than the equilibrium solution. Also, while the total costs between optimal the flow and equilibrium flow solutions shown in Figure 3 (b) are very similar, the mean cost of a road (per vehicle) may differ significantly (difference of about 25% for a demand of n = 50000), as suggested by Figure 3 (d).    Table 5 displays the first solutions for a system of 1, 2 and 3 roads with cost functions given by Table 1.
System of m roads

Part B
Here we investigate an extended 10 road model, whose parameters are given in

Comparisons for the equilibrium mathematical programs
Comparisons between solutions of the optimal flow and equilibrium flow programs are shown in Figure 4.
The results are generated using a Tabu Search heuristic at demand intervals of 100.

Concluding remarks
In this paper we presented a simplified traffic model, for which we have formulated various discrete and continuous optimal flow and equilibrium flow mathematical programs. Solutions of these programs were obtained for two scenarios involving 3 and 10 roads respectively, which were solved using exact (exhaustive search, dynamic programming), numerical (interior point methods such as fmincon) and heuristic (Tabu Search) techniques. The latter method seemed to be effective in most of the scenarios considered.
We showed that in this parallel-link setup using traditional BRP travel time function, the variance based mathematical programs performed well at high demand, however at low demand convergence was slow due to the inconsistency of the systems (6) and (9). A full investigation on a more complex network topology -one where multiple origin-destination pairs are connected by routes which share certain roads (i.e., Sioux Falls model [17]) -is required to consider how effective this method is compared with traditional methods.
Whilst the variance based method may struggle to match their speed and accuracy, it does allow for the possibility of more complex link travel time functions.
Multi-criteria optimization could also be employed as an alternative methodology. Problem (12) may be seen as a particular scalarisation (with equal weights) of the multi-objective optimisation problem min{g i (x) | i = 1, . . . , m} subject to x 1 + · · · + x m = n, x i ∈ N.