Routing in Road Networks: the toll booth problem

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1 Routing in Road Networks: the toll booth problem Fernando Stefanello, Luciana S. Buriol, Marcus Ritt Instituto de Informática Universidade Federal do Rio Grande do Sul Caixa Postal Porto Alegre RS Brazil Michael J. Hirsch Raytheon Company, Intelligence and Information Systems 300 Sentinel Drive Annapolis Junction, MD 20701, USA. Panos M. Pardalos Dept. of Industrial and Systems Engineering, University of Florida 303 Weil Hall, Gainesville, FL 32611, USA. Tania Querido Linear Options Consulting 7450 SW 86th Way, Gainesville, FL 32608, USA. Mauricio G. C. Resende Internet and Network Systems Research Center, AT&T Labs Research 180 Park Avenue, Room C241, Florham Park, NJ 07932, USA. ABSTRACT Due to population growth and the massive production of automotive vehicles, traffic congestion problems have become larger and more common. This is a reality that governments are facing everywhere, even in medium sized cities that were not used to this scenario. However, despite the growth of the number of vehicles, traffic congestion can be lessened using different strategies. One possibility, that is explored in this research, is assigning tolls to roads, inducing users to take alternative paths, and thus better distributing the traffic across the road network. This problem is called the toll booth problem and is NP-hard. We propose mathematical formulations for variations of the toll booth problem, using two piecewise linear functions to approximate the congestion cost. We test these models on a set of real-world instances, and apply a previously proposed genetic algorithm to all instances. The experimental results show that the proposed piecewise linear functions approximates the original convex function quite well, and the genetic algorithm produces high quality solutions. KEYWORDS. Transportation networks, Genetic algorithms, Toll both problem, Combinatorial Optimization.

5 2.3. Mathematical model 3 (MM3): Linear Approximate Average Link Travel Time The performance of commercial mixed integer linear programming solvers has improved considerably over the last few years, whose efficiency cannot be disregarded by the scientific community. This current state motivates the creation of linear mathematical models (approximating the original mathematical models) so that these resources can be utilized. Thus, we propose the model below which describes a piecewise linear approximation of function Φ. Decision Variables: ϕ a = cost on arc a; Where: minimize ϕ a (7) a A subject to: (3) and (4) (m i a/c a )l a + b i a ϕ a a A, i = 1,..., n (8) x k a [0, 1], ϕ a 0, l a 0 a A, k K. (9) m i a = (Y i Y i 1 )/(X i X i 1 ) b i a = Y i X i m i a with Y i = X i c a t a (1 + β a (X i ) pa )/T K for X 0 = 0 < X 1 <... < X n. The objective function (7) minimizes link costs. Constraint set (8) evaluates the partial cost on each arc, while the last constraint set (9) defines the domain of variables. The cost Φ a is approximated by a piecewise linear function ϕ a defined in constraint set (8). The linearized function and the values of X and Y are described in the next subsection Piecewise linear functions ϕ a and ϕ l a The piecewise linear function is an estimation of the cost function Φ. This linearization requires a balance between the accuracy of the solution found and computational time. A lower value of n (number of line segments composing the approximation) can lead to a poor approximation of the original objective function, while a large value of n can lead to a considerable increase in computational time since each element entails A additional constraints to the model. We describe two linearizations of the cost function Φ for a A. In both cases it is necessary to define a vector X whose values are computed as a function of ( la c a ). This vector can be arbitrarily defined according to the accuracy required for the linearization of the cost function, or according to characteristics of the set of instances. The linearization ϕ is an overestimation of Φ. The second linearization, denoted ϕ l, is an underestimation, which gives a lower bound of Φ. A representation of these three functions is depicted in Figure 1. It shows the behavior of the cost function Φ (solid line) and the linear piecewise cost functions ϕ and ϕ l for an arc with t a = 5, c a = 200, p a = 4, β a = 0.15, and

6 T K = 1000 to vector X with n = 6 for a given instance. The filled circle dots are defined by vector X and the square filled dots by vector X l φ φ l Φ Cost Utilization (l a /c a ) Figure 1. Comparison of the cost function with the linear piecewise cost function. For ϕ, the values of Y i are calculated given the values of X i as Y i = Φ(X i ). For each pair of points P i 1 = (X i 1, Y i 1 ) and P i = (X i, Y i ) we can write the linear equation of the straight line that links these two points. The slope is represented by m i a and the independent term by b i a. For ϕ l we use Φ = ta x T K + (pa+1)taβaxpa c pa a T K to calculate the slope in a point x. By convention we use the vector X of the first case to calculate the vector X l used in this case, where Xi l = X i 1+X i. We empirically defined X={0, 0.65, 1, 1.25, 1.7, 2.7, 5} 2 based on the set of instances tested. Proposition 1: if la c a X n a A, then ϕ l Φ ϕ. By construction ϕ l a Φ a, then Φ = a A Φ a a A ϕl a = ϕ l. Thus Φ ϕ l. Furthermore, if la c a X n by construction ϕ a Φ a, then ϕ = a A ϕ a a A Φ a = Φ. Thus ϕ Φ. Therefore ϕ l Φ ϕ. 3. A Hybrid Genetic Algorithm for the Toll Booth Problem In this section we describe the hybrid genetic algorithm used for solving the toll booth problem proposed in Buriol et al. (2010). Each solution is represented by two arrays w and b. Array w stores the integer arc weights, while b is a binary array indicating the set of tolls. An arc a of the network has weight equal to w a = w a b a. Each individual weight belongs to the interval [1, w max ]. Each demand is routed forward to its destination through the shortest path calculated based on the weights of the tolled arcs. Un-tolled links are considered to have zero weight. Depending on the number of tolls and the network, there can be several shortest paths of zero cost. In this case we evaluate the shortest path through the number of hop counts. Traffic at intermediate nodes is split equally among all outgoing links on shortest paths to

7 the destination. After the flow is defined, the solution is associated with a fitness value defined by the objective function Φ. The initial population is randomly generated, with arc weights selected uniformly in the interval [1, w max ]. Also, the position of K toll booths is chosen at random. The population is partitioned into three sets A, B, and C. The best solutions are kept in A, while the worst ones are in C. All solutions in A are promoted to the next generation. B other solutions are generated by crossover of one parent from A with another from B C using the random keys crossover scheme. Solutions from class C of the previous generation are replaced by new randomly generated solutions. A local search procedure is incorporated into the genetic algorithm to enhance its ability to find better-quality solutions with less computational effort. Local search is applied to each solution generated by the crossover operator. In short, the local search approach works as follows. Given a solution, the five arcs with largest congestion cost are selected. For each of these arcs, in the case there is a toll already installed on it, its weight is increased by one, in order to induce a reduction of its load. The increases are repeated until the weight reaches w max or Φ does not improve. For each of the arcs that do not currently have a toll installed, a toll is installed with initial weight one, and a toll is removed from some other link (using a circular order). The procedure stops at a local minimum when there is no improved solution changing the weights of the candidate arcs. 4. Computational results This section presents and discusses the computational experiments performed. First, we present the dataset used in the experiments. Next, we detail the experiments performed. Two sets of experiments were performed: the first explores CPLEX results for the models presented in Section 2, while the second explores the GA results. For the experiments, we used a computer with an Intel i GHz CPU, with 4GB of main memory and Ubuntu The genetic algorithms were implemented in the C programming language and compiled with the gcc compiler, version with optimization flag -03. The commercial solver CPLEX 12.3 was used to solve the mathematical models presented in Section 2. Table 1 presents details of the ten real-world instances considered in our experiments and available by Bar-Gera (2011) Minimal average link usage MM1 and linear piecewise cost MM3 The first set of experiments tests the performance of CPLEX (set up to use the primal algorithm) applied to a subset of the instances, considering the models introduced in sections 2.1 and 2.3. Table 2 presents, for each instance and model, the objective function values and the computational time (in seconds) needed to solve the problem instance. The objective function is Θ for MM1, while for MM3 the results for both piecewise linearizations are presented. For every solution found, the corresponding Φ (6) value is also calculated for the sake of comparison. From the results in Table 2, there are four main observations that can be drawn. First, minimizing the maximum utilization (Θ) is not effective in minimizing the link travel time as measured by Φ. However, we were not expecting a close relation among these functions. Second, there are instances where in any given feasible solution, at least one arc

8 Table 1. Attributes of problem instances. The columns represent the instance name, number of vertices, number of links, number of OD demand pairs, number of nodes that originate demands, and number of nodes that are destinations of demands, respectively. Instance Vertices Links Demands Origins Destinations SiouxFalls Friedrichshain Center Prenzlauerberg Center Tiergarten Center Mitte Center Stockholm MPF Center Barcelona Winnipeg ChicagoSketch Table 2. Computational results for MM1 and MM3 MM1 MM3 Instance Θ Φ Time ϕ Φ Time ϕ l Φ Time SiouxFalls Friedrichshain Center Prenzlauerberg Center Tiergarten Center Mitte Center Stockholm 9, , carries more flow than its capacity, since Θ > 1. Third, both piecewise linear functions have values close to Φ, which shows that they are good approximations. Finally, for the Stockholm instance, the objective function value is very high. This is expected since node 5 is a destination for a demand of size , but has only one incoming link, the capacity of which is 1. The remaining instances listed in Table 1 cannot be solved because of the size of the problems. The large number of constraints generated by these problem instances increase the memory requirements beyond the capacity of our test machine Computational results of genetic algorithms This section presents results of the genetic algorithm for the ten test instances. We extend the experimental study performed by Buriol et al. (2010) which presents results for four of these instances (the other six were only available recently). Moreover, we provide an analysis of the best solution for each combination of instance and K. In the first experiment we compare the results obtained between the genetic algorithm (GA) and the genetic algorithm with local search procedure (GA+LS). The stopping criteria are a time limit of 1800 seconds and a maximum number of generations (5000 for GA and 1000 for GA+LS). Table 3 shows the results obtained, averaged over five runs with different random seeds and over the seven different values of K (the number of toll booths used in this experiment are the same as shown in Table 4). Table 3 presents the number of generations and average running times for the GA and the GA+LS. Furthermore, the last column of Table 3 presents the average gap between the fitness value of both methods, showing the superiority of the GA+LS over the GA (a negative average gap means that GA+LS performed better than GA alone). On average, GA+LS spent less time and

9 found better solutions than GA on the experiments performed. We note that the only cases where the times were comparable was when both approaches reached the time limit of 1800 seconds. Table 3. Comparison of GA and GA+LS algorithms. Number of generations Run Time (s) Instance GA GA+LS GA GA+LS Av. gap SiouxFalls 5, , Friedrichshain Center 5, , Prenzlauerberg Center 5, , Tiergarten Center 5, , Mitte Center 5, , , Stockholm 5, , , MPF Center 1, , , Barcelona , , Winnipeg , , ChicagoSketch , , For the next experiments we only consider GA+LS. The stopping criteria were set to 1800 seconds or 2000 generations. We also added a third criterium: the number of generations without improvement of the fitness value, set as 5% of the maximum number of generations. Table 4 shows the average over five runs. For each K (number of installed tolls), in the first vertical half of the table we show the number of generations (gen), best solution value (best Φ) over the five runs, average fitness values (avg Φ), standard deviation (SD), and average running time in seconds. For the best solution found among the five runs, the second half of the table presents the average number of paths for each OD pair (#Path), the average number of hops among all OD shortest paths (#Hops), the average number of tolled arcs among the shortest path (#Toll), the average sum of the tariffs on a path ( Tariff), and the number of different arcs used in each OD pair (#Arcs). As K increases, the cost tends to decrease and present small oscillations. In most cases, the best solutions were not found with larger K, which is close to E (the number of arcs), but for K E or slightly larger than that. Also, we can see that the standard 2 deviation is small in most cases showing that the algorithm is robust. The column #Path highlights the number of shortest paths on average used between each OD pair. With no tolls, or with few tolls, there are on average about two shortest paths. However, as K increases, the number of shortest paths decreases. With a small K, there are alternative paths with no tolls. In this case only the hop count is used to evaluate the shortest path, increasing the possibility of paths with the same cost. In computer networking, hop count refers to intermediate devices through which a piece of data must pass between source and destination. We extend this concept to road networks where the intermediate devices are vertices which represent intersecting roads or waypoints. Thus, we show in column #Hops the average hop count for all shortest paths between OD pairs. When compared with the situation without tolls (K = 0) we observe that as the number of installed tolls increases, so does the hop counts. The column #Tolls shows the average number of tolls installed on OD shortest paths. Clearly, this value increases with K. In the column Tariff we show the average value of tariff in each path. In this problem our objective is to install K tolls such that

10 Table 4. Detailed results of GA+LS algorithm. GA+LS results Analysis of the best solution found by the GA+LS Instance K gen best Φ avg Φ SD time (s) #Path #Hops #Toll Tariff UArcs SiouxFalls Friedrichshain Center , Prenzlauerberg Center 100 1, , , , , Tiergarten Center , , , Mitte Center , , , , , Stockholm , , , , , , , , , MPF Center , , , , , , , Barcelona , , , , , , , , Winnipeg , , , , , , , , ChicagoSketch , , , , , congestion costs are minimized. Since minimizing the tariff is not part of the objective of this problem, that results in more variability in this column. By the last column we conclude that the increase of K slightly influence the average number of arcs used to send flow for an OD pair.

11 5. Conclusions In this paper we proposed various mathematical formulations for different problems of traffic assignment, using two piecewise linear functions to approximate the link travel time. We tested these models on a set of real-world instances, and we applied a previously proposed genetic algorithm for the toll booth problem to all these instances. The mathematical model MM1 provided information about the network such as the ability to handle the flow without exceeding the capacity. The results demonstrated that in some instances this constraint could not be satisfied. We also observed that the proposed piecewise linear functions closely approximate the original convex function and provided a lower bound for their value. For the toll booth problem, although being computationally demanding, the genetic algorithm with local search produced better solutions in no more computational time when compared with the standard implementation of the genetic algorithm. Also, the procedure produced high quality solutions even for large problem instances. Finally, considering that users would take the least costly path, toll setting can be used for the distribution of better the flow on the network, thereby reducing traffic congestion. In future work we intend to include shortest path constraints to the formulation, and to solve larger instances with the GA+LS algorithm. Acknowledgements This work has been partially supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) - Brazil, FAPERGS (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul) and PRH PB-17 Petrobras S.A., Brazil. References Bai, L., Hearn, D., Lawphongpanich, S. (2004). Decomposition techniques for the minimum toll revenue problem. Networks 44 (2), Bai, L., Hearn, D., Lawphongpanich, S. (2010). A heuristic method for the minimum toll booth problem. Journal of Global Optimization 48, , /s Bar-Gera, H. (2011). Transportation networks test problems. il/ bargera/tntp Beckmann, M., McGuire, C., Winsten, C. (1956). Studies in the economics of transportation. Yale University Press. Bureau of Public Roads,. (1964). Bureau of public roads: Traffic assignment manual. US Department of Commerce, Urban Planning Division. Buriol, L., Hirsch, M., Pardalos, P., Querido, T., Resende, M., Ritt, M. (2010). A biased random-key genetic algorithm for road congestion minimization. Optimization Letters 4, , /s Buriol, L. S., Resende, M. G. C., Ribiero, C. C., Thorup, M. (2005). A hybrid genetic algorithm for the weight setting problem in OSPF/IS-IS routing. Networks 46, Dial, R. B. (1999a). Minimal-revenue congestion pricing part I: A fast algorithm for the single origin case. Transportation Research Part B 33,

12 Dial, R. B. (1999b). Minimal-revenue congestion pricing part II: An efficient algorithm for the general case. Transportation Research Part B 34, Fortz, B., Thorup, M. (2004). Increasing internet capacity using local search. Computational Optimization and Applications 29 (1), Hearn, D. W., Ramana, M. V. (1998). Solving congestion toll pricing models. Equilibrium and Advanced Transportation Modeling, Hearn, D. W., Yildrim, M. B. (2002). A toll pricing framework for pricing assignment problems and elastic demands. In: Gendreau, M., Marcotte, P. (Eds.), Transportation and network analysis: current trends. Vol. 63 of Applied Optimization. Springer. Schrank, D., Lomax, T., Eisele, B. (2011) urban mobility report. Tech. rep., Texas Transportation Institute. Tsekeris, T., Voß, S. (2009). Design and evaluation of road pricing: state-of-the-art and methodological advances. Netnomics 10, 5 52, /s z. W., Wen (2008). A dynamic and automatic traffic light control expert system for solving the road congestion problem. Expert Systems with Applications 34 (4), Yang, H., Zhang, X. (2003). Optimal toll design in second-best link-based congestion pricing. Transportation Research Record: Journal of the Transportation Research Board 1857 (1),

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