Transit priority lanes in the congested road networks

Abstract

Given the advances in communication technologies and real-time traffic management, transit priority lanes are emerging as an indispensable component of intelligent transport systems. This scheme calls for giving priority to public transport. In this study, the question of interest is: Which roads can be nominated to give an exclusive lane to transit modes? Due to computational and theoretical complexities, the literature has yet to address this problem comprehensively at the network level considering various modes (public and private). Additionally, taking space away from private modes in favor of public transport may adversely affect the congestion level. To this end, inspired by the Braess Paradox, we seek mis-utilized space used by private modes to be dedicated to transit modes mainly on congested roads. To find such candidate roads, we define a merit index based on transit ridership and congestion level. The problem then becomes to find the best subset of these candidate roads to cede a lane to transit mode. It is formulated as a bilevel mixed-integer, nonlinear programming problem in which the decision variables are binary (1: to cause the respective road to have an exclusive transit lane or 0: not). The adverse effects are minimized on the upper level represented by total travel time (public and private modes) spent on the network. The lower level accounts for a bimodal traffic assignment, to consider the impact of transit priority on private modes. We then develop an efficient low-RAM-intensity branch and bound as a solution algorithm. The search for the subset is made in such a way that improved public transport is achieved at zero cost to the overall performance of the network. A real dataset from the city of Winnipeg, Canada is used for numerical evaluations.

Appendices

Appendix A: The branch-and-bound

In the following exposition, we adopted the same terminology used by (LeBlanc 1975). The strings of solutions are either (i) partial, like (0, 1, 0, 2, 2) representing the situation in which only the first three components are determined with values of 0/1 and the last two, represented by 2, are as yet unspecified or (ii) complete, representing the situation in which all projects are decided and assigned values 0 or 1 (the previously mentioned partial solution can eventually become any of the following complete solutions (0, 1, 0, 0, 0), (0, 1, 0, 0, 1), (0, 1, 0, 1, 1), (0, 1, 0, 1, 1).

1.1 Lower bounds

The algorithm initiates by calculating SO flow for the do-nothing scenario [in case of five projects it becomes (0, 0, 0, 0, 0)] which renders the lowest possible lower bound value. In the course of branching, whenever two nodes are added to the tree, it is only necessary to compute one lower bound since the other one has already been computed. For example for partial solution z _j  = (0, 1, 0, 2, 2), at node j of the tree, the branching at the fourth project will end up with two new solutions z _l  = (0, 1, 0, 0, 2) on the left side and z _r  = (0, 1, 0, 1, 2) on the right side. The lower bound of z _l which is SO flow of (0, 1, 0, 0, 0) has already been calculated when node j was added. Hence it is only necessary to calculate the lower bound of z _r which is the SO flow of (0, 1, 0, 1, 0). This process is shown graphically in Fig. 3a.

Fig. 3

How does the tree structure of the branch-and-bound algorithm work? a Proposed node selection and branching rules, b efficacy of calculating the lower bound (Bagloee et al. 2016a, b)

1.2 Node selection and branching rules

At the selected node, a free variable (or undecided project, represented by “2”) has to be selected to be assigned 0 and 1, which is called branching. There are some methods to make a move on the tree that sometimes requires solving an additional problem or retrieving the entire database to find what is—hopefully—the best node and branching. Such methods become computationally intensive, as the size of the network increases, Alternatively, based on the merit index we propose the following rules: (i) for node selection; take the deepest node on the branch which has emerged from adding a project successively. Note that at each node, two branches come out: one corresponding to deciding to add a candidate (y _a  = 1) and the other one not to add (y _a  = 0). (ii) For branching; select the very next undecided project in the row of the respective partial solutions. As described before (see condition (10)), the projects have already been sorted based on their merits. Hence, it makes sense to go deep into the tree and select the next best project for branching, hoping that the optimum solution lies there. These two simple and intuitive rules obviate any burden of retrieving the information relating to the entire tree. At each node, the algorithm just needs to move forward as much as possible through the y _a  = 1 branch. In case there is no space to move forward, the algorithm moves only one node back to the previous node and then moves through the y _a  = 0. Once it reaches a new node, the algorithm proceeds normally, meaning that it first moves through y _a  = 1 if possible, otherwise it moves through y _a  = 0. The process goes on until a termination criterion is met. Figure 3b shows the gradual buildup of the tree based on the above-mentioned rules for a case consisting of three candidates.

As the tree structure expands the algorithm does not need to remember the paths already taken, nor the paths ahead. As shown in Fig. 3b, it just needs to know the lower bounds of the nodes on the current path, plus the best solution found so far, which is a string of binary values (0/1), and the corresponding incumbent value. For example, if the current node is (11002), the next move is to process node (11001) followed by the node represented by (11000). For the third move, the algorithm moves three nodes back to reach node (10222). The navigation on the BB’s tree is, therefore, memoryless, which is highly advantageous in dealing with networks of large size. In such cases, the rapid expansion of BB tree to extend memorizing the topography and structure of the tree comes at a heavy RAM cost.

Once the algorithm reaches a new node in the tree, it then determines whether the node stands for partial solutions (some variables are “2”: free to be 0 or 1) or complete solutions (all binary variables are determined to be 0 or 1). For complete solutions, the UE flows and the total travel time (upper bound) for the corresponding networks are computed. This solution is then compared with the incumbent solution (the best solution found). The current solution is labeled as the incumbent solution if its upper bound is lower than the incumbent value. For the partial solutions; the SO flow is solved to compute the lower bound on the successors of z _l . Note that the lower bound for z _r which is the same as the predecessor node (z _j ) has already been computed.

1.3 Termination condition

On three occasions the extension of the tree at a particular node stops or freezes or is called fathoming: (1) reaching the bottom of the tree where there is no free variable, (2) no more project can be added to the branch represented by y _a  = 1 due to depleted budget, (3) the lower bound is found greater than the incumbent value, then the respective node is deleted from the list of partial solutions and the process continues until no partial solution is left (termination condition).

Appendix B: Braess Paradox example

Our conjecture is as follows: “1” means the respective road has to give away a lane as a bus lane and “0” means not, and the current mixed use lane is not appropriate because of BP, and some 1’s (actions) may improve it. Similar to the famous Braess example shown in Fig. 4, network (b) with additional road “5” is Braess-tainted compared with the network “a”. It is proven that even in the network “b” (where the BP exists) if we charge drivers the marginal costs of the roads, the Braess-tainted road “5” becomes unattractive as if it was never open to the traffic like Network “a”. Charging drivers the marginal costs results in SO traffic flow which are equivalent to the minimum possible travel time. Therefore, keeping all the roads, even the Braess-tainted roads (like network “B”) open to the traffic [that is (y _a ) = 0], and charging drivers with the marginal cost (SO flow) would result in the minimum possible travel time, which is mathematically a valid lower bound.

Fig. 4

Braess Paradox where removing link 5 yields a better traffic flow