Lane-based optimization for signalized network configuration designs

Lane-based traffic signal design has been developed for designing signal-controlled intersections. Conventional designs take on fixed configurations as exogenous inputs to design traffic signal settings. The proposed study will be an extension to merge geometrical junction arrangement and signal controls together for network configuration designs. Design methodology would be directly extended from existing lane-based design method. New Path flows variables and new flow conservation constraints are required to ensure the users’ input OD flows could be assigned onto network paths through different signal-controlled intersections. This problem is new that involves binary variables and related linear constraints which is formulated as a BMILP. Standard technique could solve the optimum solution. A four-intersection network with two approach lane settings is optimized for demonstration purposes.


Introduction
In design signalized networks, users' routes are compiled by path flows. Flow patterns in a network could be affected by signal settings and the connections of network links. Connections of network links should be based on lane marking arrows. To design networks, different researchers developed different mathematical formulations and solution algorithms with different objective functions [1][2][3][4]. Literature review in the network design is found [5]. Coordination of traffic signal settings and path-based traffic flow assignment algorithm was combined [6]. Network configurations including lane marking arrows are generally fixed by users excluding from the optimization process. With given lane marking arrows, traffic lanes could be grouped to form network links. Signal timings can be optimized using traditional stage-or phase-(group-) based methods [7]. Stage-based method was applied [8][9][10][11]. Group-based method is used [6,12,13]. Users' given network configuration may be suboptimal to serve the travel demands. Lane markings are defined as control variables in the proposed lane-based design framework to optimize the entire network link connections. signal-controlled networks  2.1 Distributing OD flows onto available network paths   ,   ,   ,   , ,  N is an arbitrary large integer figure used in the formulation.

Flows turning at intersection level
Users' given OD demand flows are input data. Inside a network, numerous users' traveling paths could be allowed. For intersection n, flows turning at intersections may be from different OD locations. Thus, turning flows equal the total of the path flows.

No lane marking for zero demand flow entering destination node
If OD flow inputs are zero, destination nodes may not be served and respective lane markings could be removed from the optimization process. Thus, respective movement turns entering these may be unnecessary. The lane marking arrows should not exist by Eq.

Removing redundant lane marking arrows
In Section 2.

Constraint sets for governing signal-controlled intersections
In lane-based design framework, we may have some 13 sets of well-established linear constraints to ensure safe traffic signal settings in designs. They include (1)

Optimization for network link connections
For optimizing the network designs, we apply the capacity maximization that has been optimized for intersections. Given turning flows are directly multiplied by the common flow multiplier  [14,15]. The optimization process maximizes the common flow multiplier so that the largest possible demand flows to enter a signal-controlled system are determined. The system's maximum degree of saturation can be attained. Using the same concept, the OD demand flows (matrix) are inputs for network study. The common flow multiplier is applied in the present study to multiply the OD demand flows until the largest OD flows can enter the system without exceeding the maximum allowable degree of saturation. Then, lane marking arrows on approach traffic lanes at intersection level could be optimized. Network link connections can then be established. Respective lane flows and path flows could be optimized as well. It is expected that all path flows can be compatible with the given OD demands satisfying the flow conservation purposes. The capacity maximization problem is a BMILP: Max , subject to Eq. (1) -Eq. (10) and the thirteen sets of lane-based constraints. Standard B&B solution technique is able to optimize this standard mathematical programming problem to achieve optimum solutions.

Case study
A four-intersection network is modeled, n=4 with 2 approach lanes, 2 ,  optimized network configurations through using optimized set of lane marking arrows. The large numerical figure N will be set to equal 10,000. Clearance times are 6.0 and minimum green duration times are 5.0 seconds. Effective green time is +1.0 second longer than the actual green time (i.e. 0 . 1  e second). Maximum allowable degree of saturation for traffic lanes is assumed to be 90%. With these inputs, the network can be optimized by maximizing the common flow multiplier. And the optimized common flow multiplier  =1.0557 with 5.57% reserve capacity (for the same sets of demand flow inputs and problem settings, if the signal-controlled network is connected by a set of suboptimal lane markings, the common flow multiplier would be as low as 0.852). Figure 1     Remarks: Lane marking does not exist when assigned lane flow = 0.0 pcu/h; 1700.0 pcu/h is the total demand flow from origin o=1; 700.0 and 300.0 pcu/h are the total 1000.0 pcu/h flows entering destination d=1.

Conclusions
In the proposed study, a lane-based formulation is developed to design the network settings. Lane marking arrows are optimized together with lane flows and signal timings to connect network links. Users' OD demand flows are key model inputs. The OD flows are then distributed onto different network links. In the case study network consisting of four intersections, the network performance could be overloaded by 14.83% if suboptimal lane marking arrows without shared lane markings are used. By using the proposed lane-based optimization model, the entire network could be improved to have +5.57% reserve capacity (with reserve capacity instead of being congested). The key contribution of the proposed optimization framework is to optimize lane marking arrows to better connect the network links for practical implementations. The proposed problem is a BMILP problem and a branch-and-bound routine has been used to solve the problem.