Lane-Based Optimization for Macroscopic Network Configuration Designs

Lane markings (arrows) at individual intersections serve as interfaces to connect upstream and downstream intersections in signal-controlled networks. Demand flows from origins to destinations may need to pass through a series of intersections. If lane markings are not well established to ban turns at intersections, then paths connecting origin and destination (OD) pairs could be inefficient. Due to indirect connections, road users need to take longer paths to reach their destinations. Conventionally, network configurations are fixed inputs for network analysis. In the present study, concepts of the lane-based designs for individual signalized intersections are extended for signal-controlled network designs. Taking OD demand flows as inputs, the proposed algorithm will optimize all lane markings and assigned lane flows on approach lanes. Paths (flows) will then be optimized by linking up the optimized lane markings across upstream and downstream intersections. Traffic signal settings at individual intersections will be optimized simultaneously bymaximizing the reserve capacity for the entireODdemand flowmatrix.The problem is formulated as a Binary-Mixed-Integer-Linear-Program (BMILP) and a standard branch-and-bound routine is applied to solve for global optimum solutions. A numerical example using a 4-intersection networkwill be given to demonstrate the effectiveness of the proposed design methodology.


Introduction
In the optimization of signal-controlled networks, users' travel routings are represented by path flows and link flows.Travel patterns from different origin and destination pairs through different paths could be influenced by traffic signal settings at individual signalized intersections and the network link connectivity.Network link connections are depending on the lane marking patterns.For network design problems, different formulations and solution methods have been developed to solve this complex multilevel programming problem adopting different design objectives, model assumptions, and application conditions [1][2][3][4].A comprehensive literature review on the road network design is given [5].Traffic signal coordination incorporating a path-based assignment algorithm using a TRANSYT traffic model was optimized [6].Signalized Cell Transmission Model (CTM) has been developed and applied to optimize the coordinated signal settings for a pair of adjacent traffic signals to avoid spillback of spatial queues and preventing gridlock due to lane closure for work zone pairs [7,8].All of these methods assumed that the network configurations and lane markings are given as fixed exogenous inputs.Based on the given set of lane markings, grouping traffic lanes into traffic streams to form a network link is performed manually and traffic signal settings can then be optimized by using either stage-or phase-(group-) based methods [9].Stage-based designs of traffic signal settings were found to influence significantly the optimized network performances [10][11][12][13][14]. Phase-or groupbased designs of traffic signal settings were found in previous studies [6,15,16].Pregrouping traffic lanes into traffic streams to form network links to generate network configuration for signalized network design analysis may not be optimal to match the demand flow patterns.We would like to relax the lane markings as design variables in the proposed lane-based design method to further enhance the network link groupings and connectivity taking the latest lane-based design concept into considerations.
Extending green duration times for the heaviest traffic stream may allow more discharges to reduce congestion.However, it may not be easily achieved as other conflicting traffic streams may then be suffering from loss of green times within a signal cycle.It may increase total delay and cause oversaturation.However, if adjacent traffic lanes occupied by other less critical turn movements with spare capacities are designed with proper lane markings, heavy traffic loadings could be evenly distributed onto approach traffic lanes.Discharge rates of the traffic streams can be increased even without extending their green duration times in a signal cycle so that conflicting traffic streams from other arms will not be adversely affected.This improvement scheme is attractive as it may be simply achieved by altering the lane marking patterns.Entire network connectivity would be improved.This design concept has been successful in designing individual signalized intersections [17].Intersection capacity has been increased significantly if lane markings and traffic signal settings are optimized in a unified framework [18][19][20][21].
In network design problems, we should ensure that feasible paths exist to connect all origin-destination (OD) demand flow pairs according to users' inputs.If the link connectivity to form a network configuration is not well designed, probably only one congesting path exists to carry the heaviest OD demand flow alone while other network links may not be fully utilized.This is possible due to fixed inputs of lane markings in conventional network designs.Indeed, optimizing the lane marking patterns across adjacent intersections could generate more feasible paths for users' route choices in traveling through a network.OD demand flows could be assigned onto these feasible paths to relieve the network congestion.The proposed methodology will be able to optimize the network configuration through optimizing the individual lane markings on approach lanes at intersection level so as to improve the overall network connectivity.It is expected that multiple OD demand flows of asymmetrical flow intensities could also be tackled.More importantly, the proposed algorithm is possible to eliminate unnecessary lane markings which may lead to redundant paths to connect some noncritical OD pairs.Overall network efficiency is improved.To optimize the control efficiency, reducing the number of conflicting movements at intersections and then reducing the loss times within a signal cycle are the new model features.Numerical example will be given to demonstrate the enhancement of lane usage through establishing proper lane markings at intersection levels to improve the link connectivity and network configuration serving the assigned flow patterns for a signal-controlled network.Usual design objective to maximize the reserve capacities of all involved intersections will be adopted for optimization.The problem is a Binary-Mixed-Integer-Linear-Program (BMILP) that could be effectively solved by standard branch-and-bound routines.

Assigning Given OD Demand Flows into Path Flows
where  and  are, respectively, the origin and destination (nodes) in a study signal-controlled network,  and  are, respectively, the total numbers of origin and destination (nodes),  is a common flow multiplier to scale the given demand flows, ℎ , is the path number connecting origin  and destination ,  , is the total number of paths to connect origin  and destination ,  , is the given demand flow from origin  to destination  that should equal the sum of all path flows, and  ,,ℎ , is the path flow from origin  to destination  using the ℎ , path.In Figure 1, for example, there is an OD pair connecting origin  and destination  with 3 paths: ℎ , − 1, ℎ where  ,,ℎ , is a binary variable to denote the existence of the ℎ , path connecting origin  and destination ,  ,,ℎ , is a path flow from origin  to destination  using the ℎ , path, and  is an arbitrary large number (e.g., 1000).Referring to the example given in Figure 1, if  ,,ℎ , −1 = 600 pcu/h,  ,,ℎ , = 400 pcu/h, and  ,,ℎ , +1 = 0 pcu/h, then correspondingly  ,,ℎ , −1 =  ,,ℎ , = 1 and  ,,ℎ , +1 = 0. Paths ℎ , − 1 and ℎ , exist for connecting the OD pair from origin  and destination  but path ℎ , + 1 does not exist.

Demand Turning Flows at Intersections from Path Flows of
Different OD Pairs.In the present formulation, users' given OD demand flows are model inputs.Inside a study signalcontrolled network, multiple paths may be available for connecting an OD pair.These path flows may pass through different intersections from arm  to arm  at intersection .
For an intersection , turning flow from arm  to arm  may involve different OD pairs depending on their used paths.Therefore, the actual demand turning flows at intersections (from arm  to arm  making a turn) would be the sum of these involved path flows.Since we do not explicitly consider traffic signal coordination in the present study, demand arrivals obtained from the path flows are assumed to reach all intersections spontaneously.In Figure 2, for example, there are two origins  and   and one destination .For a typical intersection  = 12, there are 4 arms  =  = {1, 2, 3, 4}.A left-turn movement could be denoted by (, , ) = (12, 1, 2) or (12,2,3).From the two origins, suppose 4 paths exist to enter destination .At intersection  = 25 from arm  = 4 to arm  = 3, it is a right-turn movement.3 path flows make this right-turn.Thus, demand flow  ,, should be the sum of the 3 path flows  ,,ℎ , ,  ,,ℎ , +1 , and    ,,ℎ   , .In general, to model all paths and all OD pairs, we need the following constraint set to evaluate the demand turning flows at intersections where (, , ) is a mathematical function to identify the path ℎ , from origin  to destination  when (, , ) are given as inputs.

Existence of OD Demand Flows
, ≥  , ≥  , , ∀ ∈ ;  ∈ , where  , is a binary-type variable to denote whether OD demand flow from origin  to destination  exists.When OD flow  , is nonzero as given by users,  , should equal "1."  is an arbitrary positive number.

Existence of Path
where  ,,ℎ , is a binary variable to denote the existence of the ℎ , path connecting origin  and destination .And only those ℎ , ∈   , are considered (  , ⊂  , ) as paths from the set   , will utilize the same lane marking to turn to the same destination node  from different origins .In Figure 2, assume 3 paths exist to connect origin  and destination  in which two of them ℎ , and ℎ , + 1 require a right-turn lane marking to enter destination  at intersection  = 25.Therefore, the set relationships may be given as follows: (ℎ , , ℎ , + 1) ∈   , ⊂  , .With respect to users' input demand flows for this OD pair, it may be possible that all demand flows are assigned to the path ℎ , − 1 only and thus the corresponding path flows for the other two paths do not exist; that is,  ,,ℎ , =  ,,ℎ , +1 = 0. Besides, from another origin   , its path to reach the destination  also requires a right-turn lane marking at intersection  = 25.If the path flow for this OD pair is also zero, then the corresponding    ,,ℎ   , = 0.If similar zero path flow is found for all origins to enter this destination  using the right-turn lane marking (, , ) = (25, 4, 3), this right-turn lane marking then becomes unnecessary and thus the lane marking variable  ,,, on all approach lane  (taking summation) should be prohibited.Mathematically, we need a user defined function   (, , ℎ , ) to identify all intersections  and their corresponding turns from arm  to arm , (, , ) =   (, , ℎ , ) for entering destination .

Prevention of Unnecessary Lane Markings.
Having restricted those unnecessary lane markings at intersections near a destination node whenever path flows are zero as given in Section 2.7, there may still exist some other useless lane markings at other intersections where arms are not directly connecting to destination nodes.Those lane markings are not serving any path flow which should be restricted from existence.The following constraint set is developed so that these useless lane markings could all be prohibited: where  ,,ℎ , is a binary variable to denote the existence of the ℎ , path connecting origin  and destination  and Δ ,,,,,ℎ , is an auxiliary binary variable to identify the turn from arm  to arm  at intersection  to support the ℎ , path connecting origin  and destination .In Figure 2, for example, from origin  to destination  along the ℎ , path, at intersection  = 1, traffic flows should be turning from arm  = 1 to arm  = 3 and thus Δ =1,=1,=3,,,ℎ , = 1.

Matching the Assigned Lane Flows with respect to the
Demand Turning Flows at Intersections.From Section 2.3, (3) evaluates the turning flows  ,, entering intersections  from different arm  to different arm .These demand flows at an intersection will be further assigned onto different approach lanes  according to the optimized lane marking patterns.
A set of flow conservation constraints can be established as follows:

Establishing Minimum Lane Marking(s) on Approach
Lanes.For practical designs, every approach traffic lane should be utilized to discharge traffic demand flows.At least one lane marking permitting a turn movement should be designed for each approach lane.Shared lane markings are also feasible if two or three movement turns are permitted on approach lanes at the same time: where  is an arbitrary large positive number.If  ,,, = 0, that is, the movement is prohibited, the respective lane flow  ,,, must be zero.However, if  ,,, = 1, that is, the turn movement is permitted, the assigned lane flow can take on any positive value, as long as it satisfies the demand flow relationship in (11).

Lane Markings across Adjacent Lanes.
For any pair of adjacent approach traffic lanes,  (on the left) and +1 (on the right), from arm , if a left-turn (or straight-ahead) movement to arm  is permitted at lane  + 1, then for safety reasons, the right-turn and straight-ahead (or right-turn) movement to other arms should all be prohibited along approach lane , in order to eliminate any potential internal conflict within the same arm.A mathematical function Γ(, , ) could be derived to identify a left-turn Γ(, , ) = 1, a straight-ahead movement Γ(, , ) = 2, or a right-turn Γ(, , ) = 3.This can be specified by the following constraint set: Given the binary nature of the variables, if  ,,,+1 = 1, then  ,,  , must vanish for all   ∈ [Γ(, , ) < Γ(, ,   )], that is, the turn movement (, ,   , ) is prohibited.On the other hand, if  ,,,+1 = 0, then the binary variable  ,,  , can take on any value of 0 or 1 freely.

Ranges of Operating Cycle
Length.Define the minimum and maximum cycle lengths to be  min and  max , respectively, for practical operations.A reciprocal of cycle length  = 1/ is modeled in the optimization process so as to eliminate the potential nonlinearity in signal timing variables.The feasible range of the operating cycle length can now be restrained by the following constraint set: which ensures that the operating cycle length will fall into the feasible range of ( min ,  max ).Normally,  max is 120.0 seconds in Hong Kong and  min should be computed when all traffic signal settings are operating at minimum green duration times.

Synchronization of Traffic Signal Settings for Approach
Lanes and Turns.For safety reasons, if an approach lane is designed with a shared lane marking, the involved movements should receive identical signal timings.From arm  at intersection , if a turning movement to arm  is permitted on lane , the following constraint sets can be given to synchronize the traffic signal settings ∀ ̸ =  = 1, . . .,   ; ∀ = 1, . . .,  , ; ∀ = 1, . . ., , where  is an arbitrary large positive number.If a movement (, , , ) is permitted along approach lane  from arm  to arm  at intersection , we have  ,,, = 1 and hence the values on both sides of the inequalities in (17) become zero.This forces Θ ,, =  ,, and Φ ,, =  ,, from the constraint sets.Identical traffic signal settings for movement turns and approach lanes could be achieved.On the other hand, if this movement is not permitted, the constraint sets would be ineffective because (1 −  ,,, ) is equal to "1" and hence different traffic signal settings could be operated.

Operating Ranges of Starts of Green Times.
Since the traffic signal settings at intersections are cyclic in nature, green times can be started arbitrarily within a signal cycle as long as they satisfy other conditions and constraints in the problem formulation.Numerically, all starts of green times are confined within the range of (0, 1), that is, within one full traffic signal cycle 1 ≥  ,, ≥ 0, ∀ ̸ =  = 1, . . .,   ;  = 1, . . ., .
3.9.Operating Upper and Lower Bounds of Duration of Green Times.Duration of green times operating for an approach traffic lane is subject to a minimum value, called minimum green times.Since all traffic signal timings are operating in a signal cycle, the longest green duration times should be one full signal cycle.These upper and lower bounds can be restricted as follows: where  is an arbitrary large positive number.The constraint set is effective only when the two incompatible movements are permitted and the two respective lane markings exist; that is,  ,,, =  ,  ,  ,  = 1.

Identical Flow Factors across Adjacent Approach Lanes
with Identical Lane Marking.Assigning demand traffic flows to approach lanes is based on the queuing theory, for which the degrees of saturation on a pair of adjacent lanes, with at least one common lane marking, must be identical.From Section 3.7, the signal settings on this pair of adjacent lanes must be identical.To ensure identical degrees of saturation, the flow factors (defined as the total assigned flow divided by the saturation flow) of these adjacent lanes should be equalized.Let  ,, be the flow factor on approach lane  from arm  at intersection , which can be expressed as where Γ(, , ) = 2 is a straight-ahead movement, we can show using the saturation flow prediction equation in [23]  ,, = 1  ,, ∑

Objective Function for Optimization
Usually, there are three standard objectives for optimizing traffic signal settings: (1) capacity maximization, (2) cycle length minimization, and (3) delay minimization for individual signalized intersections.In the present study, traffic signal settings are optimized based on criteria (1) because they can be effectively formulated as Binary-Mixed-Integer-Linear-Programming (BMILP) problems and standard branch-andbound routines could be applied to solve the optimum solution [19].The problem of delay minimization involves nonlinear objective mathematical function which is a highly nonlinear and nonconvex mixed-integer problem in the lanebased optimization framework [17].We may leave this hard problem in future study.
For capacity maximization, we will borrow the concept that has been widely adopted for individual intersections.We introduce the common flow multiplier,  in which the demand turning flow matrix (demand flows from different arms to different arms) is multiplied by the single common flow multiplier  [19,21].The maximization process is to keep increasing the common flow multiplier and in turn enlarging the demand flow inputs to a signal-controlled system until the system reaches the maximum degree of saturation.Similarly, in the proposed signal-controlled network system, we take on the OD demand flows (matrix) as inputs.The scaled OD demand flows by the single common flow multiplier will be entered to the modeling network system until the maximum degree of saturation is reached for all the critical approach lanes serving for different turn movements matching the optimized set of lane markings.The resultant lane markings at various intersections and approach lanes should be linked up for connections of all network links.The assigned lane flows according to the optimized set of lane markings could be combined to form the respective path flows.And all path flows should be able to match the given OD demand flow patterns as the flow conservation purposes.The capacity maximization problem can therefore be formulated as a Binary-Mixed-Integer-Linear-Program: maximize subject to constraints in ( 1)-( 27).Standard branch-andbound solution technique could be applied to solve the optimum solutions (refer to the Appendix for the general solution process).

Case Study
In this section, we would like to demonstrate the proposed formulation to design a network configuration and optimize the link connectivity.A 4-intersection network is modeled,  = 4. Figure 3 shows the general arrangement of the case study network.There are 2 approach lanes,  , = 2, and 2 exit lanes,  , = 2, for all arms at all intersections.All left-hand lane is nearside lane and right-hand lane is nonnearside lane with saturation flows of  ,,=1 = 1,965 and  ,,=2 = 2,080 pcu/h (for straight-ahead movements), respectively.Lane markings (turning arrows) are free to design on approach traffic lanes.Saturation flows will be revised according to the turning proportions obtained from the optimized lane flow results.4 nodes are modeled as origins and destinations.Users' input demand flows are tabulated in Table 1.For example, arm  = 1 at intersection  = 1 is the origin node  = 1 and destination node  = 1.
From the input demand flows in Table 1, all demand flows from origins to destinations are given.Visually, we could expect that demand flows from one origin may have two feasible paths to reach other destinations.For example, from origin  = 1 to destination  = 3, the demand flow input is 300 pcu/h and these flows could enter intersections  = 1 and then  = 2 and then  = 3 or enter intersections  = 1 and then  = 4 and then  = 3 to reach the destination depending on the provisions of lane markings.If all relevant lane markings are available depending on the optimization results, then one of the two paths or both paths could be existed to carry the demand flow inputs.In the case study, Figure 4 plots all the feasible paths that could exist to connect all OD pairs.It is worth noting that, for the OD pair from origin  = 1 to destination  = 2, the first path ℎ 1,2 = 1 is a shorter path that only passes through two intersections to reach the destination while the second path ℎ 1,2 = 2 is a longer path that needs to pass through four intersections.Similar short and long paths are found in some other OD pairs such as origin  = 2 and destination  = 1.To effectively demonstrate our proposed design method to optimize the lane marking patterns at intersections to connect all the given OD pairs, we allow three OD pairs ( = 1,  = 3), ( = 1,  = 4), and ( = 2,  = 4) with path choices and all remaining OD pairs are restricted to use the shorter path only (passing through less numbers of intersections).It would be clearer to illustrate the optimization results.Paths in solid lines in Figure 4 will be modeled and other paths in dashed lines are disabled in the case study.Turning radius is set to be 12.0 m for all left-and right-turn movements from both nearside and nonnearside lanes.All  ,,,ℎ , are all set to be 1.0 assuming all input demand flows once entering the network system will appear at all downstream intersections.The arbitrary large positive number will take on  = 10,000.Clearance times and minimum green duration times are 6.0 and 5.0 seconds, respectively.Effective green duration times are 1.0 second longer than the actual (or displayed) green duration times; that is,  = 1.0 second.Maximum degree of saturation for all approach lanes is set to be 0.9.
Based on all the above users' inputs, then the proposed formulation can be applied to optimize the common flow  multiplier to maximize the capacity of the signal-controlled network.And the optimized common flow multiplier  = 1.7106 with 71.06% reserve capacity (for the same sets of demand flow inputs and problem settings, if the signalcontrolled network is operated with a set of worse noncooperated lane markings, the common flow multiplier can be as low as 0.8517 implying that the system is overloaded by 14.83%).Figure 5 gives detailed optimization results on lane markings, assigned lane flows, and traffic signal settings.For example, at intersection  = 1, from arm  = 1, a left-turn lane marking is optimized to allow left-turn traffic to enter arm  = 2 along the nearside lane  = 1.Next to the nearside lane, there is another nonnearside lane  = 2 with a straight-ahead lane marking for traffic turning from arm  = 1 to arm  = 3.
Starts of green times and ends of green times (in seconds) are given inside brackets.Their differences (ends of green times -starts of green times) are the actual green duration times.All effective green times equal actual green times plus one second numerically due to  = 1.0 second.Numbers next to the lane marking arrows are the respective assigned lane turning flows in pcu/h.Inside the 4-intersection network, 6 shared lane markings are optimized to permit more than one turning movement along one approach lane.For each of the four intersections in the case study intersection, it consists of 3 arms (T-intersection) and there are two approach lanes and two exit lanes in each arm.Details of assigned lane flows which should be consistent with the lane markings are given in Tables 2-5.Column 1 is the origin arm of the turning flows.Column 2 is the approach lane number.Columns 3-5 give the individual assigned lane flows.Column 6 is the total lane flows that are the sum of all assigned turning flows.Column 7 is the turning proportion along the approach lanes which will be used to revise the lane saturation flows.Column 8 tabulates the revised lane saturation flows taking the lane turning proportion into account.Column 9 (=Column 6/Column 8) is the flow factor for each individual approach lane.Columns 10 and 11 are the starts of green times and effective green duration times.The last Column 12 is the degree of saturation for approach lane.To verify the proposed formulation is robust, it is necessary to examine whether the assigned lane flows can be matching with the users' given demand flow patterns in Table 1.Referring to the remarks under Tables 2-5, the total assigned lane flows from corresponding arms (and approach lanes) can be retrieved and their summation flow values can be evaluated to exactly match the given demand flows.From Table 2, for example, arm  = 1 at intersection  = 1 is the origin  = 1; therefore the row sum of the total flow from arm  = 1 is 1,000 pcu/h which is the same as the total flow from origin  = 1 given in Table 1 (the last column).Similarly, demand flows from other three origins and exit flows entering the four destinations can all be traced similarly according to the remarks under Tables 2-5.
Another set of outputs that is worth discussing is the path flows.We mentioned earlier that only three OD pairs ( = 1,  = 3), ( = 1,  = 4), and ( = 2,  = 4) are given path choices.And all other OD pairs are restricted to use their shorter paths.Figure 6 presents the optimized path flow details.All paths given in solid lines in Figure 4 are all considered in the optimization process.For OD pair ( = 1,  = 4), we provide path choices for selections.Indeed, path ℎ , = 1 is a longer path passing through four intersections and ℎ , = 2 is a shorter one and thus the optimization results for these two paths are  =1,=4,ℎ , =1 = 0.00 pcu/h and  =1,=4,ℎ , =2 = 500.00pcu/h, respectively.It shows logical result.As for the OD pair ( = 1,  = 3), the users' input demand flow is  =1,=3 = 300 pcu/h and the two path flows are  =1,=3,ℎ , =1 = 246.5 pcu/h and  =1,=3,ℎ , =1 = 53.5 pcu/h.And for the OD pair ( = 2,  = 4), the users' input demand flow is  =2,=4 = 600 pcu/h and the two path flows are  =2,=4,ℎ , =1 = 505.2pcu/h and  =2,=4,ℎ , =2 = 94.8 pcu/h.For other OD pairs, all demand flows will be assigned onto the available paths.All these path flows will then serve as demand turning flow inputs at individual intersections for optimizing the lane markings and traffic signal settings at intersection levels as shown in Tables 2-5.Overall, to model the case study network containing four signalized intersections with two approach lanes and two exit lanes for each arm, there are totally 617 variables including 405 binary-type integer variables and 212 continuous variables.Number of linear governing constraints is 1,955 for binding the 617 variables.It takes less than 5 minutes of computing time in an intel i7 CPU of 2.67 GHz with 8.0 GB memory (RAM) to reach the global optimum solution.

Conclusions
In the proposed study, we successfully extend the lane-based design concept to build a new mathematical optimization model to optimize the individual lane usage including the lane markings, assigned lane flows, and their corresponding traffic signal settings.The optimized design provides multiple paths to connect the given origin and destination pairs through different turns at different intersections according to the optimized lane markings.Link connectivity and network configuration are enhanced and optimized in a unified framework together with the assigned lane flows and traffic signal settings.It has been found that the performance of the case study network would be overloaded by 14.83% when a set of noncooperated lane markings (without shared lane marking design) is implemented.Using the proposed optimization method, the entire signal-controlled network could be improved by possessing +71.06% reserve capacity (instead of being overloaded).The main contribution of this optimization result is to fix the lane markings to build a signal-controlled network for further sophisticated network analysis.The problem is formulated as a Binary-Mixed-Integer-Linear-Program (BMILP) which has been solved by a standard branch-and-bound routine for the global optimum solution.In the example, 4-intersection network with two approach and exit lanes, 405 binary variables, 212 continuous variables, and 1,955 linear governing constraints are required.Computation time is less than 5 minutes using a Gurobi solver in an i7 CPU computer system.We conclude that the proposed method is efficient and effective to design the signal-controlled network problem.
The proposed method does not take into account users' responses to the optimized network configuration settings.The modeling results serve like a system optimal setting in which the signalized intersections in a network could be optimized if users from origins could straightly follow the lane markings making their turns at intersections to reach their destinations ideally.Actual network performance could be deviated under users' equilibrium patterns.For further investigations, nonlinear link travel times and delays at intersections should be modeled so that proper responses from users and users' equilibrium patterns for realistic route choices could be obtained for better evaluations of the optimized network configurations.

Appendix
The BMILP problems for the capacity maximization can be standardized into the integer programming problem defined as follows: where x is a vector collecting all model variables including continuous and integer components and c is a vector of coefficients multiplied by the vector of model variables to form the objective function.A and b are coefficient matrix and a constant vector for setting up the problem constraints, respectively.A standard branch-and-bound technique starts by solving the program as if all integer variables are not restricted to being integers.It turns the original problem into a general linear programming (LP).The optimal solution can be found easily by either the simplex method or other graphical methods.The solution process can stop here if all the integer variables are found to be integers in this relaxed LP problem.The other strategy is to round each noninteger variable solution obtained from the LP results to its nearest integer as the optimal solution for the original integer program.However, some of the constraints may be violated during this rounding process making the solution infeasible.Thus, the problem has to be branched into two subproblems in which the solution space is divided into two regions.It requires a branching variable, which is normally picked from one of the integer variables, whose numerical value is less near to an integer.Based on the solution found in the LP problem relaxed from the original integer program, the two nearest integers of the branching variable solution become the two new bounds, which are then added to the two subproblems, in the form of constraints.For instance, one of the integer variables,  0 , is set as the branching variable and has a numerical value of 1.45 from the LP problem.Since  0 is supposed to be an integer, then the two new bounds can be set as  0 ≤ 1 and  0 ≥ 2 (=0 or =1 for binary variables) and added to the original constraint set to form the two subproblems without removing any feasible solution space.In each subproblem, if not all the integer variables have integer solutions, then further branching is required, generating a tree of subproblems.Once a subproblem is infeasible, then no further branching is necessary along that subproblem.An optimal solution found in a subproblem though it is feasible in the full problem, is not necessarily globally optimal.However, it can be regarded as a criterion to compare and trim the rest of the tree.If the objective solutions of those subproblems waiting for further branching (i.e., infeasible solutions because not all integer variables have integer solutions) are not better than the one from a feasible subproblem (i.e., a feasible solution because all integer variables have integer solutions), then further branching for those infeasible subproblems is not necessary and should be removed from further consideration.The search continues until all branching subproblems have been solved or no further branching subproblems exist.A computer package, called the MPL modeling system, integrating a Gurobi or CPLEX solver is used to implement the branchand-bound algorithm to solve the present BMILP for the capacity maximization problem.Origin node in a study signal-controlled network :

Notations
Destination node in a study signal-controlled network : Total number of origin nodes in a study signal-controlled network : Total number of destination nodes in a study signal-controlled network ℎ , : P a t hn u m b e rf r o mo r i g i n to destination   , : Total number of paths connecting origin  and destination   , : Set of paths connecting origin  and destination    , : A subset extracted from  , (  , ⊂  , ) in which the same lane marking from arm  to arm  at intersection  will be used to connect origin  and destination  (, , ): A mathematical function to identify origin  and destination  along path ℎ , that will make a turn from arm  to arm  at intersection    (, , ℎ , ): A mathematical function to identify all the turns from arm  to arm  on lane  at intersection  for a path ℎ , connecting origin  and destination  Γ(, , ): A mathematical function to identify the turn direction(s) from arm  to arm  at intersection  in which Γ(, , ) = 1 is a left-turn, Γ(, , ) = 2 is a straight-ahead movement, or Γ(, , ) = 3 is a right-turn : An arbitrary large positive number Ψ  : Set of incompatible signal groups at intersection  Ψ  : Set of incompatible turn movements at intersection

1 1 Figure 1 :
Figure 1: Paths connecting OD pairs in an example signal-controlled network.

1 Figure 2 :
Figure 2: Converting path flows to turning flows at intersections for all OD pairs.

Figure 5 :
Figure 5: Optimized network configurations including lane markings and assigned lane (turning) flows at individual intersections.

Figure 6 :
Figure 6: Optimized path flow details for the case study network matching the given OD demand flows (in pcu/h).
, : I n t e r s e c t i o nn u m b e r : Total number of intersections in a signal-controlled network , : Arm number of an intersection   : Total number of arms at intersection  : Approach lane number  , : T o t a ln u m b e ro fa p p r o a c hl a n e sf r o ma r m  at intersection   , : Number of exit lanes on arm  at intersection  : , , and ℎ , + 1.If  , = 1,000 pcu/h, then numerically 1,000 pcu/h should equal the sum of path flows for conservation,  ,,ℎ , −1 +  ,,ℎ , +  ,,ℎ , +1 .