# Lane-Based Optimization for Macroscopic Network Configuration Designs

Authors: C. K. Wong; Yi Liu
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

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 by maximizing the reserve capacity for the entire OD demand flow matrix. 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 network will be given to demonstrate the effectiveness of the proposed design methodology.


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## Body

## 1. 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 [1–4]. 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 [10–14]. Phase- or group-based 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 [18–21].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.


## 2. Constraints for Connecting Network Links

### 2.1. Assigning Given OD Demand Flows into Path Flows

(1)
μ
T
a
,
b
=
∑
h
a
,
b
=
1
H
-
a
,
b
P
a
,
b
,
h
a
,
b
,
∀
a
∈
A
,
b
∈
B
,where a and b are, respectively, the origin and destination (nodes) in a study signal-controlled network, A and B are, respectively, the total numbers of origin and destination (nodes), μ is a common flow multiplier to scale the given demand flows, ha,b is the path number connecting origin a and destination b, H-a,b is the total number of paths to connect origin a and destination b, Ta,b is the given demand flow from origin a to destination b that should equal the sum of all path flows, and Pa,b,ha,b is the path flow from origin a to destination b using the ha,b path. In Figure 1, for example, there is an OD pair connecting origin a and destination b with 3 paths: ha,b-1, ha,b, and ha,b+1. If Ta,b = 1,000 pcu/h, then numerically 1,000 pcu/h should equal the sum of path flows for conservation, Pa,b,ha,b-1+Pa,b,ha,b+Pa,b,ha,b+1.Figure 1
Paths connecting OD pairs in an example signal-controlled network.


### 2.2. Existence of a Path Connecting an OD Pair

(2)
M
2
P
a
,
b
,
h
a
,
b
≥
M
α
a
,
b
,
h
a
,
b
≥
P
a
,
b
,
h
a
,
b
,
∀
a
∈
A
,
b
∈
B
,
h
a
,
b
∈
H
a
,
b
,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b, Pa,b,ha,b is a path flow from origin a to destination b using the ha,b path, and M is an arbitrary large number (e.g., 1000). Referring to the example given in Figure 1, if Pa,b,ha,b-1 = 600 pcu/h, Pa,b,ha,b = 400 pcu/h, and Pa,b,ha,b+1 = 0 pcu/h, then correspondingly αa,b,ha,b-1 = αa,b,ha,b = 1 and αa,b,ha,b+1 = 0. Paths ha,b-1 and ha,b exist for connecting the OD pair from origin a and destination b but path ha,b+1 does not exist.


### 2.3. 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 signal-controlled network, multiple paths may be available for connecting an OD pair. These path flows may pass through different intersections from armi to arm j at intersection n. For an intersection n, turning flow from arm i to arm j may involve different OD pairs depending on their used paths. Therefore, the actual demand turning flows at intersections (from arm i to arm j 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 Figure2, for example, there are two origins a and a′ and one destination b. For a typical intersection n=12, there are 4 arms i=j=1,2,3,4. A left-turn movement could be denoted by n,i,j=12,1,2 or 12,2,3. From the two origins, suppose 4 paths exist to enter destination b. At intersection n=25 from arm i=4 to arm j=3, it is a right-turn movement. 3 path flows make this right-turn. Thus, demand flow Qn,i,j should be the sum of the 3 path flows Pa,b,ha,b, Pa,b,ha,b+1, and Pa′,b,ha′,b. In general, to model all paths and all OD pairs, we need the following constraint set to evaluate the demand turning flows at intersections(3)Qn,i,j=∑a=1A∑b=1B∑ha,b=1H-a,bPa,b,ha,b,∀a,b,ha,b∈Fn,i,j,where Fn,i,j is a mathematical function to identify the path ha,b from origin a to destination b when n,i,j are given as inputs.Figure 2
Converting path flows to turning flows at intersections for all OD pairs.


### 2.4. Existence of OD Demand Flows

(4)
M
β
a
,
b
≥
T
a
,
b
≥
β
a
,
b
,
∀
a
∈
A
;
b
∈
B
,where βa,b is a binary-type variable to denote whether OD demand flow from origin a to destination b exists. When OD flow Ta,b is nonzero as given by users, βa,b should equal “1.” M is an arbitrary positive number.


### 2.5. Existence of Path Flows for Connecting OD Pairs

From Section2.4, when OD demand flow Ta,b exists with nonzero user’s input, then a binary variable βa,b should equal “1” numerically. Whenever βa,b=1, at least one path should exist to serve the demand flow as governed by (5). For heavy OD demand flows, multiple paths may exist to share the loadings in different network links. Equation (5) may allow the existence of multiple paths (αa,b,ha,b>1) when βa,b=1.(5)∑ha,b=1H-a,bαa,b,ha,b≥βa,b,∀a∈A;b∈B,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b and βa,b is a binary-type variable to denote the existence of an OD flow from origin a to destination b.Also from Section2.4, if users give zero flow input for an OD pair, βa,b is forced to be zero numerically. And it in turn forces αa,b,ha,b to be zero. No path should exist as governed by(6)βa,b≥αa,b,ha,b,∀a∈A;∀b∈B;∀ha,b=1,…,H-a,b.


### 2.6. Establishing Lane Markings at Intersections to Form a Path Connecting OD Pairs

(7)
∑
k
δ
n
,
i
,
j
,
k
≥
α
a
,
b
,
h
a
,
b
,
∀
n
,
i
,
j
,
k
∈
F
′
a
,
b
,
h
a
,
b
,where δn,i,j,k is a binary-type variable to denote the existence of a lane marking permitting a turn from arm i to arm j on lane k at intersection n and αa,b,ha,b is a binary variable to model the existence of the ha,b path connecting origin a and destination b. Whenever a path exists, αa,b,ha,b=1, associated lane markings for turns at intersections must exist on approach lane k so that path ha,b can be established to connect origin a and destination b. And F′(a,b,ha,b) is a user defined mathematical function to identify all movement turns from arm i to arm j on lane k at intersection n for path ha,b connecting origin a and destination b.


### 2.7. Prohibiting Lane Markings for Nonexistence of Destination

Depending on users’ OD demand flow inputs, some nodes could be serving as origins but not as destinations. Therefore, corresponding turns entering these nodes at various intersections may be unnecessary. Respective lane markings should be prohibited for existence(8)∑a=1A∑ha,b∈Ha,b′αa,b,ha,b≥∑k=1Ln,iδn,i,j,k,∀b∈B,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b. And only those ha,b∈Ha,b′ are considered (Ha,b′⊂Ha,b) as paths from the set Ha,b′ will utilize the same lane marking to turn to the same destination node b from different origins a. In Figure 2, assume 3 paths exist to connect origin a and destination b in which two of them ha,b and ha,b+1 require a right-turn lane marking to enter destination b at intersection n=25. Therefore, the set relationships may be given as follows: ha,b,ha,b+1∈Ha,b′⊂Ha,b. With respect to users’ input demand flows for this OD pair, it may be possible that all demand flows are assigned to the path ha,b-1 only and thus the corresponding path flows for the other two paths do not exist; that is, αa,b,ha,b=αa,b,ha,b+1=0. Besides, from another origin a′, its path to reach the destination b also requires a right-turn lane marking at intersection n=25. If the path flow for this OD pair is also zero, then the corresponding αa′,b,ha′,b=0. If similar zero path flow is found for all origins to enter this destination b using the right-turn lane marking n,i,j=(25,4,3), this right-turn lane marking then becomes unnecessary and thus the lane marking variable δn,i,j,k on all approach lane k (taking summation) should be prohibited. Mathematically, we need a user defined function F′(a,b,ha,b) to identify all intersections n and their corresponding turns from arm i to arm j, n,i,j=F′a,b,ha,b for entering destination b.


### 2.8. 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 Section2.7, 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:(9)Δn,i,j,a,b,ha,b=αa,b,ha,b,∀n,i,j=F′a,b,ha,b,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b and Δn,i,j,a,b,ha,b is an auxiliary binary variable to identify the turn from arm i to arm j at intersection n to support the ha,b path connecting origin a and destination b. In Figure 2, for example, from origin a to destination b along the ha,b path, at intersection n=1, traffic flows should be turning from arm i=1 to arm j=3 and thus Δn=1,i=1,j=3,a,b,ha,b = 1. Similarly, traffic flows should be turning from arm i=4 to arm j=2 at intersection n=24 and thus Δn=24,i=4,j=2,a,b,ha,b = 1. More importantly, at the same intersection n=24, no path flow turns left from arm 4 to arm 1 and therefore Δn=24,i=4,j=1,a,b,ha,b = 0. Again, the user defined mathematical function F′a,b,ha,b is to identify all turns from arm i to arm j at intersection n with n,i,j=F′a,b,ha,b.With the auxiliary variableΔn,i,j,a,b,ha,b, we can determine whether a specific lane marking δn,i,j,k turning from arm i to arm j at intersection n is required or not. The lane marking is considered to be useless if all path flows from all OD pairs do not require it along all approach lanes. The following constraint set is then introduced to prevent all useless lane markings from existence in the optimization process(10)M2∑k=1Ln,iδn,i,j,k≥M∑a=1A∑b=1B∑ha,b=1H-a,bΔn,i,j,a,b,ha,b≥∑k=1Ln,iδn,i,j,k,∀n∈N,∀i∈In,∀j∈In.In Figure2, for example, at intersection n=13, two paths exist for connecting two different origins and one destination. Both of them require a left-turn from arm i=1 to arm j=2. In this case, we have Δn=13,i=1,j=2,a,b,ha,b+1=Δn=13,i=1,j=2,a′,b,ha′,b=1 and therefore at least one left-turn lane marking on one of the approach lanes must exist; that is, ∑k=1Ln,iδn=13,i=1,j=2,k≥1. Conversely, no path exists using straight-ahead movement and right-turn movement at intersection n=13, respectively, from arm i=1 to arm j=3 and from arm i=1 to arm j=4. Numerically, Δn=13,i=1,j=3,a,b,ha,b+1=Δn=13,i=1,j=4,a,b,ha,b+1=Δn=13,i=1,j=3,a′,b,ha′,b=Δn=13,i=1,j=4,a′,b,ha′,b=0 should be required, correspondingly. In turn, M∑a=1A∑b=1B∑ha,b=1H-a,bΔn=13,i=1,j=4,a,b,ha,b=0 and it then forces ∑k=1Ln,iδn=13,i=1,j=2,k=0. Lane markings for straight-ahead and right-turn movements are useless and should be prohibited for existence.


## 2.1. Assigning Given OD Demand Flows into Path Flows

(1)
μ
T
a
,
b
=
∑
h
a
,
b
=
1
H
-
a
,
b
P
a
,
b
,
h
a
,
b
,
∀
a
∈
A
,
b
∈
B
,where a and b are, respectively, the origin and destination (nodes) in a study signal-controlled network, A and B are, respectively, the total numbers of origin and destination (nodes), μ is a common flow multiplier to scale the given demand flows, ha,b is the path number connecting origin a and destination b, H-a,b is the total number of paths to connect origin a and destination b, Ta,b is the given demand flow from origin a to destination b that should equal the sum of all path flows, and Pa,b,ha,b is the path flow from origin a to destination b using the ha,b path. In Figure 1, for example, there is an OD pair connecting origin a and destination b with 3 paths: ha,b-1, ha,b, and ha,b+1. If Ta,b = 1,000 pcu/h, then numerically 1,000 pcu/h should equal the sum of path flows for conservation, Pa,b,ha,b-1+Pa,b,ha,b+Pa,b,ha,b+1.Figure 1
Paths connecting OD pairs in an example signal-controlled network.


## 2.2. Existence of a Path Connecting an OD Pair

(2)
M
2
P
a
,
b
,
h
a
,
b
≥
M
α
a
,
b
,
h
a
,
b
≥
P
a
,
b
,
h
a
,
b
,
∀
a
∈
A
,
b
∈
B
,
h
a
,
b
∈
H
a
,
b
,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b, Pa,b,ha,b is a path flow from origin a to destination b using the ha,b path, and M is an arbitrary large number (e.g., 1000). Referring to the example given in Figure 1, if Pa,b,ha,b-1 = 600 pcu/h, Pa,b,ha,b = 400 pcu/h, and Pa,b,ha,b+1 = 0 pcu/h, then correspondingly αa,b,ha,b-1 = αa,b,ha,b = 1 and αa,b,ha,b+1 = 0. Paths ha,b-1 and ha,b exist for connecting the OD pair from origin a and destination b but path ha,b+1 does not exist.


## 2.3. 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 signal-controlled network, multiple paths may be available for connecting an OD pair. These path flows may pass through different intersections from armi to arm j at intersection n. For an intersection n, turning flow from arm i to arm j may involve different OD pairs depending on their used paths. Therefore, the actual demand turning flows at intersections (from arm i to arm j 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 Figure2, for example, there are two origins a and a′ and one destination b. For a typical intersection n=12, there are 4 arms i=j=1,2,3,4. A left-turn movement could be denoted by n,i,j=12,1,2 or 12,2,3. From the two origins, suppose 4 paths exist to enter destination b. At intersection n=25 from arm i=4 to arm j=3, it is a right-turn movement. 3 path flows make this right-turn. Thus, demand flow Qn,i,j should be the sum of the 3 path flows Pa,b,ha,b, Pa,b,ha,b+1, and Pa′,b,ha′,b. In general, to model all paths and all OD pairs, we need the following constraint set to evaluate the demand turning flows at intersections(3)Qn,i,j=∑a=1A∑b=1B∑ha,b=1H-a,bPa,b,ha,b,∀a,b,ha,b∈Fn,i,j,where Fn,i,j is a mathematical function to identify the path ha,b from origin a to destination b when n,i,j are given as inputs.Figure 2
Converting path flows to turning flows at intersections for all OD pairs.


## 2.4. Existence of OD Demand Flows

(4)
M
β
a
,
b
≥
T
a
,
b
≥
β
a
,
b
,
∀
a
∈
A
;
b
∈
B
,where βa,b is a binary-type variable to denote whether OD demand flow from origin a to destination b exists. When OD flow Ta,b is nonzero as given by users, βa,b should equal “1.” M is an arbitrary positive number.


## 2.5. Existence of Path Flows for Connecting OD Pairs

From Section2.4, when OD demand flow Ta,b exists with nonzero user’s input, then a binary variable βa,b should equal “1” numerically. Whenever βa,b=1, at least one path should exist to serve the demand flow as governed by (5). For heavy OD demand flows, multiple paths may exist to share the loadings in different network links. Equation (5) may allow the existence of multiple paths (αa,b,ha,b>1) when βa,b=1.(5)∑ha,b=1H-a,bαa,b,ha,b≥βa,b,∀a∈A;b∈B,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b and βa,b is a binary-type variable to denote the existence of an OD flow from origin a to destination b.Also from Section2.4, if users give zero flow input for an OD pair, βa,b is forced to be zero numerically. And it in turn forces αa,b,ha,b to be zero. No path should exist as governed by(6)βa,b≥αa,b,ha,b,∀a∈A;∀b∈B;∀ha,b=1,…,H-a,b.


## 2.6. Establishing Lane Markings at Intersections to Form a Path Connecting OD Pairs

(7)
∑
k
δ
n
,
i
,
j
,
k
≥
α
a
,
b
,
h
a
,
b
,
∀
n
,
i
,
j
,
k
∈
F
′
a
,
b
,
h
a
,
b
,where δn,i,j,k is a binary-type variable to denote the existence of a lane marking permitting a turn from arm i to arm j on lane k at intersection n and αa,b,ha,b is a binary variable to model the existence of the ha,b path connecting origin a and destination b. Whenever a path exists, αa,b,ha,b=1, associated lane markings for turns at intersections must exist on approach lane k so that path ha,b can be established to connect origin a and destination b. And F′(a,b,ha,b) is a user defined mathematical function to identify all movement turns from arm i to arm j on lane k at intersection n for path ha,b connecting origin a and destination b.


## 2.7. Prohibiting Lane Markings for Nonexistence of Destination

Depending on users’ OD demand flow inputs, some nodes could be serving as origins but not as destinations. Therefore, corresponding turns entering these nodes at various intersections may be unnecessary. Respective lane markings should be prohibited for existence(8)∑a=1A∑ha,b∈Ha,b′αa,b,ha,b≥∑k=1Ln,iδn,i,j,k,∀b∈B,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b. And only those ha,b∈Ha,b′ are considered (Ha,b′⊂Ha,b) as paths from the set Ha,b′ will utilize the same lane marking to turn to the same destination node b from different origins a. In Figure 2, assume 3 paths exist to connect origin a and destination b in which two of them ha,b and ha,b+1 require a right-turn lane marking to enter destination b at intersection n=25. Therefore, the set relationships may be given as follows: ha,b,ha,b+1∈Ha,b′⊂Ha,b. With respect to users’ input demand flows for this OD pair, it may be possible that all demand flows are assigned to the path ha,b-1 only and thus the corresponding path flows for the other two paths do not exist; that is, αa,b,ha,b=αa,b,ha,b+1=0. Besides, from another origin a′, its path to reach the destination b also requires a right-turn lane marking at intersection n=25. If the path flow for this OD pair is also zero, then the corresponding αa′,b,ha′,b=0. If similar zero path flow is found for all origins to enter this destination b using the right-turn lane marking n,i,j=(25,4,3), this right-turn lane marking then becomes unnecessary and thus the lane marking variable δn,i,j,k on all approach lane k (taking summation) should be prohibited. Mathematically, we need a user defined function F′(a,b,ha,b) to identify all intersections n and their corresponding turns from arm i to arm j, n,i,j=F′a,b,ha,b for entering destination b.


## 2.8. 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 Section2.7, 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:(9)Δn,i,j,a,b,ha,b=αa,b,ha,b,∀n,i,j=F′a,b,ha,b,where αa,b,ha,b is a binary variable to denote the existence of the ha,b path connecting origin a and destination b and Δn,i,j,a,b,ha,b is an auxiliary binary variable to identify the turn from arm i to arm j at intersection n to support the ha,b path connecting origin a and destination b. In Figure 2, for example, from origin a to destination b along the ha,b path, at intersection n=1, traffic flows should be turning from arm i=1 to arm j=3 and thus Δn=1,i=1,j=3,a,b,ha,b = 1. Similarly, traffic flows should be turning from arm i=4 to arm j=2 at intersection n=24 and thus Δn=24,i=4,j=2,a,b,ha,b = 1. More importantly, at the same intersection n=24, no path flow turns left from arm 4 to arm 1 and therefore Δn=24,i=4,j=1,a,b,ha,b = 0. Again, the user defined mathematical function F′a,b,ha,b is to identify all turns from arm i to arm j at intersection n with n,i,j=F′a,b,ha,b.With the auxiliary variableΔn,i,j,a,b,ha,b, we can determine whether a specific lane marking δn,i,j,k turning from arm i to arm j at intersection n is required or not. The lane marking is considered to be useless if all path flows from all OD pairs do not require it along all approach lanes. The following constraint set is then introduced to prevent all useless lane markings from existence in the optimization process(10)M2∑k=1Ln,iδn,i,j,k≥M∑a=1A∑b=1B∑ha,b=1H-a,bΔn,i,j,a,b,ha,b≥∑k=1Ln,iδn,i,j,k,∀n∈N,∀i∈In,∀j∈In.In Figure2, for example, at intersection n=13, two paths exist for connecting two different origins and one destination. Both of them require a left-turn from arm i=1 to arm j=2. In this case, we have Δn=13,i=1,j=2,a,b,ha,b+1=Δn=13,i=1,j=2,a′,b,ha′,b=1 and therefore at least one left-turn lane marking on one of the approach lanes must exist; that is, ∑k=1Ln,iδn=13,i=1,j=2,k≥1. Conversely, no path exists using straight-ahead movement and right-turn movement at intersection n=13, respectively, from arm i=1 to arm j=3 and from arm i=1 to arm j=4. Numerically, Δn=13,i=1,j=3,a,b,ha,b+1=Δn=13,i=1,j=4,a,b,ha,b+1=Δn=13,i=1,j=3,a′,b,ha′,b=Δn=13,i=1,j=4,a′,b,ha′,b=0 should be required, correspondingly. In turn, M∑a=1A∑b=1B∑ha,b=1H-a,bΔn=13,i=1,j=4,a,b,ha,b=0 and it then forces ∑k=1Ln,iδn=13,i=1,j=2,k=0. Lane markings for straight-ahead and right-turn movements are useless and should be prohibited for existence.


## 3. Constraints for Individual Signalized Intersections

### 3.1. Matching the Assigned Lane Flows with respect to the Demand Turning Flows at Intersections

From Section2.3, (3) evaluates the turning flows Qn,i,j entering intersections n from different arm i to different arm j. These demand flows at an intersection will be further assigned onto different approach lanes k according to the optimized lane marking patterns. A set of flow conservation constraints can be established as follows:(11)Qn,i,j=∑k=1Ln,iqn,i,j,k,∀i≠j=1,…,In;∀n=1,…,N.


### 3.2. 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:(12)∑∀j≠iδn,i,j,k≥1,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.


### 3.3. Assigning Lane Markings without Exceeding Downstream Exit Lane Number

For each turn movement from armi to arm j at intersection n, number of exit lanes on arm j should always be greater than or equal to the number of assigned lane markings to enter arm j. Undesirable traffic merging at downstream exit lanes may occur if number of lane markings is more than exit lane number. Constraint set in (13) should be given to control the number of lane markings:(13)En,j≥∑k=1Ln,iδn,i,j,k,∀i≠j=1,…,In;∀n=1,…,N.


### 3.4. Ensuring Consistency between Lane Markings and Assigned Lane Flows

If a traffic movement is prohibited on an approach traffic lane, the assigned lane flow of the associated turn movement should also be vanished. A linear constraint set can be defined as follows:(14)Mδn,i,j,k≥qn,i,j,k≥0,∀k=1,…,Ln,i;∀i≠j=1,…,In;∀n=1,…,N,where M is an arbitrary large positive number. If δn,i,j,k=0, that is, the movement is prohibited, the respective lane flow qn,i,j,k must be zero. However, if δn,i,j,k=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).


### 3.5. Lane Markings across Adjacent Lanes

For any pair of adjacent approach traffic lanes,k (on the left) and k+1 (on the right), from arm i, if a left-turn (or straight-ahead) movement to arm j is permitted at lane k+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 k, in order to eliminate any potential internal conflict within the same arm. A mathematical function Γn,i,j could be derived to identify a left-turn Γn,i,j=1, a straight-ahead movement Γn,i,j=2, or a right-turn Γn,i,j=3. This can be specified by the following constraint set:(15)1-δn,i,j,k+1≥δn,i,j′,k≥δn,i,j,k+1-1,∀j′∈Γn,i,j<Γn,i,j′;∀i≠j=1,…,In;∀k=1,…,Ln,i-1;n=1,…,N.Given the binary nature of the variables, ifδn,i,j,k+1=1, then δn,i,j′,k must vanish for all j′∈Γn,i,j<Γn,i,j′, that is, the turn movement (n,i,j′,k) is prohibited. On the other hand, if δn,i,j,k+1=0, then the binary variable δn,i,j′,k can take on any value of 0 or 1 freely.


### 3.6. Ranges of Operating Cycle Length

Define the minimum and maximum cycle lengths to becmin and cmax, respectively, for practical operations. A reciprocal of cycle length ζ=1/c 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:(16)1cmin≥ζ≥1cmax,which ensures that the operating cycle length will fall into the feasible range of (cmin,cmax). Normally, cmax is 120.0 seconds in Hong Kong and cmin should be computed when all traffic signal settings are operating at minimum green duration times.


### 3.7. 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 armi at intersection n, if a turning movement to arm j is permitted on lane k, the following constraint sets can be given to synchronize the traffic signal settings(17)M1-δn,i,j,k≥Θn,i,k-θn,i,j≥-M1-δn,i,j,k,M1-δn,i,j,k≥Φn,i,k-ϕn,i,j≥-M1-δn,i,j,k,∀i≠j=1,…,In;∀k=1,…,Ln,i;∀n=1,…,N, where M is an arbitrary large positive number. If a movement (n,i,j,k) is permitted along approach lane k from arm i to arm j at intersection n, we have δn,i,j,k=1 and hence the values on both sides of the inequalities in (17) become zero. This forces Θn,i,k=θn,i,j and Φn,i,k=ϕn,i,j 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-δn,i,j,k) is equal to “1” and hence different traffic signal settings could be operated.


### 3.8. 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(18)1≥θn,i,j≥0,∀i≠j=1,…,In;n=1,…,N.


### 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:(19)1≥ϕn,i,j≥gn,i,jζ,∀i≠j=1,…,In;n=1,…,N.


### 3.10. Regulating Displaying Sequence of Traffic Signal Timings

Any two signal groups (or movements),(n,i,j) and (n,i′,j′), are said to be incompatible within an intersection n, if movement paths (n,i,j) and (n,i′,j′) have intersection point at the common area of an intersection. Denote the set of incompatible signal groups as Ψ-n, which can be derived from Ψn, the set of incompatible movements. For any two incompatible signal groups (n,i,j) and (n,i′,j′) in Ψ-n, the order of signal displays is governed by a successor function [22], Ωn,i,j,i′,j′, where Ωn,i,j,i′,j′=0 if the start of green of signal group (n,i′,j′) follows that of signal group (n,i,j); and = 1 if the opposite is true. Therefore, the following constraints can be set for the successor functions:(20)Ωn,i,j,i′,j′+Ωn,i′,j′,i,j=1,∀n,i,j,n,i′,j′∈Ψ-n.


### 3.11. Provision of Clearance Times Separating Conflicting Movements

For any pair of incompatible movements, clearance time constraints are required when both movements are permitted through establishing the respective lane markings. For two incompatible traffic movements defined asun=(n,i,j,k) and vn=(n,i′,j′,k′), the following constraint set can be given to satisfy the clearance time requirements:(21)θn,i′,j′+Ωn,i,j,i′,j′+M2-δn,i,j,k-δn,i′,j′,k′≥θn,i,j+ϕn,i,j+ωn,un,vnζ,∀un,vn∈Ψn,where M 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, δn,i,j,k=δn,i′,j′,k′=1.


### 3.12. 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 Section3.7, 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 yn,i,k be the flow factor on approach lane k from arm i at intersection n, which can be expressed as(22)yn,i,k=∑∀j≠iqn,i,j,ksn,i,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.Since(23)fn,i,j,k=∑j≠iqn,i,j,k-qn,i,j′,k∑j≠iqn,i,j,k,∀j′∈Γn,i,j=2;∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N,where Γn,i,j=2 is a straight-ahead movement, we can show using the saturation flow prediction equation in [23](24)yn,i,k=1s-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.To equalize flow factors for two adjacent approach lanes with identical lane marking(s), we have(25)M2-δn,i,j,k-δn,i,j,k+1≥1s-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k-1s-n,i,k+1∑j≠i1+1.5rn,i,j,k+1qn,i,j,k+1≥-M2-δn,i,j,k-δn,i,j,k+1,∀k=1,…,Ln,i-1;∀i=1,…,In;∀n=1,…,N,where M is an arbitrary large positive number and k (on the left) and k+1 (on the right) are two adjacent approach lanes from arm i at intersection n. The constraint sets are effective only when δn,i,j,k=δn,i,j,k+1=1.


### 3.13. Maximum Acceptable Degree of Saturation

Letpn,i,k be the maximum degree of saturation on approach lane k from arm i at the intersection n in a study signal-controlled network. For an approach traffic lane k from arm i at intersection n, the degree of saturation can be given by(26)ρn,i,k=yn,i,kΦn,i,k+eζ,∀k=1,…,Ln,i;i=1,…,In;n=1,…,N,where ρn,i,k and e are the degree of saturation on lane k from arm i at intersection n and the difference between actual and effective greens (measured in time unit, usually taken as 1 second), respectively. The following constraint set can be given to ensure that the degree of saturation is below the maximum acceptable limit:(27)Φn,i,k+eζ≥1pn,i,ks-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.


## 3.1. Matching the Assigned Lane Flows with respect to the Demand Turning Flows at Intersections

From Section2.3, (3) evaluates the turning flows Qn,i,j entering intersections n from different arm i to different arm j. These demand flows at an intersection will be further assigned onto different approach lanes k according to the optimized lane marking patterns. A set of flow conservation constraints can be established as follows:(11)Qn,i,j=∑k=1Ln,iqn,i,j,k,∀i≠j=1,…,In;∀n=1,…,N.


## 3.2. 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:(12)∑∀j≠iδn,i,j,k≥1,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.


## 3.3. Assigning Lane Markings without Exceeding Downstream Exit Lane Number

For each turn movement from armi to arm j at intersection n, number of exit lanes on arm j should always be greater than or equal to the number of assigned lane markings to enter arm j. Undesirable traffic merging at downstream exit lanes may occur if number of lane markings is more than exit lane number. Constraint set in (13) should be given to control the number of lane markings:(13)En,j≥∑k=1Ln,iδn,i,j,k,∀i≠j=1,…,In;∀n=1,…,N.


## 3.4. Ensuring Consistency between Lane Markings and Assigned Lane Flows

If a traffic movement is prohibited on an approach traffic lane, the assigned lane flow of the associated turn movement should also be vanished. A linear constraint set can be defined as follows:(14)Mδn,i,j,k≥qn,i,j,k≥0,∀k=1,…,Ln,i;∀i≠j=1,…,In;∀n=1,…,N,where M is an arbitrary large positive number. If δn,i,j,k=0, that is, the movement is prohibited, the respective lane flow qn,i,j,k must be zero. However, if δn,i,j,k=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).


## 3.5. Lane Markings across Adjacent Lanes

For any pair of adjacent approach traffic lanes,k (on the left) and k+1 (on the right), from arm i, if a left-turn (or straight-ahead) movement to arm j is permitted at lane k+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 k, in order to eliminate any potential internal conflict within the same arm. A mathematical function Γn,i,j could be derived to identify a left-turn Γn,i,j=1, a straight-ahead movement Γn,i,j=2, or a right-turn Γn,i,j=3. This can be specified by the following constraint set:(15)1-δn,i,j,k+1≥δn,i,j′,k≥δn,i,j,k+1-1,∀j′∈Γn,i,j<Γn,i,j′;∀i≠j=1,…,In;∀k=1,…,Ln,i-1;n=1,…,N.Given the binary nature of the variables, ifδn,i,j,k+1=1, then δn,i,j′,k must vanish for all j′∈Γn,i,j<Γn,i,j′, that is, the turn movement (n,i,j′,k) is prohibited. On the other hand, if δn,i,j,k+1=0, then the binary variable δn,i,j′,k can take on any value of 0 or 1 freely.


## 3.6. Ranges of Operating Cycle Length

Define the minimum and maximum cycle lengths to becmin and cmax, respectively, for practical operations. A reciprocal of cycle length ζ=1/c 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:(16)1cmin≥ζ≥1cmax,which ensures that the operating cycle length will fall into the feasible range of (cmin,cmax). Normally, cmax is 120.0 seconds in Hong Kong and cmin should be computed when all traffic signal settings are operating at minimum green duration times.


## 3.7. 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 armi at intersection n, if a turning movement to arm j is permitted on lane k, the following constraint sets can be given to synchronize the traffic signal settings(17)M1-δn,i,j,k≥Θn,i,k-θn,i,j≥-M1-δn,i,j,k,M1-δn,i,j,k≥Φn,i,k-ϕn,i,j≥-M1-δn,i,j,k,∀i≠j=1,…,In;∀k=1,…,Ln,i;∀n=1,…,N, where M is an arbitrary large positive number. If a movement (n,i,j,k) is permitted along approach lane k from arm i to arm j at intersection n, we have δn,i,j,k=1 and hence the values on both sides of the inequalities in (17) become zero. This forces Θn,i,k=θn,i,j and Φn,i,k=ϕn,i,j 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-δn,i,j,k) is equal to “1” and hence different traffic signal settings could be operated.


## 3.8. 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(18)1≥θn,i,j≥0,∀i≠j=1,…,In;n=1,…,N.


## 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:(19)1≥ϕn,i,j≥gn,i,jζ,∀i≠j=1,…,In;n=1,…,N.


## 3.10. Regulating Displaying Sequence of Traffic Signal Timings

Any two signal groups (or movements),(n,i,j) and (n,i′,j′), are said to be incompatible within an intersection n, if movement paths (n,i,j) and (n,i′,j′) have intersection point at the common area of an intersection. Denote the set of incompatible signal groups as Ψ-n, which can be derived from Ψn, the set of incompatible movements. For any two incompatible signal groups (n,i,j) and (n,i′,j′) in Ψ-n, the order of signal displays is governed by a successor function [22], Ωn,i,j,i′,j′, where Ωn,i,j,i′,j′=0 if the start of green of signal group (n,i′,j′) follows that of signal group (n,i,j); and = 1 if the opposite is true. Therefore, the following constraints can be set for the successor functions:(20)Ωn,i,j,i′,j′+Ωn,i′,j′,i,j=1,∀n,i,j,n,i′,j′∈Ψ-n.


## 3.11. Provision of Clearance Times Separating Conflicting Movements

For any pair of incompatible movements, clearance time constraints are required when both movements are permitted through establishing the respective lane markings. For two incompatible traffic movements defined asun=(n,i,j,k) and vn=(n,i′,j′,k′), the following constraint set can be given to satisfy the clearance time requirements:(21)θn,i′,j′+Ωn,i,j,i′,j′+M2-δn,i,j,k-δn,i′,j′,k′≥θn,i,j+ϕn,i,j+ωn,un,vnζ,∀un,vn∈Ψn,where M 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, δn,i,j,k=δn,i′,j′,k′=1.


## 3.12. 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 Section3.7, 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 yn,i,k be the flow factor on approach lane k from arm i at intersection n, which can be expressed as(22)yn,i,k=∑∀j≠iqn,i,j,ksn,i,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.Since(23)fn,i,j,k=∑j≠iqn,i,j,k-qn,i,j′,k∑j≠iqn,i,j,k,∀j′∈Γn,i,j=2;∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N,where Γn,i,j=2 is a straight-ahead movement, we can show using the saturation flow prediction equation in [23](24)yn,i,k=1s-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.To equalize flow factors for two adjacent approach lanes with identical lane marking(s), we have(25)M2-δn,i,j,k-δn,i,j,k+1≥1s-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k-1s-n,i,k+1∑j≠i1+1.5rn,i,j,k+1qn,i,j,k+1≥-M2-δn,i,j,k-δn,i,j,k+1,∀k=1,…,Ln,i-1;∀i=1,…,In;∀n=1,…,N,where M is an arbitrary large positive number and k (on the left) and k+1 (on the right) are two adjacent approach lanes from arm i at intersection n. The constraint sets are effective only when δn,i,j,k=δn,i,j,k+1=1.


## 3.13. Maximum Acceptable Degree of Saturation

Letpn,i,k be the maximum degree of saturation on approach lane k from arm i at the intersection n in a study signal-controlled network. For an approach traffic lane k from arm i at intersection n, the degree of saturation can be given by(26)ρn,i,k=yn,i,kΦn,i,k+eζ,∀k=1,…,Ln,i;i=1,…,In;n=1,…,N,where ρn,i,k and e are the degree of saturation on lane k from arm i at intersection n and the difference between actual and effective greens (measured in time unit, usually taken as 1 second), respectively. The following constraint set can be given to ensure that the degree of saturation is below the maximum acceptable limit:(27)Φn,i,k+eζ≥1pn,i,ks-n,i,k∑j≠i1+1.5rn,i,j,kqn,i,j,k,∀k=1,…,Ln,i;∀i=1,…,In;∀n=1,…,N.


## 4. 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-and-bound 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 lane-based 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,μ in which the demand turning flow matrix (demand flows from different arms to different arms) is multiplied by the single common flow multiplier μ [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:(28)maximizeΛ=Λb,Λcμ,subject to constraints in (1)–(27). Standard branch-and-bound solution technique could be applied to solve the optimum solutions (refer to the Appendix for the general solution process).


## 5. 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,n=4. Figure 3 shows the general arrangement of the case study network. There are 2 approach lanes, Ln,i=2, and 2 exit lanes, En,j=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 s-n,i,k=1 = 1,965 and s-n,i,k=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 i=1 at intersection n=1 is the origin node a=1 and destination node b=1. From the input demand flows in Table 1, all demand flows from origins to destinations are given.Table 1
Demand OD flowsTa,b.
T
a
,
b (pcu/h)
Destination,b
Total flows from origin
1
2
3
4
Origin,a
1
—
200
300
500
1,000
2
300
—
200
600
1,100
3
100
200
—
200
500
4
200
200
100
—
500
Total flows entering destination
600
600
600
1,300Figure 3
A case study signal-controlled network with 4 signalized intersections.Visually, we could expect that demand flows from one origin may have two feasible paths to reach other destinations. For example, from origina=1 to destination b=3, the demand flow input is 300 pcu/h and these flows could enter intersections n=1 and then n=2 and then n=3 or enter intersections n=1 and then n=4 and then n=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 a=1 to destination b=2, the first path h1,2=1 is a shorter path that only passes through two intersections to reach the destination while the second path h1,2=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 a=2 and destination b=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 a=1,b=3, a=1,b=4, and a=2,b=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 σn,a,b,ha,b 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 M = 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, e=1.0 second. Maximum degree of saturation for all approach lanes is set to be 0.9.Figure 4
Feasible paths in the case study network connecting the 4 OD pairs.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 multiplierμ = 1.7106 with 71.06% reserve capacity (for the same sets of demand flow inputs and problem settings, if the signal-controlled 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 n=1, from arm i=1, a left-turn lane marking is optimized to allow left-turn traffic to enter arm j=2 along the nearside lane k=1. Next to the nearside lane, there is another nonnearside lane k=2 with a straight-ahead lane marking for traffic turning from arm i=1 to arm j=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 e=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 i=1 at intersection n=1 is the origin a=1; therefore the row sum of the total flow from arm i=1 is 1,000 pcu/h which is the same as the total flow from origin a=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.Table 2
Optimization results for intersectionn=1.
From arm,i
Lane,k
To arm,j  Assigned lane flow (pcu/h)
Total lane flow (pcu/h)
Turning proportion
Saturation flow (pcu/h)
Flow factor,yn,i,k
Start of green (s)
Effective green (s)
Degree of saturation
1
2
3
1
1
—
446.5185
0.0000
446.5185
1.0000
1746.67
0.2556
0.00
120.00
0.2556
2
—
0.0000
553.4815
553.4815
0.0000
2105.00
0.2629
0.00
78.80
0.4004
Total
—
446.5185
553.4815
1000.0000
2
1
144.0704
—
94.8377
238.9081
1.0000
1746.67
0.1368
83.80
31.20
0.5261
2
255.9296
—
0.0000
255.9296
1.0000
1871.11
0.1368
83.80
31.20
0.5261
Total
400.0000
—
94.8377
494.8377
3
1
96.5602
0.0000
—
96.5602
0.0000
1965.00
0.0491
0.00
78.80
0.0748
2
103.4398
0.0000
—
103.4398
0.0000
2105.00
0.0491
0.00
78.80
0.0748
Total
200.0000
0.0000
—
200.0000
Remarks. Lane marking is prohibited when assigned lane flow = 0.000 pcu/h; 1000.0000 pcu/h is the total demand flow from origin a=1; 400.0000 and 200.0000 pcu/h are the total 600.0000 pcu/h exit flows entering destination b=1.Table 3
Optimization results for intersectionn=2.
From arm,i
Lane,k
To arm,j  Assigned lane flow (pcu/h)
Total lane flow (pcu/h)
Turning proportion
Saturation flow (pcu/h)
Flow factor,yn,i,k
Start of green (s)
Effective green (s)
Degree of saturation
1
3
4
1
1
—
705.1623
0.0000
705.1623
0.0000
1965.00
0.3589
33.15
81.85
0.5261
2
—
0.0000
394.8377
394.8377
1.0000
1871.11
0.2110
33.15
48.13
0.5261
Total
—
705.1623
394.8377
1100.0000
3
1
134.9355
—
100.0000
234.9355
0.4256
1865.73
0.1259
86.28
28.72
0.5261
2
265.0645
—
0.0000
265.0645
0.0000
2105.00
0.1259
86.28
28.72
0.5261
Total
400.0000
—
100.0000
500.0000
4
1
200.0000
15.5795
—
215.5795
1.0000
1746.67
0.1234
0.00
28.15
0.5261
2
0.0000
230.9389
—
230.9389
1.0000
1871.11
0.1234
0.00
28.15
0.5261
Total
200.0000
246.5185
—
446.5185
Remarks. Lane marking is prohibited when assigned lane flow = 0.000 pcu/h; 1100.0000 pcu/h is the total demand flow from origin a=2; 400.0000 and 200.0000 pcu/h are the total 600.0000 pcu/h exit flows entering destination b=2.Table 4
Optimization results for intersectionn=3.
From arm,i
Lane,k
To arm,j  Assigned lane flow (pcu/h)
Total lane flow (pcu/h)
Turning proportion
Saturation flow (pcu/h)
Flow factor,yn,i,k
Start of green (s)
Effective green (s)
Degree of saturation
1
3
4
1
1
—
446.5185
38.6138
485.1323
0.0796
1945.64
0.2493
58.13
56.87
0.5261
2
—
0.0000
466.5485
466.5485
1.0000
1871.11
0.2493
58.13
56.87
0.5261
Total
—
446.5185
505.1623
951.6807
3
1
28.4705
—
200.0000
228.4705
0.8754
1771.19
0.1290
23.71
29.42
0.5261
2
271.5295
—
0.0000
271.5295
0.0000
2105.00
0.1290
23.71
29.42
0.5261
Total
300.0000
—
200.0000
500.0000
4
1
200.0000
0.0000
—
200.0000
1.0000
1746.67
0.1145
58.13
80.58
0.1705
2
0.0000
153.4815
—
153.4815
1.0000
1871.11
0.0820
0.00
18.71
0.5261
Total
200.0000
153.4815
—
353.4815
Remarks. Lane marking is prohibited when assigned lane flow = 0.000 pcu/h; 500.0000 pcu/h is the total demand flow from origin a=3; 446.5185 and 153.4815 pcu/h are the total 600.0000 pcu/h exit flows entering destination b=3.Table 5
Optimization results for intersectionn=4.
From arm,i
Lane,k
To arm,j  Assigned lane flow (pcu/h)
Total lane flow (pcu/h)
Turning proportion
Saturation flow (pcu/h)
Flow factor,yn,i,k
Start of green (s)
Effective green (s)
Degree of saturation
1
2
3
1
1
—
53.4815
256.0701
309.5516
0.1728
1923.46
0.1609
78.29
36.71
0.5261
2
—
0.0000
338.7677
338.7677
0.0000
2105.00
0.1609
78.29
36.71
0.5261
Total
—
53.4815
594.8377
648.3193
2
1
0.0000
—
340.4530
340.4530
1.0000
1746.67
0.1949
0.00
73.29
0.3191
2
0.0000
—
364.7092
364.7092
1.0000
1871.11
0.1949
0.00
73.29
0.3191
Total
0.0000
—
705.1623
705.1623
3
1
200.0000
0.0000
—
200.0000
0.0000
1965.00
0.1018
0.00
120.00
0.1018
2
0.0000
300.0000
—
300.0000
1.0000
1871.11
0.1603
0.00
73.29
0.2625
Total
200.0000
300.0000
—
500.0000
Remarks. Lane marking is prohibited when assigned lane flow = 0.000 pcu/h; 500.0000 pcu/h is the total demand flow from origin a=4; 594.8377 and 705.1623 pcu/h are the total 1,300.0000 pcu/h exit flows entering destination b=4.Figure 5
Optimized network configurations including lane markings and assigned lane (turning) flows at individual intersections.Another set of outputs that is worth discussing is the path flows. We mentioned earlier that only three OD pairsa=1,b=3, a=1,b=4, and a=2,b=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 a=1,b=4, we provide path choices for selections. Indeed, path ha,b=1 is a longer path passing through four intersections and ha,b=2 is a shorter one and thus the optimization results for these two paths are Pa=1,b=4,ha,b=1=0.00 pcu/h and Pa=1,b=4,ha,b=2=500.00 pcu/h, respectively. It shows logical result. As for the OD pair a=1,b=3, the users’ input demand flow is Ta=1,b=3=300 pcu/h and the two path flows are Pa=1,b=3,ha,b=1=246.5 pcu/h and Pa=1,b=3,ha,b=1=53.5 pcu/h. And for the OD pair a=2,b=4, the users’ input demand flow is Ta=2,b=4=600 pcu/h and the two path flows are Pa=2,b=4,ha,b=1=505.2 pcu/h and Pa=2,b=4,ha,b=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.Figure 6
Optimized path flow details for the case study network matching the given OD demand flows (in pcu/h).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.


## 6. 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.



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*Source: 1257569-2017-07-19.xml*