Reversible Lane Optimization of the Urban Road Network Considering Adjustment Time Constraints

Abstract

Reversible lanes constitute an important solutions for sustainable transportation, with the aim to solve the practical problem of reversible lane optimization of urban road networks constrained by adjustment time. Considering the relationship between the number of lanes and the capacity of sections, a mixed-integer bilevel programming model of reversible lane optimization constrained by adjustment time is constructed in order to minimize the total travel time of the system. The results show that the model can effectively obtain the optimal strategy for any number of reversible sections subject to adjustment time constraints. With the increase of the number of reversible sections that can be optimized within the adjustment time, the cumulative reduced system time increases monotonically and the road network optimization effect improves, but as a whole, the optimization effect of the newly added reversible sections in each stage shows a decreasing trend. When the number of reversible sections that can be optimized within the adjustment time reaches a certain number, increasing the number of reversible sections will have a limited further effect on the overall system. For the reversible lane optimization problem of urban road networks, only efficient reversible sections need to be optimized to achieve a good optimization effect.

1. Introduction

Reversible lanes constitute an important solutions for sustainable transportation, which can fully utilize road space, alleviate the contradiction between the lane distribution and the periodically changing tidal traffic flow, and reduce traffic congestion [1,2,3,4]. Reversible lane optimization is suitable for various types of roads and greatly increases the road capacity without any major changes to the road structure, control facilities or traffic infrastructure [5]. Therefore, reversible lane optimization has been widely used in many scenarios involving two-way traffic imbalance, including tidal traffic flow in the morning and evening rush hours [6], emergency rescue [7], and temporary road maintenance [8]. Adjusting road resources in a light traffic flow direction to a heavy traffic flow direction can not only reduce the traffic pressure of the heavy traffic flow direction but also avoid unnecessary road construction, while additionally it can greatly improve the operation efficiency of the whole traffic system [9].

Bede [10] pointed out that a reversible lane can cope with the dynamic change of traffic flow in the main crowded direction, that traffic flow density is an important reference basis for reversible lane optimization, and that the model of a reversible lane system is constructed based on a road network divided into multiple units. Waleczek [11] studied the impact of the reversible lane system on road safety and traffic flow, and pointed out that the reversible lane system is a safe, practical and intelligent traffic management tool. Wolshon [12] noted that the advantages, disadvantages and cost of various design schemes and the long-term effects of the entire transportation system should be comprehensively considered when planning reversible lanes. Wu [13] developed a hierarchical planning model based on the BPR function and the imbalance of urban traffic distribution during morning and evening rush hours. Chu [14] proposed a dynamic reversible lane scheduling management scheme based on autonomous vehicles. Concicao [15] introduced the reversible lane network design problem (RL-NDP) and embedded the traffic allocation and reversible lane decision process into a mixed integer nonlinear mathematical programming model. Mao [16] proposed a lane change control model that can determine the number of lanes and the time of lane change. Di [17] considered the tidal and asymmetric features of the traffic demand structure and studied the demand-based reversible lane design scheme in a traffic network. Mendez [18] proposed a model of adaptive reversible lanes (also known as dynamic reversible lanes) by cellular automata. Dey [19] evaluated the reversible lane setting scheme in Washington State from the perspective of infrastructure utilization, safety, and economic development.

Yue [20] demonstrated the necessity and feasibility of implementing reversible lanes during the period of the Shanghai World Expo and discussed the specific implementation scheme preliminarily. Chen [21] proposed a reversible lane suitable for a tidal traffic section by analyzing the traffic characteristics of Shuxin Road in Chengdu. In view of the problem of an unbalanced demand for two-way road traffic in large cities, Ma [22] proposed a traffic control method that can dynamically adjust the driving direction of reversible lanes according to a two-way traffic flow. Zhang [23] used the entropy method to evaluate the traffic efficiency of urban roads and put forward an evaluation model of the implementation effect of reversible lanes and related calculation methods combined with a traffic capacity model. Cai [24] introduced a method for achieving system optimization through ITS adjustment and constructed a system optimal reversible lane model based on autonomous vehicles. Shi [25] discussed the optimization points of the reversible lane setting scheme and analyzed the master and slave game relationship between traffic organizers and travelers. Zhang [26] proposed a bilevel programming model for the optimization of reversible lanes, which aims to minimize the total impedance of road networks.

The above studies are based on the assumption that all reversible sections can be adjusted in the time before the emergence of tidal traffic. However, with current technology, even with an advanced tidal lane zipper truck carrying isolation piers to redivide reversible lanes, road sections that can be optimized within the reversible lane adjustment time are limited. In order to facilitate the practical application of the theory of reversible lane optimization, we have the chance to study the reversible lane optimization of urban road networks on the basis of limited reversible sections constrained by adjustment time. Therefore, this paper takes the basic premise of maintaining normal traffic in the light traffic flow direction when implementing a reversible lane, considers the reduction of traffic capacity caused by the number of lanes, and studies the optimal setting scheme of reversible lanes with the bilevel programming method from the perspective of system optimization, with the goal of obtaining the optimal reversible lane optimization scheme for urban road networks constrained by adjustment time.

2. Bi-Level Programming Model for Reversible Lane Optimization

Note that the node set of the entire road network is $N$, the road section set is $A$, the sections for reversible lane optimization during adjustment time are $A_{rl}$, the sections without reversible lane optimization are $A_{sl}$, and the OD (origin–destination) pair set is $w$. It is assumed that section $a\in A$ has the corresponding reverse section $\overline{a}$, and the two-way section $\ddot{a}$ is composed of section $a$ and section $\overline{a}$. $x_a$ represents the flow of section $a$, $l_a$ represents the number of lanes of section $a$ after the reversible lane adjustment time, $k_a$ represents the number of lanes of section $a$ without reversible lane optimization, $s_{\ddot{a}}$ represents the original number of lanes of the two-way road section $\ddot{a}$, $c_a$ represents a single lane’s capacity of section $a$, $c_a^n$ represents each lane’s average capacity of $n$ lane section $a$, $n_a$ represents the number of lanes in each section, $C_a$ represents the capacity of section $a$, $t_a^0$ represents the free travel time of section $a$, $t_a(x_a,n_a)$ represents the travel time of each section, and its functional form adopts the BPR formula:

$$t_a(x_a,n_a)=t_a^0\left[1+\alpha {\left(\frac{x_a}{C_a}\right)}^{\beta }\right],a\in A,$$

(1)

$$C_a=n_a\cdot c_a^n,a\in A,$$

(2)

where $\alpha$, $\beta$ are undetermined parameters.

According to Xu, Yang and Wang’s research results [27,28,29], $c_a^n$ can be obtained as:

$$c_a^n=\sigma (n-1)c_a^1+u(n-2)0.935c_a^1{e}^{-0.224(n-2)/n},$$

(3)

$$\delta (n-1)=\{\begin{array}{c}1,\begin{array}{cc}& n=1\end{array}\\ 0,\begin{array}{cc}& \mathrm{others},\end{array}\end{array}$$

(4)

$$u(n-2)=\{\begin{array}{c}1,\begin{array}{cc}& n\ge 2\end{array}\\ 0,\begin{array}{cc}& \mathrm{others},\end{array}\end{array}$$

(5)

Suppose that all travelers in the road network will choose the path with the minimum travel time, that is, that the path selection behavior of travelers conforms to the first principle of Wardrop. Additionally, the traffic management department can optimize the allocation of road resources through the adjustment of the number of lanes with the aim to minimize the total travel time of the entire road network. A mixed integer bilevel programming model of the reversible lane optimization is established as follows:

Upper planning:

$$\underset{}{\min } Z={\displaystyle \sum _{a\in A_{rl}}{t}_a(x_a,l_a)\cdot x_a}+{\displaystyle \sum _{a\in A_{sl}}{t}_a(x_a,k_a)\cdot x_a,}$$

(6)

$$\mathrm{s}. \mathrm{t}. 1\le l_a\le s_{\ddot{a}}-1, \forall a\in A_{rl},$$

(7)

$$l_a+l_{\overline{a}}=s_{\ddot{a}}, \forall a\in A, \overline{a}\in A_{rl},$$

(8)

Lower planning:

$$\underset{}{\min }{\displaystyle \sum _{a\in A_{rl}}{\displaystyle {\int }_0^{x_a}{t}_a(\omega ,l_a)}d\omega }+{\displaystyle \sum _{a\in A_{sl}}{\displaystyle {\int }_0^{x_a}{t}_a(\omega ,k_a)}d\omega ,}$$

(9)

$$\mathrm{s}. \mathrm{t}. {\displaystyle \sum _k{f}_k^{rs}=d_{rs}}, \forall \left(r,s\right)\in w,$$

(10)

$$f_k^{rs}\ge 0, \forall \left(r,s\right)\in w,$$

(11)

$$x_a={\displaystyle \sum _r{\displaystyle \sum _s{\displaystyle \sum _k{f}_k^{rs}{\delta }_{a,k}^{rs}}}}, \forall a\in A,$$

(12)

where $\omega$ is the integral variable symbol, $f_k^{rs}$ is the flow on the path $k$ between OD pairs $(r,s)$, $d_{rs}$ is the travel demand between OD pairs $(r,s)$, ${\delta }_{a,k}^{rs}$ is the correlation coefficient between the path and the section; when the path $k$ passes through the section $a$, ${\delta }_{a,k}^{rs}=1$; otherwise, ${\delta }_{a,k}^{rs}=0$; Formula (7) shows that the adjustment range of the number of lanes in the section $a$ is $\left[1,s_{\ddot{a}}-1\right]$; Formula (8) is the conservation constraint of the number of lanes in the section; Formula (10) is the conservation constraint of the traffic flow; Formula (11) is the non-negative constraint of the path flow.

3. Model Solution

Model (6)~(12) is a non-linear mixed integer bilevel programming problem, which is extremely difficult to solve and includes NP-hard optimization problems [30,31,32]; furthermore, a particle swarm optimization algorithm can usually be used to solve the problem. The detailed solution steps are as follows:

Step 1: Initialization. Set the iteration times to $\gamma$, the maximum allowable iteration times to ${\gamma }_{\max }$, the particle swarm size to $m$, the dynamic delay period to $\xi$, the maximum speed to $v_{a,\max }$, the inertia weight factor to $\kappa$, the reduction coefficients to $b_1$ and $b_2$, respectively, and the acceleration coefficients to $q_1$ and $q_2$, respectively. The position $(\cdot \cdot \cdot ,{\phi }_{a,\gamma }^i,\cdot \cdot \cdot )$ of the $\gamma$th iteration of the $i(i=1,2\cdots m)$th particle corresponds to the state $(\cdot \cdot \cdot ,s_{a,\gamma }^i,\cdot \cdot \cdot )$ of the $\gamma$th iteration of the $i$th reversible lane optimization scheme, the maximum particle position of section $a$ to ${\phi }_{a,\max }$, corresponding to the maximum number of lane settings $s_{\ddot{a}}-1$ of section $a$, and the travel demand to $d_{rs}$; each feasible particle initial position ${\phi }_{a,0}^i$ and initial speed $v_{a,0}^i$ is randomly generated for sections with reversible lane optimization while considering the adjustment time constraints, the rest are sections without reversible lane optimization, and their initial position is the original number of lanes $k_a$.

Step 2: Calculate fitness. For each feasible particle, solve the lower user equilibrium traffic assignment model, and then solve the upper objective function value, namely the particle fitness, according to the section flow.

Step 3: Update the historical optimal locations of individuals and groups. For the $i(i=1,2\cdots m)$th particle, the individual extremum $pbes{t}_{a,\gamma }^i$ is updated with the position corresponding to the current optimal fitness. For particle swarm optimization, the optimal position of all $pbes{t}_{a,\gamma }$ is used to update the population extremum $gbes{t}_{a,\gamma }$. If $gbes{t}_{a,\gamma }$ does not update after $\xi$ successive iterations, set $\kappa =b_1\kappa$, $v_{a,\max }=b_2{v}_{a,\max }$.

Step 4: Update the particle velocity on the basis of $v_{a,\gamma +1}^i=\kappa v_{a,\gamma }^i+q_1{R}_1(pbes{t}_{a,\gamma }^i-{\phi }_{a,\gamma }^i)+q_2{R}_2(gbes{t}_{a,\gamma }-{\phi }_{a,\gamma }^i)$, where $v_{a,\gamma }^i$ is the velocity of the $\gamma$th iteration of the $i(i=1,2\cdots m)$th particle, $R_1$ and $R_2$ are random numbers between $(0,1)$, and every particle’s velocity should be rounded to an integer. If $v_{a,\gamma +1}^i>v_{a,\max }$, set $v_{a,\gamma +1}^i=v_{a,\max }$.

Step 5: Update the particle position according to ${\phi }_{a,\gamma +1}^i={\phi }_{a,\gamma }^i+v_{a,\gamma }^i$. If ${\phi }_{a,\gamma +1}^i$ does not meet the constraint condition $1\le {\phi }_{a,\gamma +1}^i\le {\phi }_{a,\max }$, discard it and do not update the position of the $i$th particle. If the constraint conditions are met, judge the $i$th particle. If the particle has no path connection between an OD pair, discard the new position and do not update the position of the $i$th particle; otherwise, update it to the new position.

Step 6: Terminate the inspection. If the termination condition is satisfied, stop the iteration and output the $gbes{t}_{a,\gamma }$ as the optimal adjustment scheme. Otherwise, order $\gamma =\gamma +1$, and return to step 1.

This algorithm is applicable to small- and medium-sized road networks. For readers’ better understanding of Algorithm 1, the pseudo-code is shown as follows:

Algorithm 1: Parameter initialization

FOR each particle i       FOR each dimension $\mathrm{a}$         ${\mathrm{pbest}}_{\mathrm{a},0}^{\mathrm{i}}\leftarrow \infty$         $\mathrm{Initialize} \mathrm{position} {\mathsf{\phi }}_{\mathrm{a},0}^{\mathrm{i}}$ randomly within permissible range         $\mathrm{Initialize} \mathrm{velocity} {\mathrm{v}}_{\mathrm{a},0}^{\mathrm{i}}$ randomly within permissible range       END FOR END FOR ${\mathrm{gbest}}_{\mathrm{a},0}^{}\leftarrow \infty$ $\mathrm{Iteration} \mathsf{\gamma }\leftarrow 1$ DO       FOR each particle i         $\mathrm{Calculate} \mathrm{fitness} \mathrm{value} {\mathrm{p}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}$         IF ${\mathrm{p}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}<$${\mathrm{pbest}}_{\mathrm{a},\mathsf{\gamma }-1}^{\mathrm{i}}$           ${\mathrm{pbest}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}} \leftarrow { \mathrm{p}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}$         ELSE               ${\mathrm{pbest}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}} \leftarrow { \mathrm{pbest}}_{\mathrm{a},\mathsf{\gamma }-1}^{\mathrm{i}}$         END IF       END FOR       $\mathrm{Choose} \mathrm{the} \mathrm{particle} \mathrm{having} \mathrm{the} \mathrm{best} \mathrm{fitness} \mathrm{value} \mathrm{as} \mathrm{the} {\mathrm{gbest}}_{\mathrm{a},\mathsf{\gamma }}$       IF $gbes{t}_{a,\gamma }<gbes{t}_{a,\gamma -1}$               j ← 0       ElSE         $\mathrm{j} \leftarrow \mathrm{j}+1$         IF $\mathrm{j} > \mathsf{\xi }$           $\mathsf{\kappa } \leftarrow { \mathrm{b}}_1\mathsf{\kappa }$           ${\mathrm{v}}_{\mathrm{a},\max }\leftarrow { \mathrm{b}}_2{\mathrm{v}}_{\mathrm{a},\max }$           $\mathrm{j} \leftarrow 0$         END IF        END IF       FOR each particle i         FOR each dimension $\mathrm{a}$           Calculate velocity according to the equation           ${\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }+1}^{\mathrm{i}}={\mathsf{\kappa }\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}+{\mathrm{q}}_1{\mathrm{R}}_1\left({\mathrm{pbest}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}-{\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}\right)+{\mathrm{q}}_2{\mathrm{R}}_2\left({\mathrm{gbest}}_{\mathrm{a},\mathsf{\gamma }}-{\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}\right)$           $\mathbf{IF} {\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }+1}^{\mathrm{i}}>{\mathrm{v}}_{\mathrm{a},\max }$              ${\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }+1}^{\mathrm{i}}\leftarrow {\mathrm{v}}_{\mathrm{a},\max }$           END IF           IF $1\le {\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}+{\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}\le {\mathsf{\phi }}_{\mathrm{a},\max }$ and the OD pair has path connectivity              ${\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }+1}^{\mathrm{i}}\leftarrow {\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}+{\mathrm{v}}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}$           ELSE              ${\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }+1}^{\mathrm{i}}\leftarrow {\mathsf{\phi }}_{\mathrm{a},\mathsf{\gamma }}^{\mathrm{i}}$           END IF         END FOR       END FOR             $\mathsf{\gamma }\leftarrow \mathsf{\gamma }+1$ WHILE maximum iterations or minimum error criteria are not attained RETUNE ${\mathrm{gbest}}_{\mathrm{a},\mathsf{\gamma }}$

4. Calculations and Analysis of Numerical Example

The test road network consists of four nodes and five two-way sections, which are shown in Figure 1. Additionally, there are four OD pairs with travel demand $d_{14}=$ 4590 pcu/h, $d_{41}=$880 pcu/h, $d_{23}=$ 760 pcu/h, and $d_{32}=$1140 pcu/h, respectively. Set the BPR function’s parameter value $\alpha =0.15$, $\beta =4$. Additionally, the characteristic parameters of each section, including the free travel time, single-lane capacity, and number of lanes, are shown in

Table 1. Characteristic parameters of each section.

Section	Free Travel Time (s)	Single-Lane Capacity (pcu/h)	Number of Lanes
1-2	95	650	4
2-1	95	650	4
4-2	55	650	4
2-4	55	650	4
1-3	55	700	3
3-1	55	700	3
4-3	95	700	3
3-4	95	700	3
2-3	41	700	3
3-2	41	700	3

.

According to the number of lanes in each section in the example, set ${\phi }_{1-2,\max }=7$, ${\phi }_{2-4,\max }=7$, ${\phi }_{1-3,\max }=5$, ${\phi }_{2-3,\max }=5$, ${\phi }_{3-4,\max }=5$, $v_{1-2,\max }=7$, $v_{2-4,\max }=7$, $v_{1-3,\max }=5$, $v_{2-3,\max }=5$, and $v_{3-4,\max }=5$. Meanwhile, to improve the convergence speed of the particle swarm optimization algorithm and ensure its effective convergence, set $m=100$, $b_1=0.8$, $b_2=0.8$, $q_1=2$, $q_2=2$, $\xi =5$, $\kappa =1.4$, and ${\gamma }_{\max }=100$. When the number of reversible sections that can be optimized within the adjustment time is $n(A_{rl})$=0, 2, 4, 6, 8 and 10, respectively, the total system time and optimization time ratio after reversible lane optimization is shown in Figure 2, and the number of lanes, capacity, flow, saturation and travel time of each section are shown in

Table 2. Characteristic values of each section when $n(A_{rl})$ = 0.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	4	2173	2384	1.097	115.637
2-1	4	2173	473	0.218	95.032
3-1	3	1822	407	0.223	55.020
1-3	3	1822	2206	1.211	72.743
2-3	3	1822	761	0.418	41.188
3-2	3	1822	1363	0.748	42.925
4-2	4	2173	473	0.218	55.019
2-4	4	2173	2607	1.200	72.107
4-3	3	1822	407	0.223	95.035
3-4	3	1822	1983	1.088	114.968

,

Table 3. Characteristic values of each section when $n(A_{rl})$ = 2.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	7	3625	2722	0.751	99.530
2-1	1	650	251	0.386	95.317
3-1	3	1822	629	0.345	55.117
1-3	3	1822	1868	1.025	64.115
2-3	3	1822	760	0.417	41.186
3-2	3	1822	1140	0.626	41.943
4-2	4	2173	251	0.116	55.001
2-4	4	2173	2722	1.253	75.313
4-3	3	1822	629	0.345	95.202
3-4	3	1822	1868	1.025	110.744

,

Table 4. Characteristic values of each section when $n(A_{rl})$ = 4.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	7	3625	3015	0.832	101.828
2-1	1	650	231	0.355	95.226
3-1	3	1822	649	0.356	55.133
1-3	3	1822	1575	0.864	59.597
2-3	3	1822	759	0.417	41.186
3-2	3	1822	1214	0.666	42.210
4-2	1	650	231	0.355	55.131
2-4	7	3625	3089	0.852	59.347
4-3	3	1822	649	0.356	95.229
3-4	3	1822	1501	0.824	101.569

,

Table 5. Characteristic values of each section when $n(A_{rl})$ = 6.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	7	3625	2857	0.788	100.494
2-1	1	650	358	0.551	96.313
3-1	3	1822	522	0.286	55.055
1-3	3	1822	1733	0.951	61.748
2-3	3	1822	822	0.451	41.254
3-2	3	1822	1140	0.626	41.944
4-2	1	650	419	0.645	56.428
2-4	7	3625	2857	0.788	58.181
4-3	1	700	461	0.659	97.688
3-4	5	2861	1733	0.606	96.922

,

Table 6. Characteristic values of each section when $n(A_{rl})$ = 8.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	7	3625	2565	0.708	98.581
2-1	1	650	424	0.652	97.575
3-1	1	700	456	0.651	56.482
1-3	5	2861	2025	0.708	57.073
2-3	3	1822	761	0.418	41.188
3-2	3	1822	1140	0.626	41.944
4-2	1	650	424	0.652	56.491
2-4	7	3625	2565	0.708	57.073
4-3	1	700	456	0.651	97.559
3-4	5	2861	2025	0.708	98.581

and

Table 7. Characteristic values of each section when $n(A_{rl})$ = 10.

Section	Number of Lanes	Capacity (pcu/h)	Flow (pcu/h)	Saturation	Travel Time (s)
1-2	7	3625	2552	0.704	98.500
2-1	1	650	424	0.652	97.575
3-1	1	700	456	0.651	56.482
1-3	5	2861	2038	0.712	57.120
2-3	2	1309	760	0.581	41.701
3-2	4	2341	1166	0.498	41.378
4-2	1	650	424	0.652	56.491
2-4	7	3625	2578	0.711	57.108
4-3	1	700	456	0.651	97.559
3-4	5	2861	2012	0.703	98.480

, respectively. The cumulative reduction time of the system is shown in Figure 3.

When the number of reversible sections that can be optimized within the adjustment time is 0 (when no reversible lane optimization is performed), the total time of the system is 1,074,012 s, the saturations of Sections 1-2, 1-3, 2-4, and 3-4 all exceed 1, and the saturation of Section 1-3 is the maximum, reaching 1.211, in a severe congestion state.

When the number of reversible sections that can be optimized within the adjustment time is 2, the optimal strategy is shown in Table 3, the total time of the system is 1,013,956 s, and the saturations of Sections 1-2, 1-3, and 3-4 decrease. With the larger capacity of Section 1-2, the flow of path 1-2-4 and the flow of Section 2-4 increases, and its saturation further increases to 1.253, becoming the bottleneck section of path 1-2-4. Compared to the original road network (when $n(A_{rl})$=0), the overall time optimization is 5.59%.

When the number of reversible sections that can be optimized within the adjustment time is 4, the optimal strategy is shown in Table 4, and the total time of the system is 951,475 s. With the improvement of the capacity of Section 2-4, which is no longer the bottleneck of path 1-2-4, the section’s saturation is significantly reduced. Compared to the original road network, the overall time optimization is 11.41%.

When the number of reversible sections that can be optimized within the adjustment time is 6, the optimal strategy is shown in Table 5, and the total time of the system is 941,933 s. With the increasing capacity of Section 3-4, the flow of path 1-3-4 increases, resulting in an increased saturation of Section 1-3. Meanwhile, the flow of path 1-2-4 is reduced, and the saturations of Sections 1-2 and 2-4 are further reduced. Compared to the original road network, the overall time optimization is 12.30%.

When the number of reversible sections that can be optimized within the adjustment time is 8, the optimal strategy is shown in Table 6, and the total time of the system is 929,179 s. With the increasing capacity of Section 1-3, which is no longer the bottleneck of path 1-3-4, the saturation is significantly reduced. The saturations of the originally congested Sections 1-2, 1-3, and 2-4 are further reduced, the saturations of the whole road network are lower than 0.75, and there are no excessively congested sections. Compared to the original road network, the overall time optimization is 13.49%.

When the number of reversible sections that can be optimized within the adjustment time is 10, the optimal strategy is shown in Table 7, and the total time of the system is 928,655 s. Compared to the original road network, the overall time optimization is 13.53%.

From the perspective of the cumulative reduction time of the system, with the increase of the number of reversible sections that can be optimized within the adjustment time, the cumulative reduction time of the system increases monotonically and the road network optimization effect improves.

From the perspective of the optimization efficiency, the optimization effect of reversible sections that can be optimized within the adjustment time is more obvious at the initial stage, and the optimization effect of the newly added reversible sections shows a decreasing trend as a whole in each stage.

From the perspective of the cost performance, when the reversible sections reach a certain number, as the reversible sections increase, the optimization effect is limited for the overall system. Therefore, in answer to the problem of reversible lane optimization in urban road networks, we only need to optimize efficient reversible sections to achieve a good optimization effect, and it is unnecessary to pursue reversible lane optimization for all sections.

5. Conclusions

(1).

Based on the actual situation in which the number of reversible sections that can be optimized within the adjustment time are limited, a reversible lane optimization study is conducted on the premise of ensuring normal traffic during the implementation of reversible lanes, which is conducive to the practical application of reversible lane optimization theory and will have a positive and beneficial impact on achieving sustainability goals.

(2).

With the goal of minimalizing the total travel time of the system, a mixed-integer bilevel programming model for reversible lane optimization constrained by adjustment time is established, a particle swarm optimization algorithm is designed to solve the model, and the effectiveness of the model and the algorithm is verified by numerical examples.

(3).

With an increase in the number of reversible sections that can be optimized within the adjustment time, the cumulative reduction time of the system increases monotonically and the road network optimization effect improves. However, the optimization effect of the newly added reversible sections shows a decreasing trend as a whole in each stage.

(4).

To answer the problem of reversible lane optimization in urban road networks, we only need to optimize efficient reversible sections to achieve a good optimization effect, and it is unnecessary to pursue reversible lane optimization for all sections.

(5).

The question of how to control traffic at intersections with reversible lanes is an important one; in order to obtain a better reversible lane optimization system, traffic control at intersections will be fully considered in further research.