数学建模美赛

2017 
MCM/ICM 
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Post-toll Traffic Merging 
Abstract 
The toll plaza is an important part of the expressway, which greatly changes the traffic state on the highway. It is important to study the toll plaza and design a toll plaza’s reasonable  structure. Based on the existing toll plaza model and the specific analysis of its design structure, we improve the scheme design. Comparing with the former design, we consider that the new design improves the performance of the toll plaza effectively.  
After comparing the expressway’s safety distance with the city’s, we build a mathematical model of the vehicle’s safety distance. Then we get the safety distance   formula, and obtain that the safety distance on the highway is 41.1260m. 
In order to consider the influence of the toll plaza and the traffic on the actual roads, we will carry out cellular automaton processing for the toll plaza and its surrounding. We set up a cellular automaton model of the toll plaza and its neighborhood, called the cellular automaton model. Based on the running status of the model, probability of accidents, throughput and cost, we propose a new design of the toll plaza and reestablish a better cellular automata model.  
In this paper, we choose the Dynamic Weighted Comprehensive Evaluation model. We take the average speed, the traffic density, the time occupancy rate as the evaluation index to assess the design. We get the comprehensive evaluation value, based on these results, the Borda function in decision analysis is used to synthesize all the evaluation results, and the conclusion is obtained that the new scheme can improve the traffic condition. Then, we use the same method to discuss the performance in light and heavy traffic. It is concluded that the traffic situation is improved significantly in both cases.  
 The rules for autonomous vehicles are quite different from those for ordinary vehicles. We discussed all the real possibility in the same traffic or in the varying mixed traffic. We find that the introduction of autonomous vehicles can greatly increase the utilization of edge tollbooth. However, it is common for ordinary vehicles to enter the toll channel but there are few autonomous vehicles. We suggest that the tollbooth near the edge can be replaced by less time-consuming electronic tollbooth to facilitate the autonomous vehicle traffic.
 
Content 
1. RESTATEMENT OF THE PROBLEM .................................................................................................. 2 
1.1. INTRODUCTION ............................................................................................................................................. 2 
1.2. THE PROBLEM ............................................................................................................................................... 2 
2. ASSUMPTION ...................................................................................................................................... 2 
2.1. PHRASE EXPLAIN ........................................................................................................................................... 2 
2.2. BASIC ASSUMPTIONS .................................................................................................................................... 2 
2.3. SYMBOL DESCRIPTION: ................................................................................................................................ 3 
3. MODEL................................................................................................................................................... 3 
3.1. MODEL I THE MODEL OF AUTO SAFETY .................................................................................................... 3 
3.1.1. Approach ........................................................................................................................................... 3 3.2. DEVELOPMENT OF MODEL .......................................................................................................................... 3 
3.2.1. Solution and Result ......................................................................................................................... 4 
3.3. MODEL Ⅱ THE CELLULAR AUTOMATON MODEL ..................................................................................... 4 
3.3.1. Approach ........................................................................................................................................... 4 
3.3.2. Development of Model ................................................................................................................. 5 
3.4. MODEL3 ...................................................................................................................................................... 10 
3.4.1. Approach ......................................................................................................................................... 10 
3.4.2. Comprehensive Evaluation Model of Toll Plaza Structure Optimization .................... 10 
3.4.3. Weighted Dynamic Comprehensive Evaluation Model for New ................................... 13 
3.4.4. Dynamic Weighted Comprehensive Evaluation Model for Hybrid Driving ............... 14 
3.4.5. Discussing the impact of the three schemes on the new scheme ................................ 15 
4. STRENGTHS AND WEAKNESSES.................................................................................................. 15 
4.1. STRENGTHS ................................................................................................................................................. 15 
4.2. WEAKNESSES ............................................................................................................................................... 15 REFERENCES .............................................................................................................................................. 16 
APPENDIX .................................................................................................................... 错误!未定义书签。 
Restatement of the Problem 
1.1. Introduction 
Today, the pace of life is increasing with technological advancements. This requires the traffic to provide people with a fast and safe travel platform. Multi-lane divided limited-access toll highways use “barrier tolls” to collect tolls from motorists. However, these tollbooths are usually more than the number of lanes. We are familiar with toll plaza. Toll plaza is the area of the highway needed to facilitate the barrier toll. It has the fan-out area before the barrier toll, the toll barrier itself, and the fan-in area after the toll barrier. when exiting the tollbooths in a barrier toll, vehicles must “fan in” from the larger number of tollbooth egress lanes to the smaller number of regular travel lanes. For high traffic density, the road may be busy when we leave from the toll plaza. Therefore, in order to improve the efficiency of traffic and reduce the traffic accidents, we should rationalize the number of tollbooths that in the toll plaza and reduce the average time of the vehicles passing through the toll plaza. 
1.2. The Problem 
    We know that the vehicles may be crowded in heavy traffic after vehicles paying charges and passing the toll plaza. We need to widen the road to increase the number of parallel tollbooths, but it can cause extra costs. We usually increase the number of tollbooths to improve the capacity of tollbooths. If the toll station is not in a reasonable position, the tollbooths will occur congestion, and the reliability of traffic will be poor. Determining the shape, size, and merging pattern of the tollbooth can reduce traffic jam better. In this paper, for the area of toll highway, our main aims are in the following text.  
   Consider a toll highway having L lanes of travel in each direction and a barrier toll containing B tollbooths (B > L) in each direction. Determine the shape and the size to keep the big throughput and the low cost of space. When exiting the tollbooths in a barrier toll, vehicles must “fan in” from the larger number of tollbooth egress lanes to the smaller number of regular travel lanes. But in the design, we need to consider accident prevention, throughput, and cost and so on. And find better solutions (shape, size, and merging pattern) than any in common use. At the same time, determine the performance of our solution in light and heavy traffic. When there are more autonomous vehicles are added to the traffic mix, determine the affected by the proportions of conventional tollbooths, exact-change tollbooths, and electronic toll collection booths (automated). 
2. Assumption 
2.1. Phrase explain 
Time occupancy: also known as the lane occupancy, traffic occupies the time ratio of the road. 
Vehicle density: refers to the number of vehicles per unit length of road in a moment. 
Average speed: A physical quantity that describes the speed of movement of an object. 
2.2. Basic assumptions 
In order to reduce traffic jam caused by the increased vehicles, and the tollbooth system maintain a large throughput, set the best number of tollbooths to achieve the maximum flow when needn’t queuing. To make the model more realistic, make the following assumptions about tollbooths and vehicle behavior: 
1) Vehicles through the toll system is that the vehicles from the L-lane expressway to B (B is 
 
longer than L) lane of the tollbooth, and then merged into the L-lane highway process. 
2) Do not consider the phenomenon of delay when the traffic is heavy. 
3) Each tollbooth has a toll lane, and it service only one car at any time. 
4) The Charging system work normally with a particular situation.  
5) Assume that self-driving can automatically select the tollbooth with low traffic flow 6) Assume that self-driving is only possible through electronic tollbooths. 
7) Assume that the automated tollbooths are released every 4 seconds 
8) the electronic tollbooths are released every 2 seconds 
9) Assume that the automatic tollbooths are released every 6 seconds 
2.3. Symbol Description 
Y Comprehensive evaluation value 
w Weights 
x standardized data the number of evaluation indexes. 
m the number of evaluation indexes. 
Borda Borda count, weighted evaluation method 
 
3. Model 
3.1. Model I The Model of Auto Safety 
3.1.1. Approach 
    The Vehicles on the highway may in heavy traffic or have traffic accidents. It easily causes rear-end and had heavy casualties in traffic accidents. Therefore, it is necessary to pay more attention to the setting of vehicle safety system. In the model of vehicle anti-rear-end, we establish a reasonable and applicable mathematical model to reduce the complexity of the relevant parameters, through the analysis of the actual situation. The vehicle safety system is set up, then getting the highway system that meets the actual vehicle driving. 
On the highway, in order to prevent the rear-end and causing traffic accidents, when the two vehicles driving in the same direction of the same lane, in the course of travel, they must maintain the minimum distance between the vehicle travel, the minimum safe driving distance is D. We divide the safety distance into three parts: the braking distance of vehicle, the braking distance of front vehicle, and the safety distance D. In order to ensure safe driving and improve the car rear-end system practicality, we establish the following safety distance model. 
3.2. Development of Model 
In the establishment of safety distance model, in order to avoid the occurrence of rear-end, rear vehicles B can be an emergency brake or change lanes and other means.[4] In the car-  anti-rear-end system. We calculated the minimum distance of the car in the parking or driving, when traveling in the same direction, generally maintain a similar speed and acceleration. 
We consider that the two cars to avoid the rear-end parking situation. When the car observed that the A car began to slow down, the B car began to slow down. Ensure that the two vehicles do not rear-end parking in the deceleration process. The safety gap is maintained between the two vehicles before the vehicle stops completely. The safety clearance ( S ) is maintained between the two vehicles, as shown in Figure 1:  
  
Figure 1: The Diagram for Vehicle Safety Distance  
In Figure 1, D indicates the minimum distance between the B car and the A car in order to avoid the rear-end. It is a safe distance. We give the following formula: SA  v tA  1 a tA 2
 2

SB  v tB 1 v tB 2  12 a tB 22

t  t1 t2
vA  a tA      
Here, S is the minimum distance for the car; t1 is the driver's response time, and the time is 1.4s[6] in accessing information. t2 is the deceleration time of B. The A car＇s growth time of linear deceleration is equal to B.vA is the braking speed of A car, aA is the A car's maximum deceleration;vB is the braking speed of B car, aB is the B car's maximum deceleration. The safe distance D is as follows:  
DSB SA S     
3.2.1. Solution and Result 
Usually, the highway speed limit is 75 miles per hour. That is the highway speed limit is 33.5m / s. After establishing the safe distance model, we need to determine some parameters. Therefore, we assume that the real values of several constants previously defined are given in Table 1. 
Table 1: Relevant parameters 
Minimum vehicle distanceS 2m 
Braking speed（vA  vB ） 33.5m/s 
Maximum deceleration（aA  aB ） 8.9m/s2 
When the B car stopped, the two cars must have a certain distance to make sure the safe. S is the distance. In this paper, we default S is the vehicle wheelbase, it is about 2.5m. Combined with the above formula, we can get the minimum safe distance of highway driving D is 41.1260m. 
3.3. Model Ⅱ the Cellular Automaton Model 
3.3.1. Approach 
Since there is little data about the traffic situation near the toll area, we use the Cellular Automaton model to simulate the traffic state of a toll plaza and obtain the simulation data. Using the Cellular Automata model to simulate traffic stand for dividing the road into discrete cell lattices. Each cell can be occupied by a car, it can also be free. Each cell lattice’s width and length bigger than vehicle’s slightly. Then set up the mobile rules for cell: the tolling rule; the driving rule; the random lane-changing rule; the boundary rule; and the fan rule. 
According to the length of vehicles in daily life, we defined the cell length as 6m. If consider the single-lane width (12 feet) as the width of a single cell of the United States standard, it is 3.66m. The minimum safe distance D (41.1260m) obtained by model 1 is equal to 7 cells＇length.  
The model takes American toll plaza as an example to establish the mathematical model to illustrate the shape of the toll area, the size of the design, and analyze the merger of the traffic after the charges were analyzed. 
3.3.2. Development of Model 
Cellular automaton is a kind of dynamic model with discrete space and time, it is not defined by mathematical functions, and is governed by established rules of cells. In this paper, we establish a two-dimensional cellular automaton traffic model, after the cell rules are determined, we simulate the actual situation, and output the relevant data to be studied. We use the cell to express the entire toll plaza and a section of adjacent highways: 
  
Figure 2: Schematic diagram of toll plaza 
Zone A: Vehicle entry zone 
Zone B: the driving area, when the vehicle near the C area to slow down 
Zone D: Queuing area 
Zone E: the toll area 
Release rules of Tollbooth  
As shown in Fig. 2, the blank portion indicates that there is no vehicle, the gray portion is the tollbooth, the black portion represents the cells occupied by the vehicle. The direction of the arrow indicates the direction of travel of the vehicle. Vehicles enter the toll plaza to wait until the end of the first car to pay to leave, driving into the next cell. Vehicles’ payment is completed, it still in accordance with a certain probability of random access to the next open cell. 
  
Figure 3: Release Rules of Tollbooth 
Driving rules 
When the distance between the two vehicles traveling in the same lane is 0, the vehicle will stop moving. In other cases, the vehicle will travel at the maximum possible speed。
However, when the front range is 8 cells, the vehicle travels at the maximum speed (speed limit) when no vehicle exists, vehicles in the vicinity of tollbooth need to slow down. As the vehicle driving on the road, there is some uncertainty, so the driving process need to maintain the maximum safety distance, that is, two cell length. 
  
Figure 4: Driving Rule Chart 
Figure 4 shows that when the preceding vehicle approaches the tollbooth, it slows down and stops at the tollbooth. Wherein the blank portion indicates that there is no vehicle, the black portion represents the cell occupied by the vehicle, and the arrow represents the traveling direction. 
Lane changing rules randomly 
If there is an open cell next to the cell occupied by the vehicle, and there are open cells on the corresponding diagonal. The vehicle randomly shifts the lane with a given probability and moves forward, the probability of changing the road is affected by the size of expected lane. If the cell in front of the vehicle is free, or it is in the tollbooth and the charges have been completed, the vehicle will remain on the same lane. Figure 5(a) shows the change rule described above. 
                    
   Figure 5(a): Lane Changing Rule Diagram   
If the distance between the two vehicles far exceeds the safety distance, the adjacent cells within the vehicle will randomly change lanes, and move forward, as shown in Figure 5 (b). 
  
Figure 5(b): Lane Changing Rule Diagram 
If the cells in front of the vehicle are free, the vehicle remains in the forward direction without changing lane. 
Boundary rules 
Entry rules: When the cell of the tollbooth is released, the vehicle randomly arrives at these cells according to a certain probability, and enters the charging area as shown in Figure 6 (a). 
  
Figure 6(a): Boundary Rule Diagram 
The length is the distance traveled by the vehicle at the highest speed per unit time. Since the time is discrete, each car on the expressway will pass this length area when entering the cell area. 
Leaving rule: After the vehicle arrives at the tollbooth, it enters the corresponding open cell, when the vehicle charges are finished, the vehicle enters the next open cell randomly under the certain probability, as Figure. 6 (b). 
  
Figure 6(b): Leaving Rule Diagram 
Sector Rule 
Figure 7 (a), the vehicle drives into the toll plaza. If a cell occupies a cell adjacent to each other with an open cell, and there are open cells on the corresponding diagonal. While the front is open cells, the vehicle can be changed without changing lane or into the front three open cells of the probability of 1/3. If the front cell is occupied, the probability of the vehicle passing into the front by the left and right lane changes is 1/2. 
  
Figure 7(a): Fan out rule 
When the vehicle leaves the toll area, in order to avoid the congestion, the vehicle near the left sector can only change to the next open cell in the forward or right direction. The right sector is the same as above, and the fan-in rule is shown in Figure 7 (b). 
  
Figure 7(b): Fan in rule 
Optimization 
Toll plaza is an important part of the highway, the existence of the toll plaza changed the movement state of the original traffic flow. Toll plaza contains three areas: fan out area; queuing and payment area; fan in area. Therefore, we consider the impact of these three areas when the vehicles are charged and eventually merged. The impact of these three regions must also be considered.  Fan in 
In the final analysis, the fan in is the car from more lanes into the less lane. Therefore, we consider this problem from the length of the sector. It is a sector that connects more lanes and fewer lanes. Actually, the sector provides a path to and from the lane, the longer the sector, the longer the sector, the more the path of the vehicle through the lane, the higher the confluence efficiency, but the sector of the larger, the higher the cost of construction. In order to discuss the optimal solution of the sector area, that is, in the case of a fixed number of lanes, to ensure that the fan in of basic traffic flow, Finding the optimal solution of construction cost and tollbooth quantity. As the laying of the road is laid in accordance with the area. We believe that the construction costs and sector area directly related to the area, then fan area instead of cost considerations. 
The cellular automaton of the sector zone exit is established, and the cellular rules are the same as those of the previous cellular automaton. However, when the vehicle enters the sector, we add a rule. In order to comprehensively consider the traffic flow on the road performance, so the vehicle is random access to a certain probability. Constantly changing the number of lanes left (L) and the number of more lanes (B), the corresponding vehicle density is solved by a cellular automaton. 
  
Figure 8: Simulation Run Diagram 
Table 4: Simulation Data (partial) 
L B C Z 
4 7 19 0.108482143 
4 7 20 0.122453571 
4 7 21 0.1047075 
4 7 22 0.087345 
4 7 23 0.107142857 
4 7 24 0.09 
4 7 25 0.092719286 
4 7 26 0.079365 
4 7 27 0.089923929 
4 8 11 0.193359375 
Here, L is the number of toll lanes; B is the number of tollbooth; C is the sector length; Z is the vehicle density, in units of cells / cell. The linear programming model was established to fit the data, the correlation coefficient R2= 0.7642, so it can be considered that this model can well explain the relationship between these quantities, the fitting results: 
Z0.081L0.003B0.003C0.524 
We can see that the density of 23 vehicles per kilometer in urban traffic is considered to be smooth. Since the fan area does not belong to the high-speed part of the expressway, we use this vehicle density to find the optimum solution. 23 per kilometer of about 0.142 vehicles ／cell. 
Area formula: 
C
S     (2 B 1 L)  
2 
The results are as follows 
Table 5: The parameter values 
L B C S 
2 4 15 37.5 
3 5 13 39 
4 6 13 45.5 
5 8 11 55 
 
Queues and payment areas 
At present, the charges are: conventional tollbooths; automated tollbooths; electronic tollbooths.  Conventional tollbooths mean that the vehicle experiences three states of pay, change, and bar rise, the automation charge has not changed the condition. There may be some acts of escaping charges, electronic charges, although the vehicle can leave directly, but still should be kept waiting for the bar to raise this state. 
Fan out 
As the fan into the area by the fan-out area into the traffic, the state of the vehicle entering the fan out indirectly affects the state of the fan in area, such as traffic concentrated in a few tollbooths to leave, which can cause local congestion. Congestion caused by a substantial increase in vehicle density, it will lead to the possibility of an accident, so the fan out of the guide is extremely important. 
Fan out is essentially a process of streaming, the density of the traffic flow is weakened into several shares of low density traffic, and into different toll roads. But in daily life, people are always willing to enter the distance from their nearest tollbooth, 
this is reflected in the previous cell rules, that is, the probability of entering the nearest tollbooth is far greater than the probability of entering a tollbooth which is further away from itself. Therefore, we believe that the key to optimizing the fan out process is how to distribute the traffic evenly and guide it to different areas. 
We establish a sector cellular automaton, In the sector area road cells, where the most intensive traffic, we set up a triangle, just block the most intensive toll lane, at the same time, we counted the leaving vehicle the after payment for a period of time. 
Due to the smaller density, the difference is not obvious, so we entered a larger traffic density, that is close to or greater than the congestion density of 200 vehicles /mile. In considering the effect of such a design, in order to directly related to the flow output to the fan in, we statistic and analyze the tollbooth throughput. The table below shows the toll throughput statistics for one hour. 
Table 6: Number of roads 
Lane number Before the change After the change 
1 0 0 
2 0 15 
3 2 44 
4 51 87 
5 85 92 
6 91 93 
7 93 84 
8 273 180 
Lane number from 1 to 8, where the smaller the number from the farthest distance into the lane farther. From the Table 6, we can be seen to change the traffic more dispersed to the middle of the charge lane. 
3.4. Model III 
3.4.1. Approach 
    In the analytic hierarchy process and the fuzzy comprehensive evaluation method, the weight of each evaluation index is fixed, although simple, but because the determination of the index weight is subjective, the results often do not reflect the actual situation.[2] According to the rule of influence of evaluation index on comprehensive evaluation result, we adjust the weights constantly to increase the objectivity of evaluation and reduce the influence of subjective factors on evaluation results. This method is called weighted dynamic comprehensive evaluation. 
3.4.2. Comprehensive Evaluation Model of Toll Plaza Structure Optimization 
    By using the cellular automaton in Model 2, we simulated the situation before and after the ten new schemes were put into use, the corresponding average speed, vehicle density and time occupancy rate were Recorded (see appendix for related data). We look up information, according to road conditions, the traffic state is divided into three states, crowded, general, and smooth. 
Table 7: Traffic Status Classification Table 
  Unobstructed General Congested 
Average speed >60 (45,60) <45 
Traffic density <20 (20,200) >200 
Time occupancy <10 (10,30) >30 
Standardization of Evaluation Indicators 
Since the average speed value is the maximum value, we minimize the average speed. 
That is, we carry out the reciprocal transformation. The data of average speed, traffic density, time occupancy and 'evaluation criteria' are normalized by means of range transformation. 
Table 8: Classification Interval 
  Unobstructed General Congested 
Average speed (0,0.75) (0.75,1) (1, +∞) 
Traffic density (0,0.1) (0.1,1) (1, +∞) 
Time occupancy (0,0.33) (0.33,1) (1, +∞) 
 
The data standardization values are as follows. 
Table 9: Data Standardization 
z After the change 
Average speed Traffic density Time occupancy Average speed Traffic density Time occupancy 
0.791765637 0.029074191 0.005809023 0.69881202 0.012963013 0.00259001 
0.811030008 0.0351112 0.007015218 0.713775874 0.022222356 0.004440027 
0.799360512 0.033777835 0.006748811 0.713775874 0.012993673 0.002596136 
0.793650794 0.031555616 0.006304812 0.709219858 0.019697055 0.003935471 
0.784313725 0.029333429 0.005860819 0.70323488 0.017676812 0.003531827 
0.778816199 0.028000096 0.005594419 0.693000693 0.016161661 0.0032291 
0.780640125 0.026666763 0.005328019 0.693000693 0.013131358 0.002623645 
0.775193798 0.025777877 0.00515042 0.691562932 0.011111115 0.002220001 
0.716845878 0.025333429 0.005061619 0.690131125 0.010101018 0.002018183 
0.716845878 0.025333429 0.005061619 0.690131125 0.009090921 0.001816366 
The Dynamic Weighting Function is Determine 
Partial large normal distribution function：Based on the analysis of the three evaluation indexes, we notice that the influence of the average travel speed, traffic density and time occupancy on the comprehensive evaluation is the slowest increase with the increase of the standardized index data. 
  
Figure 9: Normal Distribution Function Curve 
The variable weight function of the three evaluation indexes can be set to a large normal distribution function 
0,x i
w xi ( )    x i 2  
1e  i  ,x i

i parameter takesi   a1( )i b1( )i  , i is determined by w bi ( k( )i )  0.9(1 i m)  2
The end result is: 
1  0.375,2  0.05,3  0.15,1  0.24712,2  0.032951,3  0.09888 
Output of dynamic comprehensive evaluation results The following is a formula for comprehensive evaluation value. 
m
Y w xi ( i ) xi 
i1
Where Y is the comprehensive evaluation value, w is the weight, x is the standardized data, and m is the number of evaluation indexes. 
Through the calculation, the evaluation results are shown in Table 10. Table 10: Evaluation of Simulation Data 
Before the change After the change If the traffic situation has improved? 
0.7604665 0.5849082 Yes 
0.7876136 0.6200262 Yes 
0.7706626 0.6164179 Yes 
0.7626808 0.6100841 Yes 
0.7485415 0.5968227 Yes 
0.7398933 0.5741049 Yes 
0.7430984 0.5724159 Yes 
0.7342098 0.5678841 Yes 
0.6264099 0.5639732 Yes 
0.6264099 0.563169 Yes 
Reanalysis of the results 
In order to comprehensively consider 10 groups of comprehensive evaluation values, We have 10 groups of comprehensive evaluation value to the group as a unit, in accordance with the size of the overall index to sort, the greater the index value represents the more crowded traffic, the smaller the number of indicators on behalf of the more smooth, so we get 10 sorting results, Using the Borda function method in decision analysis, the comprehensive sequencing results of the traffic state before and after the change are determined, and the ranking rule is from big to small. The Borda function method is that the number of rows in the h-th sorted result after the g-th evaluated object dg is B dh ( g ) , where the evaluation object refers to the ability of the new scheme to optimize the road. The Borda number of the object dg to be evaluated is: 
10
B d g B dh ( g )(g 1,2) 
h1
Table 11: Comprehensive Evaluation 
  Before change After change 
Borad 10 0 
Comprehensive sorting results 1 2 
3.4.3. Weighted Dynamic Comprehensive Evaluation Model for New 
  Table 12: Performance at High Flow Rates 
High flow conditions before the change High flow conditions after the change If the traffic situation has improved? 
1.172119 1.124509 Yes 
1.123837 1.0499002 Yes 
1.220484 1.0494512 Yes 
1.220961 1.041892 Yes 
1.218412 1.0348275 Yes 
1.171633 1.046076 Yes 
1.11828 1.05188739 Yes 
1.189415 1.04495504 Yes 
1.196624 1.06067761 Yes 
1.116937 1.05440955 Yes 
Table 12: Performance at Light Flow Rates 
Light flow conditions before the change Light flow conditions after the change If the traffic situation has improved? 
0.7604665 0.5849082 Yes 
0.7876136 0.6200262 Yes 
0.7706626 0.6164179 Yes 
0.7626808 0.6100841 Yes 
0.7485415 0.5968227 Yes 
0.7398933 0.5741049 Yes 
0.7430984 0.5724159 Yes 
0.7342098 0.5678841 Yes 
0.6264099 0.5639732 Yes 
0.6264099 0.563169 Yes 
It can be seen from Table 11-12 that the comprehensive evaluation under the condition of light flow and high flow before the change is higher than the change, the result shows that the traffic condition has improved. 
3.4.4. Dynamic Weighted Comprehensive Evaluation Model for Hybrid Driving 
Development of Model 
We introduce a new evaluation index, tollbooth throughput, that is, within one hour the number of vehicles released from the tollbooth. 
Table 13: Tollbooth’s Throughput and Standardization 
  Free Common Busy 
Throughput <70 (70,150) >150 
Standardization （0，0.467） （0.467，1） （1，∞） 
Dynamic comprehensive evaluation results 
We believe that the size of the proportion of self-driving represents the ability of traffic redistribution. Assuming that all tollbooth are electronic tollbooth, the total flow rate is unchanged and the ratio is changed. The Cellular automaton is used to simulate ten sets of data. 
Table 14: Weighted Comprehensive Evaluation Data (partial) 
Before the change After the change 10:01 1:01 
0.90003 0.90003 0.82607 0.41964 
0.47492 0.90003 0.79432 0.67667 
0.63035 0.90003 0.67667 0.87874 
0.41964 0.023055 0.1566 0.023055 
0.83488 0.023055 0.054716 0.73173 
0.98892 0.99749 0.99749 0.99999 
0.029225 0.99907 0.99907 0.54478 
0.1566 0.17038 0.219 0.054716 
Re-analysis of the output results: 
Table 15: The Number of Borda 
  10:01 1: 01 
Tollbooth Number Before the change After the change Before the change After the change 
1 0 0 0 0 
2 6 1 5 0 
3 3 1 3 0 
4 5 1 6 5 
5 1 3 1 10 
6 9 9 10 8 
7 10 9 10 3 
8 7 8 4 5 
As can be seen from the above table, the introduction of self-driving vehicles flow shows the ability to optimize the peripheral toll station. For toll plaza's outermost periphery, number 1's tollbooth has the strongest optimization capability. In the proportion of smaller proportion of self-driving. Comparing with the change of the tollbooth, its optimization ability is embodied in the 1,3,5 tollbooth. Compared with the tollbooth before the change, its optimization ability is embodied in 1 ~ 5 tollbooth. 
In summary, due to the electronic tollbooths spend less time, it is recommended that less time-consuming electronic tollbooths can be placed at the tollbooths near the edge, which is convenient to the self-driving vehicle, what’s more, it improve the efficiency of the charging area. 
3.4.5. Discussing the impact of the three schemes on the new scheme 
We run the cellular automata and output ten sets of data, including time occupancy, vehicle density and average speed. the comprehensive evaluation value is obtained by using the dynamic weighted comprehensive evaluation model. We use the Borda function to synthesize all the evaluation results. The smaller the Borda number, the better the optimization. As we can see from the table. 
 
Table 16: Evaluation Results 
  Conventional tollbooths Automated tollbooths Electronic tollbooths 
Borda 1 2 7 
Sorting 3 2 1 
 
4. Strengths and Weaknesses 
4.1. Strengths 
 In the absence of data, according to the way of traffic flow, we set up a mathematical model to analyze the actual situation. 
 Using the dynamic weighted comprehensive evaluation model, the influence of subjective factors is reduced. 
 In the analysis of traffic confluence, we consider the whole structure of the toll plaza, so the optimization model is more reasonable. 
4.2. Weaknesses 
 Cellular automaton is a simulation method, it cannot fully take into account all the factors, such as serious traffic jam on the traffic block. 
 The proposed scheme did not take into account all the factors, such as the fan-out area. 
 The selected cells are too small to introduce some details. 
 The prevention of accidents is only considered as an optimization consideration. In fact, it should also be used as an evaluation index to introduce dynamic weighted comprehensive evaluation. 
 
 
 
 
Dear Sir/Madam: 
Our team has proposed an optimization scheme for the structure of the tollbooth, including: optimizing the structure of the fan-out area, guaranteeing the scheme which can keep the traffic smooth and make the cost of fan-in area as small as possible, and optimizing the charging methods. We built a mathematical model to discuss and evaluate new and existing schemes, and found that the new solution is better than the existing one. 
Specifically, the fan-out region is optimized to divide the vehicles averagely into different lanes to ensure that the traffic density of the fan-in area is relatively small, increasing the distance between the vehicles and reducing the likelihood of accidents. The way we allot the vehicle is to create a triangular obstacle in the innermost lane of the fan-out zone, directing traffic to the outside tollbooth. 
The key to optimize the fan-in area is to find the balance between the fan-shaped area and the vehicle density. We obtain the traffic density by cellular automaton simulation, and fit function to describe the function of the length of the sector, the number of lanes, the number of toll stations and the traffic density. 
In order to avoid the influence of subjective factors, we use the dynamic weighted comprehensive evaluation model to evaluate the existing and new schemes, and use the Borda function method of decision analysis to sort the two schemes and finally get the conclusion that the new scheme is better than the existing scheme. 
Considering the fact that New Jersey will have autopilot vehicles in traffic, we add a certain percentage of autopilot vehicles into the model and evaluate the impact of the autopilot, and find that autopilot is a vehicle that follows different rules. Actually it can increase the throughput of the outside toll lanes. Therefore, we propose that the outside tollbooth can be reconstructed into an electronic tollbooth to facilitate the vehicles.  
We firmly believe that our model can be used as a reference for future updates to the New Jersey Toll Plaza. 
Table 1: Optimum solution of fan - in area 
Number of lanes Number of toll stations Length(m) S(m^2) 
2 4 90 675 
3 5 78 624 
4 6 78 728 
5 8 66 990 
  
Figure 1，Fan-out area optimization 
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