About the Optimized Design of the Parking Space on the Campus of a College

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1. Introduction

Due to the rapid development of society and the improvement of people’s living standards, it is not unusual to own a private car, and college teachers are no exception. The number of college teachers who use private cars as a means of travel is increasing year by year, and it is no longer news for students to drive to school [1] [2]. The relationship between colleges and the society is getting closer and closer, and the number of social transportation facilities entering colleges is also increasing rapidly, which has caused great traffic dangers to the highly crowded campus environment. At the same time, the disorderly parking and random access of motor vehicles on the campus will also generate a large amount of exhaust gas and noise pollution, which will destroy the clean, comfortable, civilized and harmonious campus environment. Therefore, it is an urgent task to build a harmonious campus to plan campus parking and eliminate hidden dangers of traffic safety [3] [4].

2. Investigation and Analysis of Campus Parking Problems

2.1. Problems with Campus Parking

This article takes Baoding University as an example. Through field surveys, it is found that the following problems exist in parking on campus:

1) Parking spaces in some areas no longer meet the daily parking needs of teachers and students, let alone the increasing demand for parking each year. Parking saturation is now reaching most of the campus.

2) Unreasonable parking space planning [5].

3) Roadside traffic signs and markings are lacking and have no guiding effect; lack of standardized management and serious parking disorder.

This article makes a more reasonable design plan for the above problems.

2.2. Questionnaire Survey on Parking Issues

A questionnaire survey was conducted on teachers and students. A total of 1000 questionnaires were sent and 910 were retrieved. The survey results found that with the promotion of a green and low-carbon lifestyle, people will use green and low-carbon transportation on sunny days even if they own a private car. About 43% of teachers may use private cars for transportation, and about 25% of teachers are unfamiliar with the location of the parking lot, and there are still “difficult parking” situations. Generally choose to park near the destination or a short walk to the destination after parking. It is best to walk within 100 meters and less than 5 minutes.

There are still many problems with the on-campus parking lot. In addition to the problems shown above, the respondents also made the following suggestions: build a carport to reduce the sun and divide the parking area. It is shown in Table 1.

2.3. Campus Vehicle Parking Features

1) The number of cars parked on campus is mainly small motor vehicles.

2) Most parking people will choose a parking space near the destination.

3) The campus roads have a single function, mainly for non-motorized vehicles and pedestrian traffic, and small motor vehicle traffic.

4) Lack of traffic signal instructions.

3. Analysis and Forecast of Parking Demand

This paper uses the GM (1, 1) model [6] and the linear fitting model [7] to make predictions simultaneously. The actual number of parking spaces at various locations in the school from 2013 to 2018 is shown in Table 2.

Table 1. School parking area.

Table 2. Number of parking spaces on campus from 2013 to 2018 (Unit: vehicles).

3.1. Model Establishment and Solution Analysis

Based on the existing data, the GM (1, 1) model is used to make prediction analysis on the number of motor vehicles.

Let the original reference data be listed as

${x}^{\left(0\right)}=\left({x}^{\left(0\right)}\left(1\right),{x}^{\left(0\right)}\left(2\right),\cdots ,{x}^{\left(0\right)}(n)\right)$

Do an accumulation (AGO) to generate a sequence

$\begin{array}{c}{x}^{\left(1\right)}=\left({x}^{\left(1\right)}\left(1\right),{x}^{\left(1\right)}(2),\cdots ,{x}^{\left(1\right)}\left(n\right)\right)\\ =({x}^{\left(1\right)}\left(1\right),{x}^{\left(1\right)}\left(1\right)+{x}^{\left(0\right)}\left(2\right),\cdots ,{x}^{\left(1\right)}\left(n-1\right)+{x}^{\left(0\right)}(n))\end{array}$

among them,

${x}^{\left(1\right)}\left(k\right)={\displaystyle \underset{i=1}{\overset{k}{\sum}}{x}^{\left(0\right)}\left(i\right)},k=1,2,\cdots ,n$ (3.11)

Find the mean series

${z}^{\left(1\right)}\left(k\right)=0.5{x}^{\left(1\right)}\left(k\right)+0.5{x}^{\left(1\right)}\left(k-1\right),k=2,3,\cdots ,n$ (3.12)

then

${z}^{\left(1\right)}=\left({z}^{\left(1\right)}\left(2\right),{z}^{\left(1\right)}\left(3\right),\cdots ,{z}^{\left(1\right)}(n)\right)$

So the gray differential equation is established as

${x}^{\left(0\right)}\left(k\right)+a{z}^{\left(1\right)}\left(k\right)=b,k=2,3,\cdots ,n$ (3.13)

The corresponding albino differential equation is

$\frac{\text{d}{x}^{\left(1\right)}}{\text{d}t}+a{x}^{\left(1\right)}\left(t\right)=b$ (3.14)

Remember

$u={\left(a,b\right)}^{\text{T}}$, $Y={\left({x}^{\left(0\right)}\left(2\right),{x}^{\left(0\right)}\left(3\right),\cdots ,{x}^{\left(0\right)}\left(n\right)\right)}^{\text{T}}$, $B=\left(\begin{array}{cc}-{z}^{\left(1\right)}\left(2\right)& 1\\ -{z}^{\left(1\right)}\left(3\right)& 1\\ \vdots & \vdots \\ -{z}^{\left(1\right)}\left(n\right)& 1\end{array}\right)$.

Then the least squares method [8] is used to find

$J\left(\stackrel{^}{u}\right)={\left(Y-B\stackrel{^}{u}\right)}^{\text{T}}\left(Y-B\stackrel{^}{u}\right)$ (3.15)

Reaching the minimum

$\stackrel{^}{u}={\left(a,b\right)}^{\text{T}}={\left({B}^{\text{T}}B\right)}^{-1}{B}^{\text{T}}Y$.

Then solving the whitening differential equation

${x}^{\left(1\right)}\left(k+1\right)=\left({x}^{\left(0\right)}\left(1\right)-\frac{b}{a}\right){\text{e}}^{-ak}+\frac{b}{a},k=1,2,\cdots ,n-1$ (3.16)

Analyze the results by error: use the step deviation test.

First, the step ratio $\lambda \left(k\right)$ is calculated from the reference data ${x}^{\left(0\right)}\left(k-1\right)$, ${x}^{\left(0\right)}\left(k\right)$ and then the corresponding step ratio deviation $\rho \left(k\right)=1-\left(\frac{1-0.5a}{1+0.5a}\right)\lambda \left(k\right)$ is obtained using the development coefficient $a$.

If $\rho \left(k\right)<0.2$, it can be considered to meet the general requirements; if $\rho \left(k\right)<0.1$, it is considered to meet the higher requirements

3.2. Model Application

Take position 1 (in front of the stadium) as an example.

The data brought into position 1 is the original data predicting the number of parkings at that position in the next three years, then

${x}^{\left(0\right)}=\left({x}^{\left(0\right)}\left(1\right),{x}^{\left(0\right)}\left(2\right),\cdots ,{x}^{\left(0\right)}\left(6\right)\right)=\left(9,12,17,19,21,22\right)$

The grade ratio can be obtained as

$\lambda =\left(\lambda \left(2\right),\lambda \left(3\right),\cdots ,\lambda \left(6\right)\right)=\left(0.75,0.71,0.89,0.90,0.95\right)$

It can be seen that except for $\lambda \left(2\right)$, $\lambda \left(3\right)$, all other ratios fall within the tolerable coverage $\left({\text{e}}^{-\frac{2}{7}},{\text{e}}^{\frac{1}{4}}\right)$, then the series ${x}^{\left(0\right)}$ cannot be used as the data for the model GM (1, 1) for gray prediction [9], and a translation transformation is required.

Take the constant and translate it

${y}^{\left(0\right)}\left(k\right)={x}^{\left(0\right)}\left(k\right)+c,k=1,2,\cdots ,n$

which is

${y}^{\left(0\right)}\left(k\right)={x}^{\left(0\right)}\left(k\right)+4,k=1,2,\cdots ,n$

Get the sequence

${y}^{\left(0\right)}=\left({y}^{\left(0\right)}\left(1\right),{y}^{\left(0\right)}\left(2\right),\cdots ,{y}^{\left(0\right)}\left(n\right)\right)=\left(13,16,21,23,25,26\right)$

Its ratio is

${\lambda}_{y}=\left({\lambda}_{y}\left(2\right),{\lambda}_{y}\left(3\right),{\lambda}_{y}\left(4\right),{\lambda}_{y}\left(5\right),{\lambda}_{y}\left(6\right)\right)=\left(0.81,0.76,0.91,0.92,0.96\right)$

Both fall within the tolerance coverage $\left({\text{e}}^{-\frac{2}{7}},{\text{e}}^{\frac{1}{4}}\right)$, and the series ${y}^{\left(0\right)}$ can be used as the data for the model GM (1, 1) for grey prediction.

Accumulate ${y}^{\left(0\right)}$ once to generate a sequence

${y}^{\left(1\right)}=\left({y}^{\left(1\right)}\left(1\right),{y}^{\left(1\right)}\left(2\right),\cdots ,{y}^{\left(1\right)}\left(6\right)\right)=\left(13,29,50,73,98,124\right)$

Mean series

${z}^{\left(1\right)}\left(k\right)=0.5{y}^{\left(1\right)}\left(k\right)+0.5{y}^{\left(1\right)}\left(k-1\right),k=2,3,\cdots ,n$

get

${z}^{\left(1\right)}=\left({z}^{\left(1\right)}\left(2\right),{z}^{\left(1\right)}\left(3\right),\cdots ,{z}^{\left(1\right)}\left(6\right)\right)=\left(21,39.5,61.5,85.5,111\right)$

also

$Y={\left({y}^{\left(0\right)}\left(2\right),{y}^{\left(0\right)}\left(3\right),\cdots ,{y}^{\left(0\right)}\left(6\right)\right)}^{\text{T}}={\left(16,21,23,25,26\right)}^{\text{T}}$,

$B=\left(\begin{array}{cc}-{z}^{\left(1\right)}\left(2\right)& 1\\ -{z}^{\left(1\right)}\left(3\right)& 1\\ -{z}^{\left(1\right)}\left(4\right)& 1\\ -{z}^{\left(1\right)}\left(5\right)& 1\\ -{z}^{\left(1\right)}\left(6\right)& 1\end{array}\right)=\left(\begin{array}{cc}-21& 1\\ -39.5& 1\\ -61.5& 1\\ -85.5& 1\\ -111& 1\end{array}\right)$,

and so

${B}^{\text{T}}=\left(\begin{array}{ccccc}-21& -39.5& -61.5& -85.5& -111\\ 1& 1& 1& 1& 1\end{array}\right)$.

Calculated using the formula in EXCEL

${B}^{\text{T}}B=\left(\begin{array}{cc}25414.75& -318.5\\ -318.5& 5\end{array}\right)$

${\left({B}^{\text{T}}B\right)}^{-1}=\left(\begin{array}{cc}0& 0.01\\ 0.01& 0.99\end{array}\right)$

Then when the minimum value is obtained by the method of least squares,

$\stackrel{^}{u}={\left(a,b\right)}^{\text{T}}={\left({B}^{\text{T}}B\right)}^{-1}{B}^{\text{T}}Y={\left(-0.10,15.58\right)}^{\text{T}}$,

That is $a=-0.10,b=15.58$.

So the gray differential equation is established as

${y}^{\left(0\right)}\left(k\right)-0.1{z}^{\left(1\right)}\left(k\right)=15.58$

The corresponding albino differential equation is

$\frac{\text{d}{y}^{\left(1\right)}}{\text{d}t}-0.1{x}^{\left(1\right)}\left(t\right)=15.58$

Solve the above formula

${y}^{\left(1\right)}\left(k+1\right)=\left(13-\frac{15.58}{-0.1}\right){\text{e}}^{0.1k}+\frac{15.58}{-0.1}=168.8{\text{e}}^{0.1k}-155.8,k=1,2,3,4,5$

The step deviation test becomes

$\begin{array}{c}\rho \left(k\right)=1-\left(\frac{1-0.5a}{1+0.5a}\right)\lambda \left(k\right)=1-\left(\frac{1-0.5\times \left(-0.1\right)}{1+0.5\times \left(-0.1\right)}\right)\lambda \left(k\right)\\ =1-\left(\frac{1.05}{0.95}\right)\lambda \left(k\right)=1-1.11\lambda \left(k\right),\text{\hspace{0.17em}}k=2,3,4,5,6\end{array}$

when $k=2$, $\lambda \left(k\right)=0.81$, $\rho \left(k\right)=0.0984$ ; when $k=3$, $\lambda \left(k\right)=0.76$, $\rho \left(k\right)=0.1546$ ;

when $k=4$, $\lambda \left(k\right)=0.91$, $\rho \left(k\right)=-0.0131$ ; when $k=5$, $\lambda \left(k\right)=0.92$, $\rho \left(k\right)=-0.0209$ ;

when, ,;

It can be seen that all are less than 0.2, which meets the general requirements. In other words, the model can be used to predict the number of parking.

In the same way, this model can be used to get the predicted number of other locations. The calculation results using EXCEL are shown in Table 3.

Table 3. GM Forecast of Parking Lots in the Next Three Years (Unit: Vehicles).

3.3. Linear Fitting Model

Linear fitting: Knowing several discrete function values of a function, by adjusting several pending coefficients in the function, the difference between the function and the set of known points is minimized. If the pending function is linear, it is called a linear fit.

3.4. Model Application

This article knows the number of parking spaces in a campus from 2013 to 2018, that is, some discrete points are known. Then you can fit these known scattered points to find an approximate function that minimizes the total deviation from the actual number of vehicles, and then use the trend of this function to predict the parking demand in the next three years.

Take campus location 1 as an example. The number of vehicles at position 1 from 2013 to 2018 is plotted on a scatter chart, as shown in Figure 1.

From the scatter point trend in the figure, it can be seen that the image is approximately a linear function. Suppose that the points (1, 9), (2, 12), (3, 17), (4, 19), (5, 21), (6, 22) are taken in, and the calculation can be obtained, , the formula is obtained. The prediction map obtained by EXCEL is shown in Figure 2.

According to and the closer to 1, the smaller the error, and the above trend line shows that the error caused by this linear formula is extremely small and can be predicted.

Similarly, the number of parking places in other locations can be obtained by using excel, as shown in Table 4.

4. Parking Capacity Analysis

Combine Tables 3-5.

Figure 1. Number of parkings at position 1.

Figure 2. Position 1 linear fit prediction.

Table 4. Linear fitted predictions of the number of parking lots in the next three years (Unit: vehicles).

Table 5. Predicted number of parking lots in the next three years (Unit: vehicles).

It can be seen from the above table that the predicted value of GM is similar to the predicted value of linear fitting. Because the purpose of this study is to ensure the parking needs of teachers and students, as many parking spaces as possible are installed without affecting the campus road system, drainage system, landscape planning, and as close to the building as possible.

Due to the vigorous development of computer technology in recent years, the application of mathematics is no longer confined to the fields of natural science, engineering technology, etc., but has penetrated all areas of life with unprecedented depth and breadth. The rise of mathematical modeling has better turned practical problems in life into mathematical languages and solved them mathematically. This article is to solve the unreasonable problem of campus parking space planning by mathematical modeling. Find out the problem through investigation and analysis, and combine the appropriate model to analyze and predict the data. It not only predicts the number of motor vehicles in the future, but also makes effective and reasonable distribution plans for parking spaces through data. It has a certain theory for improving the campus environment, using parking resources reasonably and effectively, and eliminating potential traffic safety hazards that may arise and realistic meaning.

Funding

Supported by Hebei Higher Education Teaching Research and Practice Project (2017GJJG261); Hebei Province Education Department Hebei University Science and Technology Research Youth Fund Project (QN2017502); Baoding University Transformation Development Research Fund Project (2017Z01); Natural Science Research Project of Baoding University (2012Z11).

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