The benefits of mathematical modeling in the development of the transport system

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THE BENEFITS OF MATHEMATICAL MODELING IN THE DEVELOPMENT OF THE TRANSPORT SYSTEM

О ПОЛЬЗЕ МАТЕМАТИЧЕСКОГО МОДЕЛИРОВАНИЯ ПРИ РАЗРАБОТКЕ ТРАНСПОРТНОЙ СЕТИ

Filippova P.O, Nadezhdina N.G.

Nizhny Novgorod State University of Architecture

and Civil Engineering (NNGASU), Russia

The theme of paper is an application of mathematical models in transport infrastructure. The paper deals with the field of road construction. The paper tackles the problems of traffic jams, road accident, overload / underload of transport nodal points. The author considers the following issue: firstly, the author favours public transport; secondly, considers construction of new roads. In conclusion it is stated that these problems can be solved more efficiently with the help of mathematical modeling.

Keywords: mathematical modeling, transport infrastructure, transport capacity, traffic jams, outlays of travel, equilibrium, public and personal transport.

The paper is devoted to the analysis of the benefits of mathematical modeling in the development of the transport system. A modern city needs a constant increase of volume of traffic, improving its reliability, safety and quality. This requires an increase of expenses for improving the infrastructure of the transport system, turning it into a flexible, all-round managed, high-efficient logistics system. But the investment risk will greatly increase if the laws of development of the transport system and the laws of load balancing its area are not taken in account. Ignoring these laws leads to the environmental damage, frequent formation of traffic jams, overload / underload of some lines and transport nodal points, increase the level of accidents.

Is it possible to do without the mathematical models and numerical experiments, using only results of engineering calculations? For example, to calculate the unloading of road section, it is necessary to know how many vehicles turn right at a crossroad. Until now, vehicles have not turned right at this crossroad, so there are no data for the calculation. Specialists have to rely on generalized experts estimations. Moreover, all the time traffic adapts to control actions. The effect of the calculated unload disappears after some time, due to redistribution of the traffic flow. If the amount of congestion increase sharply on the following day, due to periodic changes or random factors, as a rules, traffic intensity reduces [1].

One might reasonably draw the conclusion that modeling is essential because of following properties of the transport system:

• the compensation of increase of transport capacity in case of development of transport system, increase of demand and his redistribution in new conditions;

• the unpredictability of drivers behavior i.e. the choice of the route, driving habits, ect;

• the influence of random factors (accidents, weather) and fluctuations connected with seasons, weekends and holidays.

Currently, on the one hand, the transport models and assessment are based, to a considerate degree, on the idea of stable and repetitive trips. Undoubtedly, such models exist, and they dominate in our views about our own lives and in our interpretations of other people's behavior.

But on the other hand, changeable and dynamically unstable factors are hidden behind superficial stability. It is surprising, but in most cases, people standing at the same time in the same queue at traffic lights crossroad for two consecutive days are not substantially the same individuals [2].

Each year, up to one third of people change their jobs, and every seventh relocate, besides people settle down to married life. Their children go to school. Some people get divorced, retire.

If, for example, the number of car owners is growing steadily by 2% a year, in fact, this means that 12% of families have increased the number of vehicles and 10% have decreased [3].

This system of stable estimations leads to the following axiom: "At first, experts predict how the traffic will increase, then the transport system is expanded in volumes, sufficient for maintenance". It is pointed out that transport planning is based on this axiom. It is believed that we can just build new roads. As soon as the load level of road become too large for comfortable traffic, it is necessary to build up their capacity.

But why the construction of new roads does not always improve the situation with traffic jams, and can even make the problem worse? Let us use the following simple model (Fig.1).

Users of transport infrastructure can be presented as players, who initially choose the type of transport, then a suitable route and try to minimize the outlays of travel. Time and comfort of travel, financial and other costs of drivers are implied by the outlays. It will lead to a regular game, the solution of which is to find a balance. Interestingly enough, the system comes to equilibrium, even if drivers are not able to assess the situation on the road accurately, but they can "learn" over time [4].

Figure 1.

mathematical modeling transport infrastructure

The paper puts forward the idea that the outlays of vehicle users are determined from the condition of balance between personal and public transport.

Let us look how public and personal transport influence each other. Figure 1 shows by graphical display, that outlays of users of personal and public transport are equal in position of balance.

Let us suppose that the construction of roads will lead to the disappearance traffic jams and to significant reduction of travel time by personal vehicle. This corresponds to the shift of schedule of outlays for personal transport "right". But in this case it will be more profitable to passengers, who have own cars, to use their cars again. It will result in increase of a number of automobilists and consequently to an increase of traffic jams and travel time. This system will come into a new equilibrium in which the outlays of using of personal and public transport re-match. On average, the benefit is much less, than expected. Available alternative is to improve the system of public transport. This corresponds to the shift of schedule of the outlays for public transport "down." In equilibrium, the outlays of personal and public transport are equal, and thus, improving public transport and bringing more people to it, new opportunity to unload road and improve the situation in the entire transport system appears. Certainly, construction of road works usually in the same way, but it is incomparably more expensive.

The following paragraph proposes the second idea that sometimes it is more profitable to close the road, rather than to build new ones. The unreasonable increase of the number of roads can not only improve the situation, but even make it worse for all road users [5].

Let us assume automobilists want to get from point Start to point End. There are two ways: through the city A and through the city B. Time of travel from point Start to the city A depends on the traffic density and is equal to the number of cars (T), divided by 100. The route from point Start to the city B does not depend on the number of cars, it is 45 minutes. Likewise, the route from point A to End takes 45 minutes and the travel time from point B to End is equal to T / 100. If A and B are not connected, the time on the route Start-A-End equals A / 100 + 45, and on the route Start-BEnd equal T / 100 + 45. If one of the ways were shorter, there would be no Nash equilibrium, so every rational driver would switch to the shorter route. Let us assume 4000 cars started from the point Start. Then looking at the equation A+B=4000, one might draw the conclusion that the system will come into balance when A = B = 2000. Therefore, regardless of the selected road vehicle will travel 2000/100+45=65 minutes (Fig.2).

Figure 2

Now let us assume that the dotted line between A and B is a new, very short way, driving on which takes about 0 minutes. In this situation, all the drivers will prefer the route Start-A to Start-B, because the route Start-A will take T / 100 = 4000/100 = 40 minutes in the worst case, while route Start-B is guaranteed to take 45. Being in point A every rational driver would prefer the short route to B, and then they will drive to the point End, because the route A-End is guaranteed to take 45 minutes and the route A-B-End will take only 0 + 40 = 40 minutes in the worst case. Thus, the travel time for each driver will be 4000/100 + 4000/100 = 80 minutes, that is, after the construction of the new road travel time increased by 15 minutes.

If the drivers agreed not to use the road between A and B, they would save this time, but as each individual driver economizes time using road A-B, then such distribution is not socially optimal. That is a paradox.

It follows from that has been said that transport infrastructure is one of the most important infrastructures that support life of cities. So optimal system planning, better traffic management, optimization of public transport route system are getting special importance. Solving such problems is impossible without mathematical modeling of transport systems.

Bibliography

1. Introduction into the mathematical modeling of transport system major:A manual for students of high schools / A.V. Gasnikov, Maples S.L., Nurminski E.A., Shamrai NB.; Applications: ML Blank, Gasnikova EV, AA Zamyatin and

MalyshevV.A., Kolesnikov A.V., Raigorodskii A.M.; Ed. A.V. Gasnikova. - M.: MIPT, 2012. - 362 p.

2. V.G.Karchik. Data modeling. Tutorial. SPb.: -PGUPS. 2010.

3. Evolutionary Implementation and Congestion Pricing / Phil Goodwin-2013.

4. V.G.Karchik. Economic-mathematical modeling. Tutorial. SPb.: -PGUPS. 2011.

5. Afanasyev L.L. Integrated transport system: A manual for students of high schools / L. Afanasyev, N.B. Ostrovsky, - M.: Transport, 2012. - 333 p.

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