1

This article discusses the optimization of the administrative model for managing urban passenger transport in route network cities, as well as the problem of optimizing traffic intervals, taking into account the time spent by passengers. At the same time, it was taken into account that the majority of passenger flows can be transported using Vehicle competing routes. With an increase in the interval of movement of vehicles along this route, the time spent by passengers increases, but the damage from transport to the urban environment is reduced, and, conversely, with a decrease in the interval of movement, the time spent by passengers is reduced, but the damage to the urban environment from the work of transport increases. The proposed model, without unnecessary restrictions and coefficients, using the economic assessment of the time of the population, allows us to calculate the optimal number of flights for any passenger flow. The above criteria will allow solving problems of high dimension corresponding to the size of any city.

vehicle

passenger traffic

traffic intensity

optimization

1. Artynov A.P. Automation of planning and management of transport systems / A.P. Artynov, V.V. Skaletsky. – M.: Nauka, 1981. – 272 p.

2. Balamirzoev A.G., Alieva Kh.R., Balamirzoeva E.R. Decision-making by passenger traffic on the choice of the route of movement // Fundamental research. - 2013. - No. 4. - P. 267–271.

3. Bolshakov A.M. Improving the quality of passenger service and the efficiency of buses / A.M. Bolshakov, E.A. Kravchenko, S.L. Chernikov. - M.: Transport, 1981. - 206 p.

4. Polak E. Numerical methods: A unified approach. – M.: Mir, 1974. – 374 p.

5. Semenova O.S. Thread optimization public transport in the urban environment / M.E. Koryagin, O.S. Semenov // Vopr. modern science and practice. University. IN AND. Vernadsky. - 2008. - T. 1 (11). – P. 70–79.

6. Himmelblau D. Applied non-linear programming. – M.: Mir, 1975. – 534 p.

In many cities, public transport consists of only one type. As a rule, these are small towns. The number of transport operators is insignificant, transportation is unprofitable, therefore, public transport is managed by the administration of the municipality, whose task is to ensure a balance between the loss of time for passengers and the damage from the operation of transport in the urban environment.

Optimization of traffic intensity of public transport on one route

To compile the model, the following initial data are required: passenger flows, i.e. the intensity of the arrival of passengers that this route is capable of transporting, as well as the total intensity of traffic of other routes competing for these passenger flows. It is also necessary to have information about the cost of one flight and the cost of a passenger hour, based on which the city system is tasked with finding the optimal interval for the movement of vehicles on a given route, ensuring the maximum efficiency of transport on the route in a specified period of time.

For the convenience of calculations, we regroup passenger flows along competing routes, i.e. Let us determine the total passenger flows carried by coalitions of competing routes:

R is the number of passenger flows carried by vehicles of this route together with coalitions of other routes;

λi - intensity of the i-th flow of passengers, transported including vehicles of this route, ;

λ is the intensity of the flow of passengers carried by vehicles of this route only;

μi - total intensity of Poisson flows of competing vehicles for the i-th flow of passengers, ;

μ is the intensity of the Poisson traffic flow of vehicles along this route;

δ - damage to the urban environment from one flight on this route.

Based on the fact that vehicle flows are Poisson, independent of each other and of passenger flows, the share of passenger traffic carried by each route is proportional to its traffic intensity, i.e. the share of the i-th flow of passengers carried by vehicles of this route is equal to

The average number of passengers transported per unit of time by vehicles of a given route is calculated by the formula

The total loss of passengers associated with waiting for vehicles is

and the damage to the urban environment from the operation of transport -

The goal of the municipality is to find optimal interval movement of vehicles along this route, providing the minimum total loss of passenger time (1) and transport damage (2):

(3)

With an increase in traffic intensity, the objective function increases indefinitely:

Therefore, it is possible to limit the traffic intensity of the GTS along the route m from above by a sufficiently large constant.

The second derivative of the objective function (3) is greater than zero:

Therefore, according to the necessary and sufficient extremum condition for μ > 0, the objective function has a global minimum provided that the first derivative is equal to zero (here and below: the asterisk denotes the optimal value of the parameter):

(4)

This section considers the problem of optimizing the interval of traffic along one route, taking into account the costs of transport and the socio-economic effect associated with passenger downtime. However, the work is mainly of theoretical interest, since in practice it is necessary to optimize the intervals of urban passenger transport along several interacting routes simultaneously.

Numerical example

Let us pay attention to an important feature of the model on small example Consider a route with a passenger flow of 1000 people. per hour, damage from 1 flight to the urban environment - 500 rubles, the average cost of a passenger hour - 50 rubles. Then we calculate the optimal number of flights:

This formula follows from (4) in the absence of competitors. The average waiting time will be 6 minutes, and the total time lost by passengers will be 100 hours (1).

Passenger traffic on the routes is different, besides, on the same route at rush hour, passenger traffic can be many times greater than in the early morning or late evening. Let's say passenger traffic drops by 4 times, to 250 passengers. Then, obviously, from the point of view of the transport operator, it is necessary to proportionally reduce the number of flights (in order to maintain profitability). Then 2.5 flights per hour will be performed, the average waiting time will be 24 minutes, the total loss of passengers will be 100 hours. Such a decision is unfair to passengers.

The model proposed in this article leads to the fact that the number of flights should be

In this case, the average waiting time will increase to only 12 minutes, and the loss of passenger traffic will be 50 hours, while the number of passengers carried per flight will drop from 100 to 50. This approach justifies the fact that even with a small passenger traffic, it is necessary to operate urban passenger flights. transport, despite the low coefficient of filling of the rolling stock.

In practice, in order to achieve a similar effect, restrictions are imposed on the maximum interval of urban passenger transport and the maximum filling factor of the rolling stock during peak hours and between peak times. This leads to approximately the same results as the model proposed in this paper. However, the model without unnecessary restrictions and coefficients, using the economic assessment of the time of the population, allows us to calculate the optimal number of flights for any passenger flow.

Optimization of traffic intervals for one type of public transport

We will construct a mathematical model for optimizing the operation of passenger transport in an urban environment. In the constructed problem, there are two criteria: the loss of time for passengers and the damage from the activities of transport. To resolve the contradictions between these characteristics, it is necessary to come to a common dimension for assessing the time of passengers and transport damage. In this model, a cost estimate is used for these characteristics, therefore, the general criterion for the efficiency of urban transport is the total cost estimate of the social significance of transportation and damage to the urban environment from the operation of passenger transport.

In order to meet the needs of each passenger in transportation, there must be routes capable of transporting a passenger between his initial and final stopping points, i.e. if

Ij > 0, then

The obvious limitation is that the intensity of vehicle flows moving along each route is not negative:

The total damage to the urban environment from the operation of urban passenger transport will be

Then the average costs of passengers waiting for transport at the i-th stop to move to the j-th, per unit of time are calculated as follows:

(7)

The objective function in this problem is the total costs of transport for the movement of vehicles along routes per unit of time (6) and the loss of time for passengers waiting (7):

(8)

Statement 1. The objective function (8) is downward convex in terms of traffic intensity over the entire existence area (5).

The left side of (8) is a simplified form of the average waiting time function

multiplied by the constant γ, and convex downwards. The right side is linear and, when added, does not affect the convexity. ◄

Assertion 2. Problem (5, 8) has a unique finite solution.

The objective function is strictly convex, and for each route l

In other words, transportation costs increase indefinitely as traffic increases. If we fix some solution , then it is in the region

so the following constraint must hold:

The set given by this constraint is convex and bounded, therefore, based on these provisions, the solution exists (statement 1), it is finite (9) and unique (statement 1). ◄

Statement 3. If (5, 8), then the damage from the work of transport to the urban environment and the loss of passengers coincide at this point.

According to the necessary extremum condition, the derivatives of the objective function in each direction are equal to zero:

(10)

Having expressed α k from (10), we substitute this expression into (8) and obtain the required result:

◄

Statement 4. If the cost of a passenger-hour in problem (8) increases by x times, then the intensity of traffic along the routes should increase by a factor of.

Let γ1 = сγ - new value passenger-hours, and - the optimal intensity of traffic on l-th route at the cost of passenger-hour γ1. Then at the optimum point the equality

(11)

in (11) we obtain (10), i.e. traffic is increasing. In a similar proportion, the waiting time for transport by passengers is reduced. ◄

Statement 5. If the damage from the operation of transport to the urban environment in problem (8) increases by x times, then the traffic intensity should decrease by a factor.

Let - the new cost of one flight on the l-th route, and - the optimal intensity of traffic on the l-th route in this case. Then at the optimum point the equality

(12)

Obviously, in this case, when substituting the expression

in (12) we obtain (10), i.e. traffic intensity is reduced. In a similar proportion, the waiting time for passengers increases. ◄

Statement 6. If the intensity of passenger traffic in problem (8) increases by x times, then the intensity of traffic should increase by a factor of.

Let - new intensities of passenger traffic, and - the optimal intensity of traffic on the l-th route in this case. Then at the optimum point the equality

(13)

Obviously, in this case, when substituting

in (13) we obtain (10), i.e., traffic intensity is reduced. In a similar proportion, the waiting time for passengers increases. ◄

To find a solution to this problem, many algorithms have been developed: the coordinate descent method, Newton's method, etc. The convexity of the criterion and its differentiability over the entire admissible region will allow solving problems of high dimension corresponding to the size of any city.

Reviewers:

Agakhanov E.K., Doctor of Technical Sciences, Professor, Head. department " Car roads, bases and foundations”, FGBOU VPO “Dagestan State Technical University”, Makhachkala;

Fataliev N.G., Doctor of Technical Sciences, Professor of the Department " Automobile transport”, FGBOU VPO “Dagestan State Agrarian University named after. MM. Dzhambulatov, Makhachkala.

The work was received by the editors on October 10, 2014.

Bibliographic link

Balamirzoev A.G., Balamirzoeva E.R., Kurbanov K.O., Gadzhieva A.M. OPTIMIZATION OF ONE TYPE OF PUBLIC TRANSPORT IN THE URBAN ENVIRONMENT // Fundamental Research. - 2014. - No. 11-3. – S. 499-503;
URL: http://fundamental-research.ru/ru/article/view?id=35549 (date of access: 12/30/2019). We bring to your attention the journals published by the publishing house "Academy of Natural History"

Our company designs and optimizes the route network of public transport, including the development integrated schemes for the organization of transport services for the population by public transport (KSOT),which allows to improve the quality of passenger service (passenger traffic):

• Suggestions for new routes to meet demand passenger traffic;
• Tariff policy optimization and much more

Passenger traffic

Urban passenger transport plays a significant role in the economy of any country, region or city, since it is route transport that is the main way to move passengers, where there is a high demand for passenger traffic.

Passenger traffic - the movement of passengers ordered by the transport network, quantitatively expressed in the volume of passengers transported by any type of public (ground, underground, air, etc.) or individual transport per unit of time (hour, day, month or year).

In connection with the increase in the level of motorization and the increase in the mobility of the population against the backdrop of insufficient rates of development of the road network of cities, the problem of optimizing passenger traffic is very acute, aimed at reducing the time or cost of travel, for example, from home to work, to university, to the store, etc. In other words, it is necessary to satisfy the demand of the passenger traffic for any type of movement as much as possible.

Optimization of work (routes, schedules) of public transport

In large metropolitan areas, public transport should serve as a “worthy” alternative to individual transport - to reduce the level of congestion on city roads, improve the ecology in the city, etc. It is for these purposes that programs are being developed aimed at optimizing the operation of urban public transport.

The main task of optimizing public transport is to increase its attractiveness for citizens. This result is mainly achieved in various ways, for example, reducing travel time for passengers, reducing the number of transfers, increasing walking distance to stopping points, keeping to the schedule, informing passengers in a timely manner about the arrival of transport, allocating public transport lanes, and much more.

Our company offers the following list of solutions to optimize the operation of public transport, aimed at improving the quality of passenger service (passenger traffic):

1. Carrying out surveys of passenger traffic on public transport routes;

2. Analysis of the current situation of urban passenger transport:

• Functional characteristics of the existing route network
• Estimating the level of route duplication
• Characteristics of the quality of public service by passenger transport routes, etc.

3. Creation of a tool to support decision-making in the field of transport complex management based on a model of public transport routes (“management” of passenger traffic)

4. Development of proposals for optimizing the operation of public transport (both for the current situation and for future years):

• Proposals for improving the transport and pedestrian service of the territory - improving accessibility (transport hub, city district, region, etc.)
• Proposals for the passage of new routes to meet the demand of passenger traffic;
• Proposals for the arrangement of additional stopping points, or their exclusion;
• Proposals for changing the public transport timetable for each day of the week (calendar use);
• Justification of the expediency of arranging dedicated lanes for public transport;
• Calculation of turnover of public transport routes;
• Tariff policy optimization and much more.

We attended a meeting of the "Quick Wins" working group at the Department of Transportation. Target - optimization and efficiency increase of SUE "Mosgortrans" by relatively simple solutions(which may be implemented in the near future).

The agenda for discussion was as follows:

The only point that caused us complete embarrassment was the equipment of the NGPT with luggage racks. Two things are not clear: where should this shelf be located and why is it needed at all? People use public transport when they travel mainly to and from work, as well as to travel around the city, while no one carries a few suitcases in their luggage, and a small flow of passengers always goes to train stations and airports by public transport.

You can evaluate and comment on the rest of the proposals yourself, I will tell you what we proposed for our part:

1. Increase in the number of rolling stock with extra large capacity(harmonicas) and tram trains. If you know bus and trolleybus routes where crowded buses run, then write to us, we are compiling a list.

2. Reducing the number of stops for main bus routes. Long routes should not stop at every stop. As an example, we discussed route No. 716 “Sokolniki - pos. Vostochny, which is duplicated for most of the route along Stromynka and Shchelkovo highway trolleybus routes. If there is a trolleybus "stopping at every pole", then the bus running on the parallel track must be a semi-express. Here, too, everything is simple - we form a list of routes and stops that we want to cancel - after that we form a list of proposals for consideration in the MGT and DT.

3. Ghost routes at intervals of half an hour or more, it is necessary to partially transfer to small-capacity buses (minibuses) with a reduction in traffic intervals. That is, instead of one large bus every 30 minutes, we propose to run three small ones, but with an interval of 10 minutes. After monitoring the passenger flow, it is possible that such routes will be rolled out and the demand for them will increase. We also believe that on such routes it is always necessary to leave at least one "big" bus that will run according to a clear schedule.

4. "Draw charts". They promised to accelerate the movement along the routes. Now there is a paradox that being late according to the schedule is not considered a violation, but arriving earlier is considered a violation! A lot of work remains to be done in this direction by the MGT. A trolleybus should not travel at a speed of 15 km per hour along an empty street simply because it has such a schedule.

5. Cancellation of extra and double stops. After the introduction of ASKP, Mosgortrans made separate stops at stops with a large passenger flow for boarding and disembarking passengers. That is, the bus actually stopped twice at the same stop, passing 5-10 meters between disembarkation and landing. This led to an even greater increase in travel time along the route. Once it was justified - it was necessary to teach people to get into the front door, but now people have already adapted to this, there is less sense in such stops. We proposed to revise them and liquidate a significant part of them.

Another problem is too frequent stops on the route: it is not uncommon for stops to be located 40-50 meters apart. Too frequent stops also slow down the movement along the route, some stops should be transferred to the "On demand" category.

6. Tram traffic lights. We are waiting for concrete proposals on problem areas, collecting tram traffic jams and where an increase in the tram cycle is required. In the same paragraph, they added a promise to deal with delusional instructions, according to which two cars in a row cannot pass the intersection, or it is possible, but it is necessary to maintain a considerable distance there, otherwise they will be deprived of the bonus.

7. Launch a website where GLONASS will show the location of buses and trolleybuses. Implement a mobile application site for iPhones and Androids.

8. In all air-conditioned buses gotta hang up the phone hotline and a short SMS number where you can quickly send a complaint about a non-working air conditioner in a particular bus.

All our proposals were approved and accepted for consideration. After analyzing the labor costs and time costs, arrangements for specific terms will be given.

What other questions were raised during the discussion:

1. The idea was put forward to make most of the stops, except for the nodal ones, "on demand". That is, what they tried to implement a few years ago and gave up - the doors should open only at the request of passengers. Well, in general, the idea may not be bad, but in our opinion, for its success, massive informational advertising is needed, as it once was with the introduction of ASKP.

2. Increasing the ticketing network - any measures that will reduce the number of ticket purchases from the driver.

3. Installing a board with the time of arrival of the shuttle bus. This is a very good, correct idea, but we see it as completely unrealistic to install a scoreboard at ALL MGT stops. On the main passenger-generating ones - it is possible, but in other places it is the mobile application that can be the solution.

4. Possible rebranding of Mosgortrans. But in order to change the sign, it is desirable to change the content, in the case of Mosgortrans, this would need to be summed up by some significant event, for example, the abolition of turnstiles. By the way, how would you suggest renaming Mosgortrans?

P.S. Members of the working group:

The transport system in the most general case is a set of employees, vehicles and equipment, elements of the transport infrastructure and the infrastructure of the subjects of transportation, including a control system aimed at the efficient movement of goods and passengers, forming a connected whole. Fig.1 Public transport overflow problem (36 Kb, 14 frames, 10 cycles)

The efficiency of the transport system cannot be considered only in terms of achieving the optimal performance of the corresponding processes within the system. The main tasks of the transport system are to meet the needs of the economy in the transportation of goods and ensure the mobility of the population. In this regard, the efficiency of the transport system will always be determined by a certain balance between the conflicting requirements of the economy and society. A striking example is the desire of the passenger that the transport approached the stop as soon as the passenger approached it, and the desire of the carrier to set such a traffic interval that the vehicles were always full and brought maximum income. Thus, to build an effective transport system, it is necessary to combine knowledge in the field of transport with economics, urban planning, geography, ecology, sociology and psychology.

In the economy, the transport industry occupies a specific position, referring to the economic infrastructure. Transport is part of the productive forces of society and is an independent branch of material production that ensures the normal operation of the economic system as a whole. It follows that the products of transport are of a material nature and are expressed in the movement of the material product of other industries.

A characteristic feature of the functioning transport systems is the cyclic nature of their work. The starting point of the working cycle of the transport system is the supply of empty rolling stock for transportation. When transporting goods, this is the supply of rolling stock for loading, for passenger traffic– Departure of the bus from the final destination to the route. Depending on the technology of transportation and organization of traffic in the process of the transport cycle, various transport processes can be performed related to loading or unloading cargo, boarding or disembarking passengers. The transport cycle ends at the moment of arrival of the empty rolling stock for loading or at the moment of the start of the route by the passenger bus.

V real conditions The implementation of the transport cycle is affected by a significant number of different disturbances, most of which are random in nature, so the main characteristics of the transport cycle, for example, its duration, are usually very unstable. In order to stabilize them, it is necessary to take measures to reduce the number of disturbances. This is, for example, the organization of a dedicated lane and priority traffic lights for urban public transport.

The transport system is a unique example of a system with the collective behavior of its subjects. In this regard, collective behavior is a powerful factor that forms the patterns of functioning of the transport system. Moreover, the processes of self-organization lead to the formation of several levels of stable functioning of the system, forming a hierarchical structure of collective adaptation with different temporal stability.

In this regard, the following three structural levels can be distinguished:

• distribution of places of formation, processing and consumption of goods, resettlement of the population;
• organization of transport processes in the network;
• formation of traffic flows in the network sections.

Transport systems occupy the most important place in providing almost all spheres of activity of the economy and society. Naturally, increasing the efficiency of their functioning is a necessary condition for the development and improvement of the economy and the quality of life of the population. Improving the efficiency of transport systems involves solving a set of interrelated tasks, many of which can be attributed to tasks of a higher level, since they go beyond narrow transport problems. Improving the management of the system of urban passenger transport (UPT) should be accompanied by tracking changes in all emerging relationships that determine the operation of the system and its effectiveness. In order to do this without conducting rather complex and potentially negative experiments in real GPT systems, the use of models is necessary. In order to maintain the accuracy and informativeness of the model, it would be reasonable to identify and include in the GPT system management model its main relations with the external environment, which include relations with consumers and the state. As domestic and foreign practice shows, it is these relationships that are the main limitations in the operation of GPT systems.

As the experience of managing the GPT system shows, in order for the system approach to be implemented in practice, it is advisable to consider various options (sections) for modeling the GPT system. One of such modeling options is considered in the work of V.V. Feenman. The author highlights engineering and economic models that allow solving the problem of improving the control of the GPT as a system.

Engineering models, public transport management cover space and specifications different types public transport. At the same time, it is proposed to create a coordinated hierarchical network of services that combines the advantages of various types of public transport. This approach finds practical application in the German-speaking area of ​​Europe. Economic models are based on the fact that public transport providers are not flexible and pay little attention to the needs of passengers. Economists believe that the solution is to change the regulatory regime. Passenger transport enterprises should be responsible for the quality of the services provided and bear the corresponding financial costs or receive corresponding profits. This approach finds support in the Anglo-Saxon parts of Europe.

Since none of these approaches alone is capable of providing the proper level of mutually beneficial transportation that suits both enterprises providing public transport services and public transport passengers. The state is also interested in the efficiency of transportation, because with the same number of passengers transported, it can receive more income in the form of taxes on the profits of carriers. In addition, in many countries, including the CIS countries, the state itself provides public transport services to the population, which, of course, requires improving its efficiency in order to save budget funds.

As V.V. Feenman notes, “the theoretical model should show how it is necessary to jointly use legislation, coordination and design of services to achieve maximum efficiency” . In other words, it is necessary to find the optimal combination of public administration and regulation of the GPT system and the development of competition in this system. At the same time, both economic flexible methods of influence and measures to directly determine the parameters of the system's activity through the establishment of schedules, schedules and traffic modes, as well as the coordination of their operational activities in the on-line mode, should be considered.

The study defines a set of requirements for the GPT system on the part of the population's demand for transportation, on the basis of which various public transport systems are evaluated. These requirements can be formulated as follows:

• urban passenger transport must ensure the integrity of the city as a geographical entity and the demand of the population for transportation must be predictable and manageable;
• for political reasons, the fare in the GPT should ensure its availability for socially unprotected segments of the population;
• performance indicators works must meet the requirements of consumers;
• the required budget subsidy must be real, stable and allocated for certain results of work, taking into account the level of quality of public service.

1. Relevance of the topic

Currently, in large cities there is a problem of overloading public transport during the so-called "rush hours". During these hours, the load on public transport increases disproportionately, which leads to the fact that it cannot cope with the passenger flow. This means that people can't get to work on time or get depressed due to traffic congestion. In turn, these factors negatively affect the productivity of workers and, as a result, the growth of the city's economy.

2. Purpose of the study

The aim of the work is to optimize the operation of public transport, ensuring a uniform load on it during the day.

Object of study: the city's passenger transportation system.

Subject of study: cost optimization for carriers and increased benefits for passengers.

As part of the master's work, it is planned to obtain relevant scientific results in the following areas:

1. Development of a model for describing the system of passenger transportation of the city.
2. Definition optimal parameters functioning of the system under given constraints.
3. Development of recommendations for the implementation of the results obtained in production practice.

3. Overview of existing approaches

The optimal planning of the operation of transport systems, which fundamentally allows to overcome most of the listed difficulties, is based on a system of interrelated mathematical models, within which it is possible to take into account such features of transport systems as the fuzziness of the available information, contradictions in the interests of partners, the multi-purpose nature of the assessment of the selected modes of operation, etc. e. Based on these models, it becomes possible to formalize optimization problems and use the appropriate mathematical apparatus.

In the work of A.E. Gorev, there are several classes of transport system optimization problems:

• The tasks of routing transportation and vehicle movement are to choose rational or optimal schemes for the movement of goods or passengers between a finite number of points.
• The tasks of loading vehicles determine the range, volume and layout of cargo during transportation.
• Traffic scheduling tasks arise during maintenance technological processes manufacturing enterprises (transportation of concrete), transportation using the "just in time" technology, when loading or unloading vehicles at large warehouses and terminals, passenger transportation.
• The tasks of planning the use of labor and technical resources in the transport hub are solved to optimize the use of general hub and specialized resources for each type of transport in order to reduce downtime of all types of resources, increase the productivity of the transport hub.
• The tasks of planning the work of transport enterprises affect the transport (transport planning) and operational (performance of maintenance, use of personnel) activities of the enterprise.
• The tasks of production and transport planning relate to logistics systems, when, according to the criterion of the minimum total costs for the production and delivery of products, a plan for the production, distribution and storage of finished products is determined in the presence of alternative sources of supply and consumption of interchangeable products.
• The tasks of determining the optimal tariffs allow maximizing the income of a transport enterprise through the implementation of a particular marketing policy.

Thus, it can be seen that the existing approaches to solving transport problems do not consider the direction of optimization chosen for this work, since they are aimed at other areas, for example, optimizing cargo transportation or building a map of possible traffic congestion areas.

4. Statement of the problem

In order to optimize public transport, this paper will consider the problem of finding the optimal number fixed-route taxis necessary for the transportation of all passengers of the city along the routes they need, taking into account the minimization of the carrier's costs and the maximization of the benefits of passengers.

The solution of this problem will allow to determine the necessary fleet to meet the need for transport for the population of the city, taking into account the interests of carriers and consumers.

To solve the problem, it would be advisable to conduct a large-scale study of passenger traffic in its dynamics over a long time (about a month) to identify all the characteristic patterns and take them into account in the planned model.

The general problem can be formulated as follows. It is necessary to build a model that allows you to determine the number of fixed-route taxis needed to transport all city passengers along the routes they need, taking into account the minimization of the carrier's costs and the maximization of passenger benefits.

It is also necessary to take into account that some of the variables can be connected not only through the functional, but also directly among themselves through some equations.

This paper considers a particular problem: it is necessary to build a model that allows you to calculate the required number of fixed-route taxis to transport the population along one route with the optimal ratio of the time of transportation of all passengers, the carrier's profit and the fare.

We introduce the following quantities:

Then the optimization problem can be formulated as follows:

where t is time;
– the number of passengers to be transported;
S is the fare;
Z is the cost of the carrier for one fixed-route taxi;
– travel time along the route;
- the number of fixed-route taxis;
- the maximum possible number of passengers in one minibus;
- weight coefficient.