Fuzzy model of transport demand

. The development of transport networks is costly and cannot solve all transport problems. In this regard, scientifically based planning of the development of the transport complex acquires an important role to improve the quality of the functioning of the transport system, such as improving the organization of traffic in various parts of the network, optimizing the system of public transport routes, creating convenient transfer hubs, etc. To solve such problems, the paper considers the development of intelligent transport systems and mathematical modeling of transport demand. The central task of developing a transport model is to determine the equilibrium state of the transport system, i.e., the problem of the distribution of flows in the network, which involves two main stages. In the first stage, one or several correspondence matrices are evaluated based on the traffic flow's initial data and behavior. The second stage consists of the distribution of traffic matrices on the graph of the transport network, i.e., in solving the problem of finding traffic flows or the problem of transport equilibrium. The problem of transport equilibrium is reduced to an optimization problem in which restrictions are imposed depending on the values of the elements of the traffic matrix. The search for new methods for solving optimization problems of weakly formalized processes is caused by the presence of processes and phenomena, which is difficult to describe by methods of classical mathematics due to their indeterminacy and unreliability, incompleteness, and fuzziness of the initial information. The mathematical model of the optimization problem is described by the objective function and constraints. Classical mathematical programming and its variations are largely a normative methodology for effective choice. Fuzzy programming highlights the natural multiplicity of inaccurately defined goals, restrictions, and their meanings.


Introduction
A transport model is a mathematical tool that allows you to build the distribution of traffic flows over a network.The main task of such mathematical models is to determine and predict the parameters of the functioning of the transport network [1].Key parameters include: -intensity of traffic flows on network elements; -traffic volumes in the public transport network; -average movement speeds; -temporary delays in movement, etc. Mathematical models used to analyze transport networks differ in the tasks to be solved, the mathematical apparatus, the initial data, and the degree of detail of the description of the movement [2].According to the review work of Shvetsov [1], there is a classification of models based on the types of problems for which they are used: -predictive models; -simulation models; -optimization models.
In the class of optimization models, optimization problems are solved routes for passenger and freight traffic, development of optimal network configurations, etc. Methods for optimizing transport networks represent a vast area of research.The foundations of this direction are outlined in [3].
The first transport models appeared in the 60s of the XX century in Great Britain; however, until the 90s of the last century, they were not popular due to the computational complexity of the algorithms used to create the models.Since the 90s, many software systems have appeared that allow solving the problem of modeling on real transport networks, which served as an impetus for the use of transport models for transport management on a city and region scale.In the 21st century, the use of transport models has become a global trend and an integral part of the management of the transport complex.
Virtually all medium and large cities in North America and Europe have developed and used such models.It is customary to build transport network models for cities and agglomerations, regions, countries, and even parts of the world.In particular, there are transport models in Germany and Switzerland, detailing the state as inland transport movements within the country and external movements from neighboring countries.There is a transport network model throughout Western Europe [4][5].
Software and computer systems, technical equipment used in traffic regulation, and solving their problems are based on visualized initial data.The number of free services used to find routes from one place to another based on web technology and in selecting travel destinations is increasing (Table 1).By visiting the service's website, a start and end address will be entered, which will return the route along with the route maps.We can offer one of the service's shortest (distance) routes.Some services consider additional factors to the route.For example, the ViaMichelin website allows users to choose the most economical or most natural way, among other options.
GoogleMaps: Google Software Engineer Barry Bnunitt has published publications about the features of the GoogleMaps software tool [6,7,8] as a reduction framework for processing multiple machines simultaneously to process thousands of large geographic data sets [9].The following suggestions, taken directly from Brumitt, provide advice on how the software tool should work: the technology and algorithms used are not public, and therefore not much is disclosed about the process.
MapQuest: In 2004, SIAM News correspondent Sarah Robinson spoke with Mark Smith, MapQuest's chief technology maker.He did not elaborate on how the algorithms developed for MapQuest worked but did provide some information about it.
Livemaps: Livemaps uses A * search and heuristic algorithms based on combined path categories [10].It should be noted that there are very efficient algorithms that identify the shortest path problems from one node to another to compute these services, where the main network can have tens of millions of nodes.These algorithms are based on angular lower boundaries [11,12], drop-down techniques [13], path hierarchies [14,15,16], parallel programming [17], geometric containers [18], and part-based hierarchical methods [19].developed.
In mathematical optimization problems, the preference between possible solutions is described using objective functions on a given set of options.The objective function values describe the evaluation of each option so that the more preferred options have larger objective function values than the less preferred ones.The set of permissible options in optimization problems is described with the help of constraints -equations or inequalities representing the necessary connections between the options.The analysis results significantly depend on how adequately various factors of real systems are reflected in the descriptions of the objective function and constraints [20][21][22].
The mathematical formulation of the objective function and constraints in optimization problems usually includes some parameters.The values of such parameters depend on many factors that are usually not included in the formulation of the problem.Trying to make the model more representative, we often introduce complex relationships, making it more cumbersome and analytically unsolvable.Often, such attempts to increase the "accuracy" of the model are useless due to the impossibility of accurately measuring its parameters.On the other hand, a model with fixed parameter values may be too coarse since these values are often chosen quite arbitrarily [4,5].

Methods
To obtain a traffic matrix by demand layers, it is necessary to solve the problem of maximizing the function for each demand layer , 1 (ln( ) max under the following restrictions .
Here ij P is likelihood of movement out of the area i to the district j ; ij x is displacement matrices from the area i in the region j .2) will be reduced to the solution of a nonlinear programming problem in the following form [4]: .
Thus, a model of transport demand is built.In general terms, a nonlinear model for optimizing weakly formalized processes is given for a fuzzy given initial information-vector ) ,..., , ( with restrictions: is a set of possible values of resources, the values of fuzzy parameters described in the form of fuzzy subsets with the corresponding membership functions.
If the membership functions of the parameters are given, then after defuzzification, one can obtain the usual problem of convex nonlinear programming: The problem of nonlinear programming in the general formulation can be modified by writing each of the nonlinear functions of this problem by the first two numbers in the corresponding expansion in a Taylor series in the neighborhood under restrictions: The convergence of these methods to the solution is guaranteed if [20]: 5) all functions appearing in the conditions are limited.This problem can be interpreted as a maximal decision-making model corresponding to the optimal solution of the fuzzy objective function with a possibility not lower than a given one.

max,
. This model allows you to determine the maximum possibility or necessity to achieve a fuzzy goal.
An algorithm for finding a solution to a nonlinear programming problem has been developed: List of variables: ) is coefficients for expressing a non-basic vector k A through the basis vector; v(i) is numbers of basic variables.
Stages and algorithms for finding a solution to a nonlinear programming problem using the linearization method.
1. Determination of the initial admissible solution of the problem.
2. Finding the gradient of the function ( 3) -( 5) at the point of a feasible solution.
We streamline the process of iterative calculations according to the inverse matrix method (we assume that the old basic solution x and basic inverse matrix 1 c B already found) with interval coefficients.When replacing the values of real coefficients with intervals and real arithmetic operations with interval-arithmetic ones, iterative calculations will be carried out in the following sequence: 3.1) Calculation of the simplex multiplier: 3) Determining the number of the variable that must be entered into the basis: ( ) max ( ).

Results and Discussion
When modeling real decision-making problems in a fuzzy form, only fuzzy descriptions of functions can be at the disposal of a researcher -mathematician F , g and the parameters included in them, as well as the set itself X .In this case, the problem of standard mathematical programming will be presented as a problem of fuzzy mathematical programming.
The computational experiment was carried out for the problem of fuzzy mathematical programming given in the following statement.
Let a nonlinear optimization model be given with fuzzy initial information: with restrictions:  .(12) Note that due to the vagueness of the description of the coefficients ij j a c , and b evaluation of any alternative X x (and, accordingly, the values of the F function when X x ) is a fuzzy subset of the numeric axis of the base set Х.
Let the values of the coefficients j c , ij a and their estimates i b are specified as the following fuzzy values: , where 01 , 0 Finding the maximum of the function: on the set D X, determined by the system of restrictions We will use the system of line-by-line restrictions whenever possible.The initial parameters will be modeled by normal distributions.Then we turn to the possibilistic formulation of the problem, which consists in optimizing the objective function (15) on the set D defined by the system Let us set the perturbations of the initial fuzzy parameters.Then instead of problem ( 15), ( 16), we get optimization problem (15)  .In this case, the distribution of the possibilities of belonging to points хX set of feasible solutions D H has the form: It is easy to see that an increase in the perturbation of the constraint parameters reduces the possibility that the points belong to хX set D H .At H=0, result 0 = 2 is reached at the point (1, 1) with a possibility of 1.The paper describes the reduction of the problem of transport equilibrium to variational inequality and optimization problem.
Methods for solving the problem of finding equilibrium flows are considered.The concept of a correspondence matrix is introduced, and various approaches to its evaluation are reviewed.The described approach for describing models of nonlinear optimization problems, based on expert representations of them in the form of fuzzy values (i.e., in the form of sets, not point values), allows the decision maker to understand the meaning (semantics) of the objective function and the constraints of the problem being solved optimization of weakly formalizable processes in the presence of various types of uncertainties in the initial information.This, in turn, makes it possible to describe the model of the problem under study (objective function and constraints) in the form of fuzzymultiple expressions, i.e., describe the problem in the form of "soft" models.
At the same time, it should be noted that the proposed approach and algorithm for solving the nonlinear optimization problem use the procedures of classical mathematical programming.Therefore, searching for a reasonable combination of ideas and algorithms of classical and fuzzy mathematical programming will make it possible to find new, more efficient methods for solving complex optimization problems under conditions of uncertainty, both in the external and internal environment of the processes being optimized.

3 . 4 )A 5 )
Calculation of coefficients to express the required vector k Determination of the number of the variable to be removed from the basis:

4 .
Determination of the calculation step. 5. Finding the components of a new feasible solution.

2
behavior of the distribution of each perturbed constraint, are displayed with the parameters b=0.5, H=0.3 in Figure1, and H=0.9 in figure2.

Table 1 .
Software tools used to find routes based on web technology simplex multiplier; DV ( i с ) is objective function coefficients for the second stage; D Conferences 365, 05013 (2023) https://doi.org/10.1051/e3sconf/202336505013CONMECHYDRO -2022 It is assumed here that the sets . By setting a smaller value H , it was possible, having made additional approximations, to come even closer to the point of the maximum value of the objective function.
* X * X on the set D H