Methodological aspects of applying mathematical modeling in making managerial decisions

. This paper considers the use of mathematical models in management. In today's world, managerial decision making is becoming an increasingly complex process and requires the use of various tools and methods. Any company makes decisions on a continuous basis, since the management process is continuous, and the content of management is determined by the content of decisions made. The author of this paper considers methodological approaches to the application of mathematical modeling in making managerial decisions based on data analysis and computational methods. Mathematical modeling helps analyze complex systems and processes, find optimal decisions, and predict the results. This paper describes the general algorithm of making a managerial decision and types of mathematical models that can be used in making managerial decisions. In addition, the paper presents an algorithm for applying mathematical modeling in making managerial decisions and considers an example of using mathematical modeling when solving a real problem. In conclusion, the author emphasizes the importance of applying a correct approach to the use of mathematical modeling in management and offers her recommendations. To achieve the maximum efficiency, it is necessary to choose the right methods and algorithms, as well as to have a sufficient amount of data for analysis. The article will be of interest to specialists in the field of management and economics, as well as to anyone who is interested in the use of mathematical methods in solving management problems.


Introduction
Decision making is a science and an art.
Each of us makes decisions on a daily basis, and the process of making a decision is influenced by a large number of factors, such as: -degree of rationality and emotionality; -mood; -external circumstances and surrounding persons; -level of intellectual development, etc.
In today's world, making managerial decisions is becoming increasingly complex and requires the use of various tools and methods. Mathematical modeling is among such tools. It allows analyzing data and forecasting various scenarios [1][2][3][4].
In fact, the development of managerial decisions is the basis of management, since decisions unite the basic functions of management into a single process: planning, organization, motivation, and control. In addition, the management entity makes decisions to set and achieve the company's goals, and one of the most important aspects of the head's activities is the organization of practical implementation of the decisions made. So, any company makes decisions on a continuous basis, since the management process is continuous, and the content of management is determined by the content of decisions made [1][2][3][4].
Since the late 1940s, decision making has been understood as the determination of the value of the Х variable, at which some performance indicator W of the system under consideration takes on the optimal (maximum or minimum) value. This value X itself is called the solution of the corresponding problem, for example, the problem of resource allocation, the transport problem, etc. These problems are usually distinguished from traditional optimization by their discrete nature that does not allow using a powerful apparatus of mathematical analysis, as well as by their applied nature associated not just with research, but with the choice of actions that directly affect the lives of many people and companies [5][6][7][8][9].
In the 1960s, it became clear that an unclear goal or rather the presence of several contradictory performance criteria prevented from solving, and even formulating decision making problems in the form of optimization statements. Various methods of multi-criteria assessment of alternatives came to the fore [5][6][7][8][9].
Along with the two above-mentioned areas that are largely related to mathematical approaches to the analysis of real systems, a more "humanitarian area" has been successfully developing. It considers the main functions of a decision in the methodology and organization of the management process.

Materials and methods
Decision making, like the communication process, affects all aspects of management. The need for control actions arises in the process of system functioning, when a problem arises as a result of modifications in external or internal conditions. The managerial decision is an act of the management system that leads to the solution of the problem ensuring normal functioning or development of the socio-economic system.
Managerial decisions are, on the one hand, a logical-thinking, emotional-psychological and organizational-legal act of choosing an alternative made by the manager within their powers at their sole discretion or with the involvement of other persons [10][11][12].
On the other hand, a managerial decision is a creative act of the management entity that determines the program of the team's activities to effectively solve the pending problem based on the knowledge of objective laws that govern the functioning of the managed system, as well as on the analysis of information about the system's state.
In other words, a managerial decision is a choice that the manager must make in order to fulfill the responsibilities arising from their position.
Any managerial decision is usually aimed at strategic planning of the company's activities; control of the activities of managers of different levels; human resources management; management of production and service activities; formation of the company's management system; management consulting; as well as the management of internal and external communications [10][11][12].
The activity of any company represents the cycle of development, selection and implementation of various managerial decisions.
A specific situation is a real state of affairs relative to the stated goal. A problem (P) is formed as a difference between the goal (G) and the corresponding situation (C).
Groups of problems are reduced to a generalized problem, which is an indicator of the effectiveness of implemented decisions. When forming a generalized problem, the total number of problems is simplified and reduced, since some problems merge into others and the most important ones are highlighted. As a rule, the solution is aimed at reducing the problem to a predetermined level. The problem can be reduced by changing the situation or by adjusting the goal. Priority is given to managerial decisions aimed at changing a particular situation, while the magnitude of the problem should gradually decrease up to the minimum permissible value [13][14].
Any management process is a continuous chain of management, a certain sequence of actions combined into stages in accordance with their qualitative content and the homogeneity of operations necessary for their implementation. The algorithm of the process of making a managerial decision can be different depending on the specifics of the problem and the company. However, in general terms, the following stages can be distinguished: 1. Problem formulation: identification of the problem to be solved and factors influencing the problem.
2. Collection and analysis of information: collection and analysis of data necessary for making a decision. This may include research, statistical data collection and analysis, surveys, etc. 3. Determination of success criteria: determination of the criteria to be used to assess decision effectiveness.
4. Development of alternative decisions: development of several alternatives for solving the problem that can be considered. 6. Decision making: selecting the best alternative based on the analysis and evaluation of each alternative. 7. Decision implementation: implementation of the selected decision and monitoring its implementation.
8. Evaluation of the results: evaluation of the decision results and analysis of decision effectiveness as a whole.
9. Decision adjustment: adjustment of the decision based on the analysis of the results and experience in its implementation, if necessary.
These stages can be repeated several times before the final decision is made. It is important to remember that the process of decision making can be complex and take much time and effort, but the right decision can lead to better results and success for the company.
Mathematical modeling is the toolkit designed to help top managers in this complex process.
Mathematical modeling is applied in practice in various domains, including decision theory. The use of mathematical models allows analyzing data, forecasting future events, and making reasoned decisions. Mathematical modeling allows anticipating various case scenarios and choosing the optimal decision based on data analysis. Mathematical modeling is used to create mathematical models that describe the system and its interaction with the environment. Models can be static or dynamic and can be used to analyze different scenarios and determine optimal decisions [5][6][7][8][9].
There are several types of mathematical models that can be used to make managerial decisions. Let's consider some of them [10][11][12][13][14]: 1. Linear models: these are models that represent a linear function using mathematical equations. They can be used to determine the optimal decision in conditions of linear constraints, such as resource or time constraints.
2. Nonlinear models: these models use nonlinear mathematical functions. They can be useful in situations where the relationships between variables are not linear.
3. Statistical models: these models use statistical methods to forecast future events and outcomes. They can be useful in forecasting trends and analyzing data.
4. Simulation models: these are models that use computer simulation to represent real processes and events. They can be used to test different scenarios and evaluate their effectiveness.
5. Optimization models: these are models that are used to find the best decision under constraints. They can be useful to determine optimal production planning or resource allocation.
6. Simulation models: these are models used to simulate complex systems with a large number of variables and relationships. They can be useful in analyzing business processes and making decisions in a context of uncertainty.
Each of the above-mentioned types of models has its advantages and disadvantages, and the choice of a suitable model depends on the specific situation and the problem that needs to be solved.

Results
Let's consider practical aspects of mathematical modeling in decision theory.
The algorithm for applying mathematical modeling in management includes the following steps: 1. Problem definition: define the problem that needs to be solved and collect all the necessary data.
2. Data collection: collect data that can be used to create a mathematical model. This may include historical data analysis, surveys, and expert assessments. Mathematical modeling allows analyzing large amounts of data and identifying relationships between various factors. This helps identify causes of problems and make decisions based on factual data.
3. Selection of a mathematical model: choose the mathematical model that best describes the system or process you are studying. For example, queue theory models can be used to analyze production, and econometric models can be used to forecast sales.
4. Model building: create a mathematical model using the selected methodology. This may include the development of equations, statistical methods, and other mathematical tools.
4. Model validation: check how accurate your model is by comparing simulation results with factual data. If the model is not accurate, make the necessary adjustments. 5. Forecasting: use the model to forecast different scenarios and their consequences. For example, you can forecast sales based on various factors such as price, competition, seasonality, etc. Mathematical models allow forecasting future events and estimating their probability. This helps make decisions based on forecasts and minimize risks.
6. Decision making: based on the results of modeling and scenario analysis, choose the optimal decision. Mathematical models allow optimizing various processes and improving results. For example, production processes optimization can result in the reduction of production time and costs [6,[12][13][14][15].
An example of practical application of mathematical modeling in decision theory is production process optimization. To gain a better understanding of the algorithm of applying mathematical modeling in management, please consider an example of using a queue theory model to analyze a production line. Suppose you have a production line that produces various products. You have noticed that queues are formed on the line at particular periods of time, which slows down the production process and increases production costs. To solve this problem, you decided to use a queue theory model.
Step 1. Problem definition and data collection: the problem is in queues that are formed on the production line, which slows down the production process and increases production costs. Collect data about the production process, including production time, number of items, material costs, and labor costs.
Step 2. Selecting a mathematical model: to solve this problem, we choose the model of queue theory. For example, you can use the queue theory model to determine the optimal number of jobs and the speed of production.
Step 3. Model building: we build a mathematical model using the formulas of the queue theory, which allow determining the optimal number of jobs and the speed of production. For example, you can develop equations that describe the production time and costs for materials and labor.
Step 4. Model validation: we check the accuracy of the model by comparing the results of modeling with factual data.
Step 5. Forecasting: we use the model to forecast different scenarios, such as the optimal number of jobs and the speed of production, which will help avoid the formation of queues.
Step 6. Decision making: based on the results of modeling, we choose the optimal decision that will help eliminate the problem of queuing on the production line.
So, the use of mathematical modeling in management allows making reasoned decisions based on data analysis and forecasting possible changes in the system.
In general, any problem can be presented in the form of "given…", "determine…". For any individual decision maker, the task of making a decision can be written as follows. Where characters describing known variables are located to the left of the slash line, and known elements of the problem are located to the right of the slash line: Sinitial problem situation; Ttime for making a decision; Rresources required for making a decision; SDpre-defined problem situation; Ha set of assumptions (hypotheses) about the development of the situation in the future, a single assumption can be used as a special case; Ga set of goals that should be achieved with the help of the decision, the decision can be aimed at achieving a single goal as a special case; Ca set of constraints; Aa set of alternative decisions (at least two alternative decisions); CRa set of criteria for choosing the best decision, a single criterion can be used as a special case; fthe preference function of the decision maker, which includes both objective criteria from the K set and personal subjective preferences of the decision maker; A*the optimal decision. In some cases, the time and resources needed to make a decision may be unknown and are subject to determination by the decision maker or system analysts. In this case it is necessary to place the T and P characters to the right of the slash line in the above formula. If the initial problem situation C was defined with the level of specificity sufficient to make a decision, then it is not required to supplement its definition, and Сд to the right of the slash line is absent.
The resources required to implement individual decisions are included in the set of constraints O. Moreover, they are taken into account in the K set as one of the main criteria.
A problem in decision theory is the difference between the actual and desired state of the decision object. The problem is always associated with certain conditions and causes of its occurrence, which are generally called a situation. For example, the problem associated with the need to update the range of products sounds different for advanced and technologically backward enterprises, its solution depends on the possibility of attracting and using the necessary resources. The combination of the problem and the situation forms a problem situation. The initial problem situation C is described conceptually and, if possible, using a set of quantitative characteristics. The description of the problem situation should end with a brief and meaningful formulation of the problem that needs to be solved.
Knowledge and experience of decision makers, system analysts and experts, the scientific, technical and information potential of the company can be used as resources Р to find (not implement) the optimal decision.
To clearly define alternatives for eliminating the problem situation, it is necessary to formulate a set of goals Ц. Real tasks are usually multi-purpose, and only in individual cases a single goal can be formulated.
It is always necessary to clearly formulate constraints O (financial, material, personnel, legal, etc.). To achieve the set of goals, a set of alternative decisions A is formed, and the only one optimal or acceptable decision A* must be chosen from them. The set of possible decisions may also include a decision of inaction, which results in the fact that the problem situation will not be eliminated. The decision shall be described conceptually and formallyusing a set of certain characteristics, which shall also include resource characteristics.
A set of criteria K is used to evaluate alternatives for achieving goals in the context of the selected situation and their ranking by priorities. One can get an absolute and relative assessment of decisions using the K criteria. Absolute assessment of decisions can be performed only in individual and very rare cases. The decisions maker selects the final best solution A* on the basis of their preference function f. Mathematical modeling can be useful in various fields such as economics, engineering, science, and medicine. It can help you make reasoned decisions based on data analysis and forecast of possible changes in the system.

Discussion
Mathematical modeling is one of the most effective tools in decision making in various fields of activity. It is used to analyze complex systems and processes, find optimal solutions, and forecast results.
In today's world, when business processes are becoming increasingly complex and volatile, mathematical modeling plays a key role in making the right decisions. For example, in economics, modeling allows determining the optimal business management strategies, forecast changes in market conditions, and assess risks [3,[6][7].
One of the most popular methods of mathematical modeling is the method of system dynamics. It allows analyzing complex systems consisting of multiple interrelated elements and forecasting their behavior in different conditions.
Another method of mathematical modeling is the method of artificial intelligence. It is used to develop computer systems that can make decisions based on the analysis of large amounts of data and forecast changes in the system. However, it should be realized that mathematical modeling is not a universal solution for all decision-making problems. It can be effective only if correct methods and algorithms have been chosen and if there is enough data to analyze.
In conclusion, mathematical modeling is an important tool in decision making in various fields of activity. It helps analyze complex systems and processes, find optimal decisions, and predict the results. However, to achieve the maximum efficiency, it is necessary to choose the right methods and algorithms, as well as to have a sufficient amount of data for analysis.