## 1. INTRODUCTION

The improvement of medical organization management (publicly funded clinics and hospitals, as well as private medical centers) is currently one of the central tasks for social development. It entails creating primary medical and social services and enabling public access to them. Currently, the management of the whole healthcare system, which refers to large medical organizations acting as integrated corporate structures or small business entities, biomedical systems, and advanced medical technologies, is reaching a new level. This reflects social demand for the development of healthcare services and the associated increase in welfare, life expectancy, and quality of life. For instance, in the Address of the President of the Russian Federation to the Federal Assembly on March 1, 2018 (1), by the end of the next decade, the country is to take a permanent position in the “80+ Club” of countries with a life expectancy exceeding 80 years. At the same time, the duration of a healthy, active, and fully able life, when a person is not limited or constrained with any type of disease, should grow at a faster pace. All this poses new challenges to Russian health care, polyclinics, and hospitals, as well as to all medical workers, whose well-organized and motivated work determines the development of healthcare organizations and the quality of medical services. Thus, economic and mathematical modeling of financial resource manage-ment is an important and relevant scientific and practical problem for the Russian economy. It is associated with the development of sound pricing policies, the instruments for administrating medical organizations, promising and effective technologies for medical care financing, structural system analysis, various internal and external environmental factors, scientifically based personnel policies, and incentives for medical and non-medical personnel.

**The main goal of the study** is to develop a methodology for mathematical modeling of financial management processes for medical organizations based on economic and mathematical models, which makes it possible to improve the quality of medical care for the population, its availability, and also to increase the efficiency of daily medical activities of organizations.

**The object of the study** involves state and commercial medical organizations as microeconomic systems that provide paid medical services in compulsory health insurance (CHI). The design solutions that were obtained in the study can be applied to managing the financial resources of most domestic medical organizations.

**The subject of the study** is the socio-economic processes and tools for managing the financial resources of medical organizations of all legal forms in the provision of medical services.

According to the Federal Law of November 21, 2011, No. 323-FZ (as amended on March 6, 2019) “On Basics of Health Protection of the Citizens in the Russian Federation”, the authors understand ** medical organizations** as a legal entity regardless of its legal form carrying out medical activities as the main (charter) type of activity based on a license issued in accordance with the legislation of the Russian Federation on licensing certain types of activity, other legal entities, regardless of the legal form, carrying out medical activities along with the main (charter) activity, as well as individual entrepreneurs engaged in medical activities.

## 2. LITERATURE REVIEW

The following researchers and healthcare experts have studied financial resource management in medical organizations: Latuha (2017a, 2017b, 2018a, 2018b), Edeleva *et al*. (2018), and Shamshurina and Voropaeva (2011). These authors mostly examined the problems of managing medical organizations with a focus on operational improvement and a higher quality of medical services. In addition, Kaneva and Breusov (2018), Breusov *et al*. (2016, 2018), Shamshurina and Voropaeva (2011) discuss personnel management, effective methods, and forms of remuneration for medical staff, work standardization, and motivation.

Gryzunova and Kiseleva (2016), Kiseleva *et al*. (2017) focus on pricing, mathematical analysis, factors affecting cash flows, efficient use of labor resources, and optimization of companies’ financial resources, including how these issues affect the incidence of tuberculosis in the Russian Federation. Gryzunova and Kiseleva (2016), Kiseleva *et al*. (2017) also consider in detail the specifics of health care financing in the Russian Federation and the mathematical modeling of financial management processes in medical organizations as integrated corporate structures and small businesses.

Some authors (Stolbov *et al*., 2011;Latuha, 2017a;Zinkin *et al*., 2013;The President of Russia, 2018) pay great attention to market mechanisms that provide medical services to the population and assess the quality of those mechanisms, as measured by the satisfaction of various social groups and the effectiveness of the patientdoctor– medical institution interaction.

Lutsenko *et al*. (2017), Baranovskaya *et al*. (2018), Loiko *et al*. (2018), Lutsenko and Baranovskaya (2018), Baranovskaya and Vostroknutov (2008) explores in detail the construction of optimal organizational management structures for both large corporations and small businesses, which is of particular interest in the development of medical services. Lutsenko *et al*. (2017) consider the influence of ecology on quality of life. At the same time, she investigates life quality’s dependence on food and on the development of the agro-industrial complex of the Russian Federation.

Edeleva *et al*. (2018) and Khalfin *et al*. (2018, 2019) study planning and analysis in medical organizations and in healthcare management as a macroeconomic system. In addition, Omelchenko *et al*. (2017), Omelchenko and Herzik (2018, 2019) develops economic and mathematical methods for the analysis and the organizational and economic modeling of businesses, including medical services.

Samoilov (2016), Sokolov and Samoilov (2015a, 2015b) proposes models to manage outpatient departments’ medical services. However, he did not describe mechanisms for their implementation that would reflect the interests of medical, non-medical, and administrative personnel, nor did he address the sources to finance organizational changes resulting from the introduction of these models into health care.

Other researchers (Mayorova *et al*., 2012, 2014, Zinkin *et al*., 2013;Omelchenko *et al*., 2017) apply mathematical modeling and tools to the management of medical organizations’ resources. These mathematical tools are meant to increase the economic efficiency of medical facilities by analyzing their financial and economic activities, managing stocks and costs, introducing advanced medical care technologies by means of new information tools and systems, and implementing those tools in health care, all of which should reduce the cost of medical services and improve their speed, quality, and availability.

Yegorova (2017) focuses on the alignment of public and private partners’ interests, including those in the healthcare sector. Yegorova (2017) also addresses the issues of multi-level management, including expert assessment methods, simulation modeling, and optimization methods. Emelyanov *et al*. (2018) proposes economic simulation modeling, system analysis, and efficient tools with elements of artificial intelligence to support management decisions. Such technologies are in demand and can be useful in the development of complex health care systems and their primary elements —clinics and hospitals.

Choi *et al*. (2017) analyze what impact the volume of medical services has on the quality of medical care and on the profitability of medical facility operation. The authors constructed a three-stage model using the leastsquares method, which reflects the simultaneous effect of the number of patients on a hospital’s quality of medical care and on its economic efficiency.

Kaneva and Breusov (2018), Breusov *et al*. (2016, 2018) considers effective mechanisms for managing medical organizations and creating incentives for medical workers. However, he explores only some aspects of introducing new resource-saving technologies and automated information systems, as well as scientific methods to analyze that technology’s performance, to increase the effectiveness of patients’ treatment, and to develop new personnel management technologies for medical organizations. Breusov does not sscientifically substantiate the methods and models of managing medical organizations to increase their efficiency, of making medical services more available to people at lower prices while increasing the volume of paid medical services, of creating material and non-economic incentives to raise labor productivity, of involving all employees in the management of a medical organization, and of acquiring advanced medical equipment and pharmaceutical products. The financing depends on the returns contributed by each department and doctor to the medical organization’s development fund.

Mayorova *et al*. (2012, 2014, 2015), Zinkin *et al*. (2013) explore the organization of effective medical care and problems in medical organizations’ management. However, her work presents no methods or models of managing medical organizations adequate for the current situation and reforms in the industry, or any decisionmaking algorithms, economic and mathematical models, or tools for the mathematical modeling of financial resources management that could be applied by the departments of medical institutions.

An effective system to finance medical organizations is examined by Kuznetsov (2016a, 2016b), Kuznetsov *et al*. (2014), Stolbov *et al*. (2011). However, he focuses on the development of personalized medicine, an individual approach to each patient that maximizes the use of scientific and technological advances, digital medicine, and neural networks, when he should focus on the most important problem in Russian medicine: to provide the necessary volume of primary medical and social care and to develop mechanisms and models to finance of this care, which could improve its quality and accessibility for all social groups.

The literature review shows a lack of papers devoted to the economic and mathematical modeling of medical organizations’ financial resources management, to the methods of scientific substantiation and decision-making, or to the mechanisms for these methods’ practical implementation in the MS Excel environment, using the add-in program Solver and MathCAD software for stability analysis.

The research goal of the article is to build a comprehensive model to manage the financial resources of medical organizations operating within the system of compulsory health insurance (CHI) and to budget their financial resources on a per capita basis to provide paid medical services. This project also aims to implement this model in MS Excel with the add-in program Solver, which analyzes the results of the economic-mathematical model’s stability in MathCAD.

## 3. METHODOLOGY

*The economic and mathematical model of financial resource management in medical organizations working in the CHI system with state funding* is presented below. Currently, Russian medical organizations receive most of their financial resources on a per capita basis. This approach tailors the amount of funding per medical organization to the number of citizens using the organization, and citizens can freely choose their organization. The allocation of funding takes into account gender and age composition as well as other factors that influence patients’ need for medical care. The financial resources of the compulsory health insurance (CHI) funds granted to a medical organization, in accordance with the annual differentiated standard and the number of citizens attached to the medical organization, are to cover all expenses of a medical institution and provide high-quality public services. The main difference between the per capita financing and the payment for each medical service is that it is not calculated for each visit of a citizen to an outpatient clinic (or the number of patients treated in a hospital) but is dependent on the number of citizens attached to the medical organization. The more citizens attached to a medical organization, the more financial resources it will receive. Medical facilities need to make economically sound decisions when distributing financial resources among its departments, which determines the relevance of research papers on this issue (Shamshurina and Voropaeva, 2011;Edeleva *et al*., 2018). The objective function that maximizes the total financial effect of the operation of a medical institution is:

where *Result _{Σ}* is the total financial result from the provision of health care financed from the budget and the CHI fund for the whole medical organization, in rubles;

*FR*is the total amount of budget financing and financial resources of the CHI fund received by the

_{j}*j*-th department of a medical organization, in rubles;

*C*is the actual total cost of the medical care provided within the CHI fund in the

_{summj}*j*-th department, in rubles; and

*m*is the number of medical departments providing medical care within the CHI fund and budget financing. Thus, the equation of the total financial result from the provision of medical care in the

*j*-th department has the form:

where *Result _{j}* is the total financial result from the provision of medical care in the

*j*-th department. The total amount of budgetary and CHI funds received by the

*j*-th department of a medical organization, taking into account the impact of the utility service factor on the increase in the number of people attached to the medical facility, is determined by the formula:

where *S* is the standard of financial resources of the CHI fund per one insured person differentiated by gender and age, in rubles; *α* is the coefficient of change in the population attached to a medical organization (a decimal quantity); *N* is the number of people served by a medical organization, people; *ε _{j}* is the coefficient of service utility of the

*j*-th department (the elasticity coefficient of demand for the services of the

*j*-th department); and

*d*is the share of budget and CHI funds given to the

_{j}*j*-th department and a decimal quantity ${\sum}_{j=1}^{m}{d}_{j}=1},{d}_{j}\ge {d}_{jmin$, where

*d*is the minimum share of budget and CHI funds received by the

_{jmin}*j*-th department). The total cost of medical care provided within the compulsory health insurance in the

*j*-th department is determined as follows:

where *β _{j}* is the coefficient of change in the volume of medical care within the CHI system of the

*j*-th department, a decimal quantity;

*V*is the volume of the

_{ij}*i*-th medical service of the

*j*-th department, units;

*C*is specific semi-variable costs of the

_{varij}*i*-th medical service of the

*j*-th department, in rubles;

*C*is semi-fixed costs of the

_{fixj}*j*-th branch, in rubles;

*n*is the nomenclature of medical services of the

_{j}*j*-th department, in units. Based on the data of the previous period, let us determine the ratio

*L*between the number of people served by the medical organization and the volume of medical care provided to the population in the CHI system in the

_{j}*j*-th department. This ratio has the following form:

where *L _{j}* is the ratio of changes in the volume of medical care provided by the

*j*-th department in the CHI system and changes in the number of people served by a medical organization, a decimal quantity. Based on equation (5), the formula for calculating the coefficient

*β*has the following form:

_{j}

It is necessary to fulfill the requirement to provide a minimum amount of medical services if among people attached to the medical organizations there are patients with chronic diseases who need constant medical care:

To model the conditions for the provision of medical care within the CHI system, it is necessary to take into account the capacity of the department of the medical organization, which is determined by the following in the equation:

where *Norm _{j}* is the standard volume of medical care provided in the CHI system per medical position of the

*j*-th department, units;

*K*is the number of doctors and nurses of the

_{j}*j*-th department. The standard amount of medical care provided in the CHI system (

*Norm*) is determined by the formula:

_{j}

where *W _{j}* is the norm of working time of a medical position of the

*j*-th department of a medical facility, minutes;

*E*is the expenditure of working time of a medical position for providing one medical service in the CHI system and performing medical procedures in the

_{avj}*j*-th department of a medical organization, minutes;

*τ*is the factor of the working time use of a medical position in the

_{j}*j*-th department of a medical facility for Diagnostics and treatment.

Thus, the **nonlinear programming problem** that optimizes the financial result of each *j*-th department and the whole medical organization depending on the number of people served by the medical organization, the volume of medical care provided within the CHI system, the total cost of medical services, and the capacity of the departments, has the form:

Objective function

Constraints

*K _{jmax}* is the maximum number of doctors and nurses in the

*j*th department. The value is determined by the ability of the medical organization to expand the staff of medical workers (as a rule, in practice for medical organizations working in the CHI system and budget financing, in the short term the possibilities to increase the number of doctors and paramedical personnel are limited and do not exceed 20% of the staffing table, i.e. 1

*K*≤ 1, 2

_{jmax}*K*, what will be taken into account in the future when modeling).

_{j}*The economic and mathematical model for managing financial resources from the provision of paid medical services*. In the context of healthcare reforms, it is increasingly necessary to expand the number of paid medical services in polyclinics and hospitals. This is primarily because the CHI rates for the services provided cannot cover all the costs of their provision.

**Building a model.** The medical facility needs to consider the conditions for reducing the prices for medical services (for example, the proposed rates do not satisfy the patient). Moreover, it seeks to maintain the same level of profit. Accordingly, the profit from the provision of the *i*-th service at the initial values of the volume of services provided and the rates is equal to:

where $Profi{t}_{i}^{0}$ is the profit from the provision of the *i*-th medical service at the base values of the rates and volume; ${B}_{i}^{0}$ is the basic rate for *i*-th medical service; ${V}_{i}^{0}$ is the volume of the *i*-th medical service at a basic rate. In accordance with the above, the profit from the provision of medical care with changed rates can be presented as follows:

where ${\delta}_{i}^{q}$ is the change of the base *i*-th rate (discount or extra charge) calculated at the *q*-th modeling step and depending on external and internal factors; ${V}_{i}^{q}$ is the volume of the provision of the *i*-th medical service calculated at the *q*-th modeling step. Transforming equation (19) and adding the value $Profi{t}_{i}^{0}$ from equation (18) into it, we obtain the dependence of the change in the *i*-th rte for a medical service on the volume of its provision:

Providing medical care to people, a medical organization very often faces another problem that is different from the above task, where it is necessary to create a pricing strategy that ensures profitability not lower than the basic one for a changed volume of medical care. The price list of a medical organization may include different changes in the volume of medical care provided and the corresponding prices. To solve this problem, let us state the rate that ensures a constant profit for a medical organization at the *i*-th modeling step as ${B}_{i}{}^{const\_q}$, which can be calculated with the following formulas:

where $Profi{t}_{i}{}^{const\_q}$ is the value of a constant level of profit with corresponding values of ${V}_{i}{}^{q}$ and ${B}_{i}{}^{const\_q}$ , in rubles. Let us use ${B}_{i}{}^{const\_q}$ in equation (20) and we will obtain the relationship between the growth in the volume of the *i*-th service and the required rate reduction:

When modeling changes in prices for paid medical services depending on their volume, it is necessary to check the capacity limits of the department:

where *Norm _{j}* is the normative workload of a medical position in the

*j*-th department of a medical facility;

*n*is the nomenclature of paid medical services provided by the

_{j}*j*-th department of a medical facility, in units. To stimulate medical personnel to increase labor productivity and the quality of medical care provided, let us reduce the change in the basic rate for the

*i*-th service (${\delta}_{i}{}^{q}$) by a certain value proportional to the rate change (denoted as

*γ*). Then the value of the rate that ensures the increase in the profit of a medical organization (${B}_{i}{}^{incr\_q}$) can be estimated as follows:

where *γ* is the coefficient of the redistribution of discounts between the patient and the medical facility.

If *γ* ≤ 0 , then this situation allows us to simulate a scenario in which a medical organization considers economic losses from establishing preferential tariff rates that reduce the level of profitability for certain categories of citizens for certain services. In other words, such a situation allows us to simulate the condition under which preferential tariffs lead to a decrease in the healthcare provider’s revenues.

According to the formula (21), we obtain:

Since *γ* ∈(0;1) , then the calculated value ${B}_{i}{}^{profit\_q}$ ensures the profit of both the patient and the medical organization. Therefore, the introduction of coefficient *γ* allows one to redistribute the profit from changes in prices for paid medical services between the healthcare provider and its patients. The profit of a medical organization from a change in the price of the *i*-th medical service is calculated by the formula:

where $Profi{t}_{i}{}^{incr\_q}$ is the values of higher profits at the *q*-th step of modeling for the *i*-th medical service, in rubles. The formula for determining the amount of the return of a medical organization from the provision of paid medical services, the prices for which are set depending on their volumes and reflect the cost of their provision, has the form:

where *n _{j}* is the nomenclature of paid medical services of the

*j*-th department of a medical facility with a variable rate; ${\u0421}_{summi}{}^{q}$ is the total cost of the

*i*-th medical service at the

*q*-th modeling step.

Considering the above, the economic and mathematical model of managing the profit from the paid medical services (**a non-linear programming problem**) takes the form:

Objective function

Constraints

Here *ε _{j}* is the coefficient of elasticity of demand at the price of the

*j*-th department of a medical facility; ${B}_{ij}{}^{l},{B}_{ij}{}^{u}$ are, respectively, the lower and upper boundaries of the price for the

*i*-th paid medical service of the

*j*-th department that is determined using data on the acceptable level of profit (cost-effective pricing methods) and carried out market research (market pricing methods), in rubles. The economic and mathematical model developed allows the modeling of prices for basic paid medical services. These prices are aimed at increasing the number of patients, as well as the profitability and returns of the medical organization.

*The comprehensive model for managing the financial resources of medical organizations that work within the CHI system and per capita budget financing and provide paid medical services*. The comprehensive management system of a medical organization is a very effective tool for making sound economic decisions aimed at improving the financial situation of medical staff and motivating them to increase labor productivity and creating development resources. Since the total financial result of the operation of a medical organization are the algebraic sum of the financial result of providing medical care within the CHI system and the profit from paid medical services, the main task of healthcare administration is the competent and efficient management of factors that influence the objective function. The model allows one to evaluate the effectiveness of the medical facility by its financial result, as well as to identify the conditions and factors for achieving a break-even point, to solve the problem of maximizing the financial result, changing rates for paid services and creating the conditions for the development of a medical organization.

**The goal of the modeling** is to determine the optimal ratio of the people attached to a medical organization and the level of paid medical services provided, at which the objective function, defined as the financial result of the operation of all departments of the medical organization reaches its maximum. The proposed economic and mathematical model allows one to consider many different modeling options and choose the most suitable one according to the criterion of maximizing the financial result. Modeling is carried out for each department of the medical organization. **The objective function** has the form:

where *FR*_{Σ} is the total financial result of a medical facility from the provision of medical services in the CHI system and paid medical services, in rubles; *FR _{oj}* is the total financial result of the

*j*-th department from the provision of medical services in the CHI system, in rubles;

*R*is the return of the

_{j}*j*-th department from the provision of paid medical services, in rubles.

**The constraints of the model** are formulas (11)-(17) of the economic and mathematical model of managing financial resources of medical organizations working in the CHI system and budget financing, and formulas (30)- (35) describe financial resources of medical organizations from providing paid medical services. Thus, after appropriate transformations, the complex economic and mathematical model for managing the financial result of the operation of a medical organization (**a non-linear programming problem**) has the following form (Baranovskaya *et al*., 2018):

Objective function

Constraints

In formulas (37)-(47) *FR _{Σ}* is the total financial result from the provision of medical services in the CHI system and the provision of paid medical services, in rubles;

*n*is the number of various medical services provided in the CHI system in the

_{CHIj}*j*-th department of the medical facility, units;

*n*is the number of various paid medical services provided in the

_{pj}*j*-th department of the medical facility, units;

*ε*is the utility coefficient of the services of the

_{CHIj}*j*-th department, the coefficient of elasticity of demand for the services of the

*j*-th department provided within the CHI system;

*ε*is the coefficient of price elasticity of demand for the paid medical services of the jth branch; ${V}_{pij}^{0}$ is the annual volume of the

_{pj}*i*-th paid medical service provided in the

*j*-th department in the base year, units;

*V*is the annual volume of the

_{CHIij}*i*-th medical service provided in the CHI system in the

*j*-th department of the medical facility, units;

*V*is the annual volume of the

_{pij}*i*-th paid medical service provided in the

*j*-th department of the medical facility, units;

*C*is the average semi-variable costs of one

_{varij}*i*-th medical service provided in the

*j*-th department of the medical facility, in rubles;

*C*is the semi-fixed annual costs of the

_{fixj}*j*-th department, in rubles; ${B}_{ij}^{0}$ is the average basic rate for the

*i*-th paid medical service provided in the

*j*-th department of the medical facility, in rubles;

*S*is the standard of financial resources of the CHI fund per one insured person differentiated by gender and age, in rubles;

*α*is the coefficient of change in the number of people attached to a medical institution, a decimal quantity;

*N*is the number of people served by a medical organization, people;

*d*is the share of differentiated per capita standard given to the

_{j}*j*-th department, a decimal quantity;

*L*is the ratio of basic changes in the volume of medical care provided by the

_{j}*j*-th department and the number of people served by a medical organization, a decimal quantity;

*K*is the number of doctors and nurses of the jth department;

_{j}*W*is the norm of working time of a medical position of the

_{j}*j*-th department of a medical facility, minutes;

*τ*is the factor of the working time use of a medical position in the

_{j}*j*-th department of a medical facility for Diagnostics and treatment;

*E*is the expenditure of working time of a medical position for providing one medical service in the CHI system and performing medical procedures in the

_{avj}*j*-th department of a medical organization, minutes;

*V*is the volume of medical care in the CHI system given to patients with chronic diseases attached to the medical facility, units;

_{CHIijchron}*V*is the volume of paid medical services given to patients with chronic diseases attached to the medical facility, units; ${B}_{ij}{}^{l},{B}_{ij}{}^{u}$ are, respectively, the lower and upper boundaries of the price of the

_{pijchron}*i*-th paid medical service of the

*j*-th department, in rubles.

## 4. RESULTS AND DISCUSSION

Figure 1 and Tables 1-2 present the mechanism of implementing the economic and mathematical model (37- 47) in MS Excel with the add-in program Solver. The simulation results given in Figure 1 and Tables 1-2 were estimated for two departments of a medical organization: the Ultrasound Diagnostics Department and the Radiology Diagnostics Department, which provide medical services covered by budget or compulsory health insurance and funded on a per capita basis and paid medical services. A fragment of the list of services provided by the Ultrasound Diagnostics Department included seven items, while the Radiology Diagnostics Department was represented by six types of medical services.

The initial data for modeling were (see formulas (37) - (47)): the norm of financial resources of the compulsory health insurance fund per one insured person, differentiated by gender and age (N in formula (3)) is 6,000 rubles; the ratio of changes in the volumes of medical care provided by the Ultrasound Diagnostics Department in the CHI system and changes in the number of people served by the medical organization is 0.15; the ratio of the change in the volume of medical care provided by the Radiology Diagnostics Department in the CHI system and the change in the number of people served by a medical organization is 0.20; the standard amount of medical care provided in the CHI system and paid medical services per medical position of the Ultrasound Diagnostics Department is 2,500 services; the standard amount of medical care provided in the CHI system and paid medical services per medical position in the Radiation Diagnostics Department is 3,500 services; the utility coefficient of the Ultrasound Diagnostics Department services, the elasticity coefficient of demand for the Ultrasound Diagnostics Department services provided to the people in the CHI system estimates 0.0098; the utility coefficient of the services provided by the Radiation Diagnostics Department, the coefficient of elasticity of demand for the services of the Radiation Diagnostics Department provided to the people in the CHI system is 0.0010; the coefficient of elasticity of demand at the price of paid medical services of the Ultrasound Diagnostics Department is 0.18; the coefficient of elasticity of demand at the price of paid medical services of the Radiation Diagnostics Department is 0.10; semi-fixed annual costs of the Ultrasound Diagnostics Department Cfix1 = 250,000 rubles; semi-fixed annual costs of the Radiation Diagnostics Department Cfix2 = 340,000 rubles; semi-variable costs per medical service of the Ultrasound Diagnostics Department and the Radiation Diagnostics Department Cvar1 and Cvar2 are given in column 9 of Tables 1 and 2, respectively; the minimum number of medical services provided in the CHI system and paid medical services to people with chronic diseases (column 5 of Tables 1 and 2, respectively); the lower and upper boundaries of the price of paid medical services of the Ultrasound Diagnostics Department and the Radiation Diagnostics Department (columns 6 and 8 of Tables 1 and 2, respectively); the maximum number of doctors for the Ultrasound Diagnostics Department is 10, and for the Radiation Diagnostics Department this figure is 16 (see formula (40) of the economic and mathematical model). As indicated above, the maximum number of doctors and paramedical personnel of the departments of ultrasound diagnostics and radiation diagnostics is determined by the capabilities of the medical organization to expand the staff of medical workers. In the short term, the ability of medical organizations to increase the number of doctors and paramedical personnel is limited and does not exceed 20% of the staffing table. The upper and lower boundaries of the price of paid medical services are 80% and 120% of the base price, respectively, i.e. for Tables 1 and 2 the values presented in column 6 equal to 80% of the values presented in column 7, and the values in column 8 are 120% of the values in column 7. The average base rate for paid medical services depends on the annual volume of these services (column 7 of column 4 in Tables 1 and 2) and is determined by the coefficient of elasticity of demand at the price of paid medical services and the constraints (45) of the economic and mathematical model.

Having tested the economic and mathematical model (37)-(47) in MS Excel using the add-in program Solver (see Figure 1), we obtained the maximum financial result from the provision of paid medical services and the services in the CHI system for two departments which estimated 113,099,277.37 rubles. At the same time, the maximum profit from the provision of paid medical services in the Ultrasound Diagnostics Department was 38,852,673.32 rubles, and in the Radiation Diagnostics Department, it was 37,797,253.60 rubles, which totals 76,649,926.92 rubles. When providing medical services in the CHI system, the maximum financial result for the Ultrasound Diagnostics Department was 18,269,965.65 rubles, and for the Radiation Diagnostics Department, it is 18,179,384.80 rubles, which in total is 36,449,350.45 rubles.

The maximum profit from the provision of paid medical services in the department of ultrasound diagnostics is determined by multiplying the annual volume of paid medical services of the department after modeling (see column 4 of Table 1) by the price of this service after modeling (column 7 of Table 1) minus the variable costs attributable to one medical service (column 9 of Table 1) multiplied by the number of these services after modeling (column 4 of Table 1). Then, the results obtained for each medical service presented in Table. 1, the conditionally constant annual costs of the ultrasound diagnostic department, which are 250,000 rubles, are summed up and subtracted from the result. Similarly, the maximum profit from the provision of paid medical services in the department of radiation diagnostics is calculated (see Table 2).

As for the financial result from the provision of medical services financed from the budget and the CHI fund, in accordance with formula (10), the increment of normative financing is calculated from the increase in the number of people attached to the medical organization depending on the growth in the volume of medical services provided. The relationship between the growth in the volume of services provided and the number of attached population is determined by the coefficient of elasticity, the coefficient of the utility of the service εj. As indicated above, for the Department of Ultrasound Diagnostics, it is equal to 0.0098, and for the Department of Radiation Diagnostics, its value is 0.0010. Further, taking into account the share of budget financing and financial resources of the CHI fund attributable to each department (ultrasound diagnostics and radiation diagnostics), the size of receipts of financial resources to each department from the provision of medical care paid from the budget and funds of the CHI fund is calculated by multiplying the number served by the medical organization population (taking into account its change depending on changes in the volume of medical services and the value of the coefficient of the usefulness of these services) to the norm of financial resources of the CHI fund per one insured and differentiated by gender and age (6,000 rubles). Finally, the conditionally constant annual costs of the department (equal to 250,000 rubles for the department of ultrasound diagnostics and 340,000 rubles for the department of radiation diagnostics) and the sum of the products of the volumes of medical services provided, paid from the budget and funds of the compulsory medical insurance, are deducted from the received amount after modeling ( see column 3 of Tables 1 and 2) for conditionally variable costs per medical service (see column 9 of Tables 1 and 2).

Thus, if we compare columns 3 and 4 of Tables 1 and 2, we can see that with similar annual volumes of medical services provided in the CHI system and paid medical services the total profit from the provision of paid medical services is more than two times higher than when providing medical services in the CHI system.

According to the objective function (37), the following parameters in the modeling are variable: the share of budget financing and the financial resources of the CHI fund allocated to each department; the annual volume of medical services in the CHI system and paid medical services; the number of doctors and paramedical staff in the Ultrasound and Radiation Diagnostics Departments. According to the simulation results, we found out that the most optimal distribution of budget funding and financial resources of the CHI fund between the Ultrasound and Radiation Diagnostics Departments has a ratio of 50% to 50%, i.e. d1 = d2 = 0.5. The required number of doctors for the Ultrasound Diagnostics Department is 10, while for the Radiation Diagnostics Department it is 15 people. Columns 3 and 4 of Tables 1 and 2 present the annual number of medical services for each nomenclature for the Ultrasound and Radiation Diagnostics Departments, respectively, in which the medical organization has the maximum financial result and the maximum objective function is achieved (37).

It should be noted that the constraints (38) - (47) of the economic and mathematical model are satisfied with the maximum of the objective function (37). For example, the patient capacity of the Ultrasound Diagnostics Department is 10 · 2,500 = 25,000 services, and the actual annual volume of this department is 11,868 (the sum of the rows in column 3 of Table 1) + 13,131 (the sum of the rows in column 4 of Table 1) = 24,999 services, i.e. condition (39) is fulfilled for the Ultrasonic Diagnostics Department. The feasibility of all other constraints of the economic and mathematical model (37)-(47) is checked in a similar manner. The annual volume of medical services in the CHI system and paid medical services obtained after simulation covers the minimum requirements for these services of patients with chronic deceases. This can be seen if we compare columns 3, 4, and 5 in Tables 1 and 2.

Figure 2 shows the results of simulating the stability of the economic and mathematical model (37)- (47) in MathCAD. Figure 2a shows the dependence of the financial result on the share of budget funding and financial resources of the CHI fund allocated to the Ultrasound Diagnostics Department and the number of doctors and paramedical personnel providing medical services. Figure 2b shows the dependence of the financial result on the annual volumes of medical services provided, fixed, and variable costs. The maximum values of the modeled parameter are shown in red, and the minimum values in blue. The results obtained in MathCAD fully comply with the results obtained in MS Excel. For example, in Figure 2a we can see that the maximum financial result from the provision of medical services in the CHI system was obtained with the same amount of budget funding and the financial resources of the CHI fund allocated to each department (d1 = d2). One can also see that the maximum financial result is achieved with the maximum number of doctors and nurses providing medical care to patients.

## 5. ANALYSIS OF THE OBTAINED RESULTS

The developed economic and mathematical model (10)-(17) allows one to manage the financial resources of medical organizations working in the CHI system and budget financing depending on the number of people served by a medical organization, the volume of medical services, their cost, and the capacity of the departments.

The proposed economic and mathematical model (29)-(35) enables the administration to coordinate the changes in the rates for paid medical services and the volume and cost of services provided. This helps to increase the efficiency of medical organizations and the availability of medical services for citizens.

The developed economic and mathematical model (37)-(47) allows one to simulate the management of financial resources of medical facilities depending on the volume of services provided by a medical institution in the CHI system and on a paid basis, and changes in the rates for paid services, which increases the income of medical organizations, doctors, non-medical personnel and enables to improve the quality of health care.

According to the simulation results presented in Tables 1 and 2, the maximum financial result of the operation of the departments of a medical organization when providing services in the CHI system, representing the value of the objective function (10), will be achieved if the share of budget funding and financial resources of the CHI fund received by the first and second departments will be the same and equal to 0.5, the number of doctors engaged in providing medical care will become the maximum allowable (see formula 13). The value of the objective function (10) is 36,449,350.45 rubles.

For paid medical services, the maximum value of the objective function (29) will be 76,649,926.92 rubles.

In the comprehensive system of managing financial resources of medical organizations (37)-(47), the maximum value of the objective function is 36,449,350.45 rubles + 76 649 926.92 rubles = 113,099,277.37 rubles.

The calculations performed in MathCAD prove the reliability of the developed comprehensive model for managing financial resources of medical organizations and the considerable potential of its practical application.

## 6. CONCLUSION

The growth of the total financial result of a single medical organization, structurally and organizationally subordinate to the Moscow Health Department, could amount to 113 million rubles after the introduction of these economic and mathematical models. The implementation of the economic and mathematical models developed in the article in all medical organizations of Moscow may lead to an increase in the total financial result estimating 154 billion rubles. The expected economic effect for Russia is 2.2 trillion rubles. The potential economic effect on the developed countries of the world is USD 1.6 trillion.