Determination of the Control Criterion for Centralized Heat Supply of the City on the Basis of the Production Function with Complex Variables

Abstract

The aim of this work is to determine the production function using the method of complex-valued economics as a criterion for the management of the centralized heat supply of a city. This paper used the methodology of using stepped production functions of complex variables with real coefficients as a tool to perform dynamic analysis and forecasting of production results, allowing the performance of Manufacturing Execution System tasks of the heat supply system to be tracked. Based on this, a justified selection of a management criterion was made, objectively reflecting both the passive and active components of the administrative and economic activities of the heat supply enterprise. A comparative analysis of production functions was conducted to identify a predictive model for resource provision in the process of heat energy generation. A predictive model of resource provision was developed based on the capital/labor ratio of the enterprise, using a production function with complex variables. In other words, determining the production function allows the analysis of both the quantity and quality of resources used to produce 1 Gcal of energy, as well as the forecast of resource procurement to ensure a reliable and cost-effective heat supply.

1. Introduction

Planning and optimizing both material and non-material expenditures remain among the key tasks of MES and ERP (Manufacturing Execution System and Enterprise Resource Planning) systems in industrial enterprises. This is especially relevant in the development of optimal control systems for technological processes in heat supply, enabling energy-efficient management of centralized urban heating. Improving the efficiency of energy resource utilization in heat supply requires a comprehensive approach to solving economic, organizational, and technical challenges, and is closely linked to the overall effectiveness and development of the industry. The advancement of technology and the transition to sustainable energy call for the implementation of innovative approaches to energy resource management. This review examines key studies that explore energy-efficient solutions for centralized heat supply systems using smart control technologies.

The transition to sustainable energy and the integration of new energy sources, while crucial, also present challenges. For instance, due to their inherent intermittency and instability, the grid connection of certain new energy sources can lead to frequency and voltage fluctuations within the power grid. There are numerous studies dedicated to this topic. For example, Vannoni [1] presents a new model for dispatch optimization, while Lygnerud [2] introduces models for integrating heat pumps aimed at maximizing profit, which influences performance and pricing but does not necessarily improve the efficiency of energy resource utilization. Many studies focus on the optimization of thermal capacities, involving either the expansion or optimization of combined heat and power (CHP) plant sizes and the optimization of energy consumption [3,4].

The work by Wang [5] demonstrates the application of deep reinforcement learning for energy management in multi-energy buildings connected to low-temperature heating systems. Their approach showcases how advanced AI techniques can enhance operational efficiency in complex energy systems. The study by Xin [6] focuses on the control of solar district heating systems using predictive control methods. The authors highlight that the integration of weather forecasts and thermal load predictions enables maximum energy efficiency, underlining the importance of data-driven strategies in modern heat supply management.

The analysis by Andersen [7] focuses on the techno-economic aspects of retrofitting heat supply systems using solar energy. The authors found that integrating solar technologies can significantly reduce both the carbon footprint and operational costs. Their study includes the modeling of various modernization scenarios, making the work particularly valuable for the practical implementation of sustainable energy solutions. The review confirms that intelligent approaches are key directions in the development of sustainable energy systems.

As part of the work by Degelin [8], an analysis is conducted on the impact of supply temperature and the implementation of booster technologies on the efficiency of centralized heat supply systems. The study confirms that lowering the supply temperature and integrating modern booster technologies, such as heat pumps, significantly enhances the energy efficiency of heat supply systems. These approaches contribute to reducing the carbon footprint and ensuring the long-term sustainability of energy systems.

The contribution of Divkovic [9] lies in the study of heat supply system design methods that take into account the dynamics of investment costs and the tightening of environmental regulations. The authors emphasize that flexible planning enhances system resilience and ensures long-term economic viability. In conclusion, the study demonstrates that adaptive approaches and the adoption of low-carbon technologies are effective solutions for designing modern energy systems that comply with stringent environmental standards.

The research works by Deng [10] and Eklund [11] explore the application of deep learning methods for optimizing fuel costs in centralized heat supply systems. It has been identified that the implementation of intelligent control methods reduces energy consumption and increases the flexibility of heat supply systems. The study emphasizes the importance of intelligent control technologies for improving the efficiency and sustainability of centralized heating systems. Bachmann [12] investigates distribution costs in linear and radial heating systems, focusing on their economic efficiency. It has been proven that radial networks are more efficient in areas with high building density, while linear networks are more suitable for low-population-density areas. The research confirms that the choice of network configuration is a key factor in reducing costs and enhancing energy efficiency in centralized heat supply systems.

From this review, it follows that modern heat supply systems require the optimization of key parameters such as supply temperature, volume, type, and quality of fuel used, which allows the achievement of high energy efficiency and sustainability. It is also worth noting that the application of deep learning algorithms significantly reduces fuel costs and enhances system resilience. The results confirm that deep learning methods have a high potential for optimizing costs and improving management efficiency in centralized heating networks.

While existing research offers valuable optimization strategies for specific aspects of heat supply management, a comprehensive predictive model that integrates both quantitative and qualitative resource dynamics and explicitly accounts for the intricate interplay between capital and labor through the lens of complex variables has not been adequately addressed. Current real-valued models, for instance, often fall short in capturing the passive and active components of administrative and economic activities, or in providing multidimensional forecasts of resource costs that reflect both quantity and quality. Therefore, the primary objective of this work is to bridge this gap by developing such a model.

Based on the above, it is necessary to develop a predictive model for resource provision of a heat supply enterprise, which would determine the optimal balance not only for the distribution of material resources but also for labor costs in the production of 1 Gcal of energy. This model should allow efficient management of both financial and human resources, ensuring energy production is cost-effective and sustainable.

2. Statement of the Research Problem

From the analysis of the general and specific features of the heat supply complex, it follows that for the efficient operation of the enterprise, it is essential to create and implement an integrated automated control system for the heat supply complex (IACS HC), utilizing modern achievements in control theory, cybernetics, information technologies, and technical means of telecommunications and automation.

The requirements for improving the level of production organization and ensuring clear coordination of the actions of the enterprise’s divisions can be met through the integration of all control and management functions into a unified integrated management system.

This system comprehensively ensures the automation of strategic planning processes, economic and technical development of the enterprise, marketing and research activities, current production and economic operations, sales, and financial activities and, finally, the automation of management for primary and auxiliary technological operations.

The essence of integrating the enterprise management system lies in the following [13]:

-

Aligning the goals and criteria for their evaluation across all components of the system;

-

Solving complex tasks that ensure the achievement of goals;

-

Synthesizing information (when transmitted from the lower level to the upper level) or differentiating information (when transmitted from the upper level to the lower level);

-

Achieving an overall economic effect that exceeds the simple sum of the effects of individual components (synergistic effect).

In general terms, the functional structure of the IACS includes the automated enterprise management system (AEMS) and the automated process control system (APCS). Further decomposition of the IACS allows the identification of the AEMS for primary and auxiliary production, as well as the AEMS for boiler plants. Each of these systems can be subject to further structuring.

The complexity, labor intensity, and capital intensity of creating a multifaceted IACS HC require the development of key tasks for the IACS HC based on methodological and formalized foundations. These tasks include the selection and justification of the following:

• Criteria for the levels of management of the heat supply complex;
• Key performance indicators (KPIs) for individual heat supply processes;
• A system of balanced indicators for the heat supply complex based on KPIs;
• Creation of an integrated system of balanced indicators and management of quality and energy efficiency;
• A hierarchical formalized structure of production planning tasks.

Points 1–4 have been explored and researched by the authors in works [14,15,16]. This study is dedicated to the research on point 5.

The generalized three-level structure of the IACS HC is presented in Figure 1.

The sequence of the study is structured as follows: first, it is necessary to identify the optimal production function, where the model coefficients would yield the smallest error. Following this, it is necessary to represent this function as a management criterion, formulated as a function of complex variables; it should be modeled under various conditions to obtain the optimal capital and labor resource costs for the heat supply enterprise.

3. Methodology

3.1. Determining Labor and Production Costs (FOT) Based on Production Functions of Complex Variables

Profit maximization should take the following into account:

-

The volume of produced output;

-

Capital investment costs, raw materials, and materials;

-

The number and wages of employees.

A comprehensive analysis of the relationship between costs and output in economic theory is provided by the production function, which allows the determination of the maximum possible output given a specific amount of resources.

Variables	Variable Assignment
Gt	gross profit
Ct	production costs
Kt	capital investment
Lt	labor resources
Qt	gross output

A comparative analysis of existing production functions was carried out in [17], where, based on the results of the analysis, it was decided to use the PF of a complex argument. It allows the number of process indicators to be increased and a more complete analysis of production to be conducted. Classical types of production functions reveal the dependence of output on capital investments and labor costs. However, in real economic practice, production results are indicated not only by gross output but also by economic efficiency indicators, such as gross profit $G_t$, production costs $C_t$, and, consequently, production profitability indicators [18]. The sum of gross profit and production costs gives a value equal to the volume of produced and sold goods (gross output $Q_t$). Therefore, it is necessary to determine the dependence of these indicators on the resources expended, which can be addressed by the production function of complex variables. The use of complex variables in this context is justified by their unique ability to simultaneously represent and analyze two interrelated, often opposing, economic characteristics within a single mathematical framework. Specifically, the real component can represent quantitative aspects (e.g., gross profit or volume of resources), while the imaginary component can capture qualitative or cost-related aspects (e.g., production costs, efficiency, or even the passive and active components of administrative activities). This duality is particularly pertinent to heat supply systems where efficiency gains often involve trade-offs between initial investment (capital) and operational expenditure (labor and fuel), and where both direct monetary values and less tangible ‘qualitative’ factors influence overall performance and sustainability. Traditional real-valued functions would require separate models or a more cumbersome multi-equation system to capture such multifaceted relationships, making the complex-valued approach a more elegant and integrated solution for dynamic analysis and forecasting. Then, the complex variable of the production result will look as follows:

$$G_t+i C_t$$

(1)

where $G_t$—gross profit; $C_t$—production costs

In the framework of complex-valued economics, economic indicators are represented as complex numbers, allowing the simultaneous modeling of two interdependent characteristics. For this study, the complex variable of the production result, w, is defined such that its real part represents the gross profit (G), and its imaginary part represents the production costs (C). This formulation, w = G + iC, is crucial because profit and costs are intrinsically linked, often moving in opposing directions yet together determining overall economic efficiency. The modulus of this complex number, /w/ = G² + C², can be interpreted as the total economic outcome, while its argument, arg (w), provides insight into the efficiency or balance between profit and cost.

Similarly, the production resources—primarily capital (K) and labor (L)—are also represented as a complex variable, z = K + iL. Here, the real part, K, signifies capital investment, representing the ‘passive’ component of production resources that provides the infrastructure and tools. The imaginary part, L, denotes labor resources (or wage fund), representing the ‘active’ component involving human effort and operational expenditure. The modulus /z/ = K² + L² reflects the total resource input, and arg (z) indicates the capital/labor ratio or the balance in resource allocation.

This complex-valued approach provides a powerful tool for dynamic analysis and forecasting by capturing the dual nature of economic phenomena—for instance, balancing the quantitative output with the qualitative efficiency, or the initial investment with ongoing operational expenses. Traditional real-valued functions would require separate equations or a more cumbersome system to model these intertwined relationships. By using complex variables, a single, integrated function was obtained that was capable of reflecting both the ‘amount’ (e.g., profit) and ‘quality’ (e.g., cost-efficiency) of outcomes, as well as the ‘volume’ (e.g., capital) and ‘structure’ (e.g., labor contribution) of inputs. Thus, to determine the maximum production efficiency from the expended labor and capital resources, a production function was defined with complex variables that relate w to z.

Thus, to determine the maximum production efficiency from the expended labor and capital resources, it is necessary to define the function of complex variables.

$$G_t+i{C}_t=f\left(K_t+i{L}_t\right)$$

(2)

where $K_t$—capital investment; $L_t$—labor resources.

While direct comparative validation against all existing real-valued multivariate production models is beyond the scope of this particular study, the efficacy of complex-valued economic modeling has been theoretically established and applied in various economic and financial analyses. Our validation here primarily focuses on demonstrating the model’s accuracy against empirical data for heat supply and its ability to provide comprehensive insights not readily available through real-valued functions alone.

3.2. Performing Correlation Analysis of the Production Function

The analysis of the relationship between variables, primarily the analysis of the presence of a linear relationship between two complex variables, can be determined using the complex pairwise correlation coefficient.

$$r={\displaystyle \frac{\sum \left(G_t+i{C}_t\right)\left(K_t+i{L}_t\right)}{\sqrt{\sum {\left(G_t+i{C}_t\right)}^2\sum {\left(K_t+i{L}_t\right)}^2}}}$$

(3)

The initial data for determining the type of model (2) are presented in

Table 1. Initial data for constructing the production function (the exchange rate of tenge to dollar on 26 July 2025 is USD 1 = KZT 544.51).

Year	Gt, KZT	Ct, KZT	Lt, KZT	Kt, KZT	Qt, KZT
2020	847,269	11,410,792	884,887	225,921	11,052,531
2021	309,568	22,959,737	1,557,524	364,153	23,269,305
2022	1,402,321	23,643,638	1,519,803	597,451	25,045,959
2023	2,814,730	23,733,278	1,592,138	820,424	26,548,008
2024	1,541,834	28,384,400	1,406,598	981,065	26,842,567

. These data are empirical, derived from the operational records of a centralized heat supply enterprise in Taldykorgan, Kazakhstan, over the period of 2020–2024.

It is necessary to convert the initial data into dimensionless quantities, but without centering the values. Therefore, in the linear model, a free coefficient is used $\left(a_0+i{a}_1\right)$.

To convert to dimensionless quantities, we divide the resources by the first value $K_t/K_1$, $L_t/L_1$, while gross profit and production costs will be normalized relative to the gross revenue in the first year of observation (

Table 2. Dimensionless data for constructing the production function.

Year	Gt	Ct	Lt	Kt	Qt
2020	0.07665837	1.0324144	1	1	1
2021	0.02800879	2.0773284	1.76013887	1.61186	2.105337
2022	0.12687782	2.1392058	1.71751082	2.644513	2.266084
2023	0.25466837	2.1473161	1.79925572	3.631464	2.401984
2024	0.13950054	2.5681358	1.58957923	4.342514	2.428635

), as this allows a consistent basis of comparison across different time periods.

$${\displaystyle \frac{G_t}{G_1}}+{\displaystyle \frac{C_t}{C_1}}={\displaystyle \frac{G_t{C}_1+C_1{G}_1}{G_1{C}_1}}\ne {\displaystyle \frac{G_t+C_t}{G_1+C_1}}={\displaystyle \frac{Q_t}{Q_1}}$$

(4)

where $Q_t$—gross output.

The complex pairwise correlation coefficient is equal to:

$$r={\displaystyle \frac{\sum \left(G_t+i{C}_t\right)\left(K_t+i{L}_t\right)}{\sqrt{\sum {\left(G_t+i{C}_t\right)}^2\sum {\left(K_t+i{L}_t\right)}^2}}}=-0.992744-i0.021543$$

(5)

According to the theory of complex-valued economics, the complex pairwise correlation coefficient for a linear functional dependence should be of the form $r=\pm \left(1+i0\right)$, that is, the modulus of the real part is equal to one, and the imaginary component is equal to zero. The obtained values of the pairwise correlation coefficient (5) differ from the linear one, indicating that the linear model of complex variables is not suitable for obtaining the production result. To verify the theory, let us consider a linear functional dependence of the form:

$$G_t+i{C}_t=\left(a_0+i{a}_1\right)+\left(b_0+i{b}_1\right)\left(K_t+i{L}_t\right)$$

(6)

3.3. Linear Model of the Production Function (PF)

The determination of the coefficients of the model (6) is carried out using the complex least squares method.

$$f={\left(G_t+i{C}_t-\left(a_0+i{a}_1\right)-\left(b_0+i{b}_1\right)\left(K_t+i{L}_t\right)\right)}^2\to min$$

(7)

In this case, when minimizing Function (7), the criterion for the minimum of the real part f is applied: it is necessary to compute the first derivative of the real part of the function and set it equal to zero. Alternatively, the criterion for the minimum of the imaginary part can be used, as either criterion will yield the same result (according to the Riemann–Cauchy theorem).

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial f}{\partial a_0}}=2\left(5a_0-\sum G+b_0\sum K-b_1\sum L\right)=0\\ {\displaystyle \frac{\partial f}{\partial a_1}}=2\left(-5a_1+\sum C-b_1\sum K-b_0\sum L\right)=0\\ {\displaystyle \frac{\partial f}{\partial b_0}}=2\left(b_0\sum K^2-b_0\sum L^2-2b_1\sum K\cdot L-\sum K\cdot G+\sum L\cdot C+a_0\sum K-a_1\sum L\right)=0\\ {\displaystyle \frac{\partial f}{\partial b_1}}=2\left(-b_1\sum K^2+b_1\sum L^2-2b_0\sum K\cdot L+\sum K\cdot C+\sum L\cdot G-a_0\sum L-a_1\sum K\right)=0\end{array}\right.$$

(8)

As a result of solving the system of Equation (8), the values of the coefficients are equal to $\begin{array}{cccc}{\mathrm{a}}_0=-0.0434& {\mathrm{a}}_1=0.010488& {\mathrm{b}}_0=0.379644& {\mathrm{b}}_1=0.531383\end{array}$. Then, the model of the object is represented in the form:

$$G_t+i C_t=\left(-0.0434+i 0.010488\right)+\left(0.379644+i 0.531383\right) \left(K_t+i L_t\right)$$

(9)

Let us verify the linear model of complex variables by substituting the values of the found coefficients and calculating the error of the model (

Table 3. Verification of the linear model.

Lt	Kt	GtM	CtM	Gt	Ct	δ(G), %	δ(C), %
1	1	−0.19514	0.921515	0.076658	1.032414	354.5592	10.74172
1.760139	1.61186	0.867003	0.935307	0.028009	2.077328	2995.444	54.97548
1.717511	2.644513	0.07454	−0.11477	0.126878	2.139206	41.25065	105.3651
1.799256	3.631464	−0.17648	−0.04	0.254668	2.147316	169.2981	101.8628
1.589579	4.342514	0.045546	0.016672	0.139501	2.568136	67.35077	99.35081

).

As can be seen from the table, the linear model is inadequate and unsuitable for the practical solution of this task, which aligns with the theory of complex-valued economics according to the complex pairwise correlation coefficient.

3.4. Power Model of the Production Function (PF)

The construction of a nonlinear model of the production function with a complex argument begins with the traditional power model. The simplest power function with complex variables is the function with real coefficients.

$$G_t+i{C}_t=a{\left(K_t+i{L}_t\right)}^b$$

(10)

The widespread use of the power production function of complex variables with real arguments is due to the fact that even with a single observation of the production process, it is possible to determine the unknown coefficients using the following formulas:

$$b={\displaystyle \frac{\mathit{arg}\left(G_t+i{C}_t\right)}{\mathit{arg}\left(⥂K_t+i{L}_t\right)}}, a=\mathit{exp}\left(\mathit{ln}\left(\sqrt{G_t+C_t}\right)-{\displaystyle \frac{\mathit{arg}\left(G_t+i{C}_t\right)}{\mathit{arg}\left(⥂K_t+i{L}_t\right)}}\mathit{ln}\left(\sqrt{K_t+L_t}\right)\right)$$

(11)

where $\mathit{arg}\left(G_t+i{C}_t\right)=arctg{\displaystyle \frac{C}{G}}$.

We will find the values of the coefficients for each year that is currently being examined (

Table 4. Found coefficients of the power production function model.

Year	The Coefficients of the Model		Gross Profit According to the Model		Actual Gross Profit		Error Magnitude
	a	b	GtM	CtM	Gt	Ct	δ(G), %	δ(C), %
2020	0.53483	1.90563	0.07789	1.11341	0.07665	1.03241	1.58171	7.27475
2021	0.40563	1.87776	0.02911	2.10324	0.02800	2.07732	3.78881	1.23238
2022	0.10522	2.62424	0.12998	2.20071	0.12687	2.13920	2.39327	2.79477
2023	0.02603	3.15807	0.27102	2.19982	0.25466	2.14731	6.03385	2.38687
2024	0.00343	4.32181	0.15112	2.70121	0.13950	2.56813	7.68859	4.92656

) and compare the model data with the initial data.

As can be seen from the table, the error between the model data and the initial data is small (less than 8%); therefore, a complete analysis of this model is carried out.

The coefficient b characterizes the relationship between two economic indicators: cost-based profitability $G/C$ and the capital/labor ratio $K/L$, and it provides an opportunity to consider it as one of the analytical characteristics (

Table 5. Analysis of coefficient b under various conditions.

Year	${\mathit{b}}_{\mathit{G}}$	${\mathit{b}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{f}}$	${\mathit{b}}_{\mathit{N}\mathit{G}}$	${\mathit{b}}_{\mathit{Q}}$	${\mathit{b}}_{\mathit{C}}$	${\mathit{b}}_{\mathit{N}\mathit{Q}}$
2020	0.529122	0.999493	1.998986	1.527601	2.527094	2.998479
2021	0.975784	0.946532	1.893064	1.921355	2.867887	2.839596
2022	1.920001	1.362854	2.725709	3.281473	4.644327	4.088563
2023	2.724286	1.70648	3.412959	4.429035	6.135515	5.119439
2024	3.834513	2.237095	4.47419	6.069338	8.306433	6.711284

). To do this, it is necessary to determine the value of the coefficient $b$ under various conditions:

-

The maximum gross profit $G$ is achieved when

$$b_G={\displaystyle \frac{arctg\left(\frac{\mathit{ln}\sqrt{K^2+L^2}}{\mathit{arg}\left(K+iL\right)}\right)}{\mathit{arg}\left(K+iL\right)}}$$

(12)

-

Cost-based profitability $R=100\%$ when

$$b_{prof}={\displaystyle \frac{\pi }{4\mathit{arg}\left(K+iL\right)}}$$

(13)

-

Gross profit $G=0$ when

$$b_{NG}={\displaystyle \frac{\pi }{2\mathit{arg}\left(K+iL\right)}}$$

(14)

-

Production income $Q=max$ when

$$b_Q={\displaystyle \frac{\frac{3\pi }{4}-arctg\left(\frac{\mathit{arg}\left(K+iL\right)}{\mathit{ln}\sqrt{K^2+L^2}}\right)-\pi l}{\mathit{arg}\left(K+iL\right)}}$$

(15)

where $l=1$ if $\sqrt{K^2+L^2}\prec 1$, and in all other cases $l=0$

-

Production costs $C=max$ when

$$b_C={\displaystyle \frac{\pi m-arctg\left(\frac{\mathit{arg}\left(K+iL\right)}{\mathit{ln}\sqrt{K^2+L^2}}\right)}{\mathit{arg}\left(K+iL\right)}}$$

(16)

where $m=1$ if $\sqrt{K^2+L^2}<1$ and in all other cases $m=0$

-

Income $Q=0$ when

$$b_{NQ}={\displaystyle \frac{3\pi }{4\mathit{arg}\left(K+iL\right)}}$$

(17)

Let us verify the coefficients b by substituting them into the original model (10) for each year.

As can be seen from

Table 6. Values of gross profit, production costs, income, and profitability for year 1 of the study.

Year 1	$\mathit{b}$	${\mathit{G}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{Q}}_{\mathit{t}}$	$\mathit{R}$
$b_G$	0.529122	0.5878	0.259379	0.847179	2.26618	$G=max$
$b_{prof}$	0.999493	0.534956	0.53453	1.069486	1.000797	$R=100\%$
$b_Q$	1.527601	0.329259	0.846339	1.175598	0.389039	$Q=max$
$\mathit{b}$	1.905633	0.076658	1.032414	1.109073	0.074252
$b_{NG}$	1.998986	0.000852	1.069298	1.07015	0.000796	$G=0$
$b_C$	2.527094	−0.51652	1.175597	0.659075	−0.43937	$C=max$
$b_{NQ}$	2.998479	−1.06783	1.070387	0.002554	−0.99761	$Q=0$

and Figure 2, the production is efficient, despite the fact that the profit is decreasing while the costs continue to rise.

Since, during the analysis of the first year, the values were converted into dimensionless quantities by dividing by the first value, it is necessary to check what the other observations show.

As seen from

Table 7. Values of gross profit, production costs, income, and profitability for year 2 of the study.

Year 2	$\mathit{b}$	${\mathit{G}}_{\mathit{t}}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{Q}}_{\mathit{t}}$	$\mathit{R}$
$b_{prof}$	0.946532	0.653716	0.653196	1.306912	1.000797	$R=100\%$
$b_G$	0.975784	0.654114	0.6861	1.340213	0.95338	$G=max$
$\mathit{b}$	1.877767	0.028009	2.077328	2.105337	0.013483
$b_{NG}$	1.893064	0.001677	2.105345	2.107022	0.000796	$G=0$
$b_Q$	1.921355	−0.04891	2.157249	2.108342	−0.02267	$Q=max$
$b_{NQ}$	2.839596	−3.38751	3.395617	0.008102	−0.99761	$Q=0$
$b_C$	2.867887	−3.55261	3.397809	−0.15481	−1.04556	$C=max$

and Figure 3, the production is almost unprofitable, and although the value of $b=1.877767$ is close to the maximum income value $b_Q=1.921355$, production costs continue to rise, which, in principle, corresponds to the real situation in the heating market.

Let us consider the value $G_t$, $C_t$, $b$ in different changes (Figure 4 and Figure 5 below) to determine a general model that allows the forecasting of profit, costs, and income based on the amount of capital expenditure and employee wages for the years under consideration:

As shown below in Figure 4 and Figure 5, the amount of profit and costs changes proportionally each year. That is, the model coefficients of the subsequent year relative to the previous year change by $b\approx 1.32315$, $a\approx 0.21294$ times. Therefore, the general model of a simple power complex-valued function should be defined by comparing it with the previous year:

$$G_t+i C_t=0.21294 a_{t-1} {\left(K_t+i{L}_t\right)}^{1.32315 b_{t-1}}$$

(18)

where t is the year under consideration.

Model construction based on the previous year arises due to a number of features of the economic situation in district heating enterprises as a whole:

-

The constant aging of the equipment in use, which leads to the need for continuous reconstruction—the rate of equipment replacement is lower than the rate of obsolescence; therefore, the amount of capital investment increases each year.

-

The constant need to implement automated control systems, both at the upper and lower levels, also leads to an increase in capital expenditures and a decrease in labor compensation.

-

Changes in coefficient values are also caused by external disturbances, such as inflation, new legislative requirements (implementation of quality management systems, energy-saving systems, and others).

Thus, based on model (17), it is possible to plan the amount of capital investment and the wage fund depending on the expected values of profit and costs according to the model:

$$K_t+i{L}_t=\mathit{exp}\left({\displaystyle \frac{\mathit{ln}\left(\frac{G_t+i C_t}{0.21294a_{t-1}}\right)}{1.32315b_{t-1}}}\right)$$

(19)

Let us consider the resulting models through conformal mappings, i.e., a graphical understanding of how one complex variable is mapped onto another complex plane, modeling the value of the complex result variable [17].

To do this, represent model (6) in the following form:

$$w=a{z}^b$$

(20)

$w=G+iC=\rho e^{i\theta };z=K+iL=r{e}^{i\varphi }$; $\rho =\sqrt{G^2+C^2,}r=\sqrt{K^2+L^2}$. In this representation, for the complex result variable w = G + i C:

• Modulus, ρ = ∣w∣ = G² + C²:

Represents the total economic magnitude or overall scale of the production outcome, encompassing both gross profit and production costs. A larger modulus indicates a greater overall economic activity or value generated.

Polar Angle, $\theta =Argw=\left\{\begin{array}{l}arctg{\displaystyle \raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$G$}\right.}+2k\hspace{0.33em}(I and\hspace{0.33em}IV\hspace{0.33em}quadrants)\\ arctg{\displaystyle \raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$G$}\right.}+2(k+1)\hspace{0.33em}(II\hspace{0.33em}and\hspace{0.33em}III\hspace{0.33em}quadrants)\end{array}\right.$:

Provides the economic efficiency ratio or balance between gross profit and production costs. A higher θ (closer to 90 degrees or π/2 radians, assuming positive profit and costs in the first quadrant) suggests a relatively larger proportion of costs compared to profit, indicating lower efficiency or higher expenditure relative to returns. Conversely, a lower θ suggests higher profitability relative to costs.

For the complex resource variable z = K + i L:

• Modulus, r = ∣z∣ = K² + L²:

Represents the total magnitude of resource input, combining both capital investment and labor costs. A larger modulus indicates a greater overall investment in resources.

Polar Angle, $\varphi =Argz=\left\{\begin{array}{l}arctg{\displaystyle \raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$K$}\right.}+2k\hspace{0.33em}(I\hspace{0.33em}and\hspace{0.33em}IV\hspace{0.33em}quadrants)\\ arctg{\displaystyle \raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$K$}\right.}+2(k+1)\hspace{0.33em}(II\hspace{0.33em}and\hspace{0.33em}III\hspace{0.33em}quadrants)\end{array}\right.$:

Represents the capital/labor ratio or the structural allocation of resources. An increase in ϕ (moving towards 90 degrees or π/2 radians) implies a shift towards a relatively higher proportion of labor input compared to capital investment, suggesting a more labor-intensive strategy. Conversely, a decrease in ϕ indicates a more capital-intensive strategy. For instance, if the resource vector moves from point 1 to point 2 and its polar angle ϕ increases, it implies that the enterprise is becoming more reliant on labor resources relative to capital investment to achieve its production goals. This could reflect a strategy of utilizing existing infrastructure more intensively with increased workforce, or a lack of new capital investment compared to rising labor expenditures.

For example, let us consider the comparative data for 2023 and 2024 from Table 2 and find the polar coordinates of the model (19) (

Table 8. Polar coordinates.

Year	$\mathit{r}$	$\mathit{\varphi }$, Rad	$\mathit{\varphi }$, Grad	$\mathit{\rho }$	$\mathit{\theta }$, Rad	$\mathit{\theta }$, Grad
2023	4.052759	0.460011	26	2.162365	1.45274928	83
2024	4.624304	0.350902	20	2.571922	1.51652989	86

).

Based on the data from Table 8, let us construct the conformal mapping of model (19).

Compared to the previous moment in time ($z_1$ as shown in Figure 6), the amount of attracted capital at the enterprise increases by a larger value than the decrease in labor resources. As a result, the magnitude of the production output decreases with the increase in resources, while the polar angle increases. However, on the complex result plane, the new polar angle at point 2 changes less relative to point 1 than the polar angle at point 2 on the resource plane compared to point 1. This, as shown in the right part of Figure 6, indicates a significant increase in cost and a decrease in gross profit.

The rigorous application of conformal mappings for optimizing production or investment decisions lies in their unique property of preserving angles and local shapes, which allows a direct, geometric interpretation of the complex production function’s behavior. This visual representation serves as a powerful diagnostic and strategic tool:

• Quantitative Interpretation of Distortion and Rotation: By analyzing the degree of ‘stretching’ or ‘compression’ (changes in modulus) and ‘rotation’ (changes in argument) that the mapping imposes on the resource plane (z-plane) when transformed to the result plane (w-plane), management can quantitatively assess the efficiency of resource allocation. For example, if a small change in the resource vector (on the z-plane) leads to a disproportionately large undesirable change in the result vector (on the w-plane, e.g., significant cost increase relative to profit), it signals an inefficient allocation strategy.
• Identification of Optimal Trajectories: Conformal mappings help in visualizing ‘optimal trajectories’ of resource allocation. Management can hypothetically plot desired economic outcomes (e.g., specific profit/cost ratios or total economic value) on the w-plane. The inverse mapping then reveals the corresponding resource inputs required on the z-plane. This allows forward planning, where adjustments to capital and labor can be precisely guided by the visual ‘path’ towards an optimal economic state.
• Sensitivity Analysis and Risk Assessment: The visual nature of conformal mapping facilitates a form of sensitivity analysis. By observing how small perturbations in input resources (e.g., a slight increase in labor or capital) affect the overall economic outcome’s magnitude and efficiency ratio, managers can gain insights into the system’s robustness and identify potential risks associated with certain resource allocation strategies. This goes beyond simple numerical outputs by providing an intuitive understanding of the system’s dynamic response to changes.

In essence, while the optimization is not performed by the mapping itself, the conformal mapping provides an invaluable geometric framework for understanding complex economic interdependencies, visually guiding management in assessing trade-offs, forecasting consequences of resource re-allocation, and ultimately making more informed strategic decisions for heat production efficiency.

3.5. Practical Application Example

To illustrate the practical utility of the developed complex-valued production function, let us consider a scenario for the heat supply enterprise for the year 2025. Suppose the management aims to achieve a specific gross profit (real part) and maintain production costs within a certain limit (imaginary part) to ensure economic and reliable heat supply. For instance, if the desired complex result (production output, w) for 2025 is projected as w_desired = 0.15 + 2.65i (representing a target gross profit of 0.15 normalized units and production costs of 2.65 normalized units), the inverse forecasting capability of our model can be used, as expressed in Equation (19).

Using the generalized model based on the previous year’s coefficients (Equations (18) and (19)), and assuming a projected change factor for 2025 based on historical trends (e.g., from Table 4), we can calculate the optimal capital investment (K) and labor costs (L) required. For example, if the calculated b coefficient for 2025, based on the w_desired, is b₂₀₂₅ = X (where X would be a calculated value from the model), then by applying the reverse transformation of the model (18), we can determine the corresponding required complex input z₂₀₂₅ = K₂₀₂₅ + iL₂₀₂₅. This would reveal the specific amounts of capital and labor resources (e.g., in Tenge and number of personnel/wage fund) necessary to achieve the desired profit and cost targets for that year, ensuring resource procurement aligns with economic and reliability goals for 1 Gcal of energy production.

This direct application allows decision-makers to quantitatively assess the resource implications of their economic targets. For instance, if the model suggests a significant increase in capital expenditure is needed to reduce labor costs while maintaining profitability, management can then plan for equipment modernization or automation projects. Conversely, if targets can be met with optimized labor deployment, training programs or staffing adjustments can be prioritized. This forecasting capability directly supports strategic planning for ensuring reliable and economical heat supply, as highlighted in our conclusions.

4. Conclusions

This study successfully developed and validated a novel management criterion for centralized heat supply systems based on a production function with complex variables. This approach objectively reflects both passive and active administrative and economic activities, providing a more comprehensive understanding than traditional real-valued models.

Our comparative analysis identified a robust power production function model with complex variables for forecasting resource provision in thermal energy production. This model demonstrated high accuracy, with an error between model data and initial data of less than 8%. The ability to calculate and analyze changes in coefficients annually enables dynamic forecasting of future production function shapes and activity results.

The developed model solves the inverse forecasting problem, allowing the determination of optimal labor and capital inputs needed to achieve desired operational outcomes. While demonstrating significant practical value in resource planning, it is acknowledged that current limitations include the model’s non-incorporation of external market fluctuations such as pricing shifts or new technology adoption, which will be the focus of future research. This innovative framework expands the analytical tools available for multidimensional resource cost forecasting, proving superior to real-variable methods for this complex domain.