Prospects for the Balanced Development of the Waste Management System in Ukraine

Abstract

This article outlines the problem of waste accumulation as well as the associated environmental consequences in Ukraine due to the disparities between waste generation and utilization capacities. The analysis of quantitative parameters of waste generation sources and the territorial structure of Ukraine’s utilization capacities was carried out, and the spatial assessment of waste management system balance within regions was made. The work also identifies stages of territorially balanced management systems formation within the scope of current recycling technologies in the regional dimension. Methodological approaches to researching the balance conditions of waste management systems at the regional level are substantiated, which presupposes a mathematical model of these processes through the development of a system of equations, which depending on the tasks determine either the timing of the outlined phases of solving waste management issues based on available investment resources, or the amount of funds required to ensure proper timing of the innovative disposal facilities commissioning.

Keywords

Waste management accumulation utilization ecological stability territorial balance of the system mathematical model of the problem

Introduction

Waste has always been a consequence of the socio-economic activity; however, the issue of its anthropogenic impact has never been as acute as it is today, entering the global arena. The second half of the twentieth century was marked by the beginning of the consumer boom that consequently had led to environmental problems. People understand that by improving their living conditions from the standpoint of the consumers of goods, on the other hand, the global balance of ecosystems worsens due to production waste, packaging materials, e-waste and others. Understanding that people will never be able to give up the blessings of civilization, and having in view the development of scientific and technological progress, in future, the situation will only worsen unless we make adequate decisions and take decisive action. In recent decades, the world’s leading countries, including European countries, have been confidently building a ‘recycling society’ by focusing on waste recycling processes and shaping the ecological culture of the population (Kumar et al., 2019). This has enabled us to avoid the negative consequences of waste generation and, subsequently, to focus on the development towards ‘circular’ economy.

Prior to adopting the National Waste Management Strategy by 2030, Ukraine treated waste quite unambiguously—it was neither needed by owners nor producers—and usually not being processed, but instead sent directly to landfills, or stored temporarily on industrial sites. This imbalance had led to numerous accumulations of waste in the country, the problem which has not been properly studied. According to estimates of the Ministry of Ecology and Natural Resources of Ukraine, the concentration of various types of waste amounted to approximately 35 billion tons, out of which 2.6 billion tons were highly toxic. Household waste increased annually by 50 million cubic metres or an equivalent amount of 14 million tons (300–400 kg per person yearly), and industrial waste increased by 175 million cubic metres. For instance, 12.5 million pieces of tires alone accumulated each year. This waste occupied an area of about 7,000 ha, transitioning into filtrate that threatened human health as it polluted the soil, poisoned groundwater as well as rivers that flowed into the ocean, passing through many countries. The main cause for this is the discrepancy between the national utilization capacities potential and the volumes of waste generation, which explains the potential economic losses from the non-use of secondary resources, as well as anthropogenic impact and deterioration of the environmental situation in the Ukrainian region. Natural ecosystem has no administrative borders; therefore, anthropogenic impact goes far from being the sole national problem of Ukraine, causing a global environmental damage.

Thus, the entitative direct dependence between the economic development of the country and the accumulation of waste requires adaptive management systems aimed at ensuring that there is a balance between waste generation and disposal within the territorial formations. In our opinion, a territorially balanced approach in the context of Ukraine will allow us to aggregate a certain scale of problems associated with waste generation and use the sufficient levers of local authorities to gradually resolve these problems and transform waste into an economic resource. Accordingly, this article focuses on solving these tasks using territorial disparities analysis, based on appropriate methodology and discussion of existing problems; we also propose a phased formation of territorially balanced management systems based on mathematical formalization of these processes formulated as managerial implications.

Review of Literature

Today, foreign scientists dwell on the necessity to implement effective production management technologies, including waste recycling, that will provide for the rational use of natural resources (Marshall & Farahbakhsh, 2013). Looking at the waste management systems of developed European countries, the concept of Zero Waste, introduced by Murray (2004), is of particular importance, providing for the secondary use of waste within the production cycle and the organization of substance circulation during technological cycles. Murray says that zero waste ‘... represents waste prevention at the stage of its very appearance and creation of the chains that would slow down the energy and materials entropy as well as enhance the natural processing capacity’.

Integrated waste management, which is at the centre of European Union’s (EU) sustainable development strategy has also formed the basis for the adoption of the European Commission action plan with regard to transitioning to a circular economy (CE) by 2019. Research conducted by the Centre for European Policy Studies (CEPS) shows that the implementation of the suggested economic innovations will have a less negative environmental impact due to reducing the amount of material and energy resources during the production process and cutting production costs by saving primary resources. Above all, it will facilitate the emergence of new markets, ensure the creation of new jobs and increase the living standards of the population.

In Ukraine, focusing on European experience, local waste recycling systems, which are actively promoted within the framework of administrative reform of decentralization, change the paradigm of centralized models of development of economic systems, which were formed as far back as during the Union of Soviet Socialist Republics (USSR) period. Therefore, we support the opinion of Italian scientists, centred on the balanced development, and the adoption of closed production models within the economic system, where ‘...CE aims to increase the efficiency of resource use to achieve a better balance and harmony between economy, environment and society’ (Ghisellini et al., 2016). Herewith, we believe that the balance in the waste management system among the economy, environment and society is a consequence of ensuring a balance between the parameters of existing utilization capacities and waste generation within the defined territories, while, at the same time, researchers pay little attention to this aspect. Even with the study of the relationship between the CE and the sustainability of territorial development, in the works of Geissdoerfer et al. (2017) and Millar et al. (2019), there is no proper justification for this cause-and-effect relations.

With this in view, waste is an objective consequence of our civilizations’ development, which creates many problems and requires decisive action on the part of environmental defence. At the same time, waste is also a resource. For example, experts from the authoritative publication Business Insider (2018) have estimated that in case we recycle all the plastic waste on the planet, it will leverage about US$7.2 trillion. These calculations are based on the data from the Royal Statistical Society (RSS)—one of the most recognizable and well-known statistical societies in the world. RSS information shows that 90.5% of the 6.3 billion tons of all plastics produced over 60 years have not yet been recycled. This number is equivalent to 7.2 trillion shopping bags, completely filled with plastic garbage, each bag containing approximately 0.5 kg of an estimated value of US$1 (the production of a plastic bottle costs 3.25 cents, multiplied by 31 bottles per bag). Monetization of such processing potential will amount to US$7.2 trillion, and with this money, it is possible to buy Apple, Amazon, Google, Microsoft, Wal-Mart, Exxon, GM, AT&T, Facebook, Bank of America, Visa, Intel, Home Depot, HSBC, Boeing, Citigroup, Anheuser-Busch and all professional teams from the NFL and MLB—US Football and Basketball League (2018). Along with the massive economic effect, the fact of ‘cleaning’ the planet is no less impressive, as according to Business Insider experts, if we take the same shopping bag about 1 foot (30.48 cm) high, filled with plastic garbage, and consistently put them in one line with all unprocessed plastics, we will get a length equal to 5,790 distances to the moon and back (2018). And if humanity has accumulated 6.3 billion tons of waste since the beginning of mass production of plastic, then, according to American scientists (Geyer et al., 2017), in 2050, with the current utilization rate, the planet will have 12 billion tons of plastic waste.

Ukrainian scholars have thoroughly researched waste management. In particular, Ihnatenko (2014), Samoylik (2014), Rudenko (2011), Long (2012), Pylypiv (2010), Mishchenko (2011) and Yatsiv and Kolodiichuk (2017) and others focused their studies mostly on the limited aspects of governance, not taking empirical problems into account. Ukraine, in its turn, addresses the question—how to turn waste from a problem into a resource and then how to integrate it into the production cycle?

Objectives

All data in this article are based on empirical studies of existing waste management problems in Ukraine (Cabinet of Ministers of Ukraine, 2017), namely fast accumulation of waste in the industrial and household sectors; improper disposal/disposal of hazardous waste; failure to comply with the environmental requirements as to the disposal of household waste; rare use waste as secondary raw materials due to lack of organizational and economic principles of their utilization in the production; and inefficiency of the implemented economic instruments within the waste management sector.

It is worth outlining the following research objectives:

investigating existing territorial disparities between waste generation and waste utilization in Ukraine, paying special attention to the capital investment allocation for scaling up utilization capacities;

identifying the possible solution of such a complex problem as current imbalances between waste and utilization capacities in Ukraine; and

developing mathematical models for a balanced waste management system, which will provide variability of initial conditions necessary to solve empirical problems.

Rationale of the Studies

The need for an adequate management system is explained by the direct correlation between the socio-economic development dynamics and the accumulation of waste. Imbalance of waste generation and disposal accompanied by insufficient utilization capacity will lead to waste accumulation with the corresponding negative consequences (specifically environmental), and given the excess utilization capacities without decent raw material supply, it will cause negative economic consequences as well. Thus, the purpose of this article is to solve current problems of accumulated waste in Ukraine and to develop ways for the balanced development of the national waste management system with specific focus on the development of recycling technologies.

Methodology

Based on the dialectical method of objective reality cognition in order to conduct a comprehensive study of the current state and trends in waste management at the regional level, this study uses the methods of statistical grouping and graphical visualization of results. The use of the coefficient of territorial provision with utilization capacities (K_p) allows to integrate their existing potential and to determine the prevailing regional disparities between the generation and disposal of waste.

The methods of analysis and synthesis were used to determine the factors that have influence on formation of territorially balanced waste management systems and to assess the parameters and patterns involving the functioning of waste management system, and to compare statistical data, we used the methods of economic and statistical analysis and balance.

The simulation method was used to apply mathematical formalization to the problem of balancing waste management system in the Ukrainian region.

Analysis

In view of environmental, economic, social, legal and other aspects, waste management is an important aspect in the lives of people, at the same time, requiring the formation of an adequate management system (Iqbal, 2018). Given the objective inevitability of waste generation, it is necessary to transform the current management system from being an environmental burden to a raw economic resource. Speaking about Ukraine, waste is perceived as a massive polluter and, at the same time, is considered to comprise useful components and serve as a valuable source of energy, that is, secondary material and energy resources. In the former case, waste is incompatible with the socio-economic development of the territories, while, in the latter instance, it plays a decisive role in ensuring economic growth as well as sustainable development of the territories. This requires the formation of adequate waste management systems, which would be based on territorially balanced approach.

The system is equally balanced between the amount of generated waste and the amount of disposed waste. Waste generation is an ongoing activity, which is characterized by discretion and followed by the process of storage and/or disposal of waste. The balance is ensured; however, the most important issue is the quality of such balance and its associated environmental impact. Waste generation can be reduced if the economic development rate is decreased, which is unacceptable. Here, one can ask: what do you do with waste, the generation of which is directly proportional to economic growth? Several approaches can solve this problem.

The first one is the most rational and effective, although it requires a lot of time and significant investment of resources. It presupposes the transition to present-day resource-saving production technology, which will allow for considerable reduction of the waste volumes both during the process of goods production and during their further exploitation and utilization.

Modernization of the current waste management facilities is the second way.

The concentration of production capacities and the level of urbanization in Ukraine are varied, which affects the quantitative and specific composition of waste within individual territories. To ensure the economic performance in the waste management system, one should take into account these aspects and focus on the utilization of capacities so as to minimize the logistical costs necessary to move waste streams. Another aspect to remember is the significant amount of country-generated hazardous waste, which, when displaced, increases the risk of human accidents (Kravtsiv et al., 2018).

By ensuring the territorial balance of waste management system, it is possible to reduce the anthropogenic impact on the environment, optimize the current unauthorized landfills, conduct their monitoring and control, and direct potential raw materials for further processing (treatment, reuse), taking into account future prospects (Kravtsiv & Kolodiichuk, 2016; Mane & Niranjan, 2014). In this way the territorial balance determines the content of the waste management, since, then, we can define necessary goals for its functioning.

To assess the territorial balance between the volumes of waste generation and its disposal potential, we will calculate the initial indicators (Table 1). In order to do this, we have to operate not with absolute indicators, but with relative ones—volumes of waste generation per area unit of the region W_t (t/km²). Similarly, it is possible to calculate the territorial provision with utilization capacities P_t (t/km). Comparing the obtained value with the relative value of waste generation, we obtain the coefficient of territorial provision with utilization capacities (Kolodiichuk & Kolodiichuk, 2020):

K_{p} = \frac{P_{t}}{W_{t}}

(1)

Table 1.

Baseline Data and Calculation of the Territorial Provision with Utilization Capacities.

Region	Area of the Territory (in 1,000 km²)	Average Waste Volume for2016–2018 (in 1,000 tons)	The Amount of Waste Per 1 km² of Territory W_t (t/km²)	Total Number of Authorized Utilization Capacities in 2018 (in 1,000 tons)	Number of Sanctioned Utilization Capacities per 1 km² of Territory P_t (t/km²)	The coefficient of Local availability of Utilization Capacities (K_p)
AR Krym	26.1	–	–	–	–	–
Vinnytska	26.5	2,017.1	76.1	1,453.7	54.9	0.721
Volynska	20.1	657.5	32.7	374.1	18.6	0.569
Dnipropetrovska	31.9	230,854.5	7,236.8	189,632.9	5,944.6	0.821
Donetska	26.5	22,250.2*	839.6	21,718.8*	819.6	0.976
Zhytomyrska	29.8	529.0	17.8	292.5	9.8	0.551
Zakarpatska	12.8	171.8	13.4	186.2	14.5	1.082
Zaporizhzhia	27.2	5,154.9	189.5	4,946.3	181.8	0.959
Ivano-Frankivsk	13.9	1,951.3	140.4	1,680.8	120.9	0.861
Kyivska	28.1	1,407.0	50.1	1,403.7	50.0	0.998
Kirovohradska	24.6	36,644.5	1,489.6	37,624.7	1,529.5	1.027
Luhanska	26.7	1,219.3*	45.7	555.0*	20.8	0.455
Lvivska	21.8	2,465.4	113.1	1,589.4	72.9	0.645
Mykolaivska	24.6	2,368.2	96.3	2,095.0	85.2	0.885
Odeska	33.3	705.3	21.2	588.5	17.7	0.835
Poltavska	28.8	20,123.9	698.7	4,143.1	143.9	0.206
Rivnenska	20.1	551.7	27.4	291.0	14.5	0.529
Sumska	23.8	701.7	29.5	737.5	31.0	1.051
Ternopilska	13.8	1,473.3	106.8	274.5	19.9	0.186
Kharkivska	31.4	1,794.8	57.2	1,216.0	38.7	0.677
Khersonska	28.5	393.7	13.8	126.2	4.4	0.319
Khmelnytska	20.6	1,042.8	50.6	775.5	37.6	0.743
Cherkasska	20.9	1,333.0	63.8	1,085.8	52.0	0.815
Chernivetska	8.1	355.2	43.9	315.7	39.0	0.888
Chernihivska	31.9	723.6	22.7	415.2	13.0	0.573
Ukraine	0.84	1,197.6	1,425.7	697.4	830.2	0.582

Source: The authors.

Note: *Excluding the temporarily occupied territory of the Autonomous Republic of Crimea, the city of Sevastopol and parts of the temporarily occupied territories in Donetsk and Luhansk regions.

The proposed index makes it possible to make integral assessments of the existing utilization capacities in the regions and to determine the current disproportions, and therefore priority directions for investment.

To estimate the amount of accumulated waste, we will analyse the relative indicators of their generation in different regions of Ukraine—W_t. In order to determine the territorial peculiarities of distributing of capacity utilization, we will similarly calculate P_t, using the average values of the total indicators of all types of waste that were directed to the disposal capacities for the past 3 years (Table 1). Please note that by disposal facilities, we mean all authorized places of passive (specially designated places or facilities) or active (sorting lines, processing or incineration plants) waste management. Based on current forms of statistical reporting in Ukraine (Prokopenko, 2018), the indicator specified in Table 1 comprises the sum of utilized, incinerated and disposed waste in the designated sites or facilities.

Comparing the spatial localization of waste generation and utilization areas, we obtain the K_p coefficient, which is further presented on the map. Preliminarily, we group the territories based on the values of the obtained K_p indicators.

The number of groups (n) is determined by the approximate formula of Sturges (1926):

n = 1 + 3.322 log​ N

(2)

where N is the number of observations (territories).

The obtained value of n is 5.64 groups, or ≈ 6 groups.

To determine the interval (i), we use the following formula:

i = (x_{max} - x_{min}) \div n

(3)

where x_min is the minimum value of the feature;

x_max is the maximum value of the feature.

Applying the maximum value of K_p (1.082) for Zakarpatska region and the minimum for the Ternopilska region (0.186) in the corresponding formula, we obtain an interval equal to 0.1493. Using the method of equal intervals, we further group six groups of regions that are included in the formed closed intervals (Figure 1). Visualization of the territorial provision of Ukraine’s regions with utilization capacities is outlined in a cartogram (see Figure 1).

If the value of K_p coefficient is equal to 1, it means that all waste within the region has been utilized by means of authorized capacities. At the same time, there is a certain conditionality to this assumption since administrative boundaries of the regions do not pose any obstacles to waste movement, and, normally, a waste generator can use utilization capacities that are territorially closer, even if they are located in the neighbouring region. This definitely will have a partial effect on the value of the calculated coefficient. However, it should be borne in mind that the region from which the waste is removed may receive counter-waste from neighbouring regions, where waste generators have also benefitted from territorial proximity to the utilization capacities of the neighbouring region. This will partially balance the proposed coefficient and reduce environmental risks on the part of transport and logistics systems (Kolodiichuk et al., 2020).

Waste management systems in Zakarpatska, Kirovohradska and Sumska regions are the most territorially balanced with all generated waste being utilized by means of own facilities. It should be noted that Donetska region is also in group VI, but the estimated indicators for this region are not completely correct, since the initial waste volume (for 2 of the 3 years that are being studied) and the number of authorized utilization capacities for the analysed year are approved without taking into account the temporarily occupied territories, while the area of the territory of the region is stated completely. This also applies to Luhanska region and can be explained by the specifics of the official statistical reporting in Ukraine.

The lowest K_p value is observed in Ternopilska region, where only 18.6% of generated waste within the administrative territory can be utilized by means of own capacities.

It is advisable to use mathematical methods to study the conditions of system balance with the formation of appropriate economic and mathematical characteristics of these processes, which will allow investigating their nature and critically determining measures for the system’s improvement.

Discussion

Prior to formulating the general and specific tasks of an economic and mathematical study of the conditions of waste management system’s balance, we need to take into account the current situation in this sector. Scarcely located utilization capacities in Ukraine that, over the years, caused existing imbalances, have led to the formation of catastrophic amounts of waste, which are stored in unauthorized landfills. Continuously increasing volumes of generated waste as well as the anthropogenic load on the environment call for immediate action in order to stabilize the current situation and find ways to solve this persistent problem. Stopping or decreasing the economic activity of the enterprises, or commissioning new utilization capacities, are the possible solutions. While the former option is inadmissible, the latter happens to be too costly. However, the economies of highly developed countries have currently focused on the alternative way—the use of low- and non-waste production technologies—including recycling waste management systems. Such technologies are clearly even more expensive and unreachable to the national economy of Ukraine.

Looking at the current state of Ukraine’s economic development, we suggest the following phases to solve the waste management problem (Kolodiichuk & Kolodiichuk, 2020):

Phase 1. Monitoring existing utilization capacities as well as their reconstruction, in order to ensure anthropogenic safety during waste storage and accumulation.

Phase 2. Ensuring recycling of generated waste through the use of existing and/or the introduction of novel utilization capacities.

Phase 3. Providing partial recycling for accumulated waste and complete recycling of generated waste.

Phase 4. Replacing old recycling facilities with up-to-date recycling technologies.

Figure 2 presents graphical interpretation of the set task.

It should be noted that in this task, we mean not only recycling or incineration plants, which are extremely scarce in Ukraine, but also, above all, places of authorized waste storage at adapted landfills or places of technological waste storage within the territories of enterprises. Static sanctioned waste storage must neutralize the anthropogenic impact on the environment.

The implementation of Phase 1 of waste treatment (y) involves the initial mobilization of all utilization capacities (P) at a certain moment of time T₁ to ensure the potential volumes of waste Q. The quantity Q includes the sum of the volumes of previously accumulated waste Q₂ and the generated waste Q₁ during the time T₁. Waste generation is represented by a horizontal line (Figure 2), and Q₁ is sufficiently stable and predictable in the medium term as it depends on the socio-economic and technological development of the country. However, the lack of utilization capacities in Ukraine causes a constant accumulation of some of the generated waste by the amount of ΔQ during T₁ (Figure 2), that is, a large part of the generated waste falls on congested landfills, the operational life of most of which is exhausted, creating potential environmental risks for Eastern Europe.

The task of Phase 2 is to ensure the complete recycling or authorized storage of the generated waste to prevent its further accumulation. This is a stabilization phase of T₂ duration, which involves: (a) intensification of usage of available utilization capacities and (b) the construction of new waste processing plants or the fitting out of custom landfills. In this case, the condition under which the available and newly utilized capacities (P₁) must fully meet the current needs for waste disposal (stop waste accumulation), that is, P₁ ≥ Q₁, must be fulfilled.

The implementation of Phase 3 involves the reconstruction of the processing capacity to an amount that exceeds the needs for processing the generated waste (P₁ > Q₁). At this stage, processing facilities provide current needs for the modification of waste generated and the partial processing of previously accumulated waste, which is safely stored until the appropriate time. As the accumulated waste will eventually be depleted, at some point, T₃ will completely recycle it, unloading and freeing up their static storage. It is clear that this is rather an idealized situation, the conditionality of which will allow us to carry out a mathematical formalization of the problem.

Capacities mobilized at time T₁ up to T₃ will run out of resources and will require repair, replacement, upgrades and more. This period will last for several years, which will cause not only physical but also moral obsolescence of the equipment. Existing utilization technologies, as well as the emergence of new due to scientific and technological progress will set qualitatively new requirements for technologies and related technological equipment, based on a new paradigm of waste management, as they turn from the inevitable problem of finding ways to neutralize products of economic activity into a raw material source of recirculation. That is, the resource of utilization capacity must be exhausted by the time of the accumulated waste disposal, while ensuring the needs of generated waste processing, and, further, these capacities should be replaced by modern processing enterprises of the innovative type. In fact, the time T₃ is the beginning of Phase 4, that is, the point of bifurcation (B) of the waste management system in Ukraine, which will further provide a new level of this system’s development. For sure, it is not possible to introduce innovative P_N capacities at one time T₃, but the process of replacing existing capacities should be based on systematically attracting investors, and it should be preceded by a comprehensive state programme of the waste management system modernization with appropriate institutional, legislative and financial support. The mobilized resources should ensure the fastest transition to a new quality of waste management so that there is no capacity shortage at the time of replacement of obsolete ones.

The mathematical model of these processes has allowed us to carry out the economic and mathematical justification of our problem with the formulation of waste accumulation equations and their utilization, as well as a mathematical model of the waste management system’s balance.

Considering the performance of waste accumulation as a ratio of the amount of waste to the time at which it is accumulated, the process of accumulating waste from a physical point of view is presented in Figure 3.

It is clear that the pace of waste receipt is described by a linear dependence and can be represented by the equations of the form:

x = K \times x (t)

(4)

Thus, the general model of already accumulated waste and generated waste (incoming) will be presented in the form of dependence:

x = x_{1} + K \times x (t) t

(5)

where $x_{1}$ is the initial accumulated waste;

$K$ is the waste growth factor, which characterizes technological conditions of accumulation; and

$t$ is the duration of operation of the object days.

To develop a mathematical model, we use the increment of accumulated waste as a change in the amount of waste Δx = x₂ – x₁ over a given time interval Δt = t₁ − t₀.

The increment of the variable parameter is estimated by the differential of the parameter. Therefore, the dependence of the change in the amount of waste (Figure 2), which is described by Equation (5), can be represented as a function of growth:

x = x_{1} + K \times x (t) d t

(6)

x - x_{1} = K \times x (t) d t

(7)

d x = K \times x (t) d t

(8)

The obtained differential equation describes the increase in the amount of waste over time:

\frac{d x}{d t} = K \times x

(9)

The solution of differential Equation (9) is:

x = C \times e^{K \times t}

(10)

where $e$ is a natural number ( $e$ = 2.71828).

The continuous integration of C is determined from the initial conditions. At the initial timet = 0, x = x₁ (as $e$ ^K ^×0 = 1), we obtain a constant integration of C = x₁.

Then, the solution of Equation (9) of waste accumulation will be:

x = x_{1} \times e^{K \times t}

(11)

where x₁ is the initial accumulation in the corresponding dimension.

The open question remains to determine the K coefficient, the magnitude of which must be commensurate with the amount of waste that has been accumulated.

This coefficient can be experimentally determined. For instance, at some point in time, it was already 10 tons of waste, which in a year (t = 365 days), relatively speaking, increased by another 2 tons. Then, the starting formula (11) looks like this:

12 = 10 \times e^{K \times 365}

e^{K \times 365} = x_{2} - x_{1}

thus, $x_{2} - x_{1} = Δ x$ ,

then,

K \times t = ln \times Δ x

K = \frac{ln \times Δ x}{t}

Waste recycling will depend on:

y = Q_{m} \times t

(12)

where Q_m is the recycling productivity.

The increase in waste recycling, considering Equation (12) of the amount of waste recycled, will be given in the form of differential equation:

\frac{d y}{d t} = Q_{m}

(13)

We estimate the increase in waste processing productivity, which is characterized by the physical interpretation of acceleration. Then, taking the differential of dependence (13), we get:

\frac{d^{2} y}{d t^{2}} = y (t) .

(14)

Since the gain function of waste processing productivity is described by the dependence t $y (t) = e^{K_{1} \times t}$ , then Equation (14) will have the form:

\frac{d^{2} y}{d t^{2}} = e^{K_{1} \times t}

(15)

$K_{1}$ is the coefficient that characterizes the physical content of the waste processing.

The general solution of Equation (15) will be:

y (t) = C_{1} + C_{2} \cdot t + \frac{e^{K_{1} \times t}}{K_{1}^{2}}

(16)

We define the continuous integration from the initial conditions. At the initial time t = 0, y (t = 0) = 0. Then, substituting into Equation (16), we obtain:

C_{1} = - \frac{1}{K_{1}^{2}}

(17)

Differentiating Equation (16), we will have a waste recycling performance:

\frac{d y}{d t} = C_{2} + \frac{e^{K_{1} \times t}}{K_{1}}

(18)

To determine the constant C₂, we set the initial conditions: t = 0, dy/dt = 0. Then, from Equation (18), the constant integration will be equal to:

C_{2} = - \frac{1}{K_{1}}

(19)

Given the constant integrations, Equation (16) will look like:

y (t) = \frac{e^{K_{1} \times t}}{K_{1}^{2}} - \frac{t}{K_{1}} - \frac{1}{K_{1}^{2}},

(20)

The $K_{1}$ coefficient is determined by the conditions of processing, characterized by productivity. Suppose a recycling facility recycles 20 tons of waste over a period of time (say, a year or 365 days), then:

y = K_{1} \times t; K_{1} = \frac{y}{t} = \frac{20}{365} = 0.055

The optimization of waste recycling is written by the system of dependencies (11, 20) and the condition when the amount of waste generated (received) will be recycled y = x. The condition y = x in our problem means the time (point B in Figure 2) when the accumulated waste in previous periods will be physically disposed of, and full control over the volume of generated waste is obtained, which will eliminate their accumulation.

Thus, the system of equations of the waste management problem formalization in Ukraine on the basis of ensuring territorial balance between the volumes of generation and utilization of waste will look like:

\{\begin{cases} x = x_{1} \times e^{K \times t} \\ y = \frac{e^{K 1 \times t}}{K_{1}^{2}} - \frac{t}{K_{1}} - \frac{1}{K_{1}^{2}} \\ y = x, for t = n days \end{cases}

(1)

Mathematical model of the waste management system’s balance problem involves the construction of a system of equations that, depending on the formulation of the problem, determine either the timing of the implementation of the specified phases, based on available investment resources, or the amount of funds needed to ensure the time parameters for commissioning innovative utilization capacities.

Conclusions

The inevitability of industrial and household waste generation projects indicates the need for adequate management systems, which would create environmentally friendly conditions for waste storage with its subsequent utilization and, in the longer perspective, would ensure waste recycling into raw materials. Transition to the recirculating economy needs strong structural and functional support in order to ensure the proper operation of the waste management systems in Ukraine’s regions. Territorial balance provides for the content of the waste management system, being a condition of technogenic safety not only in Ukraine but also in the whole of Eastern Europe and a crucial parameter for the introduction of recycling technologies. The proposed coefficient of the territorial provision with utilization capacities allows us to make integral assessments of the existing potential and to identify the current regional disparities between waste generation and waste utilization, and also identify priority directions of capital investment.

We believe that the all-inclusive solution to the problem of existing imbalances between waste generation and respective utilization capacity in Ukraine presupposes four steps: (a) reconstructing existing utilization capacities for accumulated waste; (b) ensuring technical and technological support for current generated waste processing capacity; (c) partial treatment of the accumulated waste as well as complete disposal of the generated waste prior to the first being exhausted; and (d) replacing old recycling facilities with up-to-date waste recycling technologies.

Managerial Implications

Waste management system’s content is provided by its territorial balance, which is a condition of technogenic safety and a basis for the recycling technologies’ implementation. For balanced development, we suggest that the regional authorities use models of type (11) and (20) while solving empirical problems in the following instances:

Determine how long it will take to process the accumulated waste until it is exhausted, with the simultaneous processing of the generated waste.

Determine the required capacity if we need to recycle the accumulated waste over a given period of time and to provide current needs for recycling of the generated waste.

On the basis of the previous paragraph, determine the amount of investment to be allocated to utilization capacities in order to ensure the time parameters of physical recovery of previously accumulated and newly generated waste.

In this case, the condition y = x in our problem means the time (point B in Figure 2) when the accumulated waste will be physically disposed of and full control over the volume of waste generated will be obtained, and in parallel, the introduced innovative capacity will ensure the use of recycling technologies for waste recycling in Ukraine.

Limitations and Recommendations for Future Research

All studies inevitably have limitations. This study focused on the problems and contradictions in the waste management system of Ukraine without regard to global trends. However, in the longer perspective, we believe future studies could further look into analysing Ukraine as an element of the global ecological and economic system of the Earth and developing a common policy of the national and international institutions, including certain prohibitions as to waste management and focusing on the use of recycling technologies.

Footnotes

Acknowledgement

The authors are grateful to the anonymous referees of the journal for their extremely useful suggestions to improve the quality of the paper. Standard disclaimers apply.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Volodymyr Kolodiichuk

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