SITUATION FORECASTING AND DECISION-MAKING OPTIMIZATION BASED ON USING MARKOV FINITE CHAINS FOR AREAS WITH INDUSTRIAL POLLUTIONS

G.V. Getun, Ph D. in Technical Sciences, Professor, Professor at the Department of Architectural Structures Y.P. Butsenko, Ph D. in Physics&Mathematics, Associate Professor, Associate Professor of Mathematical Analysis and Probability Theory Department V.A. Labzhynsky, Ph.D., Associate Professor, Associate Professor of Energy Processes and Systems Design Automation Department O.I. Balina, Ph D. in Technical Sciences, Associate Professor, Associate Professor of Design Information Technologies and Applied Mathematics Department I.S. Bezklubenko, Ph D., in Technical Sciences, Associate Professor, Associate Professor of the Design Information Technologies and Applied Mathematics Department

Introduction. The complexity of modern technical systems and the increase of man-made risks arising from environmental pollution requires not only the increased reliability of such systems, which can be achieved, in particular, by the use of advanced monitoring and diagnostics, but also by the availability of effective means to predict probable accidents that can occur in the operation of these systems as well as the availability of timely and optimal measures of response to the emergencies. The purpose of establishing the permanent system to monitor the faults occurrence lies in reducing the possibility of sudden unexpected events, undermining the economy of the industrial enterprise, dangerous production stops, damage to equipment and accidents to personnel, as well as to facilitate technical maintenance of equipment. Critical levels of environmental pollution should also be minimized. Although the pursuit of greater reliability and less cost may seem incompatible at first, a closer examination of this issue shows that this is not the case [1].
In Ukraine, a national methodology has been approved to predict the effects of chemical pollution on the environment [2]. In addition, in the areas where environmentally hazardous enterprises are located, specific engineering within nature complex systems are formed, which are characterized by certain trends of environmental changes, which sometimes lead to negative ecological and economic consequences. The material costs of restoring the natural equilibrium within such territories are usually extremely high. Therefore, the problem of creating adequate modeling techniques and forecasting the functioning of industrial enterprises to prevent accidents on them is a matter of first and foremost importance.
The review of existing approaches to forecasting the man-made (anthropogenic) risks. Predicting the state (condition) of the environment under the influence of dangerous man-made objects is becoming increasingly important when solving environmental problems associated with finding optimal forms of environmental safety management [3]. The most characteristic of the following tasks are the following: • environmental monitoring; • nature exploitation rationing; • industrial sites environmental impact assessment. One of the traditional approaches to predicting anthropogenic impact on the environment is the use of mathematical models that describe the processes and phenomena, which characteristic for the studied natural object [4]. The following methods are most often used in predicting the state of the environment: • dynamic systems; • time series (regressions); • Markov models. In mathematical modeling, let a natural object (water object, soil, stand, atmospheric air, etc.) be considered as a dynamic system containing n components. In this case, the mathematical model of a natural object usually takes the form of a system of differential equations [5]. 1 1 ( ,..., , ,..., , ), is the vector that characterizes the state of the natural object; -vector of external factors that affect the state of a natural object; t -time. The solution of the differential equation system is the functional that allows predicting the state of a natural object. In the absence of an adequate deterministic dynamic model, statistical forecasting methods are used. Among the statistical methods, the most common method for solving the environmental forecasting problem is regression analysis [6].
Suppose that an observation i y is the sum of a regular deterministic component and random interferences: Another statistical approach for modeling the behavior of these systems is based on stochastic modeling. Stochastic modeling does not use rigorous ratios, but expert and empirical evaluations and a universal mathematical apparatus. Stochastic modeling based on Markov finite-chain theory [7] has been successfully applied in various industries [8].
Suppose that the evolution of an ecosystem is described by the Markov chain. The transitions of the system from one state to another mean the moving a point that depicts the current state of the system from one set of phase space to another, and the corresponding system of phase space sets , is built on the basis of environmental standards. The transition probabilities matrix is the same vector at time Such a description allows you to solve at least three of the following tasks: 1. To determine system transition probabilities ( ) n ij P from the state i A to the state j A in n steps.
2. To find the vector of probabilities ( ) ( ) n P B of the system being in all possible states of the set B in n steps, if the state of the system is known at the initial moment. 3. For the specific states of the system, to determine the probabilities of getting into them in no more than n steps and stationary probabilities, which allow to determine the measure (portion) of time that the system is in these states.
The main disadvantage of existing forecasting methods is the inability to estimate the average residence time of a system in one or another set of states and the lack of attention to economic effect of its evolution.
Formulation of the task to model the behavior of engineering-within-nature complex systems. The models based on Markov finite chains have the following characteristics: • simplicity of the content; • actual environmental standards and regulations are naturally taken into account, since the phase space is built on the basis of current environmental legislation; • the possibility of reducing the set of estimated parameters to the elements of the transition matrix.
It should also be noted that the use of such an interpretation of the system evolutions eliminates the need to determine the distributions of random variables and processes that determine the state of the system.
The practical application of such models, due to the existing advanced theory of Markov chains allows us to use the following criteria of optimality: • to minimize the probabilities of system states, which are extraordinary situations, in the steady state distribution of the respective chain; • to maximize the average time of reaching the respective state; • to minimize the damage caused by the system being in "ecologically disadvantaged" states; • to maximize the economic impact of the system's operation, taking into account both profits from industrial sites, positive social shifts as well as losses related to environmental damages.
The purpose of this work is to develop a methodology for predicting the occurrence of ecological threats, to study the distribution of time spent by the engineering-within-nature complex system in safe and in unfavorable (unsafe) states, ecological and economic analysis of the consequences of its evolution.
Methods of forecasting and optimizing the economic effect on a discrete set of strategies. Let the system is responsible to control the territories, which represent an amalgamation of zones (regions) that will in future be considered as non-intersecting. The ecosystem phase space  is a direct product of l  , where l  is the set of all possible ordered sets of concentrations of harmful substances in the air and water environments of the l -th region of controlled area. The model of each of these zones is a corresponding Markov chain [9]. The natural modification (version) of the Markov property for the situation under consideration is the following: that is, the probability of finding the j -th zone in the k -th state k A is locate. Accordingly, statistical studies of the transition probabilities [10] for each zone should include a study of their dependencies on the "configuration" of the environmental situation in the adjacent zones. Note that in some cases there is even a deterministic dependence between the states of adjacent zones with a certain time lag. This is the case, for example, for the condition of air basins in the case of steady air currents, for water basins of zones settled sequentially downstream. Thus, the happening of an emergency even in one region (or vice versa, the normalization of the ecological situation in it) requires a consistent recalculation of transition probability matrices for the entire controlled area. It should be noted that such calculations often lead to controversial results (we obtain different transition matrices for the same region). To eliminate these contradictions, it is suggested that: • to create, on the basis of statistical studies, a bank of scenarios that may occur in each zone; • calculate time lags for interregional effects for each scenario; • operational management shall be carried out by using standard scenarios with simultaneous control of their adequacy to the real situation.
It should be noted, that the forms of interdependence between transition matrices for adjacent zones can be calculated in two ways: • based on available statistics; • on the basis of a correspondence model [11], based on the available information about air currents, dynamics of water reservoirs, etc.
This requires: • the previous choice of the most likely way of forming such a dependency related to the accumulated information; • statistical control of incoming information in terms of selecting the most likely hypothesis according to its available dynamics.
The first question to be solved when using this model is the question of discreteness or continuity of time (in the evolution of a chain). The choice of Markov chains with discrete time is explained by not only available practical experience of using them [9], but also to the objectively available periodicity of information inflow into the control system from its primary links.
For discrete-time chains, the problem of homogeneity arises. Homogeneous chains are much easier to investigate, but applying them requires a solid statistical justification, which is not always possible. At the same time, it is natural to use several homogeneous models adapted to the operating modes of the enterprises and the seasons. Thus, one task of considering inhomogeneous chains can be replaced by several tasks of analyzing the corresponding homogeneous chains and developing an algorithm of transition from one of them to another depending on the current state.
The next question is the choice of the phase space structure (set of states) of the chain (Fig. 1). Its solution is based on a legislative base (framework) that determines the levels of environmental pollution and the need to consider such levels for each of the regions that form the controlled system. At the same time, phase space, constructed as a direct product of "local" phase spaces, which in turn duplicate the levels of pollution in the respective zones, is, on the one hand, too cumbersome (in the presence of m regulated levels of pollution and n zones it contains n m states) and, at the same time, is not always adequate from the point of view of the system description, since it does not contain clearly identified states of "threatening". From this point of view, it is advisable to isolate "transient" states in some zones, which would mean the approaching to the critical levels of pollution, or vice versa -the tendency to decrease them. At the same time, based on economic (potential loss) and other considerations, the primary phase space containing ( ) n m  ( m  -the number of levels of contamination, taking into account the abovementioned additional) phase space should be enlarged (i.e. to combine several states into one).  The opportunity of decomposing the phase space into classes of achievability, that is constructing the states graph of the chain (Fig. 3), requires in-depth analysis. By providing a high probability of defining the "starting" state of the system, it is possible to limit yourself to a chain that is nondecomposable.
Formal (mathematical) distribution of absorbing states of the system should be consistent with their character as states of ecological catastrophe resulting from the limited resources of natural environmental purification. For such states, they determine the average time to achieve them and the probability of achieving them no earlier than a fixed time. If in this case for a random time  pre-set values, the system is considered as highly-insecure and needs structural changes (Fig. 4). For each of the other states of the chain, along with its probabilistic characteristics, environmental and economic parameters are also considered: achievable economic effect, permissibility of the corresponding environmental pollution, possibility of exploitation of natural resources and technological objects. This allows not only to set the "rating" of the states of the system from the most desirable to the extremely undesirable, but also to set optimization tasks for it, based on the possibilities of choosing the initial state and influencing the matrix of transient probabilities [10]. Given the real characteristics of the system under consideration, it is proposed to use Markov periodic chains to model its behavior [12], which gives an additional opportunity to characterize and detect emergencies as a "deviation from periodicity" in the behavior of the system. transient probabilities, both of which can be combined in series. It should be noted that changing the transition probability matrix can require significant investment (e.g., improved reliability of industrial installations, reduced emissions levels, etc.), which should be taken into account when making appropriate management decisions.
Conclusions. The article proposes a technique for forecasting emergencies that may occur when a nature & engineering complex system operates. The use of the methodology presented above will allow to increase the efficiency of functioning of enterprises, generate the balanced informed management decisions and to create software and technologies to respond the emergencies.

SITUATION FORECASTING AND DECISION-MAKING OPTIMIZATION BASED ON USING MARKOV FINITE CHAINS FOR AREAS WITH INDUSTRIAL POLLUTIONS
The paper considers the issues of predicting the situations and optimizing decision-making to improve the environmental situations in the areas with industrial pollution based on the finite Markov's chains.
The article systematizes the existing approaches to forecasting technological risks. The problems associated with the search for optimal forms of environmental safety management and approaches for predicting anthropogenic impact on the environment using mathematical models are considered. To predict the state of the environment, stochastic modeling is proposed, the basis of which is the theory of finite Markov chains. A technique for predicting and optimizing the economic effect on a discrete set of strategies has been developed. The figures show: building system states graph, determining the basic characteristics of system states, finding transition probabilities of Markov chains for non-critical states, a typical cycle of checking the model's adequacy and system quality.
Based on the analysis of existing approaches to forecasting technological risks, a methodology has been developed for forecasting and optimizing the economic effect on a discrete set of strategies. The proposed methodology allows combining economic estimates with the ability to predict the situations and optimize decision-making to improve the environmental situation in the areas of possible chemical pollution.
Using the developed methodology will increase the efficiency of the industrial enterprises, facilitate generating informed management decisions, create software and hardware ways to respond the emergencies.
The methodology for modeling engineering within nature complex systems and the optimization of decision-making based on finite Markov chains in the areas with industrial pollution will be helpful to researchers and operators of complex technical systems in predicting emergencies using environmental monitoring systems.
Keywords: complex technical systems, engineering-within-nature complex systems, nature & engineering complex systems, environmental monitoring, emergency forecasting, statistical research, decision support. The paper proposes a methodology for modeling engineering-within-nature complex systems (further, "systems"), which will be helpful for researchers and operators of complex technical systems in predicting the emergencies using environmental monitoring systems.