Introduction. When administering complex multi-parameter systems, management decisions are often made under uncertainty. There is an acute problem of reduction of the likelihood of unwanted events and mitigation of possible damage. The efficiency of predicting damage to complex systems depends directly on the quality of processing methods, systematization, and the amount of input data. It is required to improve methods for assessing and predicting damage and to develop new approaches and criteria for statistical forecasting of damage and evaluating the system reliability. The solution to such problems is complicated by a large number of indicators, data uncertainty, short series of observations, incomplete initial information, insufficiently developed scientific methodological apparatus. Existing methods for predicting damage in the systems of potentially dangerous objects do not take into account the causes of accidents that happened due to unfavorable circumstances. As a consequence, management decisions are made upon unreliable forecasting results. In this regard, an urgent scientific task is the development of methods and techniques for the formation of viable management decisions, free from this shortcoming. The major study objective is to consider a particular problem for predicting damage due to unfavorable circumstances associated with the indistinguishability of the initial data. The tasks are to consider this kind of uncertainty which includes indistinguishability of the true system condition and the real value of its quantitative characteristics; to formulate a combinatorial problem for the case when a rather dangerous composite feature is determined by the joint manifestation of two or more simple features. Materials and Methods. Under the conditions of multiple indistinguishability, the following was used as the source data: a set of indistinguishable outcomes with reliable information on the event instance and the uncertainty of assigning the event to a certain type; a family of sets having the same number of elements. The Cartesian product of the families of the corresponding sets and the actual value of the group of a compound potentially dangerous factor with a compound rather dangerous feature are taken into account. The resulting mono-element fuzzy group is presented, which is also a possible event resulting from the intersection of two necessary events. Results. It is established that the problem of predicting damage due to unfavorable circumstances corresponds to a combinatorial-type problem, which consists in enumerating all sets of arguments. The resulting range, which is an elemental group of indistinguishability, characterizes the smaller and larger possible values of the size of the group of a potentially dangerous factor with a composite rather dangerous feature. It is shown that the formulated combinatorial problems without significant changes are applicable to problems in a generalized form, when composite rather dangerous features are determined using not only the operation of intersection, but also uniting and difference; thereby, the initial groups are not necessarily the objects with simple features. Discussion and Conclusions. The results obtained are focused on the construction of analytical algorithms for establishing indistinguishability under the monitoring, modeling, forecasting state-related processes and complex dynamic multiparameter objects.

In situations where it is not possible to timely use new methods for assessing hazardous objects, traditional methods of risk assessment can be supplemented with new methods based on possible assessments of objective primary information. Evaluation of the destructive capacity and consequences of the use of high-risk systems becomes increasingly important as their structural and functional complexity grows. Despite the existence of the considered works on certain aspects, the known approaches to the consideration of the problem of indistinguishability of influencing factors are characterized by limited functionality. These capabilities do not fully provide the required level of reliability when making management decisions. In view of this, the problem of predicting equipment failure caused by an unfavourable coincidence of circumstances due to the indistinguishability of the initial data is one of the urgent tasks of risk management. Its solution is of significant theoretical and practical interest for many complex heterogeneous dynamical systems. The paper deals with a problem from the field of probabilistic modeling. The relevance of the performed research lies in the fact that the existing methods for predicting equipment failure in systems of potentially hazardous objects do not take into account the causes of accidents that occurred due to unfavourable circumstances. The aim of the work is to develop a method for the set-theoretic modeling of the occurrence of problems with equipment due to an unfavourable coincidence of circumstances, which provides a range of possible values for the size of a group with a given composite very dangerous sign. To achieve this goal, the following tasks are solved: – finding the range of possible values of the number of groups of objects modeled by the intersection of indistinguishable sets and finding the range of possible values of the number of groups of objects modeled by the union and the difference of indistinguishable sets; – finding possible values of the number of groups of objects modeled by an arbitrary composite property and generalization of the problem, features of the set-theoretic apparatus of its solution. In the course of the study, it was found that a set of objects with a compound very dangerous feature of an arbitrary type can be associated with a set representing the union of intersections (intersection of unions) of sets of objects with simple features and their complements. Moreover, the operations of intersection, union, and complement constitute a complete set of operations for the most common version of Boolean algebra of sets.