OPTIMAL STRUCTURAL RESERVATION OF TECHNICAL SYSTEMS

Dep. «Applied Mathematics and Information Technologies», Pridneprovsk State Academy of Civil Engineering and Architecture, Chernishevskiy St., 24-A, Dnipro, Ukraine, 49600, tel. +38 (067) 639 60 64, e-mail Ssemenets28@gmail.com, ORCID 0000-0002-6359-1069 Dep. «Higher Mathematics», Ukrainian State Chemical-Technological University, Gagarin Av., 8, Dnipro, Ukraine, 49005, tel. +38 (097) 940 98 56, e-mail ms.nasonova.s@gmail.com, ORCID 0000-0002-0920-7417 Dep. «Higher Mathematics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (096) 879 48 19, e-mail semga1952@gmail.com, ORCID 0000-0003-2693-3282


Introduction
Inadequate design reliability of technical systems leads to a significant increase in the share of operating costs in total expenditures for their design, fabrication and application.At the same time, the cost of operation can many times exceed the cost of designing and manufacturing the system [1,3,10].Therefore, the reliability of technical systems should be ensured, first of all, at the design stage [2,12,13].
One of the main methods for ensuring the reliability of the technical systems being designed is the method of structural redundancy [6][7][8], which in-volves the use of redundant elements in the system.The essence of structural redundancy lies in the fact that in addition to the main (reserved) object containing the minimum required number of elements for the normal performance of the system's functions, in the system structure the additional identical (in terms of performed operational functions and reliability) elements are introduced.These redundant elements are designed to perform the work functions of the main elements.Thus, a system with redundancy is a system containing redundant structural components with respect to being reserved object that perform the same work functions as the corresponding components of the main object.The redundant system remains operable after subsequent failure of any element if the number of operable elements does not become less than the minimum required number provided for by regulatory requirements for the main object.
The problem of rational choice of one of the several variants of the system composition always arises during the structural redundancy.On the one hand, to increase reliability it is desirable to provide each of the system elements with maximum possible number of redundant elements, and, on the other hand, it is impossible to design a system with too large cost, weight or dimensions.Therefore, the actual task is how to reserve the system in such a way as to provide the required level of system reliability at acceptable costs.The choice of the cost characteristic is determined by the type of the system and its purpose.For example, for aircrafts the essential factor is weight, and for the ground systemthe cost.Regardless of the physical essence, the selected cost characteristic for brevity below will be referred to as cost.Usually, the problems of optimal structural redundancy of technical systems are formulated as non-linear programming problems, and to solve them, depending on the complexity and required accuracy, the special algorithms are used.They are based primarily on the dynamic programming method [6].In this article, the initial optimization model is represented in terms of the problem of non-linear integer programming with binary variables, which greatly simplifies its numerical implementation.

Purpose
The purpose of the article is to give the designers of highly reliable technical systems that do not know modern optimization methods and programming skills, a simple mathematical tool for choosing the optimal structure of a redundant system.

Methodology
During problem definition of the optimal structural redundancy, the main object is considered as a system consisting of n of various elements that have a coherent connection [4] (see Fig. 1).As the main indicator of reliability of the redundant system, the probability of its fail-safe operation is assumed.It is believed that the failure flow of elements put into operation is described by the Poisson arrival [5].Possible variants of the main object redundancy are limited to the consideration of typical schemes of separate «cold» backup with an integer multiplicity with an ideal switch (see Fig. 2) and «hot» backup with parallel connection of elements (see Fig. 3) [7].Note that, taking into account the assumptions made, the probability of failure-free operation of the redundant group of elements, the scheme of structural reliability of which is shown in Fig. 2, is determined by the Erlangian formula [5,9] and the formula for calculating the probability of failure-free operation of a redundant group of elements, the structural scheme of reliability of which is shown in Fig. 3, has the form 1 ( ) 1 ( 1) where is the failure rate of the element put into operation; r -is the redundancy rate.
Let us introduce the following designations: k is the failure rate of the element of the k -th type put into operation; mis the maximum allowable redundancy rate of main elements; k cis the cost of one element of the k -th type; ki xis the binary variable, equal to 1, if the number of redundant elements of the k -th is equal to i , and 0 ki x  , f the number of redundant elements of the k -th type is not equal to i ; Let us consider a redundant group of elements, consisting of the elements of the k -th type.Taking into account the introduced designations, the probability of failure-free operation of a redundant group at «cold» backup is estimated by formula 00 and when using the «hot» backup scheme, this probability is represented as follows With neglect of the cost of switching equipment in case of «cold» backup, the cost of the redundant group consisting of the elements of the k -th type in both backup schemes can be found using formula 0 ( ), 1, 2,.., Taking into account the coherent connection of the main elements and the product rule, the probability of failure-free operation and the cost of the entire redundant system are determined as follows: ) where X -is the matrix describing composition of the elements of the redundant system  4) depending on the backup scheme used.
In terms of the mathematical programming problem, the following two typical statements of the task of optimal structural redundancy are possible.
The problem 1: it is required to find the composition of redundant elements (matrix elements X ) that provides the required level of system reliability for a given time at the lowest possible cost The problem 2: it is required to find the composition of redundant elements, which provides the highest possible level of system reliability for a given time T , with a restriction on its cost * 0 ( , ) max The optimization models ( 8) and ( 9) have dimensions nm  and belong to the class of prob- The optimal values of the unknowns ki x in cases of «cold» and «hot» backup are found in Tables, respectively, 2 and 3, and the corresponding structural diagrams of the system reliability are shown in Figures 4 and 5.The calculated reliability estimates and the cost of the optimal system for «cold» and «hot» backup are listed in Table 4.The reliability parameters and the cost of optimal system

Findings
The calculated models for estimating reliability, as well as the optimization models for the redundant systems according to the «hot» and «cold» separate backup schemes, have been developed.The optimal options for reserving the 7 element object at separate «cold» and «hot» backup using the Excel spreadsheet were found.

Originality and practical value
New calculation models for estimating the reliability of redundant systems are proposed, as well as the optimization models developed on their basis, which are formulated using the unknown decomposition of initial problem of structural redundancy into binary components.The obtained optimization models belong to the class of problems of non-linear mathematical programming with binary variables, for the numerical solution of which (even for a sufficiently large dimensions) the well-known packages of applied computer programs, in particular, the MS Excel spreadsheet, are well adapted.Thus, the process of solving the initially very complicated problem of optimal structural redundancy is greatly simplified and reduced to performing elementary actions in the corresponding programming interfaces.The proposed calculation models for estimating the reliability of redundant systems, the models of optimal structural redundancy, and the methodology for their formation in order to simplify further numerical implementation can be useful in solving the problems of ensuring the reliability of technical systems at the early stages of their design.

Conclusions
The results obtained in the article using the specific examples show the efficiency and sufficient generality of the considered approach to solving the problems of optimal structural redundancy.The optimization models ( 8) and ( 9) belong to the class of non-linear programming problems with binary variables and are easily implemented numerically in the operating environment of the Excel spreadsheet.The obtained calculation models for estimating the reliability of redundant systems (3) -( 6), the models for optimal structural redundancy ( 8) and ( 9), and the methodology for their formation can be used in practical problems of ensuring the reliability of technical systems at the early stages of their design.

Fig. 1 .
Fig. 1.Structural diagram of reliability of the main object

Fig. 2 .Fig. 3 .n
Fig. 2. Structural scheme of reliability at «cold» backup the probability of failsafe operation of the main element of the k -th type, over the time t