^{1}

^{2}

^{3}

^{*}

Matrix method is being proposed for qualitative evaluation of the reliability of technical systems on a finite set of structural elements. We are introducing the criteria for qualitative assessment of the reliability in the form of structural reliability of the system as the probability of the troubleproof state of this system and the significancy of the individual elements in ensuring the structural reliability of the system as a general aggregate of conditional probabilities, which compose two (2 × 2) matrices of significancy for each element. We are using chain diagrams for solving the combinatronic problems and matrices for algorithmization of calculating procedures.

The reliability of the technical system is one of the main indicators of quality and is taken into consideration in the early stages of design, which addresses different functional, assembling and construction schemes [

The method of function of the technical system can be implemented with various structures. In general, these structures determine the reliability of the designed technical system. The ability to assess the structural reliability of the individual blocks and the technical system as a whole, as well as to assess the significance of the individual blocks in ensuring the reliability of the system is the relevant problem in conceptual design, which can be solved using mathematical methods, including combinatorial, probability theory, Boolean algebra, matrix calculations [

The matrix method, which will be described further, allows both qualitative and quantitative reliability study including structural reliability as well as the importance of the individual elements in this reliability. The method is adapted to the use modern computer technologies to solve complex combinatorial problems in assessing the reliability of technical systems on an arbitrary, finite set of structural elements [

It is assumed that the structure of a technical system is given as a finite set of elements and their relationships. It is required to find the value of structural reliability and evaluate the importance of each element in ensuring the reliability of the overall system.

Structural reliability is calculated using the classical formula of calculating the probability

where N is the number of possible states of the system

The number of possible states of the system is calculated using the following formula:

where n is the number of elements in the system; S is the number of states of each element.

We assume that each element of the system can be in one of two states: operating (working) or failure, i.e.

In the case where the elements of the system have the ability to be in one of three states: working, neutral and failure

The determination of a finite set of possible states of the system

The state diagram of the system can be represented by a rectangular matrix of states, which has the dimension

State of the system in general is also a random event A, which has two possible values that are determined using the matrix of state depending on the structural scheme being considered and is presented by a row matrix

The states of the system are determined by a random state

In general the assessment of different separate elements in ensuring the structural reliability of the system is done using matrices of significance that have the following form

The conditional probabilities that compose these matrices can be calculated using formulas:

These conditional probabilities characterize to which extent operation or failure of a single element is reflected on working or failure state of the whole system, i.e. on structural reliability or unreliability of the system, as well as how important this element is to ensure working state of the system as a whole. Note that considered here conditional probabilities satisfy the following conditions:

which can be used for verification immediately by definition we get:

and also

We are considering products (intersections) of random events of the following types:

For each

From the created matrices

Basic theoretical principles of the described method are shown in [

The algorithm will be illustrated as an example on one of the finite set of structural schemes of the penta-system. Elements of the system can be connected by either series or parallel principle. Chosen structural scheme is filled with five inhomogeneous, independently working elements as shown in

Determining the number of possible states of the penta-system: 2^{5}, i.e.

?working, specified as

?failed, specified as

Chain Diagram of states of the penta-system is built (

Using chain diagram (states graph) each of the 32 possible states of the penta-system is found

Rectangular state matrix

Then the row-matrix

Here 1 corresponds to the working state of the penta-system, while 0 to failed state respectively.

Structural reliability of the specified penta-system is determined:

Here the working state of the penta-system as a whole and the corresponding possible state are considered as random events, i.e.

Matrixes of significance of each element of the penta-system

i.e.

Then, obviously, probability of a random event

1) The calculation matrix for the second element

From which one can get probabilities of multiplication of needed random events, i.e.:

Then matrixes of significances for the first element will take the following form:

2) The calculation matrix for the second element

Then:

Consequently:

i.e. elements 1 and 2 have the same significance in ensuring the reliability of the considered penta-system.

3) The calculation matrix for the second element

Then:

Consequently:

i.e. significance of the third element is lower than the significances of elements 1 and 2.

4) The calculation matrix for the second element (i = 4) is constructed. The calculation matrix then has the following form:

Then:

Consequently:

i.e. elements 3 and 4 have the same significance in ensuring the reliability of the considered penta-system.

5) The calculation matrix for the second element

Then:

Consequently:

i.e. the significance of the fifth element in ensuring the reliability of the penta-system is the highest.

The elements of the system

1)

2)

3)

Research on the reliability of technical systems on a finite set of structural elements is proposed to carry out on the basis of the developed matrix method that also allows to effectively applying modern computer technologies for solving multidimensional combinatorial problems. Algorithms for the qualitative assessment of the reliability of technical systems, structural reliability, the significance of the reliability of individual elements using chain diagrams of states, State Matrices, Matrices of Significance were developed.

VictorKravets,VladimirKravets,OlexiyBurov, (2015) Matrix Method for Determining Structural Reliability of the System and Significance of Its Elements in Terms of Reliability. Open Journal of Applied Sciences,05,669-677. doi: 10.4236/ojapps.2015.511066