PSI - Issue 59

V. Grudz et al. / Procedia Structural Integrity 59 (2024) 757–762 V. Grudz et al. / Structural Integrity Procedia 00 (2019) 000 – 000

761

5

with restrictions

    

T

  

  



( ) Iy x Iy x x k    ( )

0 ( ) x Q q    i

,

 

   

i

i

i

i

i

io

2

( ) y x y x  

( ) 0;

( ) 0; 

( ) 0, 

y x 

i y x T 

 

i

i

i

       

  



( ) y x x k 

( ) x Q q x x T x Q q x x x              ; 2 ( ) ; , i i io i

,

(11)

   

i

i

i

i

( ) y x k 

 

  

i

i i

i

io

i

i

i

2

1    ik k ij x    

( )

,

x B

x

 



2

i

j

k

i

where I – the unit matrix. For discrete values of the probability distribution law (7), the mathematical expectation of criterion (10) can be written as follows:

      

* 1   k l l  l l l  

1

T

     

  

  

  

    ( ) ( ) i i  

*     k

( ) x Q q   

,

M y x

p l   

x l

 

i

i

i

i

i

io

2

k

(12)

i

1

T

  

*

( ) x Q q x l     

.

M y x

p l   

k

 

 

i

i

i

i

io

i

2

k

1

l



i

The value ( ) i y x  means that in some cases the increase in reserves may be insufficient to cover the given needs ( ) i y x  that there may be surpluses. In order to better balance the need and opportunity in the increase of reserves, two additional terms are introduced in the objective function (10). The first term gives an estimate of the average loss in case of a shortage of stocks, the second – in case of their surplus. The specific loss rate can be determined similarly rez C . As for the penalty for excesses, given the slight deviation of the stocks from the average value and the lower relevance of this case, the last term (10) can be excluded. Problem (9), (10) can be implemented by various mathematical methods, but its implementation is very time consuming. One of the solution methods is the method of generalized gradients by Zapukhliak et al. (2019) and by Grudz et al. (2018). Having obtained the solution 0 x at the first stage, it is accepted as a zero approximation when using the following recurrence relation: of the country, and

1 x x s

s     

( ), s x s    s s x

(13)

0,1, ,



s  – size of the gradient descent step;

where

s  – normalizing factor; ( ) s x x  – the gradient of the objective function (10) from the point s x . 3. Research results and their discussion

Using the iterative procedure according to relation (13), the solutions of problem (10), (12) are found, which provide the minimum mathematical expectation of costs, taking into account the loss from a possible shortage of gas resources with a probabilistic method of their determination. Later, based on the obtained results and using them as initial data, the technical and technical and economic parameters of the development of the gas supply system of each of the regions are calculated.