PSI - Issue 59

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

760

4

V. Grudz et al. / Structural Integrity Procedia 00 (2019) 000 – 000

or

1

x

T

T

        

;

1  

,

q

Q

q

i 

 







i

i

i

io

2

2

k

k  

k

k

i 

i 

i 







i

i

i

i

;   ; x x x x   i str i i q q 

,

k q k q   

(6)

i

i

str

str

0; 0   

; x x q

0,



k

k

k

j

i

i

i

1

1

1

2

and taking into account the probability distribution function of stock growth, obtained after approximating the initial data, we have

(7)

2

,

N

p x p x         x p

0

1

2

3

4

5

i

i

i

i

i

i

i

where ( ) i daily C , ( ) i rez C – specific reduced costs for transport, production and creation of reserves, respectively; ( ) k tr C – specific operating costs for the existing system of gas pipelines; (1) (1 ) i k E  – assessment of the "freezing" of capital investments in geological exploration, assuming that the available reserves i Q will be used until the moment (1) i k ; i k – the multiplicity of stocks; N – the number of geological structures of the i -th production area, necessary to determine the probability of an increase in reserves not lower than i x ; i p – the probability of an increase in reserves is not lower 1 k x ; 0 1 5 , , ...,    – approximation coefficients. In the considered problem of mathematical programming, the growth of reserves, and accordingly, the costs of geological exploration are probable values. To solve this problem, the following procedure of sequential actions is proposed. The problem is solved with average increases in reserves in individual areas, obtained from the found distribution law (7), and average specific costs for geological exploration. At the same time, the average value of the growth of reserves is set as the upper limit i x . Thus, the solution of this problem can be represented in the form of a vector   0 1 2 , , , , x x x x q k  of zero approximation. At the second stage, given that the multiplicity and reserves are determined, as well as the configuration of the gas pipeline network is specified, the optimization criterion can be presented as follows: ( ) j tr C ,

2      ( ) j C x  ( ) i C q ( ) i C N  ( ) tr ki tr j daily i tr i          ( ) k R C x  inc

( ) i C q 

,

(8)

rez io

k

j

i

i

(1) (1 ) i k E

at fixed value of (1)

i k , with an average i k .

where

  

From constraints (5), (6) for i x we find that

T

ik k   x      ij 

 

x Q q x     

.

x k

B

i i    

(9)



2

i

i

j

i

io

i

2

Since i x is a random variable, and the problem represented by expressions (5), (6), (9) corresponds to a stochastic problem with random i B . Thus, we obtain a two-stage stochastic programming problem, which can be written as follows     2 ( ) ( ) lim lim min min ( ) ( ) , ij ij inr i i ij шо R C M y x C M y x               (10)