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

759

3

1 1 i   m n j      

   

n

t

n











jr   



2 2 z C y     

daily D y MQ D y MQ      tr

min,



(1)

ij

i

ij

i

jr

j

ij

i

1

1

1

i

r

i







provided that the production and consumption balance is fulfilled in each district

n

t

ij     

,

1 r y MQ z B  i jr

(2)

j

1

i



as well as restrictions on resources and reserves

        

t

1    1 r m i n  

,

z k L  

jr

j

j

max

,

(3)

y MQ q  

ij

i

i

2 2

2 k j   ,

1, 2, , , m

y



i 

ij

j

1

i



where ij y – the share of the reserves of the i -th district that are mined for pleasure annual needs of the j -th economic district; j L – restrictions on reserves in the j -th district; max i q – the maximum permissible (for technical reasons) annual volume of gas production at the fields of the i -th gas production district. In this case i MQ and 2 i  , they make up the matrix of statistics of the coefficients of stochastic nonlinear programming constraints. It is difficult to solve this problem in a nonlinear formulation, but after linearization of the components of the objective function and constraints, it can be represented as a stochastic linear programming problem, and after introducing mathematical expectations and variances (as written above), it can be considered as a deterministic linear programming problem. As a result of solving the given problem, the minimum of the objective function is obtained for some combinations of average values by stocks, taking into account the pre-set fluctuation range. However, in this formulation of the problem, exploration costs were taken into account as a constant component included in production costs. In fact, it is necessary to differentiate the reserves into those determined for this planning period and the increase in reserves as a result of the expansion of geological exploration. Taking into account the differentiation of reserves and considering the increase in the more distant perspective as probabilistic, it is possible to formulate the following problem of stochastic programming Wentzel (1999). It is necessary to find the value of the functional:

1      2 ( ) j C x ( ) i rez 1 tr k tr j daily i k j i k C q      

(1)

i k   

( ) k R C x 

( ) i C q C N E   ( ) (1 )

( ) i rez

inc

i

(4)

min,

(1 ) E 

1

i

under the following restrictions

 T Q q q x k q                 1 2 ik k ik j i x x q B ,

,

    

i

(5)

i

i

io

i

i

i

2