Technical science/4.Transport
c.t.s. Kabikenov S. Zh.
Kyzylbaeva E. Zh.
Kukesheva A.B.
Karaganda State Technical University, Kazakhstan
Mathematical model of management of reserves of spare parts on mining
enterprises
Calculation of the need for spare parts (units,
aggregates) by the statistical modeling method will be considered on the
example of the work of a warehouse, from which at an arbitrary time t, are
issued a random number of aggregates (units, spare parts) of one name Vt. If at
the moment t the stock X in the warehouse is sufficient, then the Vt request is satisfied completely. If the stock is
insufficient to fully satisfy the requirement, then it is satisfied only by the
amount of the stock in the warehouse. In the latter case, the enterprise bears
losses due to a deficit, the value of which is proportional to the number of
units not supplied to the technical service of aggregates, i.
e. Сdef = к(Vt – X), where к – s the loss of the enterprise due
to the idle time of one car with a deficit of units in the warehouse.
The supply of aggregates to the enterprise warehouse
from suppliers also occurs at random instants of time τ and in a
random volume Yt. Expenses for the maintenance and
storage of aggregates in the warehouse Cstor = λХ, where λ  cost of
storage and maintenance of one unit in the warehouse for the period T; X 
average stock in the warehouse for the period T.
It is necessary to find such a planned level of the
initial stock of X0 units in the warehouse, at which the total costs of the
enterprise will be minimal: (Cdef + Cstor) → min (Figure 1).
Thus, here there are four random variables: the moment
of receipt of the demand for the issue of aggregates from warehouse t; the
volume of this requirement Vt; the moment of aggregates
arrival to the warehouse from suppliers τ and the volume of this supply Yτ.
Figure 1  The dependence of the enterprises costs on the initial stock
The laws of distribution of these random variables are
established on the basis of processing information contained in the warehouse
accounting records.
To solve the problem further, we introduce the
quantities:
Vi+1 = ti+1 – ti – the duration of the interval between (i + 1)st и i issue of aggregates from a
warehouse;
μi+1 = τi+1 – τi – the
duration of the interval between (i +
1) st и i supply of aggregates from a warehouse.
Since τi and ti
are random quantities, therefore, the quantities νi + 1 and μi + 1 are
also random.
To solve this problem, the initial data are needed,
which can serve as various values of the planned level of the initial stock of Х_{о} units.
Calculations on a computer (Figure 2) with different
initial data for the entire plan period T (year) allow to simulate the actual
processes occurring at the enterprise, for which the values of random variables
are established with the help of a random number generator, distribution law of
which should correspond to the distribution law of random variables on the
given enterprise.
As a result of calculations for all selected levels of
initial stock Хо, we reveal
the dependence of total costs for the whole period on storage of the aggregate
stock and because of a deficit in case of failure to meet the requirements due
to the lack of aggregates in the warehouse.
Modeling of
the needs of the enterprise in spare parts 








Modeling of
the needs of the enterprise in spare parts 


Modelling period days
P= 
30 


Conditions for
modeling the consumption of parts 

Conditions for replenishment
modeling 



Number of
requests for spare parts N1= 
8 
How many times
to restock N4= 
7 

Min number of
requested spare parts N2= 
5 
Min quantity
of spare parts in consignment N5= 
6 

Max number of
requested parts (N3˃N2) N3= 

10 
Max quantity
of spare parts in consignment (N6˃N5) N6= 
12 

Resulrs of modeling 




Initial stock 
Losses due to deficits 
Storage costs 
Total losses 
Endofperiod stock 

Initial stock
in modeling X0= 
3 
3 
22456 
3,200 
22459,200 
59 

Step of
changing the initial stock D= 
5 
8 
17936 
8,533 
17944,500 
54 

Number of steps
for modeling X2= 
10 
13 
14040 
17,067 
14057,100 
49 

Loss of a
trucking company from idle vehicles K1= 
8 
18 
10560 
27,733 
10587,700 
44 

Spare parts
storage costs L1= 


8 
23 
7608 
40,000 
7648,000 
39 


28 
5328 
54,400 
5382,400 
34 


33 
3528 
70,400 
3598,400 
29 


38 
2448 
94,400 
2542,400 
24 

CALCULATE 











Figure 2 The program for modeling the needs of the enterprise in spare
parts
References
1. Bykov A. Optimizacija zapasov na osnove
imitacionnogo modelirovanija
[Optimization of reserves based on simulation modeling] LogistikaLogistics,
2004, no. 1.
2. Anilovich V.Ja. K raschjotu zapasnyh
chastej. Traktory i s/hozjajstvennye mashiny [To the calculation of spare parts. Tractors
and agricultural machinery] 1975, no. 1.
3. Berg A.I. Osnovnye voprosy teorii i praktiki nadjozhnosti
[Main issues of reliability theory and practice] Moscow, Sovetskoe
radio Publ., 1975. 408 p.