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Engineering
In this paper we propose economicmathematical model of labor potential optimal distribution of the region by economic sectors, which has been tested on statistical data for the Stavropol Territory. Research on the formation and use of labor potential of the region can use various methods: economic, social, mathematical, etc., that allow to evaluate the effects of the labor potential for economic development and to justify the ways of managing them [3, 4].
In the region there is allocated n sectors of the economy, which we denote conditional № 1, № 2, ..., № n. The region's economy may be in the m states: 1st, ..., me. We denote qij as income, which brings one employee to the ith economic sector of the region, if the economy will be in the jth state, i = 1, ..., n; j = 1, ..., m. Matrix {qij} is called effects matrix (in accordance with the common terminology). In addition, we denote by xi  the number of employees at the ith sector, p  the minimum income per employee of the region. Then the value of income, that one employee of the ith sector will bring to the region, if its economy is in the jth state, is .
The model, that allows to distribute optimally the labor potential by economic sectors, obviously, would be:
, (1)
, (2)
, (3)
, (4)
, , ,,
where T  transpose operation.
Dividing the variables xi, i = 1, ..., n, by p and denoting , we go from the tasks (1)  (4) to the following linear programming task (which is more convenient to use in applied research than (1)  ( 4)):
, (5)
, (6)
. (7)
Note that the results of these research overlap with the results of the research set out in [1, 2].
We use the model (5)  (7) for the optimal distribution of labor potential of the Stavropol region on its economic sectors. According to [5] in the region there are the following economic sector (see Table 1.).
Table 1
Employed population distribution in the manufacturing process of the Stavropol region by economic activity in 2009
Number 
Economic sectors 
Th. of people 
Total employment in the economy 
799,2 

including: 


1 
Mining 
4,1 
2 
Manufacturing 
148,3 
3 
Production and distribution of electricity, gas and water 
36,9 
4 
Building 
87,3 
5 
Wholesale and retail trade; repair of motor vehicles, motorcycles, household goods 
218,1 
6 
Hotels and restaurants 
25,7 
7 
Transport and communication 
93,6 
8 
Financial sector 
13,4 
9 
Realty, renting and other services with real estate 
54,8 
10 
Unemployed 
117,0 
Table 2 shows the costs of the Stavropol Territory employers for the labor resources in 2009 in the various economic sectors.
Table 2
Costs of the Stavropol Territory employers for the labor resources in 2009 (thousand rubles per employee per year)
Number 
Economic sectors 
Costs (thousand rubles) 

Total in considered sectors 
2037,6 

including: 

1 
Mining 
246,5 
2 
Manufacturing 
201,1 
3 
Production and distribution of electricity, gas and water 
198, 5 
4 
Building 
185,9 
5 
Wholesale and retail trade, repair of motor vehicles, motorcycles, household goods 
162,3 
6 
Hotels and restaurants 
103,8 
7 
Transport and communication 
273,9 
8 
Financial sector 
401,9 
9 
Realty, renting and other services with real estate 
228,7 
10 
Unemployed 
2,9 
According to the data in Tables 1, 2 and the fact, that the cost per employee is 20% of the income that he brings to the sector in which he works, it is easy to calculate the value of p = 890142,9 rubles.
Effects matrix {qij} is calculated depending on the economy state: "bad", "satisfactory", "good" (i = 1, ..., 10; j = 1, 2, 3)
. (8)
We find the solution of the task (5)  (7) with m = 3, n = 10, and the found matrix (8). We have: s1 = 0,03; s2 = 0,03; s3 = 0,02; s4 = 0,02; s5 = 0,02; s6=0,01; s7 = 0,03; s8 = 0,05 ; s9 = 0,03; s10 = 0,00004. Since p = 890142,9 rubles., Then x1 = 27295, x2 = 22276, x3 = 21979, x4 = 20587, x5 = 17968, x6 = 11496, x7 = 30335, x8 = 44507, x9 = 25329, x10 = 0 .
As a result, with a minimum income of 890,142.9 rubles per employee of the Stavropol Territory, it is necessary to plan the labor resources by economic sectors in accordance with the above values of xi, i = 1, ..., 10 (note that such planning Unemployment in the region disappears, x10 = 0).
2. Zaitseva, I.V. Mathematical model of the labor market selforganization for several economic sectors / I.V. Zaitseva, E.A. Semenchin / / Economics and Mathematical Methods, 2007.  Volume 43.  № 1.  Pp. 133  136.
3. Zaitseva, I.V. The development of the concept "working capacity" as a socioeconomic category / I.V. Zaitseva, M.V. Popov, Y.V. Vorohobina / Management / economic systems: electronic scientific journal in 2013.  № 1.  Access mode: http://www.uecs.ru/index.php.
4. Zaitseva, I.V. A systematic approach to determining the structure of the regional labor potential / I.V. Zaitseva, M.V. Popova / / System Analysis and Information Technology (SAIT 2013): Proceedings of the 15th International Scientific and Technical Conference SAIT in 2013.  Kiev, the ESC "IASA" NTU "KPI", 2013.  516 p.  Pp. 101102.
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Zaytceva I.V., Semenchin E.A. OPTIMUM DISTRIBUTION OF THE REGIONAL LABOR POTENTIAL IN ITS ECONOMIC SECTORS. International Journal Of Applied And Fundamental Research. – 2013. – № 1 –
URL: www.sciencesd.com/45224352 (06.10.2022).