Dynamic distribution of labor resources by region of investment

This article discusses the problem of the optimal distribution of labor resources by investment region, taking into account changing conditions. A deterministic model is considered, that is, both the investment region and the invested resources go into new states with a probability equal to one. Several economic agents jointly organize the process of investing labor resources in several investment regions.


Introduction
At the initial moment, many types of invested labor resources and investment regions were identified, the efficiency with which each investment resource in a particular investment region is also known. Each investment resource at any time can be in one of a finite number of states that differ in the efficiency of use of investment resources. It is necessary to invest investment resources in the investment region in an optimal way, that is, so that the total investment efficiency is maximum. By the beginning of the next period of time, depending on the decision made at the previous stage, and, possibly, for other reasons, many investment regions may change (some may appear), many labor resources invested, as well as the efficiency of a particular investment resource or another region of investment [1][2][3][4][5]. Consequently, a new situation arises in which it is necessary to solve a new problem of optimal appointment. There are many such situations. Thus, a dynamic problem arises. It can be solved by the method of dynamic programming, and as strategies at each step, various solutions to the optimal assignment problem are used.

Formal statement of the problem
We now turn to the formal statement of the problem. There are many M labor resources investing. We  [6][7][8][9]. We will evaluate the effectiveness of the resource in the investment region l size p(l). We introduce some system of numbers . We will say that the investment region is in a state ak, if the . If, before investing the invested resource, the investment region l was at the level of efficiency ak, and after the period of investment resource investment m he is in a state 2 k a , then under the efficiency of investing an investment resource m in the region of investing l we will understand the value ( To calculate the optimal income from the functioning of the system during T time periods at finite T we will use the recurrence relations of dynamic programming. Consider them [17][18][19][20][21].
Let be -maximum income from the functioning of the system during T-t time periods under optimal policy. The maximum income of the system during one period of time is determined by the formula ( The maximum income from the functioning of the system for two periods of time is delivered by the expression Where ) ( 1 1 S V -maximum income for the functioning of the system in a one-step process. To calculate the optimal income, we have the following functional relation can be set to zero, which is natural. Applying relation (5) sequentially for indicate the optimal distribution of investment labor resources by investment region at any given time [22][23]. The specified calculation method can be implemented using a computer.
3. An example of solving a dynamic problem To illustrate the described method for solving a dynamic problem, we give a simple example. Two economic agents simultaneously invest in labor resources in two investment regions. Each investment resource can be in one of two possible states. Each investment region can be in one of three possible economic conditions determined by a system of numbers . Obviously, the system can be in one of 36 possible states defined by the following table 1.   Many strategies consist of two elements: 1) the first resource is invested in the first investment region, the second resource is invested in the second investment region; 2) the first resource is invested in the second investment region, the second resource is added to the first investment region.
The efficiency of labor resources investment in investment regions is determined by the following table 2.   Using the tables above, you can determine the function of the transition of the system from one state to another. Table 5 below shows the table definition of the function   ) , ( s q F .  In table 5, in addition to the values of the function of the transition of the system from one state to another, the values of the income received are indicated. The amount of income was calculated by the formula (5). The determination of the optimal strategy in a one-step process is carried out by simple comparisons of the incomes obtained for different strategies and the choice of the strategy for which this value will be greater. Determining the optimal strategy in a two-step process is as follows: 1) for each strategy, the amount of income received during the transition and the state into which the system will go with this strategy are determined; 2) determine the amount of income received in one-step processes from the states to which the system can go when using various strategies; 3) incomes are summarized according to (5) and compared with each other. The calculation results for this example are summarized in table 6, in which, for each state of the system, the optimal strategies and the values of the received income are shown in the one-step, two-step, and three-step investment processes, respectively. If the column contains 0, then this means that investing does not make sense, since the state is final. If -is indicated, then this indicates the pointlessness of investing, since the final state is achieved in fewer steps.  Table 6 contains the solution to the problem of optimal investment. It is seen from it that the problem can be solved in no more than three time periods, and if the initial state is any, except for the 19th and 21st, then the problem can be solved in no more than two time periods. Thus, for any situation (see table  1 of the example), you can specify the time for which it is possible to solve the problem of optimal investment and optimal strategies at any time.

Conclusion
The optimal way to solve the problem of optimal investment can be implemented on a computer using simple programs, although it should be noted that the amount of required memory increases significantly with an increase in the number of investment regions, the number of investment labor resources and the number of their conditions.