# Modeling of the Optimal Allocation of Labor Resources

**Зайцева И. В. Modeling of the Optimal Allocation of Labor Resources // Taurida Journal of Computer Science Theory and Mathematics, – 2019. – T.18. – №4. – P. 59-77**

https://doi.org/10.37279/1729-3901-2019-18-4-59-77

https://doi.org/10.37279/1729-3901-2019-18-4-59-77

The article presents models of optimal distribution of labor resources. The

developed game-theoretic model of a static optimal-purpose problem is described as a game in

a normal form. The game is given a lot of workers and a lot of businesses, and the situation is

a substitution. Each substitution is one of the possible assignments of employees to enterprises.

To select an employee or an enterprise, an evaluation criterion is introduced.

The number of evaluation criteria is called the utility for the employee from the appointment

to the enterprise (the degree of satisfaction of the player’s interests), and for the enterprise —

the utility for the enterprise from the appointment of an employee (the degree of satisfaction of

the player’s interests). From the numbers of the evaluation criterion, the utility matrix is written

and the matrix of players ’ winnings in the game is built. According to the matrix, a compromise

set is built in the game and a compromise win is found, which is a guaranteed win for the least

satisfied player. An algorithm for constructing a compromise set is presented. For the algorithm,

its time estimate and complexity class are found.

This paper considers game-theoretic model of dynamic optimal assignment in the example

of the functioning of the labour market. The deterministic model of the optimal distribution of

workers by enterprises is described, taking into account the changing conditions over time. At

each moment of time, the States of the employee and the enterprise are determined. Moments of

time are moments of stationary States of the system. In each stationary state is determined by

the game in normal form. The game is a compromise situation, the optimal policy of the labor

exchange, and also calculates the income of the system from the appointments as the sum of

the functions of the winnings of all players. The functioning of the labor market as a system for

some periods of time is presented in the form of a multi-step game on the tree.

In a one-step game based on the principle of optimal compromise set there is a compromise

situation and the corresponding compromise vector of control. On the tree of a multi-step game

there is a compromise income of the system in a few steps, when the sequence of games was

realized, and a compromise path corresponding to the sequence of compromise control vectors.

The compromise income of the system and the sequence of compromise controls are found

by means of recurrent relations of dynamic programming. Thus, it is possible to specify the

optimal behavior of all participants in the labor market at any given time. The article presents

the solution of the static and dynamic problem of optimal distribution of labor resources based

on principle of optimal compromise set.

**Keywords:** *compromise set, modeling, optimality principle, distribution, labor resources.*