Investigating competition in the problems of optimal resource allocation

Abstract

Aim. The presented study aims to address the issues of parameter estimation in the problems of optimal resources allocation for the previously introduced competition indicator; to analyze the influence of dimensionality, resource constraints, and other factors on the competition indicator; to exemplify the relationship between the indicator and the extremum of the objective function, constraints, and dual estimates.Tasks. The authors consider cases when the competition indicator captures a change in the initial data that cannot be estimated on the basis of traditional indicators of analysis and estimates: the maximum of the objective function, the optimal solution, Lagrange multipliers, or dual variables; determine the relationship between the competition indicator and the optimum of the objective function and dual variables through examples and in general; show that the analysis of the results of solving the problem becomes more capacious and informative if the factor of variable “competitiveness” is applied; identify patterns between efficiency, competition, resource constraints, and dual estimates.Methods. The selected competition indicator for optimal resource allocation tasks is based on the concept of “rigorous selection” of competitors applying for resources. The indicators are calculated in full accordance with the known optimality conditions for problems of this class, making it possible to interpret the results of optimization as a measure of competition for resources.Results. The provided examples reflect linear and nonlinear functions as well as the relationship between the competition indicator and dual estimates, resource constraints, and efficiency. It is proved that the competition indicator logically fits into the traditional analysis of the results of solving the problem of linear and nonlinear programming with allowance for duality.Conclusion. The competition indicators considered in the study can be included in the standard analysis for solving the problems of optimal resource allocation, which involves finding an extremum, searching for an optimal plan, analyzing stability, limits, dual estimates, a measure of resource scarcity. As can be seen from the examples, applying the competition indicator to the analysis not only makes the analysis of the results more capacious and informative, but also makes it possible to detect patterns between competition and efficiency, similar to when the removal of barriers and restrictions in the economy leads to its revival, and the reduction of resources causes increased competition.