A method for solving the problem of optimizing the distribution of an integer resource

Abstract

The problem of optimizing the distribution of an integer resource (funds) by tasks (activities, goals) is investigated. The methods investigating this problem relate to the field of combinatorial optimization, namely, to the tasks of assigning goals. The known methods for solving this problem are numerical, selective, approximate, require a large number of iterations, do not involve checking the conditions for the existence of an integer solution, in some cases they can produce a solution not only far from optimal, but also violating the range of acceptable values of variables. The purpose of this work is to develop a new analytical method for solving the problem of the distribution of integer resources by the method of indefinite Lagrange multipliers. To do this, the allocated resources are represented as the sum of the integer and fractional parts of the number. The conditions are formulated and proved when the fractional parts of the variables of the solution of the problem are zero, that is, it is an integer. A theorem (criterion for the existence of an integer solution) is proved, which determines the necessary and sufficient conditions under which the solution of the problem exists and is found according to the algorithm developed in the article. Such conditions include the homogeneity of resources, as well as additional conditions (restrictions on integers and positivity of additional derived formula conditions of the problem). It is shown that the obtained solution of the problem corresponds to the maximum of the objective function. An algorithm for finding an integer solution to the problem of resource allocation by the method of indeterminate Lagrange multipliers is developed and a specific example is analyzed. The method described in this article can be used for the allocation of resources in industrial production, agriculture, organizational management systems, educational process, solving issues of target allocation in military affairs, building information systems, techno sphere security, emergency response, creating systems for the protection of objects and alarm systems. In this case, it is necessary to adapt it to the problems and tasks under consideration. It can also be used for the distribution of life-supporting resources: food, clothing, heat, electricity, gas, water supply.