О построении совместных систем линейных ограничений экономико-математических моделей задач с двухсторонними неравенствами

Abstract

В статье рассмотрена актуальная проблема построения совместной системы линейных ограничений для экономико–математических моделей, задач с двухсторонними ограничениями на переменные. Приведены примеры и сформулированы условия совместности линейных систем.

У статті розглянута актуальна проблема побудови спільної системи лінійних обмежень для економіко–математичних моделей задач із двосторонніми обмеженнями на змінні. Наведені приклади і сформульовані умови спільності лінійних систем.

This article deals to the actual problem of building a joint system of linear constraints for economic and mathematical models problems with bilateral constraints on the variables. In applications of the economic models of production systems, lower and upper limits of the values correspond to the minimum and maximum possible values of variables and constraints which are specified explicitly. Such a statement, compared with the traditional when variables imposed only non–negativity condition, is more common and necessary in the construction of econometric models and the solution of practical problems of management and decision–making. Building a joint system of linear constraints and bilateral inequalities carried out on the basis of verification of the fulfillment of conditions: Consistency of a system of linear constraints in Rn ( Kronecker – Capelli theorem) ; Consistency of a system of linear constraints in Rn and in X ≥ 0, by constructing and solving linear programming problem, which determines the consistency in area where X≥ 0; Consistency of a system of linear constraints in the X≥Xmin, by linear coordinate transformations (change of variables X=Xmin+Z, Z≥0 and solving linear programming problem, which determines the consistency at Z ≥ 0 and, that’s why, at the area X≥Xmin); Consistency of a system of linear constraints at X≤Xmax, by checking of the condition Xmax–Xmin=Z, Z≥0. There were given examples of solutions of the problem of determining the consistency of systems of linear constraints and restrictions on the variables in the form of bilateral inequalities.