RESTRICTIONS OF THE TYPE OF POLYHEDRAL DISPLACED CONE IN LINEAR PROGRAMMING PROBLEMS

Abstract

The paper considers the linear programming problem. A method is proposed to simplify its solution by identifying a class of constraints of a special type, due to which the desired plan will belong to a polyhedral convex cone located in a non-negative orthant. In this case, you must perform the following algorithm of actions. First, the original coordinate system is parallel transferred to the top of the selected cone. Then a transition to another space is made, which will lead to significant changes: a decrease in the number of restrictions. Next is the solution to the problem in any convenient way, for example, by the simplex-method – the most frequently used algorithm for finding solutions to linear extremal problems. One of its features is that with a large number of restrictions, its effectiveness decreases. This is a significant drawback in solving a number of problems, in particular, those of an economic nature, which, as a rule, striving to most accurately reflect the real state of affairs, impose a large number of restrictions on the desired plan. Therefore, if possible, it is better to reduce their number, even by increasing the variables, as this can happen in the proposed method for solving the selected class of problems. After finding the optimal plan, you need to return to the original space, and then to the old coordinate system. An important condition of this algorithm is the non-negativity of the cone elements. Thanks to this assumption, when a task is modified, new constraints are excluded. To track the implementation of this requirement, a condition is given in the work that guarantees its fulfillment. At the end of the work, the technique of searching for the operator (transition matrix) is described, with the help of which the task is transferred to another space. It is based on the search for vectors aligned with the generators of the cone.