Determination of the relative significance of efficiency and safety of air traffic with the aid of the inverse problem of the linear programming

Control and navigation systems


Tin P.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



This work is devoted to the solution of the inverse problem of linear programming to establish weight coefficients of the importance, effectiveness and safety of aircraft movement in echelon of landing approach. The air traffic controller knows how he should act in a concrete case, but he does not know the mathematical model of the criterion. In one case he makes alternative decisions on reduction, or increase of a distance between planes due to lateral maneuver or increase in draft. In other cases at increase of risk of air traffic, the air traffic controller prohibits entry of one of the planes into an echelon and gives command to leave to a repeated circle. Therefore expediency arises to reconstruct the criterion by separate examples of optimum behavior and then implement them in general cases. This criterion, as well as restrictions, can be presented in the form of linear convolution of parameters. The variables by nature are not negative, and selected decisions are alternative. Thus the process of selection of the most rational decision can be treated as finding of one of the vortexes of convex polyhedron. It can be done by means of a known direct problem of linear programming. The inverse problem sets the group of linear inequalities in the form of restrictions on a set of variables. The polyhedron vortex taken for the best decision with coordinates is also known. It is required to identify criterion coefficients whenever possible more precisely. The paper offers the reverse simplex method of identification of unknown criterion function of criterion in the form of linear convolution if the result of the decision at known restrictions on variables in the form of linear inequalities was set. The reconstruction of criterion function by means of the received interval estimates allows using the solution of a direct problem with new conditions — or in automatic mode, or as a prompter, reducing thereby influence of a human factor. Accuracy of the solution of the inverse problem depends on what vortexes of a polyhedron, and how many of them become known as a result of rational alternative actions of the person taking the decision that can be used for identification. The example confirming operability of the offered algorithm is presented.


flight safety, linear programming, inverse problem of parametrical optimization


  1. Gass S. Lineinoe programmirovanie (Linear programming), Moscow, FIZMATGIZ, 1961, 304 p.
  2. Lebedev G.N., Tin Pkhon Chzho. Mekhatronika, avtomatizatsiya, upravlenie, no.11, 2008, pp. 41 — 46.
  3. Lebedev G.N. Teoriya optimal’nykh system (The theory of optimal systems), Moscow, Izd-vo MAI, 1999, 137 p.

Download — informational site MAI

Copyright © 2000-2024 by MAI