Keeping order of vessels in problem of safe merging aircraft flows

Nowadays, the problem of creating an optimal safe schedule for arrival of aircraft coming in several flows to a checkpoint, where these flows join into one, is very important for air-traffic management. Safety of the resultant queue is present if there is a safe interval between neighbor arrivals to the merge point. Change of an arrival instant of an aircraft is provided by changing its velocity and/or usage of fragments of the air-routes scheme, which elongate or shorten the aircraft path. Optimality of the resultant queue is considered from the point of some additional demands: minimization of the deviation of the actual aircraft arrival instant from the nominal one, minimization of order changes in the resultant queue in comparison with the original one, minimization of fuel expenditures, etc. The optimality criterion to be minimized, which reflects these demands, is often taken as a sum of penalties for deviations of the assigned arrival instants from the nominal ones. Each individual penalty is considered in almost all papers as either the absolute value of the difference between the assigned and nominal arrival instants or a similar function with asymmetric branches (which punishes delays and accelerations of an aircraft in different ways). The problem can be divided into two subproblems: one is a search for an optimal order of aircraft in the resultant queue, and the other is a search for optimal arrival instants for a given order. The second problem is quite simple since it can be formalized in the framework of linear programming and solved quite efficiently. However, the first one is very difficult and now is solved by various methods. The paper suggests sufficient conditions for the problem, which guarantee that the order of the optimal assigned instants is the same as the order of the nominal ones and, therefore, exclude the first subproblem.