a model to solve en route air traffic flow management - ATM Seminar

A MODEL TO SOLVE EN ROUTE AIR TRAFFIC FLOW MANAGEMENTThird, it could be easily adapted to cover cases with more than one alternative route a for flights i, i∈F a ,considering these routes as additional flights and adding the corresponding sums to the left side of theconstraints (3).4. PROPOSED SOLUTIONThe LP relaxation approach is used for solving the BIP model (1−4). LP relaxation of the BIP model is solvedusing a standard LP package. If the solution is integer, it is an optimal one and the value of the objectivefunction (1) represents the minimum total delay cost. Otherwise, each flight i with value 1 for correspondingvariable x i,k is served with delay k. Then all flights with non−integer values for some of the correspondingvariables are served by applying one of the heuristic approaches described in Tosic et. al. (1995).5. NUMERICAL EXAMPLEIn this section, results of one of the conducted experiments will be presented in order to illustrate theapplication of the developed model and proposed method of solution. It is similar to the numerical exampleused in Tosic et. al. (1995) were we had N=4 ATC sectors (North=N, East=E, South=S, West=W) and T=8half−hour time periods with 162 flights in set F each intending to fly en−route through some of those foursectors. In this example we introduce two more ATC sectors, N' and E' (Figure 1) offering two alternativeroutings for the two most loaded air traffic flows, namely NE flow with 45 flights and EN flow with 26flights.Those flights are "offered" the following alternative routes: for NE flights, the alternative route is N'E' and forEN flights the alternative route is E'N'. Remaining flights of other traffic flows are not offered alternativeroutes. Therefore the initial total traffic load remains the same as in the example used in Tosic et. al. (1995),as well as sector capacities per period which were set equal to Q jt =18 flights (j=1,2,3,4, i.e. j=N,E,S,W andt=1,2,...,8). Sectors N' and E' have capacities Q 5t =Q 6t =9 (t=1,2,...,8) because of the load produced by flightsflying through those two sectors on their original route (flights not considered in this example), so those twocapacity values should be understood as remaining sector capacities.Alternative routes N'E' and E'N' are equal in length and take 1/3 of a half−hour time period (10 minutes)longer to fly than their respective original routes NE and EN. It is assumed that all flights originally in NE andEN flows will fly the same extra amount of time when flying on alternative routes for reasons of simplicity.For the same reason, we assume that the ratio of cost per unit time when the aircraft is in the air and when it ison the ground equals 4 (Andrews 1993), and this value was used for all flights. We used the same coefficientsc i as in the previous example, Tosic et. al. (1995) where the function of aircraft size given ranged from 0.1 foraircraft up to 15 seats to 4.0 for aircraft with 320 to 450 seats. We have therefore the following values forcoefficients c i '(k) and c i " in objective function (1): c i '(k)=c i ⋅k and c i "=4⋅1/3⋅c i =4/3⋅c i . Maximum permittedground delay per flight i was set at k i * =4 half−hour time periods and was assumed equal for all flights. Then,maximum permitted delay for flights i on alternative routes a was calculated as k a * =int(k i * − 1/3)=3 and wasalso assumed equal for all flights assigned to alternative routes.4