In air traffic control, incoming aircraft can be sent on holding patterns, meaning they can circle...

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In air traffic control, incoming aircraft can be sent on holding patterns, meaning they can circle above the airport to delay their landing. However, with each holding pattern cycle, the available time window for landing is offset, as shown in Fig. 1. These periodic landing windows have a period given by the holding pattern time, call it T, and vary in length depending on the size and maneuverability of the aircraft. 0 N42584 1222 350 351 L34 aircraft a2 a₁ b2 T b₁ a3 b3 a4 b4 aircraft 4 aircraft 3 aircraft 2 aircraft 1 CTAS tracks courtesy of NASA Ames feasible arrival times Figure 1. Incoming aircraft can be sent on holding patterns of length T, which create periodic windows of feasible arrival times. For concreteness, let us consider four aircrafts as shown in Fig. 1, index by i = 1, 2, 3, 4. Let [a, b] represent the feasible arrival time window, without any holding patterns. Then when considering multiple holding patterns (with the same T), the set of feasible arrival times for the four aircraft are given in Fig. 1. Let ni E Z be an integer decision variable that represents how many holding patterns we instruct for aircraft i. Then we can write the feasible landing time as aitrit stis bit nit, i = 1,2,3,4 where n; and ti are integer and continuous decision variables, respectively. Moreover, suppose we require at least A = 3 minutes separation between landings. This can be written in OR form as |ti-tj| ≥ 4, Vi,j 1,2,3,4, i # j (a) Transform the OR form into AND form using the Big-M method. (b) Extend 4 to m, and formulate the problem as a Mixed Integer Linear Program. Indicate all continuous and integer decision variables, the objective function, and constraints. In air traffic control, incoming aircraft can be sent on holding patterns, meaning they can circle above the airport to delay their landing. However, with each holding pattern cycle, the available time window for landing is offset, as shown in Fig. 1. These periodic landing windows have a period given by the holding pattern time, call it T, and vary in length depending on the size and maneuverability of the aircraft. 0 N42584 1222 350 351 L34 aircraft a2 a₁ b2 T b₁ a3 b3 a4 b4 aircraft 4 aircraft 3 aircraft 2 aircraft 1 CTAS tracks courtesy of NASA Ames feasible arrival times Figure 1. Incoming aircraft can be sent on holding patterns of length T, which create periodic windows of feasible arrival times. For concreteness, let us consider four aircrafts as shown in Fig. 1, index by i = 1, 2, 3, 4. Let [a, b] represent the feasible arrival time window, without any holding patterns. Then when considering multiple holding patterns (with the same T), the set of feasible arrival times for the four aircraft are given in Fig. 1. Let ni E Z be an integer decision variable that represents how many holding patterns we instruct for aircraft i. Then we can write the feasible landing time as aitrit stis bit nit, i = 1,2,3,4 where n; and ti are integer and continuous decision variables, respectively. Moreover, suppose we require at least A = 3 minutes separation between landings. This can be written in OR form as |ti-tj| ≥ 4, Vi,j 1,2,3,4, i # j (a) Transform the OR form into AND form using the Big-M method. (b) Extend 4 to m, and formulate the problem as a Mixed Integer Linear Program. Indicate all continuous and integer decision variables, the objective function, and constraints.

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