Case 2:

The context for this case is described in Prof. Odoni's paper on Flow Management Problems in Air Traffic Control (ATC), referred to as the FMP paper in the rest of the case.

1. Develop a network description of the ATC system that addresses the issues raised in the FMP paper.

1. Define the nodes and links (undirect) and/or arcs (directed) appropriately.
2. Classify nodes, as sources, sinks, etc. For sources, describe the supply; for sinks, describe the demand. That is, what are supply and demand in the context of the FMP paper and what generates them? Are they deterministic or stochastic? How would you estimate the values?
3. Describe the flow on links/arcs, that is, what is the flow and what generates it. What are the lower bounds and upper bounds of flow and what determines them? Are they deterministic or stochastic? How would you estimate the values?
4. What are appropriate costs/distances for the links/arcs? Are they deterministic or stochastic? How would you estimate the values?

2. Assume that the network that addresses some of the issues raised in the FMP paper is:

There is one airport (AAA) that acts as a sink and n flights that act as sources. One arc is directed from each of the flight nodes to airport AAA. Note that departures from Airport AAA are not modeled.

Assume the following.

1. There is a given time interval of interest.
2. The time interval is divided into m equal periods (e.g., 15 minutes).
3. Airport AAA's arrival capacity during each period j is deterministic and known and is denoted by K(j).
4. For each flight:
   1. Flight i's scheduled landing period is known and is denoted by S(i).
   2. Flight i's travel time is known and deterministic and is denoted by T(i).
   3. Flight i is subject to delay only at Airport AAA.
5. During some time periods, there will be arrival congestion, i.e., when there are more flights scheduled to land than available capacity.

Suppose the objective of the FAA is to minimize total delay. Formulate a mathematical program to help the FAA. State clearly any additional assumptions you have made.

3.Suppose that the network that addresses some of the issues raised in the FMP paper consists of nodes representing each airport and bidirectional links connecting each node to every other node.

Assume three airports for simplicity. We also assume the following.

1. The network is closed, so no flights come into the network or leave the network.
2. There is a given time interval of interest (e.g., 24 hours).
3. The time interval is divided into m equal periods (e.g., 15 minutes).
4. Each airport a's arrival capacity during each time period j is deterministic and known and is denoted by K(a,j).
5. Assume there are n flights.
6. For each flight i:
   1. The origin airport is known and is denoted by O(i).
   2. The destination airport is known and is denoted by D(i).
   3. The scheduled departure period is known and is denoted by L(i).
   4. The scheduled landing period is known and is denoted by S(i).
   5. The aircraft that is used is known and denoted by P(i).
7. During some time periods, there will be arrival congestion at the airports, but not on the links between airports.
8. There is a 2 period turnaround time at a given airport for all aircraft. That is, when an aircraft arrives at an airport in time period j as flight i', it cannot depart before time period j+2 as flight i''.

Suppose the objective of the FAA is to minimize total delay. Formulate a mathematical program to help the FAA. State clearly any additional assumptions you have made.