Homework/Miniproject 5: Transportation Planning

The Olympics were in Atlanta in 1996. This case outlines one aspect of the complicated logistics problems the organizers faced.

One major difficulty Atlanta had in getting the Olympics was to convince the Olympic Committee that the transportation system was up to the challenges of handling the Games (much of Atlanta's hotel capacity is in outlying areas).

During the Olympics, private automobiles will be essentially banned from the city. Instead, the organizing committee plans to transport people on buses and the city's subway line, MARTA.

In the following exercise, we will examine just a portion of the transportation plan: the northern quadrant. We will be evaluating the plan for just one four hour period. This period is in the morning when all the traffic is going from the outer periphery into the center core.

In this core, there are two main areas: the Natatorium, on Georgia Tech's campus, and the downtown Olympic Stadium area. Based on projections, the Natatorium will recieve 20,000 visitors from our quadrant and the Stadium area will recieve 90,000 people. The network for our quadrant looks like the attached figure.

The thick line represents our portion of MARTA. The thin lines are bus routes. The numbers by the bus routes represents number of people (in thousands) that can be transported along the route.

To determine the capacity of the MARTA system, we have to do some calculations. We have 40 trains to in this quadrant. Each train will run a single route during the period (though different trains may run different routes). Depending on the length of the route, the train will be able to make a differing number of trips, as given in the following table:

Each train can hold 300 people and stops at every stop along the way to pick up more people if it has capacity.

Our surveys suggest that people will enter the system in the following quantitities:

**Question 1.** Currently, we have 26 trains assigned to the
Doraville-Downtown route and 14 trains assigned to the Lenox Downtown
route. Is this sufficient to move 20,000 people to the Natatorium and
90,000 to the stadium area Downtown?

At this point we are going to do a rough ``back of the envelope'' calculation to determine if the capacity of the system is enough. Formulate a maximum flow model for this problem to determine if the capacity in this network is sufficient.

We are going to have to solve many, many of these (and the real problem is quite a bit larger), so it is important that you limit yourself to a maximum flow model. To this end, please clearly draw the network and give the capacities.

A couple of hints: This really can be modeled as a maximum flow. You may find it helpful to add an artificial source and destination node.

Solve the resulting model the linear programming formulation and Solver. Can all the people be moved in the period given? What is the bottleneck of the system?

**Problem 2.** The bus route capacities are fixed by traffic
patterns (we actually have plenty of buses to fill up the road
capacity). We can, however, change how we schedule the MARTA trains.
We could move the trains onto different routes. We must, however,
have an integer number of trains on each route. Formulate the problem
of determining how to allocate the trains to maximize the number of
people moved. For this, you might not be able to find a network flow
model (I certainly couldn't), so use an integer programming model if
necessary.

If you are still unable to transport all of the people, how many trains are needed to successfully move everyone?

**Problem 3.** What assumptions have we made about the relationship
between where people start and where they end up? How could we change
the model in order to more closely match reality? Would the resulting
model require more or fewer trains to move all of the people? (Do not
solve the model: simply words are enough for this question).

Sun Nov 10 11:56:35 EST 1996