One problem that arises in public transit authorities is that of assigning routes to garages. In its simplest form, the problem may be thought of as follows. The locations of the bus garages are given. For each route, there is an “in-service” location and an “out-of-service” location. These are the locations at which the vehicles that traverse the route begin serving the route and the locations at which they go out of service. Vehicles must deadhead, or move empty, from the garage to in-service locations at which they begin their routes (typically at the beginning of the day) and from out-of-service locations at which the routes end to the garage (at the end of the day). The figure above illustrates this situation. Two routes are shown in this figure along with two garages. Candidate assignments of routes to garage 1 are shown with solid lines, while candidate assignments to garage 2 are shown using dotted lines.

The objective in assigning vehicles from a garage to a particular route is to minimize the total deadheading distance of all vehicles subject to the following constraints:

·         the number of buses assigned out of any garage cannot exceed the capacity of the garage;

·         the numbe r of buses assigned to each route must be at least the number that are required to serve the route; and

·         the vehicle must return to its home garage (the one from which it departed for the in-service location at the beginning of the day) from the out-of-service location (at the end of the day).

(a) Formulate this problem as a linear programming problem. Clearly define all inputs and decision variables. Clearly sate the objective function and all constraints in words and in notation.

(b) Solve this problem for the following 3 garage, 12 route problem:

(c) Often, transit authorities want all buses assigned to a route to originate from the same garage. This facilitates vehicle dispatch› ing. Would the solution to the problem that you formulated in part (a) ensure that all buses assigned to a route were garaged at the same location? If so, why? If not, how can you reformulate the problem to ensure that this condition is met.

(d) Many transit authorities operate different types of vehicles. Associ› ated with a bus route will be the in-service location, the out-ofservice location, and the number and type of vehicle to be used on the route (e.g., a standard transit vehicle, a minibus, an articulated bus). Buses of different types will require different amounts of parking space at each garage. Since the garage capacity is gener› ally measured in terms of the number of standard bus parking spaces available, assigning different bus types to a garage will utilize different amounts of the capacity at the garage. For exam› ple, a large articulated bus might count as two buses. In this case, the authority must simultaneously determine how many of each vehicle type should be assigned to each garage (subject to capacity constraints) and the assignment of vehicles to routes to minimize the total deadheading distance. Ignoring the issue outlined in part (c) tha t of requiring that all vehicles assigned to a route originate from the same garage formulat e this problem as an optimization problem. Again, clearly define all notation separating inputs from decision variables. Also, clearly state the objective function and the constraints in words as well as notation.

(e) Briefly discuss why the model formulated in part (d) may or may not be solved using the same algorithm(s) that can be used for the problem of part (a).

Note: A number of additional concerns must be addressed in assigning bus routes to garages. Maze et al. (1981, 1982), Daskin and Jones (1993), and Vasudevan, Malini, and Victor (1993) all discuss models for this problem.