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CSE561 Public Transport: Operations and Service Planning

Assignment 2

DUE 17 November 2022

Problem 1. Given a region with 15 demand nodes, 4 bus lines, and 6 candidate bus stop locations as shown in Figure 1(a). The bus travel demand at each node is shown in Figure 1(b). Assume the travel distance between two adjacent zones is 5 minutes and travel can only take place in horizontal and vertical directions. A demand node is covered by a bus stop if it can be reached by the stop in 10 min. Two bus stops are to be located at this region.

a. Write out the whole mathematical formulation for this problem (how many decision variables and constraints?)

b. Show your solution procedure either by observation or using the solver. Where are the optimal bus stop locations that can cover the largest demand?

(a) Bus stop locations                                                     (b) Bus demand

Figure 1

Problem 2. A railway service operates from stop 1 to stop 8. The number of alighting and boarding passengers at each stop during the peak hour is shown in the following table. The total passenger along the whole route during peak hour is 7,070 passengers. 

For the following questions, assume a value of passenger waiting time of $90 per hour, and a cost of services of $2,500 per hour. The total round-trip time (including layovers) on this route is 140 minutes.

a. What is the optimal headway for this route?

b. What should be the train capacity to serve the demand on this route?

Hint: The load at each stop (or point) can be calculated based on the difference between the accumulated number of boarding passengers and the accumulated number of alighting passengers. It is suggested to first list the demand at each load point and find the peak load point.

Problem 3. A small transit agency operates two routes. Route 1 runs from A to B and back to A (A-B-A). Route 2 runs from A to C and back to A (A-C-A). Also, the one-way travel time between nodes is shown on the network (which includes all dwell times and required minimal layover times). Service on Route 1 operates on 15-minute headways, and service on Route 2 operates on 10-minute headways.

a. Assume the first bus on Route 1 departs at X:02 (where X is the hour of the day), and the first bus on Route 2 departs at X:00 on the clock. Create a deficit function for each of the routes separately, and for terminal A as a whole, for the first 90 minutes of operation.

b. Based on your answer to part (a), what is the minimum number of buses required on each route, if interlining is not allowed?

c. Based on your answers to parts (a) and (b), is interlining likely to have any positive impact in reducing operating costs in this case? Briefly explain your answer.

Problem 4. Consider the timetable of a service route in the table below. The terminal of this service route is at station A. The transit company has in total 5 vehicles to operate this line. The pull-in and pull-out times are 15 mins each. Given that the transit company would like to minimize the layover time between trips.

a. Provide the complete mathematical model formulation.

b. Use the mathematical model to get the optimal vehicle schedule.

Problem 5. Consider the timetable of two service routes of another transit company in the table below (between X and Y, and between X and Z). The terminal of this transit company (for crew and vehicle operation) is at station X. The minimum terminal time (including pull-out, trip, lay-over, and pull-in times) for each transit vehicle is 10 minutes. The vehicle blocks already include pull-out, trip, lay-over, and pull-in times. The transit time between X and Y is 160 minutes (same for both directions). The transit time between X and Z is 75 minutes.

a. Construct the vehicle blocks assuming the first-in-first-out vehicle assignment principle.

b. Create pieces of work for crew scheduling from these vehicle blocks. Each crew can be assigned to work for at most 8 hours/day. Crews can be switched to any route at station X.

c. Formulate an integer linear program for this crew assignment problem with the following constraints:

I. The number of trippers should be less than 3.

II. The crew size is 6.

III. The number of splits should be less than or equal to 1.

d. Based on subproblem c, assume that the cost of a straight run is 2, the cost of a split run is 3, and the cost of a tripper run is 1. What is the best crew schedule? Show the solving procedure of using a solver.