Skip to main content

Multiproduct revenue management: An empirical study of Auto Train at Amtrak


This study involves working with Amtrak, the National Railroad Passenger Corporation, to develop a revenue management model. The Revenue Management Department at Amtrak provides the sales data of Auto Train, a service of Amtrak that allows passengers to bring their vehicles on the train. We analysed the demand from the sales data and built a mathematical model to develop a pricing system for Auto Train. An algorithm was developed to calculate the optimal pricing strategy that yields the maximum revenue. We further introduced three pricing policies Myopic policy, Static-Price heuristic, and Pseudo-Dynamic heuristics, as benchmarks for our dynamic programming solution. Because Auto Train is a real-world application of multiproduct revenue management, our findings make an important contribution to the revenue management literature.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2


  1. The authors of this paper are collaborating with the Amtrak Revenue Management Department to develop a pricing system based on the current study.

  2. Adding capacity is possible but very costly. Therefore, we do not consider this possibility.

  3. We call a group of passengers who share a vehicle as a ‘party’ throughout this study.

  4. Following a consistent pattern means that the total number of reservations for each accommodation and for each train departing during Summer 2003 are almost the same.

  5. Among the 90 trains that depart in the summer of 2003, there were only five trains that did not sell out their van accommodations.

  6. This method assigns a Poisson distribution function to the data set and calculates the chi-square based on the difference between the observed values and the output of the distribution function.

  7. We also call it ‘day t’ or ‘t days before departure’.

  8. This assumption is based on the experience of employees in Amtrak Revenue Management Department.

  9. The presentation of the results for this problem is challenging due to the multi-dimensionality. Complete numerical results can be provided upon request.

  10. We calculate the average revenue per train from the original transactional data described in the second section.

  11. We subtract the coach seats and sleepers assigned to the passengers riding a van.

  12. Recall that the price for van accommodation is not subject to revenue management, we update the available passenger accommodations by subtracting the average number of passengers in a van from the original passenger accommodations.

  13. Note that a higher bucket number is associated with lower price level; for example, price bucket 4 is the cheapest price level for all accommodations.

  14. By ‘regular’ monotonic property, we mean the well-known monotonic properties in revenue management. For example, in revenue management, it has been proven that the optimal price is non-decreasing with respect to remaining time, or the optimal revenue is non-decreasing with respect to available products.


  • Campbell, K. (1996) ‘Booking and Revenue Management for Rail Intermodal Services’, Ph.D. dissertation, University of Pennsylvania.

  • Crevier, B., Cordeau, J. and Savard, G. (2006) ‘Integrated operations planning and revenue management for rail freight transportation’, 2006 optimisation days, Montreal, May 2006.

  • Gallego, G. and van Ryzin, G. (1997) ‘A multiproduct dynamic pricing problem and its applications to network yield management’, Operations Research, 45, 1, 24–41.

    Article  Google Scholar 

  • Kraft, E. (2002) ‘Scheduling railway freight delivery appointments using a bid price approach’, Transportation Research Part A: Policy and Practice, 36, 2, 145–165.

    Google Scholar 

  • Ladany, M. and Hersh, S. (1978) ‘Optimal seat allocation for flights with one intermediate stop’, Computer and Operations Research, 5, 31–37.

    Article  Google Scholar 

  • Maglaras, C. and Meissner, J. (2006) ‘Dynamic pricing strategies for multi-product revenue management problems’, Manufacturing and Service Operations Management, 8, 2, 136–148.

    Article  Google Scholar 

  • Talluri, T. (1993) ‘Airline Revenue Management with Passenger Routing Control: A New Model with Solution Approaches’, Working Paper, USAir.

  • Talluri, T. and van Ryzin, G. (2004) The Theory and Practice of Revenue Management, Kluwer Academic Publishers, Dordrecht, MA.

    Book  Google Scholar 

  • Winston, W. (1994) Operations Research Applications and Algorithm, Dexbury Press, Belmont, CA.

    Google Scholar 

  • You, P. (1999) ‘Dynamic pricing in airline seat management for flights with multiple flight legs’, Transportation Science, 33, 2, 192–206.

    Article  Google Scholar 

Download references


We thank the Amtrak Revenue Management Department for providing us a chance to study the Revenue Management of Auto Train.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Soheil Sibdari.

Additional information

1 Soheil Sibdari is an assistant professor at the Charlton College of Business, University of Massachusetts Dartmouth. He received both his MS in Economics and PhD in Industrial Engineering from Virginia Tech. His research interests include dynamic pricing, revenue management, game theory applications, and transportation planning.

2 Kyle Y. Lin is an associate professor at the Operations Research Department, Naval Postgraduate School. He received his MS and PhD in Industrial Engineering and Operations Research from UC Berkeley. His research interests include stochastic modelling, decision making under uncertainty, queueing theory, and game theory. He is an associate editor for the journal Operations Research.

3 Sriram Chellappan received his MS in Transportation Engineering from University of Maryland College Park. He is the director of decision support system at National Railroad Corporation (Amtrak). In this position he is responsible for setting strategy and goal for gathering and disseminating Business Intelligence along with the implementation of analytical models and tools.



To estimate the parameters of the fitted curve for the last 30 days and for all combinations of price buckets, we use the following steps. First, we transform the observation values in order to linearise the relationship between y and t. To this end, we transform each value of y into ln y. Secondly, we estimate the least-square regression line for the transformed data. Let β̂0, β̂1, and s e , respectively, denote the intercept of the least-squares line, the slope of the least squares line, and the standard error of the regression estimate for the transformed data. Thirdly, using these parameters, we estimate the relationship between the number of reservations and the time before departure for any price combination of car accommodation and coach seats as follows:

On the other hand, for the remaining time horizon (30 days before departure through 330 days before departure), the total number of reservations for car accommodations appears to be linear and of the form:

Using the sales data, we estimate the parameters of the demand function for each price combination. Note that from our data analysis, there is demand seasonality for trains departing on different days. For example, a train leaving on Sunday has a different demand distribution from a train leaving on Monday. To resolve this obstacle, we multiply the mean demand by the parameters listed in Table 2.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sibdari, S., Lin, K. & Chellappan, S. Multiproduct revenue management: An empirical study of Auto Train at Amtrak. J Revenue Pricing Manag 7, 172–184 (2008).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: