# Linear Programming Formulation of the Elevator Trip Origin-destination Matrix Estimation Problem

### Juha-Matti Kuusinen, Mirko Ruokokoski, Janne Sorsa, Marja-Liisa Siikonen

#### Abstract

Elevator group control dispatches elevators to passengers’ calls in a dynamic environment where new calls constantly emerge. At the moment of making a dispatching decision, it is not known when and at which floors new passengers will register new calls, what is the number of passengers waiting behind these and existing calls, and what are their destinations. Robust dispatching decisions require that future passenger traffic is forecast based on the realized passenger flow in a building. The problem is that this flow cannot be directly measured. It can, however, be estimated by finding the passenger counts for the origins and destinations of every elevator trip occurring in a building. An elevator trip consists of successive stops in one direction of travel with passengers inside the elevator. We formulate the elevator trip origin-destination matrix estimation problem as a minimum cost network flow problem. We also present a branch-and-bound algorithm for finding all solutions to the problem and study its performance based on numerical experiments.

#### References

- Baldoni-Silva, W., De Loera, J., and Vergne, M. (2003). Counting integer flows in networks. Retrieved December 17, 2012, from http://www.math.ucdavis.edu/ ˜latte/theory/totalresidue.pdf.
- Bazaraa, M., Jarvis, J., and Sherali, H. (2009). Linear Programming and Network Flows. John Wiley & Sons, Hoboken, New Jersey, 4th edition.
- Bell, M. (1983). The estimation of an origin-destination matrix from traffic counts. Transportation Science, 17(2):198-217.
- Ben-Akiva, M., Macke, P., and Hsu, P. (1985). Alternative methods to estimate route-level trip tables and expand on-board surveys. Transportation Research Record, 1037:1-11.
- Bertsimas, D. and Tsitsiklis, J. (1997). Introduction to Linear Optimization. Athena Scientific/Dynamic Ideas, LLC, Nashua/Charlestown, U.S.A., 4th edition.
- Cascetta, E. and Nguyen, S. (1988). A unified framework for estimating or updating origin/destination matrices from traffic counts. Transportation Research Part B, 22(6):437-455.
- Danna, E., Fenelon, M., Gu, Z., and Wunderling, R. (2007). Generating multiple solutions for mixed integer programming problems. In IPCO 2007, LNCS 4513, pages 280-294. Springer-Verlag.
- Fisk, C. (1988). On combining maximum entropy trip matrix estimation with user optimal assignment. Transportation Research Part B, 22(1):69-79.
- Furth, P. and Navick, D. (1992). Bus route o-d matrix generation: Relationship between biproportional and recursive methods. Tranportation Research Record, 1338:14-21.
- Kuusinen, J.-M., Sorsa, J., and Siikonen, M.-L. (2012). The elevator trip origin-destination matrix estimation problem. Unpublished manuscript submitted to Transportation Science 4.7.2012.
- Lamond, B. and Stewart, N. (1981). Bregman's balancing method. Transportation Research Part B, 15(4):239- 248.
- Li, B. (2009). Markov models for bayesian analysis about transit route origin-destination matrices. Transportation Research Part B, 43(3):301-310.
- Li, Y. and Cassidy, M. (2007). A generalized and efficient algorithm for estimating transit route ods from passenger counts. Transportation Research Part B, 41(1):114-125.
- Lundgren, J. and Peterson, A. (2008). A heuristic for the bilevel origin-destination matrix estimation problem. Transportation Research Part B, 42(4):339-354.
- Maher, M. J. (1983). Inferences on trip matrices from observations on link volumes: a bayesian statistical approach. Transportation Research Part B, 17(6):435- 447.
- Nguyen, S. (1984). Estimating origin-destination matrices from observed flows. In Florian, M., editor, Transportation Planning Models, pages 363-380. NorthHolland, Amsterdam.
- Siikonen, M.-L., Susi, T., and Hakonen, H. (2001). Passenger traffic flow simulation in tall buildings. Elevator World, August:117-123.
- Tsygalnitsky, S. (1977). Simplified methods for transportation planning. Master's thesis, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge.
- Yoneda, K. (2007). Elevator trip distribution for inconsistent passenger input-output data. Decision Making in Manufacturing and Services, 1(1-2):175-190.
- Zuylen, H. V. and Willumsen, L. (1980). The most likely trip matrix estimated from traffic counts. Transportation Research Part B, 14:281-293.

#### Paper Citation

#### in Harvard Style

Kuusinen J., Ruokokoski M., Sorsa J. and Siikonen M. (2013). **Linear Programming Formulation of the Elevator Trip Origin-destination Matrix Estimation Problem** . In *Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-8565-40-2, pages 298-303. DOI: 10.5220/0004338502980303

#### in Bibtex Style

@conference{icores13,

author={Juha-Matti Kuusinen and Mirko Ruokokoski and Janne Sorsa and Marja-Liisa Siikonen},

title={Linear Programming Formulation of the Elevator Trip Origin-destination Matrix Estimation Problem},

booktitle={Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2013},

pages={298-303},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004338502980303},

isbn={978-989-8565-40-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - Linear Programming Formulation of the Elevator Trip Origin-destination Matrix Estimation Problem

SN - 978-989-8565-40-2

AU - Kuusinen J.

AU - Ruokokoski M.

AU - Sorsa J.

AU - Siikonen M.

PY - 2013

SP - 298

EP - 303

DO - 10.5220/0004338502980303