Skip to main content
SpringerLink
Log in

Optimizing Network Flows with Congestion-Based Flow Reductions

  • Published:
Networks and Spatial Economics Aims and scope Submit manuscript

Abstract

When optimizing traffic systems using time-expanded network flow models, traffic congestion is an important consideration because it can decrease both the discharge traffic flow rate and speed. One widely used modeling framework is the Cell Transmission Model (CTM) (see Daganzo, Transp Res-B 28(4):269–287, 1994, Transp Res-B 29(2):79–93, 1995), which is implemented in a linear program (LP) in Ziliaskopoulos (Transp Sci 34(1):37–49, 2000). While the CTM models the reduction in speed associated with congestion and the backward propagation of congestion, it does not properly model the reduction in discharge flow from a bottleneck after the onset of congestion. This paper discusses this issue and proposes a generalization of the CTM that takes into account this important phenomena. Plainly, an optimization that does not consider this important negative result of congestion can be problematic, e.g., in an evacuation setting such an optimization would assume that congestion does not impact network clearance time, which can result in poor evacuation strategies. In generalizing the CTM, a fairly simple modification is made, yet it can have significant impacts on the results. For instance, we show that for the generalized CTM the traffic holding (a result of the linearization of the CTM flow constraints) plays a more harmful role, which thus requires a scheme to eliminate traffic holding. In this paper, we propose a mixed binary program to eliminate traffic holding, along with methods to improve solvability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. A slightly different equation is used for links that enter a merge cell or leave a diverge cell (see Daganzo 1994, for details). Also if i is a source, or j a sink, we can eliminate their respective terms.

  2. (See Kalafatas and Peeta 2006; Dixit et al. 2008) recommend setting the δ-parameter to 1 if Q i /(N i  − Q i ) is in the range of 1/6–1/4 for freeway modeling.

  3. INTEGRATION has been validated against standard traffic flow theory (Rakha and Crowther 2002, 2003; Dion et al. 2004) and been utilized for the evaluation of real-life applications (Rakha et al. 1998, 2005).

References

  • Banks JH (1990) Flow processes at a freeway bottleneck. Transp Res Rec 1287:20–28

    Google Scholar 

  • Bertini RL, Leal MT (2005) Empirical study of traffic features at a freeway lane drop. J Transp Eng 131(6):397–407

    Article  Google Scholar 

  • Bish DR, Sherali HD (2012) Aggregate-level demand management in evacuation planning. Eur J Oper Res. doi:10.1016/j.ejor.2012.07.036

  • Bish DR, Sherali HD, Hobeika AG (2011) Optimal evacuation strategies with staging and routing. Manuscript, Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA

  • Carey M (1987) Optimal time-varying flows on congested networks. Oper Res 35(1):58–69

    Article  Google Scholar 

  • Cassidy MJ, Bertini RL (1999) Some traffic features at freeway bottlenecks. Transp Res-B 33(1):25–42

    Article  Google Scholar 

  • Chalmet LG, Francis RL, Saunders PB (1982) Quickest flows over time. Manage Sci 28(1):86–105

    Article  Google Scholar 

  • Chamberlayne EP (2011) The use of dynamic network flows with congestion for evacuation planning. PhD Dissertation, Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, Virginia

  • Chamberlayne EP, Rakha HA, Bish DR (2011) Modeling the capacity drop phenomenon at freeway bottlenecks using the INTEGRATION software. Manuscript, Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA

  • Chiu Y, Zheng H, Villalobos J, Gautam B (2007) Modeling no-notice mass evacuation using a dynamic traffic flow optimization model. IIE Trans 39(1):83–94

    Article  Google Scholar 

  • Chung BD, Yao T, Xie C, Thorsen A (2011) Robust optimization model for a dynamic network design problem under demand uncertainty. Netw Spat Econ 11(2):371–389

    Article  Google Scholar 

  • Chung BD, Yao T, Zhang B (2012) Dynamic traffic assignment under uncertainty: a distributional robust chance-constrained approach. Netw Spat Econ 12(1):167–181

    Article  Google Scholar 

  • Chung K, Rudjanakanoknad J, Cassidy MJ (2007) Relation between traffic density and capacity drop at three freeway bottlenecks. Transp Res-B 41(1):82–95

    Article  Google Scholar 

  • Daganzo CF (1994) The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp Res-B 28(4):269–287

    Article  Google Scholar 

  • Daganzo CF (1995) The cell transmission model, part 2: network traffic. Transp Res-B 29(2):79–93

    Article  Google Scholar 

  • Dion F, Rakha HA, Kang YS (2004) Comparison of delay estimates at under-saturated and oversaturated pre-timed signalized intersections. Transp Res-B 38(2):99–122

    Article  Google Scholar 

  • Dixit VV, Ramasamy S, Radwan E (2008) Assessment of I-4 contraflow plans: microscopic versus mesoscopic simulation. Transp Res Rec: J Transp Res Board 2041:89–97

    Article  Google Scholar 

  • El-Metwally M, Rakha HA (2011) Comparison of queue discharge rates from time-dependent and time-independent bottlenecks. Manuscript, Charles E. Via, Jr. Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA

  • Gomes G, Horowitz R (2006) Optimal freeway ramp metering using the asymmetric cell transmission model. Transp Res-C 14:244–262

    Article  Google Scholar 

  • Hall FL, Agyemang-Duah K (1991) Freeway capacity drop and the definition of capacity. Transp Res Rec 1320:91–98

    Google Scholar 

  • Han LD, Yuan F, Urbanik T (2007) What is an effective evacuation operation? J Urban Plann Dev 133(1):3–8

    Article  Google Scholar 

  • Kalafatas G, Peeta S (2006) An exact graph structure for dynamic traffic assignment: formulation, properties, computational experience. In: 86th annual meeting of the transportation research board

  • Kerner BS, Klenov SL (2006) Probabilistic breakdown phenomenon at on-ramp bottlenecks in three-phase traffic theory. Transp Res Rec: J Transp Res Board 1965:70–78

    Article  Google Scholar 

  • Li Y, Waller ST, Ziliaskopoulos AK (2003) A decomposition scheme for system optimal dynamic traffic assignment models. Netw Spat Econ 3:441–455

    Article  Google Scholar 

  • Lighthill MJ, Whitham GB (1955) On kinematic waves. I. Flood movement in long rivers. Proc R Soc Lond, A Math Phys Sci (1934–1990) 229:281–316

    Article  Google Scholar 

  • Lin WH, Wang C (2004) An enhanced 0-1 mixed-integer LP formulation for traffic signal control. IEEE Trans Intell Transp Syst 5(4):238–245

    Article  Google Scholar 

  • Liu Y, Lai X, Chang GL (2006) Cell-based network optimization model for staged evacuation planning under emergencies. Transp Res Rec 1964:127–135

    Article  Google Scholar 

  • Lo HK (2001) A cell-based traffic control formulation: strategies and benefits of dynamic timing plans. Transp Sci 35(2):148–164

    Article  Google Scholar 

  • May AD (1990) Traffic flow fundamentals. Prentice Hall, New Jersey

    Google Scholar 

  • Nagel K, Wagner P, Woesler R (2003) Still flowing: approaches to traffic flow and traffic jam modeling. Oper Res 51(5):681–710

    Article  Google Scholar 

  • Nie Y (2011) A cell-based Merchant-Nemhauser model for the system optimum dynamic traffic assignment problem. Transp Res-B 45:329–342

    Article  Google Scholar 

  • Papageorgiou M (1998) Some remarks on macroscopic traffic flow modelling. Transp Res-A 32(5):323–329

    Google Scholar 

  • Papageorgiou M, Diakaki C, Dinopoulou V, Kotsialos A, Wang Y (2003) Review of road traffic control strategies. Proc IEEE 91(12):2043–2067

    Article  Google Scholar 

  • Rakha HA, Crowther B (2002) A comparison of the Greenshields, Pipes, and Van Aerde car-following and traffic stream models. Transp Res Rec 1802:248–262

    Article  Google Scholar 

  • Rakha HA, Crowther B (2003) Comparison and calibration of FRESIM and INTEGRATION steady-state car-following behavior. Transp Res-A 37:1–27

    Article  Google Scholar 

  • Rakha HA, Flintsch AM, Ahn K, El-Shawarby I, Arafeh M (2005) Evaluating alternative truck management strategies along Interstate 81. Transp Res Rec 1925:76–86

    Article  Google Scholar 

  • Rakha HA, Van Aerde M, Bloomberg L, Huang X (1998) Construction and calibration of a large-scale microsimulation model of the Salt Lake area. Transp Res Rec 1644:93–102

    Article  Google Scholar 

  • Richards PI (1956) Shock waves on the highway. Oper Res 4(1):42–51

    Article  Google Scholar 

  • Sbayti H, Mahmassani HS (2006) Optimal scheduling of evacuation operations. In: TRB 2006 annual meeting, Washington DC

  • Schnhof M, Helbing D (2007) Empirical features of congested traffic states and their implications for traffic modeling. Transp Sci 41(2):135–166

    Article  Google Scholar 

  • Shen W, Nie Y, Zhang HM (2007) Dynamic network simplex method for designing emergency evacuation plans. Transp Res Rec 2022:83–93

    Article  Google Scholar 

  • Tuydes H, Ziliaskopoulos A (2006) Tabu-based heuristic approach for optimization of network evacuation contraflow. Transp Res Rec 1964:157–168

    Article  Google Scholar 

  • Ukkusuri SV, Waller ST (2008) Linear programming models for the user and system optimal dynamic network design problem: formulations, comparisons and extensions. Netw Spat Econ 8(4):383–406

    Article  Google Scholar 

  • Van Aerde M, Rakha H (2007a) INTEGRATION © RELEASE 2.30 FOR WINDOWS: user’s guide—volume II: advanced model features. M. Van Aerde & Assoc., Ltd., Blacksburg, VA

  • Van Aerde M, Rakha H (2007b) INTEGRATION © RELEASE 2.30 FOR WINDOWS: user’s guide—volume I: fundamental model features. M. Van Aerde & Assoc., Ltd., Blacksburg, VA

  • Xie C, Lin DY, Waller ST (2009) A dynamic evacuation network optimization problem with lane reversal and crossing elimination strategies. Transp Res-E 46(3):295–316

    Article  Google Scholar 

  • Yao T, Mandala SR, Chung BD (2009) Evacuation transportation planning under uncertainty: a robust optimization approach. Netw Spat Econ 9(2):171–189

    Article  Google Scholar 

  • Yazici MA, Ozbay K (2008) Evacuation modelling in the United States: does the demand model choice matter? Transp Rev 28(6):757–779

    Article  Google Scholar 

  • Ziliaskopoulos AK (2000) A linear programming model for the single destination system optimal dynamic traffic assignment problem. Transp Sci 34(1):37–49

    Article  Google Scholar 

Download references

Acknowledgements

The computational work was performed with support from the Simulation and Optimization Laboratory in the Grado Department of Industrial and Systems Engineering at Virginia Tech. This work has been partially supported by the National Science Foundation under Grant Number 0825611 and 1055360 and the Mid-Atlantic University Transportation Center (MAUTC). The authors also thank two anonymous reviewers for their constructive suggestions that have helped improve this paper.

Author information

Authors and Affiliations

  1. Grado Department of Industrial and Systems Engineering (0118), Virginia Tech, Blacksburg, VA, 24061, USA

    Douglas R. Bish

  2. Charleston District, US Army Corps of Engineers, 69A Hagood Avenue, Charleston, SC, 29403, USA

    Edward P. Chamberlayne

  3. Charles E. Via, Jr. Department of Civil and Environmental Engineering (0105), Virginia Tech, Blacksburg, VA, 24061, USA

    Hesham A. Rakha

Authors
  1. Douglas R. Bish
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edward P. Chamberlayne
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hesham A. Rakha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Douglas R. Bish.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bish, D.R., Chamberlayne, E.P. & Rakha, H.A. Optimizing Network Flows with Congestion-Based Flow Reductions. Netw Spat Econ 13, 283–306 (2013). https://doi.org/10.1007/s11067-012-9181-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11067-012-9181-3

Keywords

Search

Navigation