# Mode Choice Change under Environmental Constraints in the Combined Modal Split and Traffic Assignment Model

## Abstract

**:**

## 1. Introduction

_{2})). Among these emission pollutants, CO is typically considered an important measurement because CO is the most critical pollutant and other pollutants have a similar pattern to CO emission [22]. In addition, although the portion of carbon monoxide (CO) is very small in greenhouse gas, it has important indirect effects on global warming [23].

## 2. Environmental Constraint

_{a}= 5 km.

## 3. Combined Modal Split and Traffic Assignment Problem with the Environmental Constraints

#### 3.1. Review of the Combined Models

#### 3.2. Combined Mode and Route Choices

- Mode-choice probability$$\begin{array}{l}{P}_{m|rs}=\frac{{q}_{m}^{rs}}{{q}^{rs}}=\frac{\mathrm{exp}\left(-\tau {w}_{m}^{rs}\right)}{{\displaystyle \sum _{n\in \mathrm{M}}\mathrm{exp}(-\tau {w}_{n}^{rs})}},\forall rs\in RS,m\in M\\ {}^{*}{w}_{m}^{rs}=-\frac{1}{\theta}\mathrm{ln}{\displaystyle \sum _{k\in {\mathrm{K}}_{m}^{rs}}\mathrm{exp}\left(-\theta {c}_{km}^{rs}\right)},\forall rs\in RS,m\in M\end{array}$$
- Route choice probability$${P}_{k|m}=\frac{{f}_{km}^{rs}}{{q}_{m}^{rs}}=\frac{\mathrm{exp}\left(-\theta {c}_{km}^{rs}\right)}{{\displaystyle \sum _{k\in {\mathrm{K}}_{m}^{rs}}\mathrm{exp}\left(-\theta {c}_{km}^{rs}\right)}},\forall m\in M,k\in {K}_{m}^{rs}$$

#### 3.3. Mathematical Programming Formulation with Environmental Constraints

## 4. Solution Algorithm

#### 4.1. Dual Variable Adjustment

#### 4.2. Solution Procedure

- Initial primal variables: ${\left(f\right)}^{n}and{\left(x\right)}^{n}=0$;
- Initial dual variables: ${\left(\mathsf{\lambda}\right)}^{n}and{\left(\mathsf{\rho}\right)}^{n}=0$;
- Initial path set: ${\left(K\right)}^{n}=\varnothing $;
- Compute: ${v}_{a}({\overline{g}}_{a})$ with given ${\overline{g}}_{a}$, using Equation (4).

- ${\left({q}_{m}^{rs}\right)}^{i}=\mathrm{exp}\left(-\tau \left({\left({\lambda}_{}^{rs}\right)}^{i}-{\psi}_{m}^{rs}-\frac{1}{\theta}\mathrm{ln}{\displaystyle \sum _{k\in {K}_{m}^{rs}}\mathrm{exp}\left(-\theta \left({c}_{km}^{rs}+{\displaystyle \sum _{a\in \overline{A}}{\left({\rho}_{a}\right)}^{i}{\delta}_{ka}^{rs}}\right)\right)}\right)\right)$;
- ${\left({f}_{km}^{rs}\right)}^{i}=\mathrm{exp}\left(-\theta \left({c}_{km}^{rs}+\left(\frac{1}{\tau}-\frac{1}{\theta}\right)\mathrm{ln}{\left({q}_{m}^{rs}\right)}^{i}+{\left({\lambda}_{}^{rs}\right)}^{i}+{\displaystyle \sum _{a\in \overline{A}}{\rho}_{a}{\delta}_{ka}^{rs}}\right)\right)$;
- ${\left({x}_{a}^{m}\right)}^{i}={\displaystyle \sum _{rs\in RS}{\displaystyle \sum _{k\in {K}_{m}^{rs}}{\left({f}_{km}^{rs}\right)}^{i}{\delta}_{ka}^{rs}}}$.

- ${\beta}_{a}^{}=-\frac{1}{\theta}\mathrm{ln}\left(\frac{{v}_{a}^{m}({\overline{g}}_{a}^{m})}{{x}_{a}}\right)$ and ${\pi}_{}^{rs}=-\frac{1}{\tau}\mathrm{ln}\left(\frac{{q}^{rs}}{{\displaystyle \sum _{m\in \mathrm{M}}{q}_{m}^{rs}}}\right)$.

- ${\left({\lambda}^{rs}\right)}^{i+1}={\left({\lambda}^{rs}\right)}^{i}+{\pi}^{rs}$ and ${\left({\rho}_{a}\right)}^{m+1}=\mathrm{Max}\left\{0,{\left({\rho}_{a}\right)}^{m}+{\beta}_{a}\right\}$.

- ${\left({q}_{m}^{rs}\right)}^{i+1}=\mathrm{exp}\left(-\tau \left({\left({\lambda}^{rs}\right)}^{i+1}-{\psi}_{m}^{rs}-\frac{1}{\theta}\mathrm{ln}{\displaystyle \sum _{k\in {K}_{m}^{rs}}\mathrm{exp}\left(-\theta \left({c}_{km}^{rs}+{\displaystyle \sum _{a\in \overline{A}}{\left({\rho}_{a}\right)}^{i+1}{\delta}_{ka}^{rs}}\right)\right)}\right)\right)$;
- ${\left({f}_{km}^{rs}\right)}^{i+1}=\mathrm{exp}\left(-\theta \left({c}_{km}^{rs}+\left(\frac{1}{\tau}-\frac{1}{\theta}\right)\mathrm{ln}{\left({q}_{m}^{rs}\right)}^{i+1}+{\left({\lambda}^{rs}\right)}^{i+1}+{\displaystyle \sum _{a\in \overline{A}}{\left({\rho}_{a}\right)}^{i+1}{\delta}_{ka}^{rs}}\right)\right)$;
- ${\left({x}_{a}^{m}\right)}^{i+1}={\displaystyle \sum _{rs\in RS}{\displaystyle \sum _{k\in {K}_{m}^{rs}}{\left({f}_{km}^{rs}\right)}^{i+1}}}{\delta}_{ka}^{rs}$.

## 5. Numerical Results

#### 5.1. Small Network

#### 5.2. Winnipeg Network

^{−8}. The total computational effort to reach RMSE of 1 × 10

^{−8}is less than 12 s (e.g., TCO < 200 case). As decreasing the emission restriction values, computational time grows, because more links are approached to the restricted values.

## 6. Conclusions

_{2}), methane (CH

_{4}), nitrous oxide (N

_{2}O), and ozone (O

_{3})) can be explored and combined into the proposed model.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- EPA (Environmental Protection Agency). U.S. Greenhouse gas Emissions and Sink. 2021. Available online: https://www.epa.gov/sites/production/files/2021-02/documents/us-ghg-inventory-2021-main-text.pdf (accessed on 16 March 2021).
- EPA (Environmental Protection Agency). Basic Information about Carbon Monoxide (CO) Outdoor Air Pollution. 2021. Available online: https://www.epa.gov/co-pollution/basic-information-about-carbon-monoxide-co-outdoor-air-pollution (accessed on 16 March 2021).
- Kheirbek, I.; Haney, J.; Douglas, S.; Ito, K.; Matte, T. The contribution of motor vehicle emissions to ambient fine particulate matter public health impacts in New York City: A health burden assessment. Environ. Health
**2016**, 15, 89. [Google Scholar] [CrossRef] [Green Version] - Kim, S. Decomposition analysis of greenhouse gas emissions in Korea’s transportation sector. Sustainability
**2019**, 11, 1986. [Google Scholar] [CrossRef] [Green Version] - Kyoto Protocol. 1992. Available online: https://en.wikipedia.org/wiki/Kyoto_Protocol (accessed on 14 March 2020).
- Paris Agreement. 2016. Available online: https://en.wikipedia.org/wiki/Paris_Agreement (accessed on 14 March 2020).
- The Korea Herald. 2019. Available online: https://en.wikipedia.org/wiki/Paris_Agreement (accessed on 18 March 2021).
- Reuters. 2018. Available online: https://www.reuters.com/article/us-germany-emissions-factbox-idUSKCN1NK28L (accessed on 18 March 2021).
- Auto.com. 2015. Available online: https://auto.economictimes.indiatimes.com/news/passenger-vehicle/cars/ngts-ban-on-diesel-run-vehicles-over-10-years-old-in-delhi-hits-used-cars-hard/46902519 (accessed on 18 March 2021).
- The Korea Herald. 2018. Available online: http://www.koreaherald.com/view.php?ud=20180121000206 (accessed on 18 March 2021).
- Xu, X.; Chen, A.; Cheng, L. Reformulating environmentally constrained traffic equilibrium via a smooth gap function. Int. J. Sustain. Transp.
**2015**, 9, 419–430. [Google Scholar] [CrossRef] - Florian, M. A traffic equilibrium model of travel by car and public transit modes. Transp. Sci.
**1977**, 11, 166–179. [Google Scholar] [CrossRef] - Abdulaal, M.; LeBlanc, L.J. Methods for combining modal split and equilibrium assignment models. Transp. Sci.
**1979**, 13, 292–314. [Google Scholar] [CrossRef] - Fernandez, E.; de Cea, J.; Floria, M.; Cabrera, E. Network equilibrium models with combined modes. Transp. Sci.
**1994**, 28, 182–193. [Google Scholar] [CrossRef] - Garcia, R.; Marin, A. Network equilibrium with combined modes models and solution algorithms. Transp. Res. Part B
**2005**, 39, 223–254. [Google Scholar] [CrossRef] - Ryu, S.; Chen, A.; Choi, K. Solving the combined modal split and traffic assignment problem with two types of transit impedance function. Eur. J. Oper. Res.
**2017**, 257, 870–880. [Google Scholar] [CrossRef] - Hearn, D.W.; Ribera, J. Bounded flow equilibrium problems by penalty methods. In Proceedings of the IEEE International Conference on Circuits and Computers, New York, NY, USA, 1–3 October 1980. [Google Scholar]
- Tam, M.L.; Lam, W.H.K. Maximum car ownership under constraint of road capacity and parking space. Transp. Res. Part A
**2000**, 34, 145–170. [Google Scholar] [CrossRef] - Tam, M.L.; Lam, W.H.K. Balance for car ownership under user demand and road network supply conditions-case study in Hong Kong. J. Urban Plan. Dev.
**2004**, 130, 24–36. [Google Scholar] [CrossRef] - Li, Z.C.; Huang, H.J.; Lam, W.H.; Wong, S.C. A model for evaluation of transport policies in multimodal networks with road and parking capacity constraints. J. Math. Model. Algorithms
**2006**, 6, 239–257. [Google Scholar] [CrossRef] - Ryu, S.; Chen, A.; Xu, X.; Choi, K. A dual approach for solving the combined distribution and assignment problem with link capacity constraints. Netw. Spat. Econ.
**2014**, 14, 245–270. [Google Scholar] [CrossRef] - Li, Y.; Tan, T.; Li, X. A log-exponential smoothing method for mathematical programs with complementarity constraints. Appl. Math. Comput.
**2012**, 218, 5900–5909. [Google Scholar] [CrossRef] - Green House Online. 2021. Available online: http://www.ghgonline.org/otherco.htm (accessed on 18 March 2021).
- Larsson, T.; Patriksson, M. Side constrained traffic equilibrium models-analysis, computation and applications. Transp. Res. Part B
**1999**, 33, 233–264. [Google Scholar] [CrossRef] [Green Version] - Bell, M.G.H. Stochastic user equilibrium assignment in network with queues. Transp. Res. Part B
**1995**, 29, 125–137. [Google Scholar] [CrossRef] - Yang, H.; Lam, W.H.K. Optimal road tolls under conditions of queuing and congestion. Transp. Res. Part A
**1996**, 30, 319–332. [Google Scholar] - Larsson, T.; Patriksson, M.; Rydegren, C. A column generation procedure for the side constrained traffic equilibrium problem. Transp. Res. Part B
**2004**, 38, 17–38. [Google Scholar] [CrossRef] - Chen, A.; Zhou, Z.; Ryu, S. Modeling physical and environmental side constraints in traffic equilibrium problem. Int. J. Sustain. Transp.
**2011**, 5, 172–197. [Google Scholar] [CrossRef] - Ferrari, P. Road pricing and network equilibrium. Transp. Res. Part B
**1995**, 29, 357–372. [Google Scholar] [CrossRef] - Yang, H.; Bell, M.G.H. Traffic restraint, road pricing and network equilibrium. Transp. Res. Part B
**1997**, 31, 303–314. [Google Scholar] [CrossRef] - Chen, X.; Kim, I. Modelling rail-based park and ride with environmental constraints in a multimodal transport network. J. Adv. Transp.
**2018**, 2018, 1–15. [Google Scholar] [CrossRef] [Green Version] - Wallace, C.E.; Courage, K.G.; Hadi, M.A.; Gan, A.G. TRANSYT-7 F User’s Guide; University of Florida: Gainesville, FL, USA, 1988. [Google Scholar]
- Yin, Y.; Lawphongpanich, L. Internalizing emission externality on road networks. Transp. Res. Part D
**2006**, 11, 292–301. [Google Scholar] [CrossRef] - Nagurney, A.; Qiang, Q.; Nagurney, L.S. Environmental impact assessment of transportation networks with degradable links in an era of climate change. Int. J. Sustain. Transp.
**2010**, 4, 154–171. [Google Scholar] [CrossRef] [Green Version] - Chen, A.; Xu, X. Goal programming approach to solving the network design problem with multiple objectives and demand uncertainty. Expert Syst. Appl.
**2012**, 39, 4160–4170. [Google Scholar] [CrossRef] - Chen, L.; Yang, H. Managing congestion and emissions on road networks with tolls and rebates. Transp. Res. Part B
**2012**, 46, 933–946. [Google Scholar] [CrossRef] - Ng, M.W.; Lo, H.K. Regional air quality conformity in transportation networks with stochastic dependencies: A theoretical copula-based model. Netw. Spat. Econ.
**2013**, 13, 373–397. [Google Scholar] [CrossRef] - Yang, Y.; Yin, Y.; Lu, H. Designing emission charging schemes for transportation conformity. J. Adv. Transp.
**2014**, 48, 766–781. [Google Scholar] [CrossRef] - Xu, X.; Chen, A.; Cheng, L. Stochastic network design problem with fuzzy goals. Transp. Res. Rec.
**2013**, 2399, 23–33. [Google Scholar] [CrossRef] - Szeto, W.Y.; Wang, Y.; Wong, S.C. The chemical reaction optimization approach to solving the environmentally sustainable network design problem. Comput. Aided Civ. Infrastruct. Eng.
**2014**, 29, 140–158. [Google Scholar] [CrossRef] [Green Version] - Boyce, D. Forecasting Travel on congested urban transportation networks: Review and prospects for network equilibrium models. Netw. Spat. Econ.
**2007**, 7, 99–128. [Google Scholar] [CrossRef] - Yao, J.; Chen, A.; Ryu, S.; Shi, F. A general unconstrained optimization formulation for the combined distribution-assignment problem. Transp. Res. Part B
**2014**, 59, 137–160. [Google Scholar] [CrossRef] - Tan, H.; Du, M.; Jiang, X.; Chu, Z. The combined distribution and assignment model: A new solution algorithm and its applications in travel demand forecasting for modern urban transportation. Sustainability
**2019**, 11, 2167. [Google Scholar] [CrossRef] [Green Version] - Wu, Z.X.; Lam, W.H.K. A combined modal split and stochastic assignment model for congested networks with motorized and non-motorized Modes. Transp. Res. Rec.
**2003**, 1831, 57–64. [Google Scholar] [CrossRef] - Canterella, G.E. A general fixed-point approach to multimode, multiuser equilibrium assignment with elastic demand. Transp. Sci.
**1997**, 31, 107–128. [Google Scholar] [CrossRef] - Kitthamkesorn, S.; Chen, A.; Xu, X.; Ryu, S. Modeling mode and route similarities in network equilibrium problem with go-green modes. Netw. Spat. Econ.
**2016**, 16, 33–60. [Google Scholar] [CrossRef] - Florian, M.; Nguyen, S. A combined trip distribution, modal split and trip assignment model. Transp. Res.
**1978**, 12, 241–246. [Google Scholar] [CrossRef] - Boyce, D.E.; Daskin, M.S. Urban Transportation. In Design and Operation of Civil and Environmental Engineering Systems; ReVelle, C., McGarity, A., Eds.; Wiley: New York, NY, USA, 1997. [Google Scholar]
- Wong, K.I.; Wong, S.C.; Wu, J.H.; Yang, H.; Lam, W.H.K. A combined distribution, hierarchical mode choice, and assignment network model with multiple user and mode classes. In Urban and Regional Transportation Modeling; Lee, D.H., Ed.; Edward Elgar Publishing: Cheltnham, UK, 2004; pp. 25–42. [Google Scholar]
- Oppenheim, N. Urban Travel Demand Modeling; John Wiley & Sons, Inc.: New York, NY, USA, 1995. [Google Scholar]
- Yang, C.; Chen, A.; Xu, X. Improved partial linearization algorithm for solving the combined travel-destination-mode-route choice problem. J. Urban Plan. Dev.
**2013**, 139, 22–32. [Google Scholar] [CrossRef] - Beckmann, M.J.; McGuire, C.B.; Winsten, C.B. Studies in the Economics of Transportation; Yale University Press: New Haven, CO, USA, 1956. [Google Scholar]
- Fisk, C. Some developments in equilibrium traffic assignment. Transp. Res. Part B
**1980**, 14, 243–255. [Google Scholar] [CrossRef] - Bell, M.G.H.; Iida, Y. Transportation Network Analysis; Wiley: New York, NY, USA, 1997. [Google Scholar]
- Emme/4 Software. INRO Consultants, Montréal. 2013. Available online: https://www.inrosoftware.com/ (accessed on 10 March 2014).
- Bekhor, S.; Toledo, T. Investigating path-based solution algorithms to the stochastic user equilibrium problem. Transp. Res. Part B
**2005**, 39, 279–295. [Google Scholar] [CrossRef]

**Figure 1.**Procedure for combined modal split and traffic assignment (CMA) model with environmental constraints.

**Figure 10.**Emission-values change map with different restriction values (

**a**) ${g}_{a}^{m}\left({x}_{a}^{m}\right)\cdot {x}_{a}^{m}\le \infty $, (

**b**) ${g}_{a}^{m}\left({x}_{a}^{m}\right)\cdot {x}_{a}^{m}\le 1000$, (

**c**) ${g}_{a}^{m}\left({x}_{a}^{m}\right)\cdot {x}_{a}^{m}\le 300$, (

**d**) ${g}_{a}^{m}\left({x}_{a}^{m}\right)\cdot {x}_{a}^{m}\le 100$.

Constraint | Link 1 | Link 2 | Link 3 | Link 4 | Link 5 | Link 6 | Link 7 | |
---|---|---|---|---|---|---|---|---|

Without constraint | Flow | 43.20 | 23.75 | 19.45 | 17.65 | 25.55 | 8.00 | 11.52 |

TCO | 71.18 | 7.29 | 11.87 | 29.01 | 41.99 | 38.01 | 54.76 | |

TCO $\le 70$ | Flow | 42.50 | 23.36 | 19.13 | 17.37 | 25.13 | 8.17 | 11.76 |

TCO | 70.00 | 7.16 | 11.67 | 28.53 | 41.30 | 38.82 | 55.89 | |

TCO $\le 60$ | Flow | 36.47 | 20.05 | 16.42 | 14.45 | 22.02 | 9.94 | 12.62 |

TCO | 60.00 | 6.12 | 9.99 | 23.75 | 36.18 | 47.23 | 60.00 | |

TCO $\le 50$ | Flow | 30.41 | 16.72 | 13.69 | 11.31 | 19.10 | 10.52 | 10.52 |

TCO | 50.00 | 5.09 | 8.32 | 18.59 | 31.38 | 50.00 | 50.00 | |

TCO $\le 40$ | Flow | 24.34 | 13.38 | 10.96 | 9.23 | 15.11 | 8.42 | 8.42 |

TCO | 40.00 | 4.07 | 6.65 | 15.17 | 24.82 | 40.00 | 40.00 | |

TCO $\le 30$ | Flow | 18.26 | 10.04 | 8.22 | 7.05 | 11.21 | 6.32 | 6.32 |

TCO | 30.00 | 3.05 | 4.99 | 11.59 | 18.41 | 30.00 | 30.00 | |

TCO $\le 20$ | Flow | 12.17 | 6.69 | 5.48 | 4.78 | 7.39 | 4.21 | 4.21 |

TCO | 20.00 | 2.03 | 3.33 | 7.86 | 12.14 | 20.00 | 20.00 |

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**MDPI and ACS Style**

Ryu, S.
Mode Choice Change under Environmental Constraints in the Combined Modal Split and Traffic Assignment Model. *Sustainability* **2021**, *13*, 3780.
https://doi.org/10.3390/su13073780

**AMA Style**

Ryu S.
Mode Choice Change under Environmental Constraints in the Combined Modal Split and Traffic Assignment Model. *Sustainability*. 2021; 13(7):3780.
https://doi.org/10.3390/su13073780

**Chicago/Turabian Style**

Ryu, Seungkyu.
2021. "Mode Choice Change under Environmental Constraints in the Combined Modal Split and Traffic Assignment Model" *Sustainability* 13, no. 7: 3780.
https://doi.org/10.3390/su13073780