Trip Mode Split 1 - Combined Trip Distribution and Modal-Split Models

Trip Mode Split 1 - Combined Trip Distribution and Modal-Split Models

1. Introduction

1.1 Modal Split or Travel Mode Choice

Input : O-D trip matrix \([T_{ij}]_{z \times z}\) generated by a trip generation model covers more than one travel mode such as auto and public transport modes

Typical urban mobility modes:

• Motorized modes (auto mode and public transport modes)

• and active mobility modes (walking and bicycling).

Aims : to split or distribute personal trips \(T_ij\) from zone \(i\) (i.e., origin \(i\)) to zone \(j\) (i.e., destination \(j\)), which are obtained in the trip distribution phase, into various travel modes used by travelers.

Modal split ( or travel model choice) is typically performed after trip distribution.

1.2 About Singapore

• Cost of Getting and Operating a Car in Singapore, source

  • Certificate of Entitlement (COE)
  • Open Market Value (OMV)
  • Additional Registration Fee (ARF)
  • Excise Duty
  • Registration Fee
  • Vehicular Emissions Scheme (VES)
  • Other Charges (IU fee, car plate fee, dealer's commission, and etc.)

• Facts and usage statistics about public transport in Singapore, source

  • Commute Time
  • Waiting Time
  • Journey Distance
  • Number of Transfers

1.3 Importance of Travel Mode Choice Modelling

• Travel mode choice model is probably one of the most important models in urban mobility modelling
• The issue of travel mode choice, is probably the single most important element in transport planning and policy making.
• Travel mode choice affects the general travel efficiency with which we can travel in urban areas, the amount of urban space devoted to transport functions, and whether a range of choices is available to travelers.
• It is important to develop tangible models, which are sensitive to those attributes of travel that influence individual choice of travel mode.

1.4 Factors Influencing Travel Mode Choice

• Characteristics of trip-makers (i.e., travelers)
• Characteristics of trips
• Characteristics of transport facilities (including both quantitative and qualitative factors), (i.e., travel modes).

(1) Characteristics of Trip-Makers (i.e., Travelers)

• Car availability and/or ownership
• Possession of a driving license
• Household structure (young couple, couple with children, retired, singles, etc.)
• Decisions made elsewhere, for example the need to use a car at work, take children to school, etc.
• Income
• Residential density

(2) Characteristics of Trips

Trip purpose : For example, the trip to work is normally easier to undertake by public transport than other trips because of its regularity and the adjustment possible for a long run.

Time of day : when the trip is undertook. Late trips are more difficult to accommodate by public transport.

(3) Characteristics of Transport Facilities (i.e., Travel modes)

• Quantitative factors such as:

  • Relative travel time: in-vehicle time, waiting time and walking time by each mode
  • Relative monetary costs (fares, fuel and direct cost)
  • Availability and cost of parking

• Qualitative factors such as

  • Comfort and convenience
  • Reliability and regularity
  • Protection and security

1.5 Two Types of Travel Mode Choice Models

• Aggregate travel mode choice models: Based on zonal (and inter-zonal) information.

  • Proportion or percentage of travelers/commuters using each mode

• Disaggregate travel mode choice models: Based on household and/or individual information.

  • Probability of individual travelers/commuters using each mode

2. Combined Trip Distribution and Modal-Split Models

2.1 Empirical Modal-Split Models

• Suppose: \(k=0,1\) two trip modes: \(k=0\) represents private car and \(k=1\) represents transit

  \(T_{ij}^{0}\) ans \(T_{ij}^{1}\) are the trip counts of the two modes for zone \(i\) to zone \(j\)

  \(C_{ij}^{0}\) and \(C_{ij}^{1}\) generalized travel cost by private car and transit

  $\beta $ non-negative parameter

\[P_{ij}^{i} = \frac{T_{ij}^{0}}{T_{ij}^{0} + T_{ij}^{1}} = \frac{T_{ij}^{0}}{T_{ij}} = \frac{1}{1 + \exp \left[-\beta (C_{ij}^{0} - C_{ij}^{1}) \right]} \]

• Two limit scenarios:

  \(\beta \rightarrow \infty\) : All or nothing choice

  \(\beta \rightarrow 0\) : 50% vs 50%

(2) Binary logit choice model

\[P_{ij}^{0} = \frac{T_{ij}^{0}}{T_{ij}^{0} + T_{ij}^{1}} = \frac{T_{ij}^{0}}{T_{ij}} = \frac{\exp (-\beta \, C_{ij}^{0})}{\exp (-\beta \, C_{ij}^{0}) + \exp (-\beta \,C_{ij}^{1})} \]

where \(P_{ij}^{0}\) is the proportion of trips traveling from zone \(i\) to zone \(j\) via mode 0.

(3) Basic Properties of Logit Model

• It generates an S-shaped curve (similar to some of the empirical diversion curves)

• If \(C_1 = C_2\), then \(P_1 = P_2 = 0.5\)

• If \(C_1 >> C_2\), then \(P_1 \rightarrow 1.0\)

• The model can easily be extended to multiple travel modes:

  \[P_{ij}^{k} = \frac{\exp (-\beta \, C_{ij}^{k})}{\sum_{l=1}^{K} \exp (-\beta \,C_{ij}^{l})} \]

2.2 Combined Trip Distribution and Modal Split Model

• Combined O-D distribution and modal-split model: Given \(O_i\) , \(D_j\) , estimate OD matrix by travel mode.

• Given the macroscopic states, estimate the most likely O-D matrix distribution by different travel modes.

(1) Trip Distribution Model

The classical trip distribution model

\[\begin{array}{cl} \displaystyle \max_{T_{ij}^{k}} H \left(T_{ij}^{k} \right) = & \displaystyle \sum_{i,j,k} \left( T_{ij}^{k} \ln T_{ij}^{k} - T_{ij}^k \right) \\ \text{s.t.} & \sum_{jk} T_{ij}^{k} - O_i = 0 \\ & \sum_{ik} T_{ij}^{k} - D_j = 0 \\ & \sum_{ijk} T_{ij}^{k} \, C_{ij}^{k} - C = 0 \\ \end{array} \]

where \(C\) is the total expenditure in travel in the system.

By Lagrange multiplier approach, we have:

\[T_{ij}^{k} = A_i O_i B_j D_j \exp(-\beta \, C_{ij}^{k}) \]

(2) Combined Trip Distribution and Modal Split Model

\[T_{ij}^{k} = A_i O_i B_j D_j \exp(-\beta \, C_{ij}^{k}) \quad \text{and} \quad P_{ij}^{k} = \frac{\exp (-\beta \, C_{ij}^{k})}{\sum_{l=1}^{K} \exp (-\beta \,C_{ij}^{l})} \]

• In this model, \(\beta\) plays double roles:

  • It acts as the parameter controlling dispersion in mode choice.

  • It acts as the parameter controlling dispersion in the choice between destinations at different travel costs from the origin.

• The model is lack of flexibility. A more practical combined trip distribution/modal-split model should be developed.

(3) Modified Logit-based Mode Choice Model

\[T_{ij}^{kn} = A_i^n \cdot O_i^n \cdot B_j^n \cdot D_j^n \cdot \exp(-\beta_n \, K_{ij}^{n}) \cdot \frac{\exp (-\lambda_n \, C_{ij}^{k})}{\sum_{l=1}^{K} \exp (-\lambda_n \,C_{ij}^{l})} \]

• where \(K_{ij}^{n}\) is the composite cost of traveling from zone \(i\) to zone \(j\) as perceived by person type \(n\).

  \(\beta_n\): control O-D trip distribution

  \(\lambda_n\): control modal split

(3) How to Determine Composite Cost: \(K_{ij}^n\)

Composite cost may be specified in different ways:

• It could be taken to be the minimum of costs incurred by different travel modes.

• It could be taken to be the weighted average of these:

  \[K = \sum_k P_k C_k \quad \text{where} \quad \sum_k P^k = 1.0 \]

  where \(i\) , \(j\) and \(n\) are omitted for simplicity.

Remark : For average method, the inequality \(\sum_k P_k C_k \geq \min_{k} \{ C_{k} \}\) always holds. Thus, using the average is not a good method. Because it is nonsensical as the introduction of a new option, even if it is more expensive, should not increase the composite costs; at worst they should remain the same. The use of the wrong composite costs will lead to misspecified models.

(4) Corrected Composite Cost

It has been shown that the only the following corrected specification, consistent with the prevailing rational choice behavior is :

\[K_{ij}^{n} = - \frac{1}{\lambda_n} \ln \left[ \sum_k \exp \left(- \lambda_n C_{ij}^{kn} \right) \right] \]

where the parameter \(\lambda_n\) must satisfy: \(\beta_n \leq \lambda_n\)

• Properties of the Corrected Composite Cost

  • \(\displaystyle K_{ij}^{n} \leq \min_k \{ C_{ij}^{kn}\}\)

  • \(\displaystyle \lim_{\lambda_n \rightarrow \infty} K_{ij}^{n} = \min_{k} \{ C_{ij}^{k} \}\), this is "all-or-nothing" travel mode choice

  • \(\displaystyle \frac{\mathrm{d} K_{ij}^{n}}{\mathrm{d} C_{ij}^{k}} = P_{ij}^{k}\)

3. Multimodal-Split Models

3.1 Model Structures for Travel Mode Choice Involving More Than Two Modes

Classical example: the red-blue bus problem

• Suppose the travel costs of car \(C_c\) and bus \(C_B\) are equal: \(C_c = C_B\)

  the travel costs of red bus \(C_{RB}\) and blue bus \(C_{BB}\) should be equal: \(C_B = C_{RB} = C_{BB}\)

  \(\Pr_{C}\), \(\Pr_{RB}\), and \(\Pr_{BB}\) are the probabilities of choosing the car, red bus, and blue bus, respectively.

(1) N-way Structure

• The logit model of a N-way structure can be applied when the travel mode choice set (available travel modes) includes distinct modes.

• Two (or more) similar or identical travel modes (blue and red buss, or even and odd number buses) should be treated as a single travel mode.

• For the red-blue bus problem:

  \[\Pr{}_C = \frac{\exp(-\beta \, C_C)}{\exp(-\beta \, C_{C}) + \exp(-\beta \, C_{RB}) + \exp(-\beta \, C_{BB})} = \frac{1}{3}, \quad \Pr{}_{RB} = \frac{1}{3}, \quad \Pr{}_{BB} = \frac{1}{3} \]

(2) Hierarchical or Nested Structure

• The travel modes which have common elements are taken together in a primary split.

• After they have been 'separated' from the uncorrelated option, they are subdivided in a secondary split.

For the red-blue bus problem:

• The first layer:

  \[\Pr{}_C = \frac{\exp(-\lambda_1 \, C_C)}{\exp(-\lambda_1 \, C_C) + \exp(-\lambda_1 \, \tilde{C}_B)}, \quad \Pr{}_B = 1 - \Pr{}_C = \frac{\exp(-\lambda_1 \, \tilde{C}_B)}{\exp(-\lambda_1 \, C_C) + \exp(-\lambda_1 \, \tilde{C}_B)}, \]

  where \(\tilde{C}_B\) is the composite cost of choosing bus:

  \[\tilde{C}_B = -\frac{1}{\lambda_2} \ln \left[ \exp(-\lambda_2 \, C_{RB}) + \exp(-\lambda_2 \, C_{BB}) \right] \]

  and \(C_C\):

  \[C_C = \tilde{C}_C = -\frac{1}{\lambda_1} \ln \left[ \exp(-\lambda_1 \, C_{C}) \right] = C_C \]

• The second layer:

  \[\Pr{}_{R|B} = \frac{\exp(-\lambda_2 \, C_{RB})}{\exp(-\lambda_2 \, C_{RB}) + \exp(-\lambda_2 \, C_{BB})}, \quad \Pr{}_{B|B} = 1 - \Pr{}_{R|B} = \frac{\exp(-\lambda_2 \, C_{BB})}{\exp(-\lambda_2 \, C_{RB}) + \exp(-\lambda_2 \, C_{BB})}, \]

• We have \(\Pr{}_{R|B} = \Pr{}_{B|B} = \frac{1}{2}\)

• If \(C_C = \tilde{C}_B\), then \(\Pr{}_C = \Pr{}_B = \frac{1}{2}\), and

  \(\Pr{}_{RB} = \Pr{}_{R|B} \Pr{}_B = \frac{1}{4}\),

  \(\Pr{}_{BB} = \Pr{}_{B|B} \Pr{}_B = \frac{1}{4}\)

4. Calibration of Binary Logit Models

By introducing the modal penalty \(\delta\) (assumed to be associated to the second mode), we have:

\[P_{ij}^{0} = \frac{\exp(-\lambda \, C_{ij}^0)}{\exp \left(-\lambda \, C_{ij}^0 \right) + \exp \left[-\lambda \, (C_{ij}^1 + \delta) \right]} = \frac{1}{1+ \exp \left[-\lambda (C_{ij}^1 + \delta - C_{ij}^0) \right]} \]

4.1 Simple Linear Regression Method

We have:

\[\frac{P^0}{1-P^0} = \frac{1}{\exp \left[-\lambda (C^1 + \delta - C^0) \right]} = \exp \left[\lambda (C^1 + \delta - C^0) \right] \]

and taking logarithms of both sides and rearranging, we have:

\[\ln \left( \frac{P^0}{1-P^0} \right) = \lambda (C^1 - C^0) + \lambda \delta \]

We have observed data for \(P\) and \(C\), and therefore the only unknowns are \(\lambda\) and \(\delta\). Thus, we can use linear regression method to estimate the coefficient \(\lambda\) and intercept \(\lambda \delta\).

5. Applications of Mode Choice Models

5.1 Evaluation of the impact of Congestion Pricing on Modal Split

5.2 Evaluate Bus Lane Scheme

Reference

[1] J. de D. Ortúzar S. and L. G. Willumsen, "Chapter 6 Modal Split and Direct Demand Models", in Modelling Transport, Fourth edition. Chichester, West Sussex, United Kingdom: John Wiley & Sons, 2011, p.p. 207-223.