Mathematical Modelling of
Travel Behavior
Theory and Advanced Applications
Dr. Yu Jiang
LUMS | January 2026
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"The Art of Japanese Swordsmanship"
Motivation
Understanding network flow dynamics
Motivation
Toy Three-Link Network: Constant Costs
OD Demand: 4 units from (1) to (3).
- Costs: $t_1 = 2, t_2 = 1, t_3 = 3$
- Path 1 Cost: $2 + 3 = 5$
- Path 2 Cost: $1 + 3 = 4$
- » Everyone chooses Path 2.
Motivation
Link Performance Function
Time
($t_a$)
Flow
($x_a$)
- Travel Time ($t_a$): Function of flow.
- Congestion: Delay increases rapidly as flow approaches capacity.
-
$$t_a = t_0 \left[ 1 + \alpha \left( \frac{x_a}{C_a} \right)^\beta \right]$$
Theory
Mathematical Foundations of Equilibrium
User Equilibrium
Wardrop's First Principle
"The journey times on all used routes are equal and minimal."
$$C_r^w(\mathbf{f}^*) - \pi^w \begin{cases} = 0, & \text{if } f_r^{w*} > 0, \\ \ge 0, & \text{if }
f_r^{w*} = 0. \end{cases}$$
Where $\pi^w = \min_j C_j^w(\mathbf{f})$ represents the minimum travel cost for OD pair $w$.
User Equilibrium
Optimization Problem
$$\min \sum_a \int_0^{x_a} t_a(\omega)d\omega$$
Subject to:
- Demand: $\sum_k f_k^{rs} = q_{rs}$
- Conservation: $x_a = \sum_{rs} \sum_k f_k^{rs} \delta_{ak}$
- Non-negativity: $f_k^{rs} \ge 0$
Stochastic User Equilibrium
Utility-Based Choice
$$U_{ik} = V_k + \epsilon_{ik}$$
- $V_k$: Systematic utility.
- $\\epsilon_{ik}$: Random error term.
The Logit Model
$$P_k = \frac{\exp(\theta V_k)}{\sum_j \exp(\theta V_j)}$$
Applications
Network Design
Infrastructure Planning
- Question: Which links to build?
-
Bilevel Problem:
- Leader: Traffic Planner ($\mathbf{y}$)
- Follower: Travelers ($\mathbf{x}$)
Algorithms
01
Exact Methods
02
Heuristics, Meta-Heuristics, Matheuristics
03
Machine Learning
"There's no art to find the mind's construction in the face."
— William Shakespeare
We are building that missing art
through mathematical modelling.
Smarter City, Smoother Trip
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