Unlock Holiday Efficiency: Mathematical Modeling Tips for Travel, Crowds, and Budgets

National Day is one of China's longest holidays, attracting billions of trips, consumption, and entertainment. Behind the festivities lie numerous practical problems that can be tackled with mathematical modeling, from travel‑route optimization to crowd‑flow prediction and personal budgeting.

1. Travel‑Route Optimization

Tourists planning a multi‑city trip over seven days face a classic optimization problem: minimize total travel cost or maximize overall benefit.

1.1 Traveling Salesman Problem (TSP) Model

Given n cities, the decision variable x_{ij} indicates whether the traveler moves directly from city i to city j . The objective is to minimize total travel distance, subject to constraints that each city is visited exactly once and subtour elimination (MTZ constraints) is enforced.

1.2 Multi‑Objective Optimization Model

Real‑world travel considers distance, time, cost, and scenic scores. By assigning weights to each objective, the multi‑objective problem can be transformed into a single‑objective weighted sum, though the choice of weights is subjective and may not capture all Pareto‑optimal solutions. Advanced methods such as ε‑constraint or NSGA‑II can provide a more comprehensive solution set.

2. Scenic‑Area Crowd‑Flow Prediction

Managers need to forecast visitor numbers to allocate resources, control crowds, and ensure safety.

2.1 ARIMA Time‑Series Model

The ARIMA(p,d,q) model captures temporal patterns in tourist counts. Parameters are estimated via maximum likelihood or least squares, and model order is selected using AIC or BIC criteria.

2.2 Multiple Linear Regression Model

A regression model relates predicted visitor count Y to factors such as weather index, ticket price, holiday type, historical visitor numbers, and regression coefficients, with random error assumed to be white noise. Model fit is evaluated by the coefficient of determination (R²).

3. Holiday Spending Budget Optimization

Visitors aim to maximize holiday experience within a limited budget using utility theory and optimization.

3.1 Utility Maximization Model

Assuming n spending categories (accommodation, food, tickets, shopping, etc.), let x_i be the amount spent on category i and U_i(x_i) the corresponding utility, often modeled as a logarithmic function to reflect diminishing marginal utility. The total utility is the sum of individual utilities, subject to a budget constraint ∑x_i = B and non‑negativity constraints. The problem can be solved with Lagrange multipliers.

3.2 Dynamic Programming Approach

Dividing the holiday into T days, the state V(t,b) represents the maximum achievable utility from day t onward with remaining budget b . The recursion and boundary conditions allow backward induction to obtain the optimal daily spending plan.

4. Traffic Congestion Analysis and Prediction

National Day traffic jams are common. Queueing theory and traffic‑flow models help analyze congestion mechanisms and propose mitigation measures.

4.1 Queueing Theory Model

Model a highway toll plaza as an M/M/1 queue: arrivals follow a Poisson process with rate λ, service times are exponentially distributed with rate μ. System utilization ρ = λ/μ must be less than 1 for stability. Standard formulas give average number of vehicles, waiting time, and queue length.

4.2 Macroscopic Traffic‑Flow Model

The continuity equation relates traffic density k (vehicles/km), flow q (vehicles/h), and speed v (km/h). Greenshields’ linear speed‑density relationship ( v = v_f (1 - k/k_j) ) yields a quadratic flow‑density function, whose maximum identifies optimal density and speed for maximum road capacity.

5. Scenic‑Area Capacity and Safety Management

Capacity calculations ensure safety by considering spatial, resource, ecological, and psychological limits.

5.1 Spatial Capacity Model

Daily spatial capacity = (available area / per‑person space) × (effective open hours / average visit time). A safety factor (0.7–0.8) adjusts the theoretical maximum to a realistic safe limit.

5.2 Emergency Evacuation Time Estimation

A simplified formula estimates evacuation time based on total crowd size, total exit width, average walking speed, and maximum density. The model assumes uniform, panic‑free flow and is useful for quick, rough planning; detailed safety plans require cellular automata, social‑force, or agent‑based simulations.

Mathematical modeling not only solves concrete problems but also cultivates systematic thinking: decomposing complex issues, abstracting and quantifying them, discovering patterns amid uncertainty, and seeking optimal solutions under constraints. By applying these methods to National Day scenarios, readers can appreciate the beauty and practical value of mathematics.