写4个数学建模的经典案例及代码程序

数学建模是一种将实际问题转化为数学语言的过程，通过解决这些问题来得出量化解决方案。以下是四个经典数学建模案例以及简化的示例代码：

1. **人口增长模型** - 梯度增长模型（如 logistic growth model）

import numpy as np

def population_growth(initial_population, growth_rate, carrying_capacity):
    time = np.linspace(0, 50, 100)  # 时间范围
    population = initial_population * (carrying_capacity / (1 + growth_rate * np.exp(-time)))
    return population

# 示例用法
initial_pop = 1000
growth_rate = 0.05
cap_carrying = 10000
population_series = population_growth(initial_pop, growth_rate, cap_carrying)

2. **经济预测模型** - 线性回归模型（如股票价格预测）

from sklearn.linear_model import LinearRegression

# 假设我们有历史股票数据
data = {'days': [1, 2, 3, ..., 100], 'prices': [10, 11, 12, ..., 20]}
model = LinearRegression()
model.fit(data['days'].reshape(-1, 1), data['prices'])

# 预测未来一天的价格
future_day = 101
predicted_price = model.predict([[future_day]])

3. **物流路线优化** - 贪心算法或遗传算法（如旅行商问题）

def tsp(city_list):
    n = len(city_list)
    best_path = [city_list[0]]
    current_city = city_list[0]
    total_distance = 0
    
    for _ in range(n - 1):
        min_distance = float('inf')
        next_city = None
        for i in range(1, n):
            if i != current_city and distance_matrix[current_city][i] < min_distance:
                min_distance = distance_matrix[current_city][i]
                next_city = i
        
        best_path.append(next_city)
        total_distance += min_distance
        current_city = next_city
    
    best_path.append(best_path[0])  # 关回起点
    total_distance += distance_matrix[current_city][0]

    return best_path, total_distance

# 使用示例，假设distance_matrix是一个二维数组存储城市之间的距离
best_route, shortest_distance = tsp(city_list)

4. **电力需求预测** - 时间序列分析（如 ARIMA 或 LSTM）

from statsmodels.tsa.arima_model import ARIMA
import pandas as pd

# 假设有电力消耗时间序列数据
data = pd.read_csv("electricity_consumption.csv")
model = ARIMA(data['consumption'], order=(1, 1, 1))
results = model.fit()
forecast = results.forecast(steps=10)[0]

print("未来10天的预测电力需求:", forecast)