基于超市商品补货策略的分析——以2023年全国数学建模竞赛C题为例
Analysis on Supermarket Product Replenishment Strategy—A Case Study of Problem C in the 2023 National Mathematical Modeling Competition

由于蔬菜类商品保质期较短，易变质会影响其销售价值，商超通常会选择每天都补货。蔬菜品种众多，商家在进货前无法准确得知具体商品及其进货价格。本文将基于已知数据和蔬菜的时令性即商超销售空间的限制，从销量、进价、售价与商品相互间的关系进行市场需求分析，通过预测未来商品的销量，制定合理的补货和定价决策，确保商超利益最大化。通过系统聚类将辣椒类、花叶类、水生根茎类、食用菌、花菜类、茄类六个品类分成两组，根据每月销量绘制折线图来判断并描述其分布规律。其次分析不同品类的相互关系，先绘制散点图初步观察，再通过计算皮尔逊相关系数来判断不同品类有无相关性，可知六个品类间花菜类和花叶类相关性最强。其次为预估未来几天的各品类的日补货总量，借助前一个月数据进行预估，将品类销量按日汇总后，先检验时间序列的稳定性，再分别求得其自相关系数(ACF)和偏自相关系数(PACF)，通过对自相关图和偏自相关图的分析，得到最佳的阶数p、q，检验模型，再利用ARIMA模型进行时间序列预测分析，得出未来七天的预估销售量，最后根据补货量与销售量的关系得出日补货总量。
Due to the short shelf life and perishable nature of vegetable products, their sales value can be af-fected, so supermarkets often choose to restock daily. With a wide variety of vegetable varieties, merchants are unable to accurately determine the specific products and their purchase prices be-fore stocking. This article will analyze market demand based on known data and the seasonality of vegetables, as well as the limitations of supermarket sales space. By predicting future product sales, reasonable restocking and pricing decisions can be made to maximize the interests of supermar-kets. Using systematic clustering, six categories of vegetables, namely chili peppers, leafy vegeta-bles, aquatic roots and tubers, edible mushrooms, cauliflower, and tomatoes, are divided into two groups. Line charts of monthly sales volume are plotted to determine and describe their distribu-tion patterns. Furthermore, the relationships between different categories are analyzed. Scatter plots are first plotted for initial observation, and then the Pearson correlation coefficient is calcu-lated to determine the correlation between different categories. It is found that cauliflower and leafy vegetables have the strongest correlation among the six categories. Next, the daily total re-stocking volume for each category in the coming days is estimated using data from the previous month. After aggregating the sales volume by category on a daily basis, the stability of the time se-ries is tested, and the autocorrelation coefficient (ACF) and partial autocorrelation coefficient (PACF) are calculated. Through the analysis of the autocorrelation plot and partial autocorrelation plot, the optimal order of p and q is determined. The model is then tested, and the ARIMA model is used for time series forecasting analysis to obtain the estimated sales volume for the next seven days. Finally, the daily total restocking volume is determined based on the relationship between restocking vol-ume and sales volume.