Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 8, No. 3, 2023 

 

147 

Pricing	and	Replenishment	Decision	of	Vegetable	
Products	
Le Fang*, Kun Zhang 

Stony Brook Institute at Anhui University, Anhui University, Anhui, 230039, China 
*Corresponding author: lefang18909664046@163.com 

 

Abstract: Most vegetable products have a short shelf life and need to be restocked according to demand after being sold on 
the same day. Therefore, it is of great significance to accurately price and make effective replenishment decisions for vegetable 
products under unknown specific unit and purchase prices. This article analyzes the sales information and wholesale prices of 
various types of vegetables. Using the system clustering method in cluster analysis, the Euclidean distance and shortest distance 
are obtained based on the characteristic values between different individual products, and the sales volume correlation is 
classified according to their strength. Then, the chi square test and Fisher exact probability test are used to obtain the sales volume 
correlation based on the characteristic values between two different categories. The paper constructs a linear programming model 
with the goal of minimizing daily transportation costs. 

Keywords: Chi-square test, ARIMA model, Logistic regression, Grey prediction, BP neural network. 
 

1. Introduction 
In fresh food supermarkets, ordinary vegetable products 

have a short shelf life and their appearance may deteriorate 
over time. Most varieties must be sold on the same day, so 
daily restocking is essential[1]. The variety of vegetable 
supply is relatively abundant from April to October, and the 
sales volume of vegetable products is closely related to the 
time, so reasonable replenishment and pricing decisions are 
particularly important. 

Based on the product information of six vegetable 
categories sold in a supermarket, this paper establishes a 
mathematical model for different product sales details, 
wholesale prices, and recent loss rates from July 1, 2020 to 
June 30, 2020 and determine the distribution pattern and 
interrelationship of sales volume between different single 
products or categories of vegetables. Then we predict the 
daily replenishment total and pricing strategy for each 
vegetable category from July 1, 2023 to July 7, 2023, to 
maximize the revenue of supermarkets. In order to optimize 
the replenishment and pricing decisions of vegetable products, 
we propose some opinions and reasons for the relevant data 
that still needs to be collected by supermarkets. 

Section I is the Introduction. Section II gives the 

preliminary work for the model. Section III and Section IV 
establish the model and analysis the specific problem. Section 
V, the Conclusion. 

2. Preliminary 

2.1. Assumption  
1.Neglecting the impact of missing daily sales flow details 

for some products on the calculation of results. 
2.The selected indicators have no significant impact on 

each other. 
3.Neglecting the impact of other unconsidered factors. 
4.Specific vegetables are only produced in specific cities. 
The general model of artificial neural network consists of 

four basic elements, which are:  

2.2. The determination of the number of 
network layers 

BP neural network is back propagating, mainly composed 
of three parts: input layer, middle layer and output layer. The 
number of nodes in the input and output layers is relatively 
easy to determine, but the determination of the number of 
nodes in the hidden layer is a very important and complex 
problem. 

 
Table 1. Symbols Notation 

Symbols Notation 
λ eigenvalue 

d(xi, xj)  Euclidean distance 
D(Gp, Gq) Minimum distance 

X2 Chi square statistical value 
v Chi Square Statistical Value Degrees of Freedom 

Tij Normalized data 
pij The daily selling price for the i-th category on the j-th day 
uij The i-th category, the daily purchase price on the j-th day 
Mij Total sales of i-th category on day j 
g(z) Logistic function 
Iij  Profit contribution on day j for the i-th category 

z(i) Group (i) operators in GM (1,1) 

p(k) Grade ratio deviation value 



 

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3. Model Establishment 

3.1. Establishment of Principal Component 
Analysis Models for Each Category 

1.Construct a sample matrix y with a size of 4 * n. 
 

       𝑦

𝑦 𝑦 ⋯ 𝑦
𝑦 ⋮ ⋮ ⋮
𝑦 ⋮ ⋮ ⋮
𝑦 𝑦 ⋯ 𝑦

𝑦 , 𝑦 , ⋯ , 𝑦     (1) 

 
2.Standardize the y-matrix. Calculate the mean and 

standard deviation of the matrix separately by column. 
 

               𝑋 ∑ 𝑋                 (2) 

 

                   𝑆
∑

               (3) 

 
3.The original sample matrix has been standardized to 
 

      𝑌

𝑌 𝑌 ⋯ 𝑌
𝑌 ⋮ ⋮ ⋮
𝑌 ⋮ ⋮ ⋮
𝑌 𝑌 ⋯ 𝑌

𝑌 , 𝑌 , ⋯ , 𝑌      (4) 

 
4.Calculate the principal component contribution rate and 

cumulative contribution rate[2]. Select the representative item 
of this category to represent the current category, with no 
more than 3 items in total. 

3.2. The solutions 
Build a system clustering model and use SPSS software to 

solve it. By clustering big data, it is found that there is a 
certain correlation between the sales volume of different 
vegetable items. After solving the model, it can be concluded 
that the significant level of correlation between cauliflower 
and the other four categories is 0.213, which is greater than 
0.1. Therefore, cauliflower has a correlation with the other 
four categories, but the correlation is weak. 

This paper uses the entropy weight method[3] to solve the 
weight vectors of each indicator 𝜔 , 𝜔 , ⋯ , 𝜔 . The results 
of obtaining weights using the entropy weight method are 
shown in the Table 1. 

 
Table 1. The weights of each vegetables 

Category name 𝜔  𝜔  𝜔  
Florifolias 0.34 0.33 0.33 

florescent vegetables 0.501 0.499 - 
Aquatic rhizomes 0.4963 0.5037 - 

Solanaceae 0.48 0.52 - 
Edible fungi 0.51 0.49 - 
Chili peppers 0.433 0.567 - 

 
Establish a logistic regression mathematical model, and the 

specific solving steps are as follows. 
                      

𝜃 𝜃 𝑚 ⋯ 𝜃 𝑚 ∑ 𝜃𝑚 𝜃 𝑚       (5) 
 
To control the output between 0 and 1, select the Logistic 

function. 

            𝑔 𝑧                   (6) 

 
The Loss Function of Logistic Regression[4] is 
 

     𝐽 𝜃 ∑ 𝑢 𝑙𝑜𝑔 ℎ 𝑚 1
𝑢 𝑙𝑜𝑔 1 ℎ 𝑚                (7) 

 

3.3. White noise processing 
By substituting the data again, the predicted results of the 

model increase steadily, and the residual is also white noise. 
All other classes pass the white noise test, that is, the model 
passes the white noise test. Perform ARIMA model[5] 
predictions for different vegetable categories, and the 
predicted results are shown in the following figure. 

 

 
Figure 1. Forecast results(a) 

 

 
Figure 2. Forecast results(b) 

 

 
Figure 3. Forecast results(c) 

 
By comparing the predicted value with the actual value, it 

was calculated that=0.892, indicating that the model has good 



 

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predictive accuracy and can be used for prediction. Using the 
ARIMA model, predict the total daily replenishment volume 
and pricing strategy from July 1 to July 7, 2023. 

3.4. Predicting the prices 
To predict the replenishment volume and pricing strategy 

of individual products on July 1, 2023, grey GM (1,1) is used 
due to the limited sample data. To improve the accuracy of 
problem prediction, a BP neural network model needs to be 
combined. Therefore, this article constructs a combined 
mathematical model based on grey GM (1,1)[6] and BP 
neural network[7] for prediction. 

4. Discussions 
1. The logistic regression model is a non parametric model 

that does not require assumptions about the distribution of 
data and has strong adaptability and flexibility. 

2. The time prediction series ARIMA model is relatively 
flexible and can achieve different types of predictions and 
analysis by adjusting the parameters in ARIMA (p, d, q). 

3. The combination of grey GM (1,1) prediction model and 
BP neural network can comprehensively utilize their 
respective strengths to improve the accuracy and robustness 
of prediction. 

4. The logistic regression model is sensitive to the presence 
of multicollinearity between features and requires measures 
such as feature selection and dimensionality reduction. 

5. The grey GM (1,1) prediction model and BP neural 
network combined model are more complex than the single 
model, which requires more computing resources and time to 
train and adjust model parameters. 

5. Conclusion 
This paper analysis the prediction of variable vegetables’ 

price based on the history data through PCA, BP neural 
network,etc. Based on the results, the model provides some 
useful strategy for businesses to make the profit biggest. We 
discuss the change of the prices and productions in different 

angles in the specific times, which is helpful for the managers 
to make up more economical plans. In addition to time series 
prediction analysis, ARIMA models can also be used in fields 
such as financial data analysis, meteorological prediction, and 
signal processing. The combination of grey GM (1,1) 
prediction model and BP neural network model can also be 
used in fields such as sales prediction, traffic flow prediction, 
and production planning. 

Acknowledgment 
The authors gratefully acknowledge the financial support 

from xxx funds. 

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