A study on vegetable commodity replenishment considering single item quantity limitations—based on gray prediction and linear regression modeling

: Data elements are beginning to play an important role in the selling of vegetable items in supermarkets. In order to get the best pricing and replenishment strategy under the limitation of the number of vegetable items, this paper firstly calculates the profit contribution rate of each item and sorts out the top 33 available items, and prioritizes the items with the top profit contribution rate for replenishment. Finally, a linear programming model is developed to solve the pricing strategy for maximizing the revenue of the superstore under the constraints of display quantity and weight of individual products.


Introduction
As a necessity of life, the pricing of vegetables has a significant impact on the protection of residents' lives and the improvement of hypermarket revenue.As a large-scale supply and marketing center radiating a region, the sales strategy of superstores for basic necessities such as vegetables directly affects the quality of livelihood protection.From the perspective of profitability, supermarkets can increase revenue by raising prices, but too high a price will lead to a decrease in sales volume; they can also increase revenue by supplying sufficient quantities, but too much backlog of vegetable goods will lead to losses.Therefore, rational pricing and replenishment decisions are important keys to maintaining a high revenue balance.In order to obtain the optimal commodity pricing for superstores, Yang Shuai (2022) obtained multi-commodity retail chain pricing as well as shelf allocation strategies based on the food decay model combined with the market demand function [1]; Gu Sihong (2023) proposed a multi-stage dynamic pricing model during the freshness decline period based on a linear programming model taking more into account the characteristics of consumable goods such as vegetables [2].For the replenishment strategy, Bao Jufang (2019) introduced a strong adaptive three-parameter weibull distribution model to study the replenishment strategy of perishable goods [3]; Zhu Hongyu (2020) took into account the impact of shelf display volume on consumer demand in physical sales on the basis of this model to establish a dual-channel supply chain inventory model for fresh agricultural products that takes into account the characteristics of non-instantaneous deterioration and shelf display volume [4].Throughout the previous research, it is found that for the short-term prediction of market prices, the commonly used methods are econometric analysis represented by regression analysis, mathematical and statistical prediction represented by time series, and intelligent analysis based on BP neural network and gray prediction [5].These pricing prediction models are less explored for the replenishment prediction of fresh vegetables and other fresh products, while this paper indirectly predicts the price through the prediction of demand, using a small cross-domain prediction based on the unit of days, and in order to be closer to the reality of the situation, the rate of attrition and the limitations of the selling space and weight will be included in the study, and maximize to meet the market demand for various types of vegetable commodities.

Data preparation
The data used in the model comes from the NCAA Mathematical Modeling 2023 C problem.In this paper, we use the pricing and replenishment data of previous vegetable food products of the superstore provided in the mathematical modeling competition to calculate the average contribution rate of each vegetable item from June 24 to 30, 2023, and select the top 33 items based on the contribution rate.

Gray prediction model establishment
Predicting the replenishment quantity of an item on a future day based on the daily sales quantity of the item in the recent week belongs to the future prediction of small samples.Gray prediction is a model based on a small sample of data for prediction, it is dealing with less eigenvalue data, do not need the sample space of the data is large enough, it can solve the problem of less historical data, the completeness of the sequence as well as the low reliability of the original data can be generated to get the regularity of the original data to get the regularity of the stronger generation of the sequence [6].
Therefore, the gray prediction model is chosen to predict the daily replenishment of a single product on July 1st.
GM(1,1) gray prediction model is a single variable prediction of the first-order differential equation model, its discrete time response function is approximately exponential law, the establishment of GM(1,1) model [7]is: Let X (0) = X (0) (1), X (0) (2), ⋯ , | X (0) (n) be the original nonnegative time series, X (0) (t) be the cumulative generating series,i.e.X (1) The whitened differential equation for the GM(1,1) model is: +  (1) = In the above equation, a is the parameter to be identified, also known as the development coefficient; u is the endogenous variable to be identified, also known as the gray role.Let the vector  ̂= (   ), and find  ̂= (  ) −1    by the least squares method, where ( (1) (1) +  (1) (2)) ( (1) (2) +  (1) ( (1) ( − 1) +  (1) The discrete time response function for gray prediction is obtained as: X (1) (t + 1) is the resulting cumulative predicted value, and reducing the predicted value is: (1) ( + 1) −  (1) (), ( = 1,2,3, … , ) Before building the gray prediction model GM(1,1), the time series is subjected to a rank-ratio test.If the level ratio test, it means that the series is suitable for the construction of gray model, if not through the level ratio test, the series will be "shifted to transform", so that the new series to meet the level ratio test [8].Gray prediction model to be tested to determine whether it is reasonable, only through the test model can be used for prediction, the system is mainly through the a posteriori difference than the C value of the gray prediction model to test.

Model solving
Gray prediction was performed using SPSSPRO and the results of the grade ratio test were obtained as shown in Table 1  ) , the data are suitable for modeling.If it does not pass the rank-ratio test, then the sequence is transformed into a "level shift", so that the sequence after the level shift meets the rankratio test.From the analysis of the above table, we can get that all the level ratio values of the shifted series are located in the interval(0.779,1.284),which indicates that the shifted series is suitable for constructing the gray prediction model.
Then the gray model is constructed to calculate the development coefficient, gray role quantity, and the value of the posterior difference ratio shown in Table 2.

Development coefficient a
Gray role quantity b posteriori difference ratio C value -0.049 24.571 0.21 The gray prediction model can be constructed from the development coefficient and gray role quantity.The development coefficient indicates the development pattern and trend of the series, and the gray role quantity reflects the change relationship of the series.The a posteriori difference ratio can verify the accuracy of gray prediction, the smaller the a posteriori difference ratio, the higher the accuracy of gray prediction [9].Generally, if the C value of the a posteriori difference ratio is less than 0.35, then the model accuracy is high, the C value is less than 0.5, which indicates that the model accuracy is qualified, the C value is less than 0.65, which indicates that the model accuracy is basically qualified, and if the C value is greater than 0.65, it indicates that the model accuracy is unqualified.Analyzing the table, we get that the a posteriori difference ratio value is 0.21 and the model precision is high [10].
The results of the model fit are shown in Table 3: 16.9 16.887 0.013 0.078 The average relative error of the model is 8.902%, which means that the model fits well.Finally, the daily replenishment volume of broccoli on July 1 was obtained as 20.489 kg.Similarly, the sales volume of other optional single items on July 1 is obtained, and according to the replenishment volume = sales volume / (1 -wastage rate), the replenishment volume of each single item on July 1 is obtained as shown in Table 4.

Regression modeling
Analyzing the sales flow from 2020 to 2023, it was found that the correlation between the sales volume of each vegetable item is weak, and vegetables belong to the necessities of life, so it is possible to price each item separately, and when the revenue of each individual item is maximized, the total revenue is also maximized.
As an example, the demand price curve of millet peppers is analyzed.Plot the scatter plot of average sales price and sales volume of millet peppers and fit a linear relationship between the two as in Figure 1.
Figure 1: Average selling price and sales scatter plot of millet pepper The linear back-analytic model is established as: where β0, β1, σ2 are all unknown parameters independent of x, where β0, β1, are called regression coefficients.Now we get n independent observations (yi, xi),i=1,...,n, from (1) we get The regression equation is obtained according to equation (7)(8)(9):

Regression coefficient hypothesis testing
When H0 above is rejected, βj is not all zero, but it is not excluded that several of them are equal to zero.So further m+1 tests are made as follows (j=0,1,…,m): ̂~(  ,  2   ) Cjj is the(j,j)element in(X T X) -1 with s2 instead of 2,when  0 () holds For a given , accept  0 The t-value for the regression coefficient test was calculated for y=37.88-2.44xand obtained as shown in Table 5. , then the original hypothesis is rejected and the regression coefficient passes the test.

Linear Programming Pricing Model Establishment
Take the average wholesale price of each optional single product from June 24 to 30, 2023 as the wholesale price of single product on July 1, and record the attrition rate of each single product.Assuming Xi is the price of individual item i,Ci is the cost of individual item i,Qi,is the replenishment quantity of individual item i, Si is the attrition rate of individual item i,and n is the number of selected individual items, the total revenue is: The constraints are as follows: (1) The total number of saleable individual items is controlled to be 27-33, i.e., 27≤n≤33.
(2) The order quantity of each single item satisfies the minimum display quantity of 2.5 kg, i.e., Q_i≥2.5.
(3) To try to meet the premise of the market demand for various types of vegetable goods, then the selection of single products should be divided into 6 types of vegetables.Combining the above analysis, the mathematical model is established as follows:

Model solving
The model was solved using Lingo software to determine what the selling price of the millet peppers could be when the revenue could be maximized.The result was obtained that when the price of millet peppers is 8.74, the revenue of millet peppers on that day can reach the maximum value of 105.83 ￥.
Similarly, under the constraints, the July 1 single-item replenishment and pricing are obtained as shown in Table 6.
When the superstore's replenishment and pricin on July 1 satisfy the above table, it can make the maximum revenue of 1,848.90￥yuanfrom vegetable items on that day.

Conclusion
In this paper, for the problem of pricing and replenishment strategy under the quantity restriction of vegetable items in supermarkets, a method based on GM(1,1) model and linear programming is proposed based on gray prediction and linear regression modeling.By calculating the profit contribution rate, the top 33 sellable items are organized, and the items with the highest profit contribution rate are prioritized for replenishment.The established GM(1,1) model and attrition rate are utilized to predict the sales volume of individual items, and a replenishment plan is developed based on the demand.Finally, the pricing strategy for revenue maximization is solved under the constraints of single-item display volume and weight.The experimental results show that under the total number of saleable single items in a single day is 27-33 and the order quantity of each single item meets the minimum display quantity of 2.5 kg, this paper derives the maximum revenue of the superstore to be 1890.90¥.The method proposed in this paper can significantly improve the revenue

Table 1 :
below: Grade ratio test result table

Table 3 :
Model fitting results

Table 4 :
Individual Product July 1 Sales and Replenishment

Table 5 :
Regression coefficient test table

Table 6 :
Pricing for Available Individual Items