This article takes cross-border e-commerce platform operation receipts as the research object. It uses fractional differential equations to construct a market demand function jointly affected by channel radiation range, price, and price differences. The study results show that the heterodyne effect of cross-border e-commerce guarantees the competitive position of the market incumbents of the cross-border e-commerce platform. The article also illustrates the effectiveness and practicality of the cross-border e-commerce supply and demand fractional differential matching decision-making method.

#### Keywords

- Fractional differential equation
- Cross-border e-commerce
- Pricing
- Optimal model
- Platform operation

#### MSC 2010

- 34A08

The product's short shelf life and its high risk of spoilage are the typical characteristics of fresh products that are different from normal temperature products. This also makes the cold chain dual-channel optimization unable to follow the previous channel optimization strategies directly. On the one hand, consumers have higher requirements for the quality and safety of the cold chain as food. The differences between online and physical sales in terms of product quality visualization and return and exchange services can easily affect the distribution of channel demand [1]. On the other hand, the excessive overlap of the target markets of different channels, the cross-effects of the quality of cold chain products, and channel prices have made channel conflicts and channel competition increasingly severe. The difficulty of dual-channel management optimization has increased.

The article analyzes the three cold chain and dual-channel models of non-converged dual-channel, converged dual-channel decentralized decision-making, and converged dual-channel integrated decision-making. We use profit maximization as the basis for decision-making to jointly optimize channel pricing and preservation investment levels. Based on this example, we explore the scope of application of the three modes [2]. The article expects to ensure channel food safety, ease channel conflicts, and promote consumption upgrade based on enriching cold chain dual-channel integration theoretical research. This assists in realizing healthy competition in the consumer market and the sustainable development of the multi-channel cold chain under the background of the mobile Internet.

We use superscript

Model symbols and meanings.

Symbol | meaning | Symbol | meaning |
---|---|---|---|

A | Potential market demand | C2 | Additional service costs for retailers serving online consumers |

X | The total area of system area | Ce | Additional unit costs are borne by network channels when non-uniform pricing |

Y | Physical channel radiation area | D1 | Market demand generated outside the Y range (network demand) |

β | The proportion of consumers shopping online within the physical radiation range 0<β<1 | D2 | Network demand generated in the Y range |

α | Demand impact factor of physical channel fresh-keeping input, a>0 | D3 | Entity demand generated in Y range |

b1 | Channel price elasticity, b1≥b2 | π | profit |

b2 | The demand influencing factor of the price difference between channels | P1 | Online channel selling price (decision variable) |

k | Fresh-keeping input cost coefficient | P2 | Physical channel selling price (decision variable) |

θ | Deterioration rate of cold chain products | s | Retailer's fresh-keeping investment level (decision variable), s ∈ [1,100] |

w | Physical channel commodity wholesale price | P | Commodity selling price when the channel price is consistent (decision variable) |

c1 | Manufacturer’s basic unit cost (including production and inventory costs, etc.) | μ | Unit compensation was given to retailers by the manufacturer (decision variable) |

Non-converged dual channels implement non-uniform pricing, and consumers within the physical radiation range

From the hypothesis, it can be seen that there is no crossover of inventory and service between the non-converged dual channels, and the retailer's profit comes from physical sales [4]. The profits of manufacturers are mainly composed of online sales profits and entity wholesale profits. The profits of each entity in the cold chain are as follows:

The cold chain dual-channel optimization is an _{2} with respect to

_{2} has a maximum value. Therefore, the reaction function of the retailer's physical channel selling price _{2} available to

We can get
^{N}^{N}

We substitute equations (4) and (5) into equation (2) and find the first derivative of

If and only if
_{1}. We assume that

In,

_{1} = ^{2} · (_{2} · (^{2} − _{2} · (^{2} − _{2}) + 2 _{1} − _{2}) + _{1} − _{2})(_{2} _{1} − _{2}_{1}

We bring formula (7) into formula (5) and formula (4) to obtain the retailer's best physical pricing
^{N*}

Converged dual channels implement unified pricing, so the price difference between channels is zero. Therefore, online channels do not need to bear additional costs. The three types of consumer demand under _{1} − _{2} = 0, _{e}

In the converged dual channel, the network demand in the range of

We use the reverse induction method to find that
^{I}

We can take Eq. (11) into Eq. (9) and find the derivative of
^{I}

If and only if
^{I}

We bring Eqs. (13) and (12) into Eq. (11) and find the derivative of

In,

_{1} = ^{4} + 6_{1}_{1}^{2})

_{2} = 2_{1}^{2} (

_{3} = 3(_{1} − _{2} −_{1} − 3_{2} − 4_{1}

_{4} = 10_{2} − 5_{1} + _{2} − 8_{1} − 3_{1}

We bring equation (14) into equation (13) and equation (12) successively to obtain the retailer's best pricing and preservation investment decision under the integrated dual-channel decentralized decision-making. We bring the above parameters into equations (8) and (9), respectively, to obtain the optimal profit

The integrated dual-channel integrated decision-making aims to maximize the overall profit of the cold chain system [7]. During integration, the decision-making variables are determined by three factors: channel pricing, fresh-keeping investment level, and producer compensation strategy. The three market requirements for integrated decision-making are:

We find the first and second derivatives of ^{IT}^{IT}^{IT}^{IT}

We bring equation (16) and equation (17) into equation (15) to obtain the optimal profit ^{IT*}

A certain large cherry planting/production and processing company was initially limited to online channels. Later, it expanded to enter the offline market at the same time. Based on the investigation of various data that occurred in big cherry circulation, the relevant parameters are simulated as follows [8]. The radiation range of the physical store within 25km covered by the system is roughly 18km. The potential market size is 200. The deterioration rate of fresh agricultural products is 0.05. Within the scope of physical radiation, 50% of consumers choose online channels to purchase fresh agricultural products. Other parameter settings are _{1} =1.5, _{2} = 0.5, _{1} = 8, _{2} = _{e}

Optimization results of different dual-channel types.

Non-converged dual-channel | Integrated dual-channel decentralized decision-making | Integrated dual-channel integrated decision-making | |
---|---|---|---|

P1 | 25.89 | 32.35 | 27.79 |

P2 | 45.28 | ||

s | 19.49 | 11.07 | 12.65 |

μ | / | 5.72 | / |

πr | 639.39 | 579.39 | / |

πm | 872.69 | 824.39 | / |

Total cold chain profit | 1512.08 | 1403.78 | 150.4.60 |

By comparing the optimization results of non-integrated dual-channel, integrated dual-channel decentralized decision-making, and integrated dual-channel integrated decision-making in Table 2, the following results can be obtained:

1) The non-converged dual-channel system has the largest total profit. The second is integrated decision-making. The system of integrated decentralized decision-making has the lowest profit. 2) There is a significant difference in commodity prices between physical and online channels in a non-converged dual-channel supply chain. 3) The sales of converged integrated decision-making products are higher than the prices of converged decentralized decision-making products. _{1} < ^{I}^{IN}_{2}. 4) The fresh-keeping investment level of fusion integrated decision-making is slightly higher than that of fusion decentralized decision-making ^{I}^{IN} s

Combining the electric field demand distribution in Figure 1, it can be seen that the physical channel demand of fusion decentralized decision-making is slightly higher than that of a non-converged entity when the cost/cost is the same. Still, the physical price of non-converged unit commodities is much higher than that of converged decentralized decision-making. This caused the profit of the non-converged system to be higher than that of the converged system [9]. In addition, although the profit difference between the two is relatively small, the total demand for an integrated electric field for integrated decision-making is 85. This is higher than the total demand of the non-converged market 70 and is also significantly higher than the total demand of the convergent decentralized decision-making market 63. Therefore, the analysis results in Table 2 and Figure 1 are non-integrated and integrated decision-making through competitive prices. Non-converged dual channels use the high fresh-keeping investment to satisfy the physical consumption experience. The significant price difference between physical channels and online channels stimulates online consumption to achieve high profitability. The performance of integrated decentralized decision-making in terms of price, preservation level, and cold chain profit indirectly proves that the integrated development of dual channels should be based on highly integrated decision-making.

There are many new varieties involved in actual production and life. There are obvious differences in the decay holdings, even for consumer groups of the same type of goods in different geographical areas [10]. The model selects three parameters: commodity deterioration rate

We select a floating value of 0.05 units to analyze the impact of the difference in new product types on dual-channel optimization. We compare the differences in the optimization results of the three dual-channel decision-making modes at

It is found from Figure 2 that the increase in the spoilage rate of the three types of dual-channel decision-making commodities will cause an increase in channel prices and a decrease in the total profit of the system [11]. The increase in spoilage rate means an increase in the number of goods spoiled in the same environment. To ensure the market supply, retailers and manufacturers have to order/produce more goods to cope with the high spoilage rate. This increases the cost of corruption. To maintain the profit of the system, the cold chain dual channels will appropriately increase the unit price and reduce the level of investment in preservation.

When the area of the system is determined, the change in the radiation range of the physical channel will directly affect the fluctuation of the market demand of each channel. This may trigger changes in the dual-channel optimization strategy [12]. Therefore, analyze the optimization results of the three decision-making modes of dual channels when

With the increase in the radiation range of physical channels, the differences between the three optimization channels are significant. In the case of non-convergence, the online price decreases with the increase of Y, while the physical price increases. The price difference between channels gradually increases. Retailers’ fresh-keeping investment also increases significantly with Y, and the channel price of integrated decentralized decision-making increases with Y. The increase has been increasing slowly [13]. Its investment in preservation has also increased slightly, while the channel price and investment level in preserving the integrated decision-making remain unchanged. When the radiation range of the physical channel does not exceed 50% of the total area of the area where the system is located, the profitability of the integrated dual-channel integrated decision-making system is relatively strong. When the radiation range of the physical channel exceeds 50%, the non-converged dual-channel decision-making is more conducive to the increase of system profits. The increase of Y in Figure 3 makes the difference between _{1} and A gradually increase, which leads to retailers’ profits gradually being higher than those of manufacturers. This also verifies from the side that even some consumers have a preference for online channels. The expansion of physical sales radiation will also reduce the “barrier-free” advantage of online sales in geographic space.

The model has not well reflected the difference in the storage cost of fresh products per unit and the superiority of the integrated dual-channel in intensive inventory management. Therefore, considering inventory cost differences, cold chain dual-channel optimization will become one of the future research directions.

#### Model symbols and meanings.

Symbol | meaning | Symbol | meaning |
---|---|---|---|

A | Potential market demand | C2 | Additional service costs for retailers serving online consumers |

X | The total area of system area | Ce | Additional unit costs are borne by network channels when non-uniform pricing |

Y | Physical channel radiation area | D1 | Market demand generated outside the Y range (network demand) |

β | The proportion of consumers shopping online within the physical radiation range 0<β<1 | D2 | Network demand generated in the Y range |

α | Demand impact factor of physical channel fresh-keeping input, a>0 | D3 | Entity demand generated in Y range |

b1 | Channel price elasticity, b1≥b2 | π | profit |

b2 | The demand influencing factor of the price difference between channels | P1 | Online channel selling price (decision variable) |

k | Fresh-keeping input cost coefficient | P2 | Physical channel selling price (decision variable) |

θ | Deterioration rate of cold chain products | s | Retailer's fresh-keeping investment level (decision variable), s ∈ [1,100] |

w | Physical channel commodity wholesale price | P | Commodity selling price when the channel price is consistent (decision variable) |

c1 | Manufacturer’s basic unit cost (including production and inventory costs, etc.) | μ | Unit compensation was given to retailers by the manufacturer (decision variable) |

#### Optimization results of different dual-channel types.

Non-converged dual-channel | Integrated dual-channel decentralized decision-making | Integrated dual-channel integrated decision-making | |
---|---|---|---|

P1 | 25.89 | 32.35 | 27.79 |

P2 | 45.28 | ||

s | 19.49 | 11.07 | 12.65 |

μ | / | 5.72 | / |

πr | 639.39 | 579.39 | / |

πm | 872.69 | 824.39 | / |

Total cold chain profit | 1512.08 | 1403.78 | 150.4.60 |

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