## 3 Model Description and Assumptions
### Assumptions
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(P_{T}\) & Retail prices of agricultural products in the offline channel, a decision variable for retailers \\ \(Q_{i}\) & Market demand for agricultural products in channel \(i\) \\ \(n\) & Cross-price demand sensitivity coefficient, elasticity of market demand with respect to retail prices of substitutes \\ \(m\) & Price demand sensitivity coefficient for agricultural commodities, elasticity of market demand with respect to retail prices \\ \(k_{T}\) & Demand sensitivity coefficient of logistics service level, elasticity coefficient of market demand with respect to logistics service level \\ \(C_{O}\) & Online channel logistics cost \\ \(C_{T}\) & Offline channel logistics cost \\ \(k\) & Cost-effectiveness of logistics services, the smaller the value of k, the better the benefits of logistics services per unit cost \\ Decision variables \\ \(\beta_{O}\) & Logistics service levels in online channels, decision variables for producers \\ \(\beta_{T}\) & Logistics service levels in the offline channel, a decision variable for retailers \\ Objective value \\ \(\pi_{C}\) & Overall profitability of the supply chain \(\pi_{C}=\pi_{S}+\pi_{r}\) \\ \(\pi_{S}\) & Producers’ profits \\ \(\pi_{r}\) & Retailer’s profit \\ tempt & The upper corner of the reaction functions as an intermediate variable in the Stackelberg game \\ \(C\) & The lower corner of the agricultural supply chain model under centralised decision-making, used to describe the expression for substitution \\ \(C\) & The upper corner label of the single-channel agricultural supply chain model under centralised decision-making for describing the optimal solution \\ \(CD\) & The upper corner label of the dual-channel agricultural supply chain model under centralised decision-making is used to describe the optimal \\ \(F\) & The lower corner of the agricultural supply chain model under decentralised decision-making for describing alternative expressions \\ \(FS\) & The upper corner label of the single-channel agricultural supply chain model under decentralised decision-making, used to describe the optimal solution \\ \(FD\) & The upper corner label of the dual-channel agricultural supply chain model with decentralised decision-making, used to describe the optimal solution \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._We can denote the offline channel market demand for agricultural products in the dual-channel model by \(Q_{T}^{D}\) and the online channel market demand for agricultural products in the dual-channel model by \(Q_{O}^{D}\), then the demand functions are, respectively:  
\[Q_{T}^{D}=sa+k_{T}\beta_{T}-mP_{T}+nP_{O} \tag{2}\]  
\[Q_{O}^{D}=(1-s)a+k_{T}\beta_{O}-mP_{O}+nP_{T} \tag{3}\]  
Assumption 3: In the offline and online channels, retailers and producers provide logistics services for agricultural products, determine the level of logistics services and bear the cost of logistics services. The cost of logistics services increases as the level of logistics services for agricultural products increases. In the offline channel, the retailer purchases agricultural products from producers and subsequently sells them to consumers. The retailer is responsible for bearing the costs associated with the logistics of the agricultural products, including warehousing and transportation. The retailer is responsible for paying the costs of logistics services for agricultural products, determining the level of service provided by logistics services in the offline environment, and bearing the costs associated with the provision of logistics services in this context. In the context of offline channels, producers engage in the direct sales of agricultural products via e-commerce platforms. In order to facilitate the analysis, this paper does not take into account the e-commerce platform's pre-invested costs associated with the construction of logistics lines and other sunk costs.