## 5 A Dual-Channel Agricultural Supply Chain Model under Decentralised Decision-Making  
Under decentralised decision-making, producers and retailers make decisions with the goal of maximising their own profits. The producer is the dominant player in the Stackelberg game and will fully estimate the reaction function of the retailers' followers generated by its own decision; it sets the wholesale price, online sales price, and online logistics service level on this basis, and the retailers as followers set the offline sales price and offline logistics service level according to the producer's decision-making results. Consider a supply chain model of agricultural products with single-channel sales in the traditional mode under decentralised decision-making, where each member of the supply chain makes decisions with the objective of optimising its own profit. Agricultural leading enterprises are the producers, and only through the offline channel to sell agricultural products to community supermarkets or large supermarkets, with supermarkets as retailers, agricultural products are sold to consumers. Under decentralised decision-making, the producer's profit function is:  
\[\pi_{\mathrm{S}}=w\mathrm{Q}_{\mathrm{T}} \tag{28}\]  
\[\pi_{\mathrm{S}}=w(\mathrm{s}a+k_{\mathrm{T}}\beta_{\mathrm{T}}-mP_{\mathrm{ T}}) \tag{29}\]  
The retailer's profit function is:  
\[\pi_{r}=(P_{\mathrm{T}}-w)\mathrm{Q}_{\mathrm{T}}-C_{\mathrm{T}} \tag{30}\]  
\[\pi_{r}=(P_{\mathrm{T}}-w)(\mathrm{s}a+k_{\mathrm{T}}\beta_{\mathrm{T}}-mP_{ \mathrm{T}})-\frac{1}{2}k\beta_{\mathrm{T}}^{2} \tag{31}\]  
The overall profit function of the supply chain is expressed as:  
\[\pi_{\mathrm{C}}=\pi_{\mathrm{S}}+\pi_{r} \tag{32}\]  
\[\pi_{\mathrm{C}}=\mathrm{P}_{\mathrm{T}}(sa+k_{\mathrm{T}}\beta_{\mathrm{T}}- mP_{\mathrm{T}})-\frac{1}{2}\mathrm{k}\beta_{\mathrm{T}}^{2} \tag{33}\]  
The first part of the equation represents the revenue gained through the sale of agricultural products, while the second part represents the cost paid for the provision of logistics services. In the context of the supply chain's overall profit function, the producer's wholesale price \(w\) is no longer a decision variable. Rather, it solely influences the distribution of profits within the supply chain, specifically between the producer and the retailer. The optimal decision result under centralised decision-making can be used as a basis for evaluating the extent to which the supply chain is coordinated under decentralised decision-making. Since the producer is the core firm in the supply chain, under decentralised decision-making, consider the Stackelberg game, where the producer is the leader and makes the decision first, and the retailer is the follower and makes its own optimal decision based on the decision-making behaviour of the producer. Under decentralised decision-making, the retailer's decision variable is \((P_{\mathrm{T}}\ \beta_{\mathrm{T}}\ )\), the retailer's profit function is the Hessian matrix \(\mathrm{H}= \begin{bmatrix}-2\mathrm{m}&\mathrm{k}_{\mathrm{T}}\\ \mathrm{k}_{\mathrm{T}}&-k\end{bmatrix} \), where \(\mathrm{H}1=[-2\mathrm{m}]\), \(\mathrm{det}(\mathrm{H}1)<0\), by assumption \(2\mathrm{km}-\mathrm{k}_{\mathrm{T}}^{2}>0\), so \(\mathrm{det}(\mathrm{H}1)>0\), the retailer's profit function is the joint concave function on \((P_{\mathrm{T}}\ \beta_{\mathrm{T}}\ )\), and there is an optimal solution.