## 4 A Dual-Channel Agricultural Supply Chain Model under Centralised Decision-Making  
Under centralised decision-making, the leading producers of agricultural products in the supply chain and the community superstores make decisions at the same time, and both take the optimal profit of the supply chain as the decision-making goal. Firstly, we consider an ideal single-channel agricultural supply chain model (CS), where each member of the supply chain makes decisions with the goal of optimising the overall profit of the supply chain, and at the same time determines the level of logistics services and the retail price of agricultural products in the supply chain. As a producer, the leading agricultural enterprise sells agricultural products to community supermarketsor large-scale superstores through offline channels, and the supermarkets, as retailers, sell agricultural products to consumers. Under centralised decision-making, the producer's profit function is:  
\[\pi_{S}=wQ_{T} \tag{6}\]  
\[\pi_{S}=w(sa+k_{T}\beta_{T}-mP_{T}) \tag{7}\]  
The retailer's profit function is:  
\[\pi_{r}=(P_{T}-w)Q_{T}-C_{T}, \tag{8}\]  
\[\pi_{r}=(P_{T}-w)(sa+k_{T}\beta_{T}-mP_{T})-\frac{1}{2}k\beta_{T}^{2} \tag{9}\]  
So, the overall supply chain profit function is:  
\[\pi_{C}=\pi_{S}+\pi_{r} \tag{10}\]  
\[\pi_{C}=P_{T}(sa+k_{T}\beta_{T}-mP_{T})-\frac{1}{2}k\beta_{T}^{2} \tag{11}\]  
By calculating the concavity of the Hessian matrix judgement function, it can be concluded that there exists a unique optimal solution. The optimal decision and the overall optimal profit of the CS under centralised decision-making is:  
\[\left\{ \begin{array}{l}P_{T}^{*}=\frac{abs}{2 km-k_{T}^{2}}\\ \\ \beta_{T}^{*}=\frac{abs}{2 km-k_{T}^{2}}\\ \\ \pi_{C}^{*}=\frac{a^{2}k^{2}}{2\left(2 km-k_{T}^{2}\right)}\end{array} \right. \tag{12}\]  
Among them, \(\left(2 km-k_{T}^{2}\right)>0,a-2 mw>0\). In the dual-channel agricultural supply chain model (CD), different from the CS, the leading agricultural products enterprise, as a producer, can sell agricultural products to community supermarkets or large-scale superstores through offline channels and can also sell agricultural products directly to consumers on the e-commerce platform through online channels. Under centralised decision-making, the producer's profit function is:  
\[\pi_{S}=wQ_{T}+P_{O}Q_{O}-C_{O} \tag{13}\]  
\[\pi_{S}=w(sa+k_{T}\beta_{T}-mP_{T}+nP_{O})+P_{O}((1-s)a+k_{T}\beta_{O}-mP_{O}+ nP_{T})-\frac{1}{2}k\beta_{O}^{2} \tag{14}\]  
The retailer's profit function is:  
\[\pi_{r}=(P_{T}-w)Q_{T}-C_{T} \tag{15}\]  
\[\pi_{r}=(P_{T}-w)(sa+k_{T}\beta_{T}-mP_{T}+nP_{O})-\frac{1}{2}k\beta_{T}^{2} \tag{16}\]  
So, the overall supply chain profit function is:  
\[\pi_{C}=\pi_{S}+\pi_{r} \tag{17}\]  
\[\pi_{C}=P_{T}(sa+k_{T}\beta_{T}-mP_{T}+nP_{O})-\frac{1}{2}k\beta_{T}^{2}+ \tag{18}\]  
First, we can solve the Hessian matrix to determine the concavity of the overall profit function of the supply chain and to determine whether there is an optimal solution to the function. Under centralised decision-making, the decision variable of the overall supply chain profit function is \((P_{T}\beta_{T}P_{O}\beta_{O})\), and the Hessian matrix of the overall supply chain profit function about \(P_{T}\), \(P_{O},\beta_{T},\beta_{O}\) is \(\mathrm{H}= \begin{bmatrix}-2\mathrm{m}&2\mathrm{n}&k_{T}&0\\ 2\mathrm{n}&-2\mathrm{m}&0&k_{T}\\ k_{T}&0&-\mathrm{k}&0\\ 0&k_{T}&0&-\mathrm{k}\end{bmatrix} \). We can solve the Hessian matrix of each order of the principal subequations, \(H1=[-2m]\), \(\det(\mathrm{H}1)<0\), \(\mathrm{H}2= \begin{bmatrix}-2\mathrm{m}&2\mathrm{n}\\ 2\mathrm{n}&-2\mathrm{m}\end{bmatrix} \), \(\mathrm{H}3= \begin{bmatrix}-2\mathrm{m}&2\mathrm{n}&k_{T}\\ 2\mathrm{n}&-2\mathrm{m}&0\\ k_{T}&0&-\mathrm{k}\end{bmatrix} \), and \(\mathrm{H}4= \begin{bmatrix}-2\mathrm{m}&2\mathrm{n}&k_{T}&0\\ 2\mathrm{n}&-2\mathrm{m}&0&k_{T}\\ k_{T}&0&-\mathrm{k}&0\\ 0&k_{T}&0&-\mathrm{k}\end{bmatrix} \).