## 4 A Dual-Channel Agricultural Supply Chain Model under Centralised Decision-Making
where \(m\) is the price elasticity coefficient, which indicates that when the selling price of channel \(i\) rises by 1 unit, the demand in channel \(i\) falls by m units, and \(n\) is the cross-price elasticity, which indicates that when the price of channel \(i\) rises by 1 unit, the demand in channel \(i\) falls by \(n\) units due to channel substitution effects. The former is the direct effect of price on demand and the latter is the indirect effect of price on demand, thus \(m>n\), \(det(H2)>0\). For the principal subforms of the Hessian matrix, \(\det(\mathrm{H}3)=4 kn^{2}-2m\Big{(}2 km-\mathrm{k}_{T}^{2}\Big{)}\), \(\det(\mathrm{H}4)=\Big{(}2 km-\mathrm{k}_{T}^{2}+2 kn\Big{)}\left(2 km-\mathrm{k }_{T}^{2}-2 kn\right)\). It is known that \(\mathrm{m}>\mathrm{n}\), then when \(2 km-\mathrm{k}_{T}^{2}>2\mathrm{kn}\), \(\det(\mathrm{H}3)<0\) and \(\det(\mathrm{H}4)>0\), the Hessian matrix is negatively definite, and an optimal solution exists for centralised decision-making. Let \(\frac{\partial\tau_{C}}{\partial P_{T}}=0\), \(\frac{\partial\tau_{C}}{\partial P_{O}}=0\), \(\frac{\partial\tau_{C}}{\partial\beta_{T}}=0\), \(\frac{\partial\tau_{C}}{\partial\beta_{O}}=0\), the optimal decisions of the dual-channel agricultural supply chain under centralised decision-making are: \[P_{T}^{*}=\frac{ak\big{(}s\big{(}2 km-k_{T}^{2}\big{)}+2(1-s)kn\big{)}}{\mathrm{ A}_{C}}\] (19) \[P_{O}^{*}=\frac{ak\big{(}(1-s)\big{(}2 km-k_{T}^{2}\big{)}+2 skn \big{)}}{\mathrm{A}_{C}}\] (20) \[\beta_{T}^{*}=\frac{ak_{T}\big{(}s\big{(}2 km-k_{T}^{2}\big{)}+2(1- s)kn\big{)}}{\mathrm{A}_{C}}\] (21) \[\beta_{O}^{*}=\frac{ak_{T}\big{(}(1-s)\big{(}2 km-k_{T}^{2}\big{)}+2 skn\big{)}}{\mathrm{A}_{C}}\] (22)  
Among them, \(A_{C}=4k^{2}m^{2}-4k^{2}n^{2}-4 kk_{T}^{2}m+k_{T}^{4}=\big{(}2 km-k_{T}^{2}+2 kn \big{)}\big{(}2 km-k_{T}^{2}-2 kn\big{)}\), \(0<s<1\), \(2 km-k_{T}^{2}>2 kn\). We can substitute the above optimal decisions into the demand function and the overall profit function of the supply chain. \[\pi_{C}^{*}=\frac{\mathrm{a}^{2}\mathrm{k}\Big{(}\big{(}2\mathrm{s}^{2}-2 \mathrm{s}+1\big{)}\Big{(}2\mathrm{km}-\mathrm{k}_{T}^{2}\Big{)}+(1-\mathrm{s })4\mathrm{kns}\Big{)}}{2\mathrm{A}_{C}} \tag{23}\]  
\[\pi_{S}^{*}=\frac{ak\big{(}2n(1-s)\big{(}wk_{T}^{2}+ak\big{)}+a(1-s)^{2}\big{(} 2 km-k_{T}^{2}\big{)}\big{)}}{2\mathrm{A}_{C}} \tag{24}\]  
\[\pi_{\mathrm{r}}^{*}=\frac{\mathrm{ak}\big{(}2 wk_{T}^{2}(\mathrm{ms}+\mathrm{ ns}-\mathrm{n})+\mathrm{s}^{2}(2\mathrm{km}-\mathrm{k}_{T}^{2}-2\mathrm{kn}) \big{)}}{2\mathrm{A}_{C}} \tag{25}\]  
\[Q_{\mathrm{T}}^{*}=\frac{\mathrm{ak}\Big{(}\mathrm{ms}\Big{(}2\mathrm{km}- \mathrm{k}_{T}^{2}\Big{)}+\mathrm{n}\Big{(}(1-\mathrm{s})\mathrm{k}_{T}^{2}-2 \mathrm{skn}\Big{)}\Big{)}}{\mathrm{A}_{C}} \tag{26}\]\[\mathrm{Q}_{\mathrm{C}}^{*}=\frac{\mathrm{ak}\Big{(}\mathrm{m}(1-\mathrm{s})\Big{(}2 \mathrm{km}-\mathrm{k}_{\mathrm{T}}^{2}\Big{)}+\mathrm{n}\Big{(}\mathrm{sk}_{ \mathrm{T}}^{2}-2(1-\mathrm{s})\mathrm{kn}\Big{)}\Big{)}}{\mathrm{A}_{\mathrm{C }}} \tag{27}\]  
When a retailer is unprofitable, the retailer will exit the market. In order to conform to a realistic scenario and ensure that the retailer's profit \(>0\), let \(as-2 mw>0\).