## 5 A Dual-Channel Agricultural Supply Chain Model under Decentralised Decision-Making
Associating \(\frac{\partial\tau_{\text{e}}}{\partial P_{\text{O}}}=0\), \(\frac{\partial\tau_{\text{e}}}{\partial\beta_{\text{O}}}=0\), \(\frac{\partial\tau_{\text{e}}}{\partial u}=0\), the solution is:  
\[P_{\text{O}}^{*}=\frac{akm\left(2m(1-s)\left(2 km-k_{T}^{2}\right)+ns\left(4 km- k_{T}^{2}\right)\right)}{\text{A}_{\text{F}}} \tag{53}\]  
\[\beta_{\text{O}}^{*}=\frac{ak_{\text{T}}m\left(2m(1-s)\left(2 km-k_{T}^{2} \right)+ns\left(4 km-k_{T}^{2}\right)\right)}{\text{A}_{\text{F}}} \tag{54}\]  
\[\text{w}^{*}=\frac{\text{a}\left(\left(n+\text{m}s-\text{n}s\right)\left(k_{ \text{T}}^{2}-2\text{k}m\right)^{2}-\text{k}k_{\text{T}}^{2}\text{n}^{2}s \right)}{\text{A}_{\text{F}}} \tag{55}\]  
\[\beta_{T}^{*}=\frac{\text{a}\text{f}_{\text{T}}\left(\left(1-s\right)\text{ k}_{\text{T}}^{2}\text{m}+s\left(\text{m}^{2}-\text{n}^{2}\right)\left(2\text{ k}m-k_{\text{T}}^{2}\right)\right)}{A_{\text{F}}} \tag{56}\]  
\[\text{P}_{\text{T}}^{*}=\frac{\text{a}(1-s)\text{n}\left(\left(2\text{k}m-k_ {\text{T}}^{2}\right)^{2}+\text{k}m\text{k}_{\text{T}}^{2}\right)}{A_{F}}+ \frac{\text{ams}\left(\left(k_{\text{T}}^{2}-2\text{k}m\right)\left(k_{\text {T}}^{2}-3\text{k}m\right)-2\text{k}^{2}\text{n}^{2}\right)}{A_{F}} \tag{57}\]  
Substituting the above results into the demand function and the profit function of the supply chain and each subject, the optimal solution is obtained as follows:  
\[ \begin{split}\pi_{\text{C}}^{*}=\frac{\text{a}^{2}\text{k}m \left(\text{m}\left(2\text{k}m-k_{\text{T}}^{2}\right)\left(3\left(s-\frac{2} {3}\right)^{2}+\frac{2}{3}\right)+2\text{ns}\left(1-s\right)\left(4\text{k}m- k_{\text{T}}^{2}\right)+2\text{k}n^{2}\text{s}^{2}\right)}{2A_{\text{F}}}\\ +\frac{\text{a}^{2}\text{k}\left(2\text{k}m-k_{\text{T}}^{2} \right)\left(s\left(\text{m}^{2}-\text{n}^{2}\right)\left(2\text{k}m-k_{ \text{T}}^{2}\right)+\left(1-s\right)\text{k}_{\text{T}}^{2}\text{m}m\right)^ {2}}{2A_{\text{F}}^{2}}\end{split}  \tag{58}\]  
\[\pi_{\text{S}}^{*}=\frac{\text{a}^{2}\text{k}m\left(\text{m}\left(2\text{k}m- k_{\text{T}}^{2}\right)\left(3\left(s-\frac{2}{3}\right)^{2}+\frac{2}{3} \right)+2\text{ns}\left(1-s\right)\left(4\text{k}m-k_{\text{T}}^{2}\right)+2 \text{kn}^{2}\text{s}^{2}\right)}{2A_{\text{F}}} \tag{59}\]  
\[\pi_{\text{T}}^{*}=\frac{\text{a}^{2}\text{k}\left(2\text{k}m-k_{\text{T}}^{2} \right)\left(s\left(\text{m}^{2}-\text{n}^{2}\right)\left(2\text{k}m-k_{\text{ T}}^{2}\right)+\left(1-s\right)\text{k}_{\text{T}}^{2}\text{m}n\right)^{2}}{2A_{ \text{F}}^{2}} \tag{60}\]  
\[\text{Q}_{\text{T}}^{*}=\frac{\text{a}\text{k}m\left(s\left(\text{m}^{2}-n^{2} \right)\left(2\text{k}m-k_{\text{T}}^{2}\right)+\left(1-s\right)\text{k}_{\text{ T}}^{2}mn\right)}{A_{\text{F}}} \tag{61}\]  
\[\text{Q}_{\text{O}}^{*}=\frac{\text{a}\text{k}m\left((1-s)\left(2\text{m}^{2} \left(2\text{k}m-k_{\text{T}}^{2}\right)-\text{n}^{2}\left(4\text{k}m-k_{\text{ T}}^{2}\right)\right)+2\text{k}ns\left(\text{m}^{2}-\text{n}^{2}\right)\right)}{A_{ \text{F}}} \tag{62}\]  
where, according to the basic Assumption 4, \(2\text{k}m-k_{\text{T}}^{2}>2\text{k}n\), and \(\text{A}_{\text{F}}=8\text{k}^{2}\text{m}^{4}-8\text{k}^{2}\text{m}^{2}\text{n}^{2}- 8\text{k}\text{k}_{\text{T}}^{2}\text{m}^{3}+4\text{k}\text{k}_{\text{T}}^{2} \text{m}^{2}\text{n}^{2}+2\text{k}_{\text{T}}^{4}\text{m}^{2}-\text{k}_{\text{T}}^ {4}\text{n}^{2}=(2\text{k}m-k_{\text{T}}^{2})^{2}(\text{m}^{2}-\text{n}^{2})+ \text{m}^{2}\)\((2\text{k}m-k_{\text{T}}^{2}-2\text{kn})(2\text{k}m-k_{\text{T}}^{2}+2 \text{kn})\).