## 5 A Dual-Channel Agricultural Supply Chain Model under Decentralised Decision-Making
Solving for \(\frac{\partial\pi_{\mathrm{T}}}{\partial\mathrm{P}_{\mathrm{T}}}=0,\ \frac{ \partial\pi_{\mathrm{T}}}{\partial\mathrm{P}_{\mathrm{T}}}=0\) yields the retailer's reaction function for \(P_{\mathrm{T}},\ \beta_{\mathrm{T}}\) as follows:  
\[P_{T}^{\mathrm{tempt}}=\frac{ksa-w\mathrm{k}_{\mathrm{T}}^{2}+kmw}{2 km-\mathrm{ k}_{\mathrm{T}}^{2}} \tag{34}\]  
\[\beta_{T}^{\mathrm{tempt}}=\frac{\mathrm{k}_{\mathrm{T}}(sa-mw)}{2 km-\mathrm{ k}_{\mathrm{T}}^{2}} \tag{35}\]  
Substituting the retailer's reaction function with respect to \(P_{\mathrm{T}},\ \beta_{T}\) into the producer's profit function yields:  
\[\pi_{\mathrm{S}}=\mathrm{km}w\Bigg{(}\frac{sa-mw}{2 km-\mathrm{k}_{\mathrm{T}} ^{2}}\Bigg{)} \tag{36}\]  
Under decentralised decision-making, the producer's decision variable is \(w\). The first- and second-order derivatives of the producer's profit function \(\pi_{\mathrm{S}}\) with respect to the wholesale price \(w\) are:  
\[\frac{\partial\pi_{\mathrm{S}}}{\partial\mathrm{w}}=\frac{mk(sa-2 mw)}{2 km- \mathrm{k}_{\mathrm{T}}^{2}} \tag{37}\]  
\[\frac{\partial\pi_{\mathrm{S}}}{\partial\mathrm{w}^{2}}=\frac{-2\mathrm{km}^ {2}}{2 km-\mathrm{k}_{\mathrm{T}}^{2}}<0 \tag{38}\]  
Substituting the optimal solution \(w^{*}\) into \(P_{T},\ \beta_{T}\), then:  
\[\mathrm{P}_{\mathrm{T}}^{*}=\frac{3 akms-a\mathrm{k}_{\mathrm{T}}^{2}s}{2m \Big{(}2 km-\mathrm{k}_{\mathrm{T}}^{2}\Big{)}}<0 \tag{39}\]  
\[\beta_{\mathrm{T}}^{*}=\frac{ak_{\mathrm{T}}s}{2\Big{(}2 km-\mathrm{k}_{ \mathrm{T}}^{2}\Big{)}} \tag{40}\]  
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:  
\[\pi_{\mathrm{C}}^{*}=\frac{3a^{2}ks^{2}}{8\Big{(}2 km-\mathrm{k}_{\mathrm{T}} ^{2}\Big{)}} \tag{41}\]  
\[\pi_{\mathrm{S}}^{*}=\frac{ks^{2}a^{2}}{4\Big{(}2 km-\mathrm{k}_{\mathrm{T}}^{ 2}\Big{)}} \tag{42}\]  
\[\pi_{r}^{*}=\frac{ks^{2}a^{2}}{8\Big{(}2 km-\mathrm{k}_{\mathrm{T}}^{2}\Big{)}} \tag{43}\]  
\[\beta_{T}^{\mathrm{tempt}}=\frac{\mathrm{k}_{\mathrm{T}}(sa-mw)}{2 km-\mathrm{ k}_{\mathrm{T}}^{2}} \tag{35}\]0  
It is assumed that the producer is a leading agricultural product company with a core position in the supply chain, with product pricing rights, and can choose both online and offline channels to distribute agricultural products. In addition to wholesaling to brick-and-mortar superstores, the producer can also sell agricultural products directly through online platforms and bear the logistics costs such as warehousing and transport by itself. Under decentralised decision-making, producers and retailers no longer take the overall profit maxim of the supply chain as a common goal; they consider their own profit maxim \(\pi_{\mathrm{S}},\ \pi_{r}\) and formulate the optimal strategies. Under decentralised decision-making, the profit function of a dual-channel supply chain producer is:  
\[\beta_{T}^{\mathrm{tempt}}=\frac{\mathrm{k}_{\mathrm{T}}(sa-mw)}{2 km-\mathrm{ k}_{\mathrm{T}}^{2}} \tag{35}\]1  
\[\beta_{T}^{\mathrm{tempt}}=\frac{\mathrm{k}_{\mathrm{T}}(sa-mw)}{2 km-\mathrm{ k}_{\mathrm{T}}^{2}} \tag{35}\]2  
where the first part represents the revenue earned by the producer from selling the produce to the retailer, and the second part represents the revenue earned by the producer from selling the produce directly through the online channel, as well as the cost incurred in providing the logistics services of the online channel.