# Pricing Strategies of Logistics Distribution Services for Perishable Commodities
## 3 Model Construction of the Optimal Decision under Different Pricing Strategies
### Optimal Dynamic Pricing Model  
Dynamic pricing model differs from the fixed pricing model. It determines different distribution pricing according to the status of perishable goods at different times. Dynamic pricing is more akin to the competition-oriented pricing method [25], and adopt price differences and other means of operation to outcompete competitors and seize market share. The dynamic pricing model considers the decay of perishable goods during the process of distribution, the interests of consumers which are more reasonable, and adjusts the distribution pricing during the process of distribution. This will attract customers due to the more reasonable cost performance ratio, and then improves the overall interests of logistics service providers.  
Under the dynamic pricing model, the average profit of logistics providers in a distribution cycle can be expressed as follows.  
\[\pi_{D}=\frac{1}{T}\bigg{[}\int_{0}^{T}(P_{D}(t)-C)Q-hI_{Q}(t,T)dt-W\bigg{]} \tag{8}\]  
Here, \(I_{Q}(t,T)\) is the quantity of preservation of deteriorating commodities with the increase of delivery time. \(I_{Q}(t,T)=\int_{t}^{T}Q(s,P)ds\). \(I_{Q}(t,T)\) is related to the delivery price during delivery time.  
\[\text{Suppose }f(t,P)=(P_{D}(t)-C)\text{Q}.\]  
\[\text{Suppose }g(t,P)=hI_{Q}(t,T).\]  
Then \(\pi_{D}=\frac{1}{T}\Big{[}\int_{0}^{T}f(t,P)-\int_{t}^{T}g(s,P)dsdt-W\Big{]}\). Because the distribution pricing is based on the variable distribution time which is divided into \(n\)\(\Delta t\) (\(\Delta t\to 0\)). In the dynamic pricing model, the average profit can be defined as follows.  
\[\pi_{D}=\lim_{\Delta s,\Delta t\to 0}\frac{1}{T}\Bigg{\{}\frac{\Upsilon}{t_{i}} \Bigg{[}f(t_{i},P_{t_{i}})\Delta t-\sum_{s_{i}=t_{i}}^{T}g(s_{i},P_{s_{i}}) \Delta s\Bigg{]}\Delta t-W\Bigg{\}}\]  
For the first order partial derivative of the above formula for \(P_{t_{i}}\), we substitute \(\Delta t=\frac{T}{n}\) and \(t_{i}=i\Delta t\), and we get \(\frac{\partial\pi_{D}}{\partial P_{t_{i}}}=\frac{1}{T}\bigg{\{}\frac{ \partial f(t_{i},P_{t_{i}})}{\partial P_{t_{i}}}-\frac{\partial g(t_{i},P_{t_ {i}})}{\partial P_{t_{i}}}t_{i}\bigg{\}}\). Because \(\frac{\partial f(t,P)}{\partial P}-\frac{\partial g(t,P)}{\partial P}t=0\), the optimal dynamic pricing is expressed as follows.  
\[P_{D}(t)^{*}=\frac{kl+C+V_{t}}{2}+\frac{ht(T-t)}{2} \tag{9}\]  
From formula (9), it can be seen that the optimal dynamic pricing largely depends on the value of perishable goods at time \(t\). The value of perishable goods decreases with delivery time, and the value of \(P_{D}(t)^{*}\) decreases with the increase of delivery time.  
**Property 5**.: _If \(\lambda^{2}V_{0}<h\), \(\lambda T>2\), with the delivery time, the optimal distribution pricing in the dynamic pricing model gradually increases and then decreases, and the extent of the reduction increases with the increase of the speed of commodity decay and the increase of storage cost._  
**Proof.**\(\frac{\partial P_{D}(t)^{*}}{dt}=-\lambda V_{t}-ht+\frac{hT}{2}\), suppose \(h(t)=-\lambda V_{t}-ht+\frac{hT}{2}\). For \(h(t)\), the second derivative can be obtained, \(h(t)=-\lambda^{3}V_{0}e^{-\lambda t}<0\). \(h(t)\) is monotonically decreasing. \(h(t)=\lambda^{2}V_{0}e^{-\lambda t}-h\), \(\lambda^{2}V_{0}<h,h(0)=\lambda^{2}V_{0}-h<0\), and then \(h(t)<0\), so \(h(t)\) is also monotonically decreasing. If \(h(0)=-\lambda V_{0}+\frac{Th}{2}\)\(>\)\(-\lambda V_{0}+\frac{\lambda^{2}V_{0}T}{2}=\lambda V_{0}\left(\frac{\lambda T}{2}-1 \right)>0\), so \(\frac{\partial P_{D}(t)^{*}}{dt}\) is greater than \(0\) at first and then less than \(0\) with the increase of \(t\). The optimal distribution pricing gradually rises and then decreases. With the increase of \(\lambda\) and \(h\), the distribution price declines further. \(\square\)