# Pricing Strategies of Logistics Distribution Services for Perishable Commodities
## 3 Model Construction of the Optimal Decision under Different Pricing Strategies
### Optimal Dynamic Pricing Model with Constrained Time  
The dynamic pricing model with time constraints is based on the dynamic pricing model, which takes into account the demand of consumers for distribution time and meets consumers' strict requirements.  
For the fixed pricing model with time constraints, the original planned delivery time is \(t_{1}\), and the consumer expected time is \(t_{2}\). Suppose \(t_{0}=t_{1}-t_{2}\). The variable cost of unit logistics is \(c(l)\).  
\[c(l)={k_{1}}{t_{0}}^{2} \tag{10}\]  
where \(k_{1}\) is the time cost variation coefficient. In the dynamic pricing model with time constraints, the average profit of the logistics service provider in the distribution cycle can be expressed as follows.  
\[\pi_{TD}=\frac{1}{t_{2}}\bigg{[}\int_{0}^{t_{2}}\big{(}P_{TD}(t)-C-c(l)Q-hQ_{Q} (t,T)dt-W\big{)}\bigg{]} \tag{11}\]  
Here, \(Q_{Q}(t,T)\) is the preserved quantity of perishable goods at time \(t\). At this point, the following definition can be given.  
\[Q_{Q}(t,T)=\int_{t}^{T}Q(s,P)ds \tag{12}\]  
\[\text{Suppose }f(t,P)=(P_{D}(t)-C-c(l))Q\]  
\[\text{Suppose }g(t,P)=hQ_{Q}(t,T)\]  
Then  
\[\pi_{TD}=\frac{1}{t_{2}}\bigg{[}\int_{0}^{t_{2}}f(t,P)-\int_{0}^{t_{2}}g(s,P) dsdt-W\bigg{]}\]  
Since price of dynamic pricing model with the time-constrained is variable, the delivery time is divided into \(n\)\(\Delta t\) (\(\Delta t\to 0\)). The average profit in the dynamic pricing model can be expressed as follows.  
\[\pi_{TD}=\underset{\Delta s,\Delta t\to 0}{\text{lim}}\frac{1}{t_{2}}\bigg{\{} \bigg{\{}\sum_{t_{i}}^{t_{2}}\bigg{[}f(t_{i},P_{t_{i}})\Delta t-\sum_{s_{i}=t_{ i}}^{t_{2}}g(s_{i},P_{s_{i}})\Delta s\bigg{]}\Delta t-W\bigg{\}}\]  
Solving the first-order partial derivative of \(P_{t_{i}}\) for the equation mentioned above, and substituting \(\Delta t=\frac{T}{n}\) and \(t_{i}=i\Delta t\), we can obtain:  
\[\frac{\partial\pi_{D}}{\partial P_{t_{i}}}=\frac{1}{t_{2}}\bigg{\{}\frac{ \partial\mathscr{f}\big{(}t_{i}P_{t_{i}}\big{)}}{\partial P_{t_{i}}}\Delta t -i\frac{\partial\mathscr{g}\big{(}t_{i}P_{t_{i}}\big{)}}{\partial P_{t_{i}}} \Delta t^{2}\bigg{\}}=\frac{1}{t_{2}}\bigg{\{}\frac{\partial\mathscr{f}\big{(} t_{i}P_{t_{i}}\big{)}}{\partial P_{t_{i}}}-\frac{\partial\mathscr{g}\big{(}t_{i}P_{t _{i}}\big{)}}{\partial P_{t_{i}}}t_{i}\bigg{\}}.\]  
Because \(\frac{\partial\mathscr{f}\big{(}t,P\big{)}}{\partial P}-\frac{\partial\mathscr{ g}\big{(}t,P\big{)}}{\partial P}t=0\), the optimal dynamic pricing is expressed as follows.  
\[P_{TD}(t)^{*}=\frac{kl+C+V_{t}+{k_{1}}{t_{0}}^{2}}{2}+\frac{ht_{0}}{2} \tag{13}\]  
**Property 6**: _The shorter the expected delivery time, the higher the distribution price._  
**Proof.** The shorter the time of consumer expects, the larger the value of \(t_{0}\). Formula (13) shows that \(P_{TD}(t)^{*}\) is a function of \(t_{0}\), and the first-order partial derivative can be obtained for \(t_{0}\), \(\frac{\partial P_{TD}^{*}}{\partial t_{0}}={k_{1}}{t_{0}}+\frac{ht}{2}>0\), which is always true, namely, \(P_{TD}(t)^{*}\) is an increasing function on \(t_{0}\). \(\square\)