# Pricing Strategies of Logistics Distribution Services for Perishable Commodities
## 3 Model Construction of the Optimal Decision under Different Pricing Strategies
### Optimal Fixed Pricing Model with Time Constraints  
The fixed pricing model with time constraints is based on the fixed pricing model and considers the demand of consumers for distribution time, which meets the demand of consumers who have strict time constraints. Consumers are divided into ordinary consumers and strategy-based consumers. Strategy-based consumers refer to the distribution of perishable goods with specific requirements for time and price [24]. Strategic consumers often have certain requirements for delivery time. This paper considers a fixed pricing model with time constraints in order to realize the personalized service of perishable goods logistics distribution.  
For the fixed pricing model with time constraints, the original planned delivery time is \(T_{1}\), and the consumer expected time is \(T_{2}\), let \(T_{0}=T_{1}-T_{2}\). The variable cost of unit logistics is \(C(l)\).  
\[C(l)=k_{0}{T_{0}}^{2} \tag{4}\]  
Here, \(k_{0}\) is the coefficient of time-cost variation, \(C(l)\) is a quadratic function of \(T_{0}\). The greater the difference between the expected time of consumers and the original planned time, the greater the increase of the unit logistics distribution cost of perishable goods. Therefore, the linear relationship between \(C(l)\) and \(T_{0}\) is not used. The first derivative of \(T_{0}\) for \(C(l)\) is \(C(l)=2k_{0}T_{0}\). The results show that the rate of increase of distribution pricing is twice the original one. This is also in line with reality, namely the shorter the distribution time, the higher the increasing rate of distribution pricing. The relationship is more acceptable by consumers. The average profit of the logistics provider is defined as follows.  
\[\pi_{TF}=\frac{1}{T_{2}}\bigg{[}\int_{0}^{T_{2}}(P-C-C(l))Q-hQdt-W\bigg{]} \tag{5}\]  
Substituting (1) in to (5), the following can be derived:  
\[\pi_{TF}=[P-C-C(l)-h]\frac{kl-P}{s}+\frac{P-C-C(l)-h}{sT_{2}}\int_{0}^{T_{2}}V_ {t}dt-\frac{W}{T_{2}} \tag{6}\]  
Substituting \(V_{t}=V_{0}e^{-\lambda t}\) and formula (4) into (6), if \(\frac{\partial\pi_{TF}}{\partial P}=0\) is satisfied, the distribution pricing makes the logistics service provider obtaining maximum profit, which can be shown in formula (7).  
\[{P_{TF}}^{*}=\frac{kl+C+{k_{0}}{T_{0}}^{2}+h}{2}+\frac{V_{0}-V_{T_{2}}}{2 \lambda T_{2}} \tag{7}\]  
**Property 4**.: _The larger the difference between the expected time and the planned delivery time, the higher the distribution price._  
**Proof.** From (7), the larger the difference between the expected time and the planned delivery time, the greater the value of \(T_{0}\). \({P_{TF}}^{*}\) is a quadratic function of \(T_{0}\). For \({P_{TF}}^{*}\), the first order partial derivative of \(T_{0}\) is obtained by \(\frac{\partial{P_{TF}}^{*}}{\partial T_{0}}=k_{0}T_{0}>0\) that is constant, here \({P_{TF}}^{*}\) is an incremental function of \(T_{0}\). \(\square\)  
From (7), it can be seen that the pricing of the distribution service for perishable goods is determined by many factors, such as consumer service satisfaction, storage cost of perishable goods and distribution time difference. These factors have a positive effect on distribution pricing, which provides a reference for logistics service providers to formulate distribution pricing for perishable goods.