# Pricing Strategies of Logistics Distribution Services for Perishable Commodities
## 3 Model Construction of the Optimal Decision under Different Pricing Strategies  
### Optimal Fixed Pricing Model  
The fixed pricing model means that the price is fixed and does not change even the decay of perishable goods and distribution time change [23]. Under the fixed pricing model, \(P\) is a fixed value, and the average profit of the logistics service provider is defined as follows. \[\pi_{F}=\frac{1}{T}\bigg{[}\int_{0}^{T}(P-C)Q-hQdt-W\bigg{]} \tag{2}\]  
where \((P-C)Q\) represents the total revenue of the logistics service provider considering the cost, \(hQ\) represents the storage cost of deteriorating items in the distribution process, \(W\) represents some additional expenses incurred in the delivery process, \(\pi_{F}\) represents the average net profit of logistics providers in the fixed pricing model. Substituting (1) into (2), the following formula can be obtained:  
\[\pi_{F}=\frac{(kl+C+h)P-ckl-hkl-P^{2}}{s}-\frac{W}{T}+\frac{(P-C-h)}{sT}\int_ {0}^{T}V_{t}dt\]If \(\frac{\partial\pi_{F}}{\partial\mathbf{P}}=0\), the distribution price makes the profit of logistics provider maximum. Correspondingly, it will be transferred as follows. \[P_{F}{}^{*}=\frac{kl+h+C}{2}+\frac{\int_{0}^{T}V_{t}dt}{2T}\]  
Substituting the value of the item at time \(T\), formula will be obtained. \[P_{F}{}^{*}=\frac{kl+h+C}{2}+\frac{V_{0}\big{(}1-e^{-\lambda T}\big{)}}{2 \lambda T} \tag{3}\]  
However, under the fixed pricing model, logistics service providers do not take into account the delivery time of perishable goods, which easily reduces customer satisfaction because of delivery time delay and leads to a reduction in the number of customers, and ultimately reduces the overall profit of logistics service providers. **Property 1**: _Under the fixed pricing model, the value of perishable goods decreases faster and the price for logistics distribution decreases._  
**Proof. ** It can be known from (3) that \(P_{F}{}^{*}\) is an expression about the decay parameter \(\lambda\), and the first-order partial derivative of \(\lambda\) is solved for (3), \(\frac{\partial P_{F}{}^{*}}{\partial\lambda}=\frac{\big{[}(\lambda T+1)e^{- \lambda T}-1\big{]}V_{0}}{2T\lambda^{2}}\), with \(0<\lambda<1\), \(t>0\). The order \(\mathrm{f}(\lambda)=(\lambda T+1)e^{-\lambda T}-1\), \(\frac{\partial\mathrm{f}(\lambda)}{\partial\lambda}=-\lambda T^{2}e^{-\lambda T }<0\) is established. Therefore, \(\mathrm{f}(\lambda)\) is a monotone decreasing function of \(\lambda\), \(\mathrm{f}(0)=0\), That is, \(\mathrm{f}(\lambda)<0\) is established with \(0<\lambda<1\). Also, \(\frac{\partial P_{F}{}^{*}}{\partial\lambda}<0\) is established with \(0<\lambda<1\), so \(P_{F}{}^{*}\) decreases with the increase of \(\lambda\). \(\square\)  
Property 1 reveals the relationship between distribution price and value decay rate of deteriorating commodities under a fixed pricing model. The value decay rate of perishable goods increases while the price of logistics distribution decreases. Therefore, the logistics service providers should select suitable perishable commodities with the decay index according to the scheduled delivery time for the maximum profit. **Property 2**: _The higher the satisfaction of the consumers' logistics service, the higher the price of the logistics distribution._  
**Proof. ** It can be shown from formula (3) that \(P_{F}{}^{*}\) is an expression about consumer service satisfaction \(l\), and the first-order partial derivative of \(l\) is adopted on formula (3); \(\frac{\partial P_{F}{}^{*}}{\partial l}=\frac{k}{2}>0\) is always true. In other words, \(P_{F}{}^{*}\) is a monotonically increasing function on \(l\). \(\square\)  
**Property 3**: _The higher the cost of preservation of perishable goods, the higher the price of logistics and distribution._  
**Proof. ** It can be seen from formula (3) that \(P_{F}{}^{*}\) is an expression about the preservation cost \(h\) of a perishable commodity per unit time, and the first-order partial derivative of \(h\) is adopted on (3).