We have considered a single retailer inventory model of seasonal products. Deterioration is reduced by preservation technology investment. The decision variable is the selling price of the product and the preservation technology investment parameter *u*. If \(\theta\) is the deterioration rate and *f*(*u*) is the proportion of reduced deterioration rate by investing preservation technology costs, then the inventory at time *t* is shown in Fig. 1 and follows this differential equation:

$$\frac{\partial I\left( t \right)}{\partial t} + \theta \left( {1 - f\left( u \right)} \right)I\left( t \right) = - D\left( {p,t} \right)\quad {\text{where}}\; D\left( {p,t} \right) = \alpha - at - \beta p,\;0 \le t \le T$$

(1)

The boundary condition *I(T)* = *0*, leads

$$I\left( t \right) = \frac{{\left( {\alpha - at - \beta p} \right)\,e^{{\theta \left( {1 - f\left( u \right)} \right)\left( {T - t} \right)}} }}{{\theta \left( {1 - f\left( u \right)} \right)}} - \frac{\alpha - at - \beta p}{{\theta \left( {1 - f\left( u \right)} \right)}} - \frac{a}{{\theta^{2} \left( {1 - f\left( u \right)} \right)^{2} }} + \frac{{ae^{{\theta \left( {1 - f\left( u \right)} \right)\left( {T - t} \right)}} }}{{\theta^{2} \left( {1 - f\left( u \right)} \right)^{2} }}$$

(2)

The on-hand inventory *Q* will be

$$Q = \frac{{\left( {\alpha - aT - \beta p} \right)e^{{\theta \left( {1 - f\left( u \right)} \right)T}} }}{{\theta \left( {1 - f\left( u \right)} \right)}} - \frac{\alpha - \beta p}{{\theta \left( {1 - f\left( u \right)} \right)}} - \frac{a}{{\theta^{2} \left( {1 - f\left( u \right)} \right)^{2} }} + \frac{{ae^{{\theta \left( {1 - f\left( u \right)} \right)T}} }}{{\theta^{2} \left( {1 - f\left( u \right)} \right)^{2} }}$$

(3)

The total profit *TP*(*T, p, u*) of the season can be formulated as

$$\begin{aligned} & TP\left( {T, \, p, \, u} \right) = {\text{ Sales revenue }}\left( R \right) \, {-}{\text{ Purchasing cost }}\left( {c_{p} } \right) \, {-}{\text{ Inventory holding cost }}\left( {c_{h} } \right) \\ & \quad {\text{Preservation cost }}\left( {I_{o} } \right) \, - {\text{ Replenishment cost }}(K) \\ \end{aligned}$$

(4)

### Sales revenue

The total revenue in time *T* can be formulated as

$$R = p\left( {\alpha T - \frac{{aT^{2} }}{2} - \beta pT} \right)$$

### Purchasing cost

According to Eq. (3), we know the order quantity. Thus, the total purchasing cost can be formulated as

### Inventory holding cost

The formulation of the total inventory holding cost is

$$c_{h} = - \frac{{h\left( {\alpha T - \frac{{aT^{2} }}{2} - \beta pT} \right)}}{x} - \frac{{h\left( {\alpha - \beta p} \right)}}{{x^{2} }}\left( {1 - e^{xT} } \right)$$

### Preservation cost

Preservation technology investment depends on the cycle length. For the inventory cycle *T*, the preservation technology investment cost is

### Replenishment cost

The replenishment cost is

by Eq. (4), the total profit function per unit time is

$$\begin{aligned} & TP\left( {T,\,p,\,u} \right) = \frac{p}{T}\left( {\alpha T - \frac{{aT^{2} }}{2} - \beta pT} \right) - \frac{c}{T}\left[ { - \frac{\alpha - \beta p}{x} - \frac{a}{{x^{2} }} + \frac{{\left( {\alpha - aT - \beta p} \right)e^{xT} }}{x} + \frac{{ae^{xT} }}{{x^{2} }}} \right] \\ & \quad + \frac{h}{Tx}\left( {\alpha T - \frac{{aT^{2} }}{2} - \beta pT} \right) + \frac{h}{{Tx^{2} }}\left( {\alpha - \beta p} \right)\left( {1 - e^{xT} } \right) - u - \frac{K}{T},\quad {\text{where}}\,x = \theta \left( {1 - f\left( u \right)} \right) \\ \end{aligned}$$

(5)

###
**Proposition 1**

*There exists a unique p* that maximizes the profit function*
\(TP\left( {T,\,p,\,u} \right)\)
*for fixed T and optimal u.*

###
*Proof*

The first and second partial derivatives of the profit function \(TP\left( {T,\,p,\,u} \right)\) with respect to *p* are as follows:

$$\frac{{dTP\left( {T,\,p,\,u} \right)}}{dp} = \alpha - \frac{aT}{2} - 2\beta p - \frac{c}{T}\left( {\frac{\beta }{x} - \frac{{\beta e^{xT} }}{x}} \right) - \frac{h\beta }{x} - \frac{h\beta }{{Tx^{2} }}\left( {1 - e^{xT} } \right)$$

Let \(\frac{{dTP\left( {T,\,p,\,u} \right)}}{dp}\) be zero and solve for optimal *p**, we have

$$p* = \frac{\alpha }{2\beta } - \frac{aT}{4\beta } - \frac{1}{2xT}\left( {c + \frac{h}{x}} \right)\left( {1 - e^{xT} } \right) - \frac{h}{2x}$$

(6)

At point *p* = *p*,* we have

$$\frac{{\partial^{2} TP\left( {T,\,p,\,u} \right)}}{{\partial p^{2} }} = - 2\beta < 0$$

Thus, *p** is the optimal market price that maximizes the profit function for fixed *T* and optimal *u*.

###
**Proposition 2**

*The profit function*
\(TP(T,\,p,\,u)\)
*is concave in the replenishment cycle T.*

###
*Proof*

The first and second partial derivatives of the profit function \(TP(T,\,p,\,u)\) with respect to *T* are as follows, where \({\text{x }} = \, \theta \left( {1 - f(u)} \right)\)

and

$$\begin{aligned} \frac{{\partial^{2} TP(T,\,p,\,u)}}{{\partial T^{2} }} & = \frac{ac}{{T^{3} }}\left( {\frac{\alpha - \beta p}{x} + \frac{a}{{x^{2} }}} \right) \\ & \quad - \frac{c}{x}\left[ {\left( {\alpha - \beta p - aT} \right)\frac{{x^{2} }}{T} - 2\left( {\alpha - \beta p} \right)\frac{x}{{T^{2} }} + \frac{2}{{T^{3} }}\left( {\alpha - \beta p} \right)} \right]e^{xT} \\ & \quad - \frac{ac}{{x^{2} }}\left( {\frac{{x^{2} }}{T} - \frac{2x}{{T^{2} }} + \frac{2}{{T^{3} }}} \right)e^{xT} \\ & \quad - \frac{h}{{x^{2} }}\left( {\alpha - \beta p} \right)\left[ {\frac{{x^{2} }}{T} - \frac{2x}{{T^{2} }} + \frac{2}{{T^{3} }}\left( {1 - e^{ - xT} } \right)} \right]e^{xT} < 0 \\ \end{aligned}$$

Hence, the total profit function is concave in *T*. Thus, there exists a unique optimal replenishment time *T*
^{*} that maximizes \(TP(T,\,p,\,u)\), and the optimal *T*
^{*} can be obtained by solving \(\frac{\partial TP(T,\,p,\,u)}{\partial T} = 0\).

###
**Proposition 3**

*For any given feasible p and T, there exists a unique optimal preservation technology investment u*
^{*}
*that maximizes TP*(*p,T,u*).

###
*Proof*

The first and second partial derivatives of the profit function *TP*(*p,T,u*) with respect to *u* are

$$\begin{aligned} & \frac{\partial TP(p,T,u)}{\partial u} = \left( {\frac{{c\left( {\alpha - \beta p} \right)}}{T\theta } + \frac{{h\left( {\alpha - \frac{aT}{2} - \beta p} \right)}}{\theta }} \right)\frac{f'(u)}{{\left( {1 - f(u)} \right)^{2} }} + \frac{2ac}{{T\theta^{2} }}\frac{f'(u)}{{\left( {1 - f(u)} \right)^{3} }} \\ & \quad c\left( {\alpha - aT - \beta p} \right)\frac{{f'(u)e^{{\theta \left( {1 - f(u)} \right)T}} }}{{\left( {1 - f(u)} \right)}} - \left( {\frac{c}{T\theta }\left( {\alpha - aT - \beta } \right) - \frac{{h\left( {\alpha - \beta p} \right)}}{\theta } - \frac{ac}{\theta }} \right)\frac{{f'(u)e^{{\theta \left( {1 - f(u)} \right)T}} }}{{\left( {1 - f(u)} \right)^{2} }} \\ & \quad - \frac{2ac}{{T\theta^{2} }}\frac{{f'(u)e^{{\theta \left( {1 - f(u)} \right)T}} }}{{\left( {1 - f(u)} \right)^{3} }} - \frac{{2h\left( {\alpha - \beta p} \right)}}{\theta }\frac{f'(u)}{{\left( {1 - f(u)} \right)^{2} }} - 1 \\ & \quad - \frac{{f'^{2} (u)}}{{\left( {1 - f(u)} \right)^{3} }}\frac{{\left( {\alpha - \beta p} \right)\left( {c - hT} \right)}}{T\theta }e^{{\theta \left( {1 - f(u)} \right)T}} \\ \end{aligned}$$

Since *f*′*(u)* > 0 and *f* ″*(u)* < 0, it is clear from the above equation that \(\frac{{\partial^{2} TP(p,T,u)}}{{\partial u{}^{2}}} < 0\).

Hence, the total profit is a concave function of the preservation cost.