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.