Best pricing and optimal policy for an inventory system under time-and-price-dependent demand and backordering

Abstract

In this paper, we study an inventory system for products where demand depends on time and price. Shortages are allowed and are fully backordered. We suppose that the demand rate is the product of a power time pattern and a three-parametric exponential price function. The objective is to determine the economic lot size, the optimal shortage level and the best selling price to maximize the total profit per unit time. We present an efficient procedure to determine the optimal solution of the inventory problem for all possible scenarios. This procedure is illustrated with several numerical examples. A sensitivity analysis of the optimal inventory policy with respect to the parameters of the demand rate function is also given. Finally, the main contributions of this paper are highlighted and future research directions are introduced.

Appendix

Proof of Lemma 1

For a fixed value p, the function P(Q, B, p) is twice-differentiable on the region

$$\begin{aligned} \Lambda =\left\{ (Q,B):Q>0,0\le B\le Q\right\} \end{aligned}$$

The first partial derivatives are

$$\begin{aligned} \frac{\partial P(Q,B,p)}{\partial Q}= & {} \frac{A\alpha e^{-\beta p^{\gamma }} }{Q^{2}}-\frac{h+\pi }{n+1}\left( 1-\frac{B}{Q}\right) ^{n}\left( 1+\frac{nB }{Q}\right) +\frac{\pi }{n+1} \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial P(Q,B,p)}{\partial B}= & {} (h+\pi )\left( 1-\frac{B}{Q}\right) ^{n}-\pi \end{aligned}$$

(24)

Thus, the second partial derivatives are given by

$$\begin{aligned} \frac{\partial ^{2}P(Q,B,p)}{\partial Q^{2}}= & {} -\frac{2A\alpha e^{-\beta p^{\gamma }}}{Q^{3}}-\frac{n\left( h+\pi \right) }{Q}\left( 1-\frac{B}{Q} \right) ^{n-1}\left( \frac{B}{Q}\right) ^{2} \\ \frac{\partial ^{2}P(Q,B,p)}{\partial B^{2}}= & {} -\frac{(h+\pi )n}{Q}\left( 1- \frac{B}{Q}\right) ^{n-1} \\ \frac{\partial ^{2}P(Q,B,p)}{\partial Q\partial B}= & {} \frac{(h+\pi )n}{Q} \left( 1-\frac{B}{Q}\right) ^{n-1}\frac{B}{Q} \end{aligned}$$

If we prove that the determinant of the Hessian matrix is positive for all $(Q,B)\in \Lambda$, the first assertion follows, because $\partial ^{2}P(Q,B,p)/\partial Q^{2}<0$ for all $(Q,B)\in \Lambda$.

Indeed, the determinant of the Hessian matrix is

$$\begin{aligned} \frac{2A\alpha \left( h+\pi \right) ne^{-\beta p^{\gamma }}}{Q^{4}}\left( 1- \frac{B}{Q}\right) ^{n-1} \end{aligned}$$

To prove the second assertion, it is sufficient to show that the point $(Q_{p}^{*},B_{p}^{*})$ given by (10) and (11) belongs to $\Lambda$ and that $\left. \frac{\partial P(Q,B,p)}{\partial Q} \right| _{(Q_{p}^{*},B_{p}^{*})}=\left. \frac{\partial P(Q,B,p)}{ \partial B}\right| _{(Q_{p}^{*},B_{p}^{*})}=0$, which is immediate. $\square$

Proof of Lemma 2

The second derivative of f(x) is

$$\begin{aligned} f^{\prime \prime }(x)=x^{-(2+1/\gamma )}\left( \frac{\beta \theta e^{\beta x/2}\left[ (\beta \gamma x-2)^{2}+4\gamma \right] }{4\sqrt{\alpha }\gamma }+ \frac{\left( \gamma +1\right) \beta c}{\gamma }\right) +\frac{2}{x^{3}} \end{aligned}$$

(25)

Note that $f^{\prime \prime }(x)>0$ for all $x>0$. Moreover, $\lim _{x\rightarrow 0^{+}}f(x)=\infty$ and $\lim _{x\rightarrow \infty }f(x)=\infty$. Therefore, the function f(x) is strictly convex and it attains its minimum at point $x_{1}$ given by (17). $\square$

Proof of Theorem 1

• 1. If $x_{1}\le c^{\gamma }$, then $f^{\prime }(p^{\gamma })>0$ for $p>c$ and, therefore, $f(p^{\gamma })>f(c^{\gamma })>0$ for all $p\ge c$. Since sign$(G^{\prime }(p))=$sign$(f(p^{\gamma }))$, we see that the function G(p) is strictly increasing on $(c,\infty )$.

• 2. If $x_{1}>c^{\gamma }$ and $f(x_{1})\ge 0$, then it is obvious that $f(p^{\gamma })>f(x_{1})\ge 0$ for all $p\ne x_{1}^{1/\gamma }$ and we conclude as in the previous case.

• 3. If $x_{1}>c^{\gamma }$ and $f(x_{1})<0$, then there exist two roots $x_{0}$ and $\widetilde{x}$ of the equation $f(x)=0$, with $c^{\gamma }<x_{0}<x_{1}<\widetilde{x}$, such that the function f(x) is positive on $(c^{\gamma },x_{0})$, negative on $(x_{0},\widetilde{x})$ and positive on $( \widetilde{x},\infty )$. Thus, the function G(p) is strictly increasing on $(c^{\gamma },x_{0})$, strictly decreasing on $(x_{0},\widetilde{x})$ and strictly increasing on $(\widetilde{x},\infty )$. Therefore, G(p) attains its minimum at $p^{*}=x_{0}^{1/\gamma }$ or $p^{*}=\infty$. Comparing the values $G(x_{0}^{1/\gamma })$ and $\lim _{p\rightarrow \infty }G(p)=0$, we obtain the optimal selling price.

$\square$

Proof of Lemma 3

From (21), we have

$$\begin{aligned} f_{1}^{\prime }(x)=\beta \left( \frac{\beta \theta e^{\beta x/2}}{2\sqrt{ \alpha }}-1\right) \end{aligned}$$

(26)

and

$$\begin{aligned} f^{\prime \prime }(x)=\frac{\beta ^{3}\theta e^{\beta x/2}}{4\sqrt{\alpha }} >0 \end{aligned}$$

(27)

The rest of the proof follows from $\lim _{x\rightarrow -\infty }f_{1}(x)=\infty$ and $\lim _{x\rightarrow \infty }f_{1}(x)=\infty$. $\square$

Proof of Theorem 2

This follows by the same method as in the proof of Theorem 1. $\square$

Proof of Lemma 4

• 1. Substituting $p_{1}$ given by (22) in the function $f_{1}(x)$ given by (21), we have $f_{1}(p_{1})=\beta (c-p_{1})+3$ and, therefore $f_{1}(p_{1})\ge 0$ is equivalent to $p_{1}\le c+3/\beta$.

• 2. Taking into account that $f_{1}(p_{0})=0$, it is verified that

  $$\begin{aligned} \theta =-\sqrt{\alpha e^{-\beta p_{0}}}\left( c-p_{o}+\frac{1}{\beta } \right) . \end{aligned}$$

  Substituting this value into the expression (19) of the function, G becomes

  $$\begin{aligned} G(p_{0})=\alpha e^{-\beta p_{0}}\left( c-p_{o}+\frac{2}{\beta }\right) \text { .} \end{aligned}$$

  Thus, $G(p_{0})<0$ if and only if $p_{o}>c+2/\beta$.

$\square$