An inventory model with uncertain demand and lost sales reduction under service level constraint

Abstract

In practice, the quantity received may not match with the quantity ordered, due to various reasons such as rejection during inspection, damage or breakage during transportation, human errors in counting etc. Subsequently, the managers often must make decisions under uncertain quantity received circumstances. This paper explores the feasibility of reducing ordering cost and lost sales caused by stock-out. In this study, we examine the continuous review inventory model with shortages include the case where the quantity received is uncertain in which the lead time and lost sales rate are decision variables and also with service level constraint. Here we consider the lead time crashing cost is a function of negative exponential lead time. The objective of this study is to minimize the joint expected total cost by simultaneously optimizing the order quantity and lost sales rate. An efficient algorithm for finding the optimal solution is developed and numerical examples are given to illustrate the results.

Appendix

We want to prove the Hessian Matrix of $JETC(Q,\tilde{\beta },L)$ at point $(Q^*,\tilde{\beta }^*)$ for fixed $L\in [L_e,L_s]$ is positive definite. We first obtain the Hessian matrix ${\mathbf {H}}$ as follows:

$$\begin{aligned} {\mathbf {H}}=\left[ \begin{array}{c c} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {Q^2}} &{} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {Q}\partial {\tilde{\beta }}}\\ \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }}\partial {Q}}&{} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }^2}}\end{array} \right] \end{aligned}$$

where

$$\begin{aligned} \frac{\partial ^2{JETC(Q,\tilde{\beta },L)}}{\partial {Q^2}}= & \frac{2(u+vln L)D}{\alpha Q^3}+\frac{h \sigma _0^2}{\alpha Q^3}+\frac{D(\pi +\pi _0\tilde{\beta })\sigma \sqrt{L}\Psi (k)}{\alpha Q^3}+\frac{2D\delta {\mathrm{e}}^{-\gamma L}}{\alpha Q^3}\\ \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {Q}\partial {\tilde{\beta }}}= & -\frac{D \pi _0 \sigma \sqrt{L}\Psi (k)}{\alpha Q^3}.\\ \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }}\partial {Q}}= & -\frac{D \pi _0 \sigma \sqrt{L}\Psi (k)}{\alpha Q^3}\\ \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }^2}}= & {} -\frac{\theta c_1}{\tilde{\beta }^2}. \end{aligned}$$

Then we proceed by evaluating the principal minor determinant of ${\mathbf {H}}$.

The first principal minor determinant of ${\mathbf H}$ is

$$\begin{aligned} |H_{11}|=\frac{\partial {JETC(Q,L,n,\theta )}}{\partial {Q^2}}= & \frac{2D}{Q^3}\left[ (u+vL)+\frac{S}{n}+c_T+R(L)+\frac{D{t_c^2}({C_b}{I_c}-{C_s}{I_d})}{2}\right. \nonumber \\&\left. -\,{t_c}{C_s}{I_d}\sigma \sqrt{L}\psi (k)\right] >0 \end{aligned}$$

The second principle minor determinant of ${\mathbf H}$ is:

$$\begin{aligned} |H_{22}|= & \left[ \begin{array}{c c} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {Q^2}} &{} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {Q}\partial {\tilde{\beta }}}\\ \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }}\partial {Q}}&{} \frac{\partial ^2{JETC}(Q,\tilde{\beta },L)}{\partial {\tilde{\beta }^2}}\end{array} \right] \\&\Rightarrow \frac{2D}{\alpha Q^3}\bigg [(u+vL)+\frac{{\sigma _0^2}h}{2D}+\frac{(\pi +\pi _0\tilde{\beta })\sigma \sqrt{L}\Psi (k)}{2}+\delta {\mathrm{e}}^{-\gamma L}\bigg ] \frac{\theta c_1}{\tilde{\beta }^2} \\&\quad - \,\bigg (\frac{-D\pi _0\sigma \sqrt{L}\Psi (k)}{\alpha Q^2}\bigg )^2\\&\Rightarrow \frac{2D}{\alpha Q^3}\bigg [(u+vL)+\frac{{\sigma _0^2}h}{2D}+\frac{(\pi +\pi _0\tilde{\beta })\sigma \sqrt{L}\Psi (k)}{2}+\delta {\mathrm{e}}^{-\gamma L}\bigg ] \frac{\theta c_1}{\tilde{\beta }^2} \\&>\bigg (\frac{-D\pi _0\sigma \sqrt{L}\Psi (k)}{\alpha Q^2}\bigg )^2 \end{aligned}$$

Therefore $|H_{22}|>0$. We see that all the principal minors of the Hessian Matrix are positive. Hence, the Hessian Matrix ${\mathbf {H}}$ is positive definite at $(Q^*,\tilde{\beta }^*)$