Advance payment based vendor–buyer production inventory model with stochastic lead time and continuous review policy

Abstract

In this paper, a multi-cycle vendor–buyer supply chain production inventory model is conceptualized. Products in the system are considered to be deteriorated at a constant rate. Before replenishment of the order quantity, the buyer paid the purchase price in advance to the vendor in multiple time with an interest. In the lead time of buyer, the market demand is stochastic. Vendor’s production cost of the item depends on the production rate, labor and technology used for the production. During the production period and shipment of quantity, a green-house gas $(CO_2)$ is emitted at a constant rate to the environment. Finally, numerical examples and sensitivity analysis are provided to illustrate the presented model and also corresponding discussions and conclusion are presented.

Appendix

When x is exponentially distributed with $p.d.f.\,\,f(x) = \lambda e^{-\lambda x}$, the expression of $F(\frac{1}{\theta }log(1+\frac{r\theta }{D}))$ $J_{11}$, $J_{22}$, $J_{33}$ and $J_{44}$ are

$$\begin{aligned} F({\frac{1}{\theta }log(1+\frac{r\theta }{D})}) = & \int _{\infty }^{\frac{1}{\theta }log(1+\frac{r\theta }{D})} f(x)dx\\ = & \int _{0}^{\frac{1}{\theta }log(1+\frac{r\theta }{D})} \lambda e^{-\lambda x}dx\\ = & 1-\bigg (1+\frac{r\theta }{D}\bigg )^{-\frac{\lambda }{\theta }} \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{0}^{\frac{1}{\theta }log(1+\frac{r\theta }{D})}e^{{-\theta }x}f(x)dx = & \frac{\lambda }{\lambda +\theta } \Bigg (1-(1+\frac{r \theta }{D})^{-\frac{\lambda +\theta }{\theta }}\Bigg )\\ \int _{\frac{1}{\theta }log(1+\frac{r\theta }{D})}^{\infty }e^{{-\theta }x}f(x)dx = & \frac{\lambda }{\lambda +\theta } \Bigg (1+\frac{r\theta }{D}\Bigg )^{(-1-\frac{\lambda }{\theta })}\\ \int _{\frac{1}{\theta }log(1+\frac{r\theta }{D})}^{\infty }xf(x)dx = & \frac{1}{\lambda {\theta }}\Bigg [\theta +{\lambda } log(1+\frac{r\theta }{D})\Bigg ] \Bigg (1+\frac{r\theta }{D}\Bigg )^{-\frac{\lambda }{\theta }}\\ \int _{\frac{1}{\theta }log(1+\frac{r\theta }{D})}^{\infty }x^2f(x)dx = & \Bigg (1+\frac{r\theta }{D}\Bigg )^{-\frac{\lambda }{\theta }} \Bigg [\frac{1}{{\lambda }^2}+\{\frac{1}{\lambda }+\frac{1}{\theta }log(1+\frac{r \theta }{D})\}^2\Bigg ]\\ \end{aligned}$$

Also, the expression of $F(\frac{r}{D})$, $J^1_{11}$, $J^1_{22}$, and $J^1_{33}$ are

$$\begin{aligned} F(\frac{r}{D}) = & \int ^{\frac{r}{D}}_{\infty }f{x}dx\\ = & \int ^{\frac{r}{D}}_{0}\lambda e^{-\lambda x}dx\\ = & 1-e^{-\frac{r\lambda }{D}} \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{0}^{\frac{r}{D}}xf(x)dx = & \frac{1}{\lambda }-(\frac{1}{\lambda }+\frac{r}{D})e^{-\frac{r\lambda }{D}}\nonumber \\ \int _{\frac{r}{D}}^{\infty }xf(x)dx = & (\frac{1}{\lambda }+\frac{r}{D})e^{-\frac{r\lambda }{D}}\nonumber \\ \int _{\frac{r}{D}}^{\infty }x^2f(x)dx = & \{\frac{1}{{\lambda }^2}+(\frac{1}{\lambda }+\frac{r}{D})^2\}e^{-\frac{r\lambda }{D}}\nonumber \\ \end{aligned}$$