A periodic review inventory system when received items are of imperfect quality under two supply mode

This paper makes an attempt to develop a periodic review inventory system with imperfect quality of items. In this system, there are two resupply modes, namely regular mode and emergency supply mode. After reviewing the rest quantity at the review epoch decisions are taken that which supply mode should be used. And hence, the order is placed. Regular supply case has a higher lead time and shorter ordering cost compared to emergency supply case. Items are received with imperfectness which is random. After receiving the items, an immediate screening process has been started and imperfect items are separated. This problem is analyzed through the order-up-to RH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_H$$\end{document} inventory control system framework. Hence, there must be suitable units of items that raise the inventory position to RH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_H$$\end{document}. In this circumstance, it can be shown that either the regular supply case is used or there is an indifferent inventory position E at the review position such that, a higher quantity than E carries out the regular supply case and a lesser quantity than E carries out emergency supply mode. Thus models are developed and optimized through some analytical methods. Finally, numerical examples are taken and concluded through analytical discussions.