An integrated vendor–buyer inventory model with imperfect items and planned back orders

In this paper, we develop a mathematical model to determine an integrated vendor–buyer inventory policy, where the vendor’s production process is imperfect and produces a certain number of defective items with a known probability density function. The vendor prepares for the repeating flow of orders of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {Q_{\mathrm{P}}}=nQ $$\end{document} from the buyer by producing items in batches of size QP and planning to have each batch delivered to the buyers in n deliveries, each with a lot of Q units. Once the buyer receives the items, a 100 % screening process is conducted. We assume the screening process and demand take place simultaneously. We also assume that shortages are allowed and are completely back ordered. The objective is to minimize the total joint annual costs incurred by the vendor and the buyer. The expected annual integrated total cost is derived and a solution procedure is provided to find the optimal solution. Numerical examples show that the integrated model gives an impressive cost reduction in comparison to an independent decision by the buyer. The results also show that even though there is a cost associated with each back order, it is profitable for the company to have planned back orders if customers are willing to wait for the next delivery when a shortage occurs.