Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach

<p style='text-indent:20px;'>In this paper, a new approach was applied to a single-item single-source (<inline-formula><tex-math id="M1">\begin{document}$ SISS $\end{document}</tex-math></inline-formula>) system at the "<inline-formula><tex-math id="M2">\begin{document}$ EOQ-type $\end{document}</tex-math></inline-formula>" mode considering imperfect items and uncertainty environment. The mentioned method was intended to produce an optimum order/production quantity as well as taking care of imperfect processes. The imperfect proportion of the received lot size was described by an imperfect inspection process. That is, two-way inspection errors may be committed by the inspector as separate items. Thus, this survey was aimed to maximize the benefit in the traditional inventory systems. The incorporation of both defects and defective classifications (Type-<inline-formula><tex-math id="M3">\begin{document}$ I\&amp;II $\end{document}</tex-math></inline-formula> errors) was illustrated, in a way that the defects were returned by the consumers. Moreover, this inventory model had an extra step in the scope of inspection; which occurred after the rework process with no inspection error. To get closer to the practical circumstances and to consider the uncertainty, the model was formulated in the fuzzy environment. The demand, rework, and inspection rates of the inventory system were considered as the triangular fuzzy numbers where the output factors of the inventory system were obtained via nonlinear parametric programming and Zadeh's extension principle. Finally, this scenario was illustrated through a mathematical model. The concavity of the objective function was also calculated and the total profit function was presented to clarify the solution procedure by numerical examples.</p>