Optimal production lot sizing with backlogging, random defective rate, and rework derived without derivatives

This paper is concerned with the algebraic derivation for the production lot size problem with backlogging, random defective rate, and rework. Conventional approaches for solving optimal production lot size are by using the differential calculus on the productioninventory cost function with the need to prove optimality first. Recent articles proposed the algebraic approach to the solution of the classic economic order quantity (EOQ) and economic production quantity (EPQ) models without reference to the use of derivatives. This paper extends it to an EPQ model taking backlogging, random defective rate, and rework into consideration. This note demonstrates that optimal lot size for such an imperfect quality EPQ model can be derived without derivatives. The expected production-inventory costs can also be obtained immediately.