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

Several researchers have recently derived formulae for economic production quantities (EPQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this note is twofold. First, this study extends earlier results to the EPQ formula, taking the imperfect rework process into account. Second, the algebraic complete-squares and perfect-squares methods can readily derive optimal expressions from an objective function in a more simple, direct, and natural manner than the algebraic method of unity decomposition adopted earlier. A numerical example has been solved to illustrate the solution procedure, and some remarks are made to conclude the note.