Mathematical analytic techniques and the complete squares method for solving an inventory modelling problem with a mixture of backorders and lost sales

Abstract

Use of the calculus approach to search for the optimal solution will be rather complex and difficult for people that lack knowledge of calculus. Hence some researchers frequently adopt algebraic methods in order to simplify or bypass the calculus approach or the mathematical analytic techniques of calculus. However, the algebraic method has its own limits of application such that the algebraic method without using derivatives may cause its questionable validity and accuracy. This paper will overcome the shortcomings of the complete square method for the solutions procedure used earlier by Leung (Int J Prod Econ 113:275–281, 2008). So, the main purpose of this investigation is based upon the following motivation: By noticing some imperfections in Leung’s proofs based upon the complete squares method used by Leung (2008), this paper not only reveals the fact that both (A) and (B) can not be completely achieved by an algebraic approach, but it also shows that the mathematical analytic techniques of calculus can satisfactorily achieve both (A) and (B). Consequently, this paper corrects and improves the above-mentioned earlier investigation by Leung (2008), thereby reiterating the well-established fact that the mathematical analytic techniques do guarantee accuracy as well as dependability of the inventory modelling problems considered by (for example) Leung (2008).