A general approach for the derivation of optimal solutions without derivatives

Abstract

In recent years, there have been a number of approaches in the literature that determine closed form optimal solutions to some classical inventory problems and their various extensions without using derivatives. Earliest of such approaches are based on algebraic manipulations with some a priori assumptions about the optimal solutions or based on properties of the quadratic function, whereas more recent approaches are based on problem specific assumptions such as an infinite planning horizon and integer decision variables, and analysing limiting behaviour of the decision variables and the objective function. None of the approaches is extendable to other similar optimisation problems. In this paper, we propose a general approach without any a priori assumptions of the optimal solution and without any problem specific assumptions. We demonstrate the proposed approach by applying it to some classical problems such as the EOQ and EPQ, with and without planned backorders.