مقدار تولید اقتصادی (EPQ) با کمبود مشتق جبری

Previously, in several papers and textbooks, the classical economic order quantity (EOQ) and the economic production quantity (EPQ) formulas for the shortage case, have been derived using differential calculus and solving two simultaneous equations (derived from setting the two first partial derivatives to zero) with the need to prove optimality conditions with second-order derivatives. In a previous original piece of work, a new approach to find the EOQ with backlogging using some slight algebraic developments appeared. This paper extends the mentioned algebraic approach to the EPQ formula taking shortages into consideration within the case of only one backlog cost per unit and time unit. The final expressions provide the same formulas that are available in the classic textbooks on inventory theory.

Economic lot size models have been studied extensively since Harris [1] first presented the famous economic order quantity (EOQ) formula. Much of the literature on inventory theory contains the basic models of EOQ with/without shortages, and economic production quantity (EPQ) with/without shortages. To obtain the final expressions of these models for determining the economic order/manufacturing quantity require a mathematical methodology which is experienced as complicated for many undergraduate students. However, some authors have presented several approaches to find the EOQ without shortage in a simpler manner. For example, Thierauf and Grosse [2, pp. 187–193] presented a tabular, a graphical and an algebraic approach as alternatives to the differentiation approach. In a recent paper, Grubbström and Erdem [3] showed that the formulae for the standard EOQ with backlogging could be derived without the classical technique of optimisation, in other words, without differential calculus.

In this paper, an extension of Grubbström and Erdem's algebraic procedure has been proposed in order to find the optimal values of Imax and B of the classical static inventory models without using differential calculus. The algebraic procedure presented in this paper for the derivation of the model of EPQ with shortages, should be considered as a more accessible approach to ease the learning of basic inventory theory for students who lack knowledge of calculus.