The economic production quantity with shortage derived algebraically

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 derivates to zero) with the need to prove optimality conditions with second-order derivates. 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.