On a stochastic extension of the EOQ formula (Q760335)

The author considers an inventory model under the following conditions: the demand rate k and the replenishment rate m, \(k\leq m\), \(r=k/m\), are constant; the lead time X for stock delivery is a random variable, \(0<\alpha \leq X<\beta\), \(Ex=\mu\), Var X\(=\sigma^ 2\). A simple modification of the classical (deterministic) economic quantity (EOQ) formula is given where replenishment cost per order \(c_ 3\) is substituted by \(c'\!_ 3=c_ 3+(c_ 1+c_ 2)\sigma^ 2k[2(1- \mu)]^{-1},\) where \(c_ 1\) is the holding cost, \(c_ 2\) is the shortage cost. The modification is valid if there is only one order per cycle. Inequalities for parameters garanteeing this supposition are shown.