Approximated closed-form minimum-cost solution to the (r, q) policy with complete backordering and further developments

This study considers the continuous-review (r, q) inventory policy with complete backordering, a deterministic and constant lead time, and a Gaussian lead-time demand. A closed-form solution to the first-order conditions of optimality is not available, so this system can only be optimized using an iterative method. However, an iterative optimization approach may still limit the application of the model itself in a practical context with numerous items and the need to frequently recalculate the control variables for the respective inventory policies. In fact, if a closed-form optimal solution is not available, the computational effort required by an iterative method may not be negligible and its implementation can actually be technically difficult. Thus, to overcome the characteristic limitations of an iterative optimization approach, we develop a closed-form near-optimal solution in this study, which is derived via an ad hoc logistic approximation of the standard normal loss function. We then provide an alternative formulation of the expected total cost function. In particular, by introducing a parameter for the number of admissible stockouts per time unit, we can ensure the functional dependence of the service level on the order quantity. This allows us to optimize the order quantity, where the safety stock and the expected shortage are considered by the constraint mentioned above. We present numerical tests to investigate the efficiency of the approximated solution method and to study the sensitivity of the new cost model with respect to some fundamental parameters.