A nonhomogeneous quasi-birth-death process approach for an <tex-math id="M1">$ (s, S) $</tex-math> policy for a perishable inventory system with retrial demands

In this paper, an $ (s, S) $ continuous inventory model with perishable items and retrial demands is proposed. In addition, replenishment lead times that are independent and identically distributed according to phase-type distribution are implemented. The proposed system is modeled as a three-dimensional Markov process using a level-dependent quasi-birth-death (QBD) process. The ergodicity of the modeled Markov system is demonstrated and the best method for efficiently approximating the steady-state distribution at the inventory level is determined. This paper also provides performance measure formulas based on the steady-state distribution of the proposed approximation method. Furthermore, in order to minimize the system cost, the optimum values of s and S are determined numerically and sensitivity analysis is performed on the main parameters.