This paper deals with the inventory replenishment problem for deteriorating items with normally distributed shelf life, continuous time-varying demand, and shortages under the inflationary and time discounting environment. The reasons of choosing normal are twofold: it is one of the most important probability phenomena in the real world due to the classical central limit theorem, and it is also one of the most commonly used lifetime distributions in reliability contexts. The problem is formulated as a dynamic programming model and solved by numerical search techniques. The solutions of the model determine the optimal replenishment schedule over a finite planning horizon so that the present worth of the future costs associated with the system is minimized. In the extensive experiments, we validate the model, demonstrate the optimal replenishment schedule and lot-size, and carry out a comparative study to ascertain its contribution. In addition, sensitivity analysis was provided to help identify the most crucial factors that affect system performance. The experimental result shows that the deteriorating problem solved by an appropriate model (i.e. the proposed normal model) can save the total cost up to 2% approximately. It also identifies that the magnitudes of purchase cost per unit and demand rate are the most significant parameters that affect the replenishment decisions and cost.