A Memory-Dependent Inventory Model with Fuzzy Price-Dependent Demand Under Backlogged Shortages

The fractional calculus significantly contributes to accounting and inventory control for the memory effect. This study introduces an inventory model with a price-influenced demand pattern and a complete backlog shortage, including the memory effect under a fuzzy environment. The proposed lot-sizing model uses the fractional-order derivative to include the memory effect. The primary objectives of this paper are summarized as follows: (i) the impact of the fractional rate of change in inventory level on the fractional inventory system is established, and (ii) price-dependent demand is considered as a triangular fuzzy number, and the total average cost is calculated with the help of two different defuzzification methods, such as signed distance (SD) method as well as the graded mean integration (GMI) method. The impact of the fractional-order rate of change in the proposed inventory system is verified through numerical examples and graphical displays. The unit shortage cost is sensitive to the strong and poor memory effects of SD and GMI methods. The results indicate that, in the SD method and the memory-less case, businesses run a long time to obtain the minimum total average cost, whereas, in the GMI method and the memory-less case, businesses run comparatively significantly little time to obtain the minimum total cost. Finally, some suggestions are given for making another type of fractional model.