An inventory model for partial backlogging items with memory effect

Inventory control is considered one of the most widely documented topics in the reality. Fractional derivatives and integration is the part of fractional calculus. Fractional calculus is the generalized part of ordinary calculus. The memory of physical phenomena is a highly concerning topic but it is neglected with describing in terms of integer-order differential equation. To discuss the memory of the inventory model, fractional derivative tools are considered. A fractional derivative at any point gives the previous marginal output and current point output for any current point input. In this model, a shortage is considered, and during the shortage period, demand is partially backlogged. Depending on the low partial backlogging rate and high partial backlogging rate, the result of the memory effect varies on the total average cost and optimal ordering interval. Moreover, in order to show the relationship between fractional models and ordinary classical model, two types of memory indices have been considered: (i) differential memory index and (ii) integral memory index. For a certain memory effect, minimized total average cost is the same for low partial backlogging rate and high partial backlogging rate.