The feature of the fractional order derivative and fractional order integration is one of the important tools to realize the beauty of the fractional calculus. Fractional order derivative and integration has a long history like classical calculus but its users are much less compared to the classical calculus. The purpose of this paper is to study an inventory model with linear type demand rate under the fuzzy environment. This paper also wants to introduce the memory effect property of fractional order derivative which can help to setup the model more authentic. Two advantages have been included to the model (i) memory effect,(ii) fuzzy environment. Here, the fractional order model is defuzzyfied using (i) signed distance method,(ii) graded mean integration method. Fuzzification can close to the reality with incorporating uncertainty behavior of some economic parameters of the inventory system and fractional order can explain the memory phenomena. For this problem due to illustrate defuzzification, set up cost, holding cost per unit, per unit cost are assumed as triangular fuzzy numbers. Fractional order derivative and integration are applied to develop the whole work. It is known that fractional calculus is a valuable tool to describe memory phenomena. Fractional order is established as the index of the memory. In this paper, depending on strength of memory, memory phenomena considered in two steps(i) long memory,(ii) short memory. The proposed fuzzy models and technique lastly have been illustrated. Results of two defuzzyfications are compared with graphical presentations. This present studies can help to moderate the classical fuzzy inventory model. From the numerical studied it is observed that in long memory effect, profit is good compared to the low memory effect or memory less system.

The purpose of this paper is to establish the memory effect in an inventory model. In this model, price dependent demand is considered during the shortage period. Primal geometric programming is introduced to solve the minimized total average cost and optimal ordering interval. And finally we have taken a numerical example to justify the memory effect of this type inventory system. From the result it is clear that the model is suitable for short memory affected business i.e. newly started business.

For real market studies of any business, inclusion of memory or past experience in inventory model has great impact. Memory means it depends on the past state of the process not only current state of the process. Indeed, the inventory system is an appropriate example as a memory affected system. Presence of long past experiences or short past experiences of any company or shop has different importance on increasing or decreasing profit. The description of the memory dependent inventory model is more appropriate process compared to the memory less inventory model. Depending on demand rate, a comparison between the minimized total average costs of different numerical example has been presented. Fractional order derivative and integration have been used to establish the model. Our considered numerical example establishes that if linear type demand rate is only time proportional, profit of the business is high compared to the linear type demand rate.