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Fractional order derivative, Fractional order inventory model under fuzzy environment, Long memory effect, Short memory effect
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.
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