International Journal of Mathematical Archive-8(2), 2017, 1-14 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 8(2), Feb. – 2017 1 GENERALIZED FRACTIONAL ORDER EOQ MODEL WHEN DEMAND IS STOCK DEPENDENT ASIM KUMAR DAS 1* , TAPAN KUMAR ROY 2 Department of Applied Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West-Bengal, India, 711103. (Received On: 12-01-17; Accepted On: 13-02-17) ABSTRACT In this article, we have developed a fractional order EOQ model, where demand is not assumed to be constant or time dependent. In many classical EOQ model it has been taken that demand may occur as stock dependent. Our objective in this article is to describe a generalized fractional order EOQ model with stock dependent demand. Keywords—Fractional differentiation, Fractional Integration, Fractional Differential Equation, Set up Cost, Holding Cost, Economic Order Quantity. 1. INTRODUCTION In recent years considerable interest in fractional calculus has been stimulated by the applications it finds in different areas of applied sciences like physics and engineering, possibly including fractal phenomena. Now there are more books of proceedings and special issues of journals published that refer to the applications of fractional calculus in several scientific areas including special functions, control theory, chemical physics, stochastic processes, anomalous diffusion, archeology. Several special issues appeared in the last decade which contain selected and improved papers presented at conferences and advanced schools, concerning various applications of fractional calculus. Already since several years, there exist two international journals devoted almost exclusively to the subject of fractional calculus: Journal of Fractional Calculus (Editor-in-Chief: K.Nishimoto, Japan) started in 1992, and Fractional Calculus and Applied Analysis (Managing Editor: V. Kiryakova, Bulgaria) started in 1998. Recently the new journal Fractional Dynamic Systems has been announced to start in 2010. The authors believe that the volume of research in the area of fractional calculus will continue to grow in the forthcoming years and that it will constitute an important tool in the scientific progress of mankind. Only recently, fractional calculus was applied to classical EOQ model to generalize this model in operation research. In a previous papers [4-5] we have discussed how the fractional calculus can utilizes to develop the classical EOQ model to generalize EOQ model in operation research. In particular, we have seen fractional calculus has a potentiality to apply this concept in any other EOQ model. In this sense we represent the more generalize EOQ model using the broad concept of fractional calculus where the model is based on stock dependent demand. Here we have applied the concept of derivative/integrals with an emphasis on Caputo and Riemann-Liouville fractional derivatives [2], [13] and have some interesting results and ideas [23] that demonstrate the generalized EOQ based inventory model. Fractional derivatives and fractional integrals have interesting mathematical properties that may be utilized to develop our motivation. In this article, first we give a short description on general principles, definitions and several features of fractional derivatives/integrals and then we review some of our ideas and findings in exploring potential applications of fractional calculus in inventory control model. II. A SHORT DESCRIPTION ON FRACTIONAL DIFFERENTIAL CALCULUS The origin of fractional calculus goes back to Newton and Leibniz in the seventieth century. S.F Lacroix was the first to mention in some two pages a derivative of arbitrary order in a 700 pages text book of 1819. Corresponding Author: Asim Kumar Das 1* Department of Applied Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West-Bengal, India, 711103.