Arbitrary-order economic production quantity model with and without deterioration: generalized point of view

Mostafijur Rahaman,Sankar Prasad Mondal,Soheil Salahshour,Ali Akbar Shaikh,Norazak Senu,Ali Ahmadian

doi:10.1186/s13662-019-2465-x

Abstract

The key objective of this paper is to study and discuss the application of fractional calculus on an arbitrary-order inventory control problem. Using the concepts of fractional calculus followed by fractional derivative, we construct different possible models like generalized fractional-order economic production quantity (EPQ) model with the uniform demand and production rate and generalized fractional-order EPQ model with the uniform demand and production rate and deterioration. Also, we show that the classical EPQ model is the particular case of the corresponding generalized fractional EPQ model. This greatly facilitates the researcher a novel tactic to analyse the solution of the EPQ model in the presence of fractional index. Furthermore, this attempt also provides the solution obtained through the optimization techniques after using the real distinct poles rational approximation of the generalized Mittag-Leffler function.

Highlights

The journey of the concept of fractional calculus (FC) has started in the seventeenth century
8 Conclusion In this paper, we realize that classical economic production quantity model (EPQ) may be generalized as a fractional-order EPQ model with and without deterioration
It is being perceived that holding costs and total average costs for non-fractional cases are the particular cases of generalized holding costs and generalized total average costs

Summary

Introduction

The journey of the concept of fractional calculus (FC) has started in the seventeenth century. (iii) Application of a real distinct pole rational approximation to optimize the inventory problem with Mittag-Leffler function This was not used by earlier researchers of the fractional-order inventory model. The differential equation of fractional order α, due to memory (0 ≤ α ≤ 1), corresponding to the generalized EPQ model, is dα q(t) dtα = K – D, for 0 ≤ t ≤ t1, dα q(t) dtα = –D, for t1 ≤ t ≤ T, With the initial condition q(0) = 0 and boundary condition q(T) = 0. The differential equation of fractional order α, due to memory (0 ≤ α ≤ 1), corresponding to the generalized EPQ model, is dα q(t) dtα + θ1q(t) = K – D, for 0 ≤ t ≤ t1, dα q(t) dtα + θ2q(t) = –D, for t1 ≤ t ≤ T. (iv) The total average cost of the system during the entire circle is given by TAPα,β (T )

Model 1

Conclusion