Title: Deterministic and stochastic optimal inventory control with logistic stock-dependent demand rate

Authors: A. Tsoularis

Addresses: Business Rose Bowl, Leeds Metropolitan University, Portland Crescent, Leeds LS1 3HB, UK

Abstract: It has been suggested by many supply chain practitioners that in certain cases inventory can have a stimulating effect on the demand. In mathematical terms this amounts to the demand being a function of the inventory level alone. In this work we propose a logistic growth model for the inventory dependent demand rate and solve first the continuous time deterministic optimal control problem of maximising the present value of the total net profit over an infinite horizon. It is shown that under a strict condition there is a unique optimal stock level which the inventory planner should maintain in order to satisfy demand. The stochastic version of the optimal control problem is considered next. A bang-bang type of optimal control problem is formulated and the associated Hamilton-Jacobi-Bellman equation is solved. The inventory level that signifies a switch in the ordering strategy is worked out in the stochastic case.

Keywords: logistic growth models; Euler-Lagrange equation; Legendre condition; Hamilton-Jacobi-Bellman equation; Wiener process; Itó conditions; optimal inventory control; stock dependent demand; optimal control; modelling.

DOI: 10.1504/IJMOR.2014.057851

International Journal of Mathematics in Operational Research, 2014 Vol.6 No.1, pp.41 - 69

Received: 25 Aug 2012
Accepted: 25 Oct 2012
Published online: 28 Jun 2014 *

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