A note on a robust inventory model with stock-dependent demand

Abstract

We investigate an inventory model with stock-dependent demand where larger pile of stock displayed leads the customer to purchase more. The dependency of demand on the inventory level is modelled as a monomial function whose shape and scale parameters are stochastic. We present a linear regression-based method for constructing ellipsoidal representations of the parameter uncertainty, which are subsequently incorporated into the inventory model under the robust optimisation framework. We show that the resulting robust optimisation model can be transformed into an equivalent convex programme, and also prove that a robust optimal inventory replenishment policy is of the base-stock type. Through a numerical illustration of the proposed approach and a performance analysis based upon Monte Carlo simulation, we demonstrate that robust optimal order decisions exhibit a unique advantage over deterministic ones.

Keywords:

• Stock-dependent demand
• inventory model
• robust optimisation
• periodic-review model
• convex optimisation

Acknowledgement

The author is grateful to three anonymous referees for their most constructive comments and suggestions on a previous version of this paper.

Notes

1. The positive definiteness of M is required to ensure a solid ellipsoid.

2. The box lots in this paper are generated by MATLAB R2017a, in which three quartiles (Q1, Q2 and Q3) are indicated by two end lines and one middle line of a box. Data outside the 1.5 interquartile range are recognised as outliers and denoted by cross signs. Averages are indicated by diamonds within boxes.