Stability analysis of a constrained inventory system
Abstract
The stability of a production and inventory system where negative orders are forbidden is investigated via an eigenvalue analysis of a piecewise linear model and a simulation study. The Automatic Pipeline, Variable Inventory and Order Based Production Control System is adopted. All classes of dynamic behaviour in nonlinear systems can be observed in this stylised model with only one constraint. Exact expressions for the asymptotic stability and Lyapunovian stability boundaries are derived when the replenishment lead-time is both one and two periods long. Asymptotically stable regions in the nonlinear system are identical to the stable regions in its linear counterpart. However, regions of bounded fluctuations that continue forever exist in the parameter plane. Simulation reveals an intriguing and delicate structure within these regions. Our results show that ordering policies have to be both designed properly and use accurate lead-time information to avoid such undesirable behaviour.