Stability analysis of constrained inventory systems with transportation delay

Abstract

Stability is a fundamental design property of inventory systems. However, the often exploited linearity assumptions in the current literature create a major gap between theory and practice. In this paper the stability of a constrained production and inventory system with a Forbidden Returns constraint (that is, a non-negative order rate) is studied via a piecewise linear model, an eigenvalue analysis and a simulation investigation. The APVIOBPCS (Automatic Pipeline, Variable Inventory and Order Based Production Control System) and EPVIOBPCS (Estimated Pipeline, Variable Inventory and Order Based Production Control System) replenishment policies are adopted. Surprisingly, all kinds of non-linear dynamical behaviours of systems can be observed in these simple models. Exact expressions of the asymptotic stability boundaries and Lyapunovian stability boundaries are derived when actual and perceived transportation lead-time is 1 and 2 periods long respectively. Asymptotically stable regions in the non-linear Forbidden Return systems are identical to the stable regions in its unconstrained counterpart. However, regions of bounded fluctuations that continue forever, including both periodicity and chaos, exist in the parametrical plane outside the asymptotically stable region. Simulation shows a complex and delicate structure in these regions. The results suggest that accurate lead-time information is essential to eliminate inventory drift and instability and that ordering policies have to be designed properly in accordance with the actual lead-time to avoid these fluctuations and divergence. Keywords: Inventory; logistics; system dynamics; complexity theory The introduction and motivationOne of the main objectives when designing an inventory system is to maintain its stability and robustness in the face of exterior disturbances. Since the introduction of control theory and system dynamics approaches to the field of inventory control (Simon, 1952;Forrester, 1961), many works have studied this problem. However, the significance of results obtained are frequently limited by both the uncertainty and complexity of the system structure. Often omitted factors in inventory system design include saturation (logistics constraints) and mis-specified delays (lead-times).In previous supply chain stability studies (Riddalls and Bennett, 2002;Nagatani and Helbing, 2004;Warburton et al., 2004;Disney, 2008), linear inventory system models are usually adopted. Linearity assumptions include infinite capacity, ignoring inventory limitations and return restrictions. This has greatly limited the applicability of published results, and has failed to explain many business phenomena (Riddalls et al., 2000). For instance, to maintain linearity of the commonly investigated IOBPCS (Inventory and Order Based Production Control System) models, order rates are permitted to take negative values. This means that all participants in a supply chain are allowed to return excess product freely. Specifically, a negative order rate value leads to a ...

Controller Design for a Class of Delayed and Constrained Systems: Application to Supply Chains

“…for all t ≥ 0 and i = 1, .., N. In addition, if conditions (28) and (29) are verified, the constraints (23), (24) and (25) …”

“…For such issues, simulation based analyses are the mostly used, such as in [3,7,8,29,2,17], where the impact of constrained production capacity is studied. In [27,28] an analytical investigation is presented for the forbidden-return case, which corresponds to the constraint of non-negativity on the replenishment orders only. These studies pointed out that considering capacity constrained on the supplying devices, removes the linearity assumption of the model and hence com-plex dynamics behaviours are revealed.…”

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“…However, their analytical framework disregards the presence of discontinuous nonlinearities. On the other hand, Wang and Disney (2012) given by a non-negative constraint on the replenishment order.…”

Insights on Partial Information Sharing in Supply Chain dynamics

“…More specifically, they adopt specific types of Order-Up-To named Automatic Pipeline Variable Inventory and Order Based Production Control System policies (APVIOBPCS) [25][26][27]. If a generic echelon merely receives, as external data, the order placer by the downstream adjacent member, he adopts a classical APVIOBPCS.…”