Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.