BioND — Dynamics of Biological Networks

Revealing instabilities in a generalized triadic supply network: A bifurcation analysis

Daniel Ritterskamp, Güven Demirel, Bart MacCarthy, Lars Rudolf, Alan Champneys, and Thilo Gross
Chaos 28, 073103, 2018.

Abstract

Figure.

Supply networks are exposed to instabilities and thus a high level of risk. To mitigate this risk, it is necessary to understand how instabilities are formed in supply networks. In this paper, we focus on instabilities in inventory dynamics that develop due to the topology of the supply network. To be able to capture these topology-induced instabilities, we use a method called generalized modeling, a minimally specified modeling approach adopted from ecology. This method maps the functional dependencies of production rates on the inventory levels of different parts and products, which are imposed by the network topology, to a set of elasticity parameters. We perform a bifurcation analysis to investigate how these elasticities affect the stability. First, we show that dyads and serial supply chains are immune to topology-induced instabilities. In contrast, in a simple triadic network, where a supplier acts as both a first and a second tier supplier, we can identify instabilities that emerge from saddle-node, Hopf, and global homoclinic bifurcations. These bifurcations lead to different types of dynamical behavior, including exponential convergence to and divergence from a steady state, temporary oscillations around a steady state, and co-existence of different types of dynamics, depending on initial conditions. Finally, we discuss managerial implications of the results.

Supply networks are commonly exposed to fluctuations in demand and supply, which makes it crucial to understand their stability properties. We here investigate the impact of the supply network structure on the stability of inventory dynamics. We show that a serial supply chain that would otherwise be stable may be destabilized if a first tier supplier also supplies parts to another first tier supplier, forming a triadic supply network. Using bifurcation analysis, we show that temporary oscillations as well as co-existence of stable and unstable dynamics can occur, depending on the inventory control policies and the existence of capacity and part availability limitations on production.

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