Supply and Production Networks: From the Bullwhip Effect to Business Cycles

Abstract

Network theory is rapidly changing our understanding of complex systems, but the relevance of topological features for the dynamic behavior of metabolic networks, food webs, production systems, information networks, or cascade failures of power grids remains to be explored. Based on a simple model of supply networks, we offer an interpretation of instabilities and oscillations observed in biological, ecological, economic, and engineering systems.We find that most supply networks display damped oscillations, even when their units -and linear chains of these units -behave in a nonoscillatory way. Moreover, networks of damped oscillators tend to produce growing oscillations. This surprising behavior offers, for example, a new interpretation of business cycles and of oscillating or pulsating processes. The network structure of material flows itself turns out to be a source of instability, and cyclical variations are an inherent feature of decentralized adjustments. In particular, we show how to treat production and supply networks as transport problems governed by balance equations and equations for the adaptation of production speeds.The stability and dynamic behavior of supply networks is investigated for different topologies, including sequential supply chains, "supply circles", "supply ladders", and "supply hierarchies". Moreover, analytical conditions for absolute and convective instabilities are derived. The empirically observed bullwhip effect in supply chains is explained as a form of convective instability based on resonance effects. An application of this theory to the optimization of production networks has large optimization potentials. 33Networks of Interacting Machines Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/25/15. For personal use only.

Modelling supply networks with partial differential equations

“…We relate the found results with the counterpart in real supply networks. Then we show that SC3 reproduces the well-known Bullwhip effect (see [8], [11], [12], [17], [18], [20]); i.e., under certain conditions (delays in adaptation of production or delivery rates), the oscillations in delivery and in the resulting inventories (stock level of the products) grow from one producer to the next upstream one, leading to instability with respect to perturbation in the production rate. The latter confirms that SC3 appears to be the more realistic modelling choice.…”

Modeling production networks with discrete processes by means of communities of autonomous units

“…It follows the concept of supply and production networks as studied, for example, in [5,9], but we replace continuous production processes by stepwise processing.…”

“…3. The introduced notion is oriented on the concepts of production networks in [5,9]. Many variations and extensions are possible, like lower bounds for input, production and output in addition to the upper bounds, or variable distribution rates rather than invariant ones.…”

Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

“…In the last years various continuous models were proposed (e.g. [9,13,14,18]), some in which the dynamics on the arcs is governed by partial differential equations (see [1][2][3][4][15][16][17]). …”