Networks of Interacting Machines

The following sections are included:

• PREFACE

• CONTENTS

A review of continuum models for production flows involving a large number of items and a large number of production stages is presented. The basic heuristic model is based on mass conservation and state equations for the relationship between the cycle time and the amount of work in progress in a factory. Heuristic extensions lead to advection diffusion equations and to capacity limited fluxes. Comparisons between discrete event simulations and numerical solutions of the heuristic PDEs are made. First principle models based on the Boltzman equation for a probability density of a production lot, evolving in time and production stages are developed. It is shown how the basic heuristic model constitute the zero order approximation of a moment expansion of the probability density. Similarly, the advection diffusion equation can be derived as the first order Chapman-Enskog expansion assuming a stochastically varying throughput time. It is shown how dispatch policies can be modeled by including an attribute in the probability density whose time evolution is governed by the interaction between the dispatch policy and the capacity constraints of the system. The resulting zero order moment expansion reproduces the heuristic capacity constraint model whereas a first order moment will lead to multiphase solutions representing multilane fluxes and overtaking of production lots. A discussion on the similarities and differences of industrial production networks and biological networks is also presented.

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 non-oscillatory 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.

After describing the general supply-demand management problem with examples from the semiconductor industry, attention is restricted to the core manufacturing problem. Using a mathematical approach to this nonlinear stochastic combinatorial financial optimization problem, an outer loop for addressing the planning parts of the problem and an inner loop to manage the execution aspects are proposed. The outer loop provides a material release plan generated by a linear programming formulation (LP) and inventory safety stock targets generated by a dynamic programming formulation (DP) to the inner loop to guide execution. Portions of the nonlinearity and stochasticity inherent in the problem are addressed by the outer loop that requires iterative convergence between the LP and the DP. The inner loop is formulated from the perspective of model predictive control (MPC) and integrates concepts from optimal control and stochastic control. Results are presented to demonstrate the ability of the inner loop to track material release and safety stock targets while improving delivery performance in the face of both supply and demand stochasticity. A simulation module is also described that supports the other components of the system by validating their efficacy before application in the real world.

This contribution deals with the modelling of manufacturing systems for control. First the concept of effective process times is introduced as a means to arrive at relatively simple discrete event models of manufacturing systems based on measured data. Secondly, a control framework is presented. Thirdly, a validation study is presented which shows that the currently available PDE-models for describing manufacturing systems need further improvement. Finally some criteria are specified which a PDE-model should at least meet in order to be considered valid.

A dynamical model of an adaptive production network is proposed as an extension of von Neumann's static model. Sets of input products are jointly converted into sets of output products by production processes. The rate of production of a given production process is limited by the minimum quantity of its set of input commodities. Switching between different choices of the minimum input commodity leads to the appearance of cyclic behaviour. The generation of complex oscillations with multiple timescales is also shown when several processes are combined to form a chain or a network. This model is primarily applied to understanding the non-equilibrium behaviour of a production economy but we also discuss its relevance to the understanding of some of the complex features of biological cellular organization and behaviour.

Statistical behavior of chemical reaction dynamics in a reproducing cell is studied. Zipf law in rank-abundance relationship in chemical compositions and log-normal distribution of abundances over many cells are universally found for a cell that exhibits recursive production, and theoretically explained. Regulation of such large abundance fluctuations, as well as relevance of the statistics to biological and social systems is briefly discussed.

Biological cells can be viewed as networks formed by molecular machines. The coordination in operations of individual machines in such networks is not externally imposed, but develops internally as a result of interactions between the machines, through a self-organization process. As an example, we consider a population of allosteric enzymes with product activation and show that this network can undergo spontaneous transitions to different synchronous regimes corresponding to just-in-time production. Analyzing a toy model of cross-coupled neural networks, we find furthermore that similar synchronization transitions are possible when the dynamics of individual elements is highly complex.

The chemical reaction is inherently stochastic. Such stochasticity is particularly prominent in small systems, such as biological cells. In this chapter, we show how gene expression and signal transduction reactions are noisy based on the recent theoretical and experimental studies.

Dynamic aspects of biological information in the cell behavior and morphogenesis by the plasmodium of the true slime mold Physarum polycephalum are studied. Particular emphasis is put on the self-organization of cell shape and size in relation to intelligent functioning by the organism. The transduction of sensed information, integration of multiple signals, decision making, biocomputing, emergence of the rhythm etc. are shown related to a transition among dynamic dissipative structures on a metabolic network or among spatiotemporal order in a collective dynamics of coupled oscillators. Importance of global order in the amoeboid cell as a whole in cell functioning is clarified.

Life without information is not life. From the genetic blueprint in our DNA to the world-wide Internet, information and its dynamic counterpart communication shape our world. However, we live under the limited information horizon, in the sense that information is often imperfect and communication is always finite. Complex networks opens for a quantitative frame to discuss these issues, both through model building, through comparative studies of topologies and through studies of signaling in real networks. In this paper we explore a model for evolution of social networks, and discuss the general interplay between communication, specific signaling and overall topologies in various complex networks.