Recent Papers

by Dieter Armbruster

Production Systems and other OR related stuff

Hongmin Li, D. Armbruster, K.Kempf: A Population-Growth Model for Multiple Generations of Technology Products, accepted MSOMS 12/2012 [abstract] [pdf]

Modeling production planning and transient clearing functions, Logist. Res. DOI 10.1007/s12159-012-0087-8 (2012). [abstract] [pdf]

The production planning problem: Clearing functions, variable leads times, delay equations and partial differential equations in: Decision Policies for Production Networks, D. Armbruster, K.G. Kempf (eds), p. 289 - 303, Springer Verlag (2012). [abstract] [pdf]

Continuous Dynamic Models, Clearing Functions, and Discrete-Event Simulation in Aggregate Production Planning in: New Directions in Informatics, Optimization, Logistics, and Production , TutORials in Operations Research ONLINE, Pitu Mirchandani, Tutorials Chair and Volume Editor J. Cole Smith, Series ed., 103126 (2012). [abstract] [pdf]

A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math. {\bf 71 (4), p 1070 - 1087 (2011). [abstract] [pdf]

Control of continuum models of production systems, IEEE Trans. Automatic Control {\bf 55} (11), p 2511 - 2526 (2010). [abstract] [pdf]

A Hyperbolic Relaxation Model for Product Flow in Complex Production Networks, Discrete and continuous dynamical systems Supplement 2009, pp. 790 - 799 [abstract] [pdf]

Controlling a re-entrant manufacturing line via the push-pull point, International Journal of Production Research, 46 (16), 4521 - 4536 (2008) [abstract] [pdf]

Kinetic and fluid model hierarchies for supply chains supporting policy attributes, Bulletin of the Inst. Math., Academica Sinica 2:433-460, 2007. [abstract] [pdf]

Bucket Brigades with Worker Learning, European Journal of Operational Research, 176, 264- 274 (2007) [ abstract] [pdf]

Multiscale analysis of re- entrant production lines: An equation-free approach, Physica A, 363(1), 1-13, 2006 [abstract] [pdf]

A Model for the Dynamics of large Queuing Networks and Supply Chains, SIAM J. Applied Mathematics 66(3) pp. 896-920. 2006 [abstract] [pdf]

Bucket Brigades when Worker Speeds do not Dominate Each Other Uniformly, European Journal of Operational Research, 172,(1), 213-229, 2006 [ abstract] [pdf]

A continuum model for a re-entrant factory, Operations research 54(5), 933-950, 2006 [abstract] [pdf]

Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. on Multiscale modeling and Simulation, 3(4), pp 782 - 800, (2005) [abstract] [pdf]

Kinetic and fluid model hierarchies for supply chains SIAM Multiscale Model. Simul. 2(1), pp 43-61 2004 [abstract] [pdf]

Control and Synchronization in Switched Arrival Systems, Chaos 13 (1), 128-137 (2003) [abstract] [pdf]

Periodic orbits in a class of re-entrant manufacturing systems, Mathematics of Operations Re- search 25(4), p. 708 - 725, 2000 [abstract] [pdf]

Pattern Formation

Localized Solutions in Parametrically Driven Pattern Formation , Phys. Rev. E 68, 016213 (2003) [abstract] [pdf]

Dynamics of polar reversals in spherical dynamos, Proceedings of the Royal Society A, London, 459, 577-596 (2003). [ abstract] [pdf]

Dynamical Systems and Networks

Basketball Teams as Strategic Networks PLoS ONE 7(11) (2012): e47445. doi:10.1371/journal.pone.0047445 [abstract] [pdf]

Design of Robust Distribution Networks Run by Third Party Logistics Service Providers, Advances in Complex Systems, 15 (5) (2012) 1150024 [abstract] [pdf]

Structural Properties of third-party logistics networks, in: Proceedings of the 2nd International Conference on Dynamics in Logistics (LDIC 2009), Ed. H.-J. Kreowski et al, Bremen, 2010. [abstract] [doc]

Information and material flows in complex networks, short survey, Physica A 363(1), Pages xi-xvi, 2006 [abstract] [pdf]

Autonomous Control of Production Networks using a Pheromone Approach, Physica A, 363(1), 104-114, 2006 [abstract] [pdf]

Does synchronization of networks of chaotic maps lead to control?, Chaos 15 014101, 2005 [abstract] [pdf]

Noisy Heteroclinic networks, Chaos 13 (1), 71-79 (2003) [ abstract] [pdf]

Perturbed on-off intermittency, Phys. Rev. E. 64 016220-1 - 9, (2001) [ abstract] [pdf]

Noise and O(1) Amplitude Effects on Heteroclinic Cycles, Chaos, 9(2) p.499-506, (1999) [abstract] [pdf]

Data Analysis

Analyzing the dynamics of cellular flames [abstract]

Symmetry and the Karhunen Loéve analysis [abstract]

Simulation and Modeling in Biology

Evolution of uncontrolled proliferation and the angiogenic switch in cancer, Mathematical Biosciences and Engineering 9(4), 843- 876, (2012). [abstract] [pdf]

Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions, International Journal of Biomathematics, 4(3), p 255 - 274, (2011) [abstract] [pdf]

Dispersal effects on a discrete two- patch model for plant-insect interactions, Journal of Theoretical Biology 268, 84-97 (2011). [abstract] [pdf]

Dynamic simulations of single molecule enzyme networks, J. Phys. Chem. B, 2009, 113 (16), 5537-5544 [abstract] [pdf]

Dynamics of a plant- herbivore model. Journal of Biological Dynamics, 2(2), 89-101, 2008 [abstract] [pdf]

Current preprints

Teun Adriaansen, Dieter Armbruster, Karl Kempf and Hongmin Li: An agent model for the High-End Gamers market, preprint ASU 2010 [abstract] [pdf]

On the stability of queueing networks and fluid models, preprint 2010 [abstract] [pdf]

Biologically inspired mutual synchronization of manufacturing machines, preprint 2010 [abstract] [pdf]


A Population-Growth Model for Multiple Generations of Technology Products
Hongmin Li, D. Armbruster, K.Kempf

In this paper, we consider the demand for multiple successive generations of products and develop a population growth model that allows demand transitions across multiple product generations, and takes into consideration the effect of competition. We propose an iterative descent method for obtaining the parameter estimates and the covariance matrix, and show that the method is theoretically sound and overcomes the difficulty that the units-in-use population of each product is not observable. We test the model on both simulated sales data and Intel’s high-end desktop processor sales data. We use two alternative specifications for product strength in this market – performance, and performance/price ratio. The former demonstrates better fit and forecast accuracy, likely due to the low price-sensitivity of this high-end market. In addition, the parameter estimate suggests that, for the innovators in the diffusion of product adoption, brand switchings are more strongly influenced by product strength than within-brand product upgrades in this market. Our results indicate that compared to the Bass model, the Norton-Bass model, as well as the Jun-Park choice-based diffusion model, our approach is a better fit for strategic forecasting which occurs many months or years before the actual product launch. Keywords: Product Transitions, Forecasting, Multiple-generation Demand Model, Diffusion

Modeling production planning and transient clearing functions,
Dieter Armbruster, Jasper Fonteijn, Matt Wienke

Abstract: The production planning problem, that is, to determine the production rate of a factory in the future, requires an aggregate model for the production flow through a factory. The canonical model is the clearing function model based on the assumption that the local production rate instantaneously adjusts to the one given by the equilibrium relationship between production rate (flux) and work in progress (wip), for example, characterized by queueing theory. We will extend current theory and modeling for transient clearing functions by introducing a continuum description of the flow of product through the factory based on a partial differential equation model for the time evolution of the wip-density and the production velocity. It is shown that such a model improves the mismatch between models for transient production flows and discrete event simulations significantly compared to other clearing function approaches.

Continuous Dynamic Models, Clearing Functions, and Discrete-Event Simulation in Aggregate Production Planning,
Dieter Armbruster, Reha Uzsoy

Abstract: TWe consider the problem of representing the capabilities of production systems to convert inputs into outputs over time in the face of deterministic demand. The fact that planning models generally operate in discrete-time while the factory being modeled operates in continuous time suggests the use of multimodel methods combining an optimization model that plans the releases into the plant over time with a simulation model that assesses the impacts of these releases on performance. We discuss several such schemes that have been implemented using discrete-event simulation, present recent computational results, and discuss their strengths and weaknesses. We then consider two alternative approaches that have emerged in the recent literature: the use of discrete-time linear programming models based on nonlinear clearing functions, and methods using systems of coupled partial differential equations based on transport phenomena. We identify the relationships between queueing models, clearing functions, and transport model-based methods, and we discuss future research directions.

The Production Planning Problem: Clearing Functions, Variable Leads Times, Delay Equations and Partial Differential Equations,
Dieter Armbruster

Determining the production rate of a factory as a function of current and previous states is at the heart of the production planning problem. Different approaches to this problem presented in this book are reviewed and their relationship is discussed. Necessary conditions for the success of a clearing function as a quasi steady approximation are presented and more sophisticated approaches allowing the prediction of outflow in transient situations are discussed. Open loop solutions to the deterministic production problem are introduced and promising new research directions are outline

A scalar conservation law with discontinuous flux for supply chains with finite buffers,
Dieter Armbruster, Simone Goettlich, Michael Herty

An aggregate continuum model for production flows and supply chains with finite buffers is proposed and analyzed. The model extends earlier partial differential equations that represent deterministic coarse grained models of stochastic production systems based on mass conservation. The finite size buffers lead to a discontinuous clearing function describing the throughput as a function of the work in progress (WIP). Following previous work on stationary distribution of WIP along the production line, the clearing function becomes dependent on the production stage and decays linearly as a function of the distance from the end of the production line. A transient experiment representing the breakdown of the last machine in the production line and its subsequent repair is analyzed analytically and numerically. Shock waves and rarefaction waves generated by blocking and reopening of the production line are determined. It is shown that the time to shutdown of the complete flow line is much shorter than the time to recovery from a shutdown. The former evolves on a transportation time scale, whereas the latter evolves on a much longer time scale. Comparisons with discrete event simulations of the same experiment are made.

Control of continuum models of production systems Michael La Marca,
Dieter Armbruster, Michael Herty, and Christian Ringhofer

A production system which produces a large number of items in many steps can be modelled as a continuous flow problem. The resulting hyperbolic partial differential equation (PDE) typically is nonlinear and nonlocal, modeling a factory whose cycle time depends nonlinearly on the work in progress. One of the few ways to influence the output of such a factory is by adjusting the start rate in a time dependent manner. We study two prototypical control problems for this case: i) demand tracking where we determine the start rate that generates an output rate which optimally tracks a given time dependent demand rate and ii) backlog tracking which optimally tracks the cumulative ∂t demand. The method is based on the formal adjoint method for constrained optimization, incorporating the hyperbolic PDE as a constraint of a nonlinear optimization problem. We show numerical results on optimal start rate profiles for steps in the demand rate and for periodically varying demand rates and discuss the influence of the nonlinearity of the cycle time on the limits of the reactivity of the production system. Differences between perishable and non-perishable demand (demand vs. backlog tracking) are highlighted.

A HYPERBOLIC RELAXATION MODEL FOR PRODUCT FLOW IN COMPLEX PRODUCTION NETWORKS
Ali Unver, Christian Ringhofer and Dieter Armbruster

This paper presents a continuum - traffic flow like - model for the flow of products through complex production networks, based on statistical information obtained from extensive observations of the system. The resulting model consists of a system of hyperbolic conservation laws, which, in a relaxation limit, exhibit the correct diffusive properties given by the variance of the observed data.

Controlling a re-entrant manufacturing line via the push-pull point
Dominique Perdaen, Dieter Armbruster, Karl Kempf, Erjen Lefeber

A reduced model of a re-entrant semiconductor factory exhibiting all the important features is simulated, applying a push dispatch policy at the beginning of the line and a pull dispatch policy at the end of the line. A commonly used dispatching policy that deals with short-term fluctuations in demand involves moving the transition point between both policies, the push–pull point (PPP), around. It is shown that, with a mean demand starts policy, moving the PPP by itself does not improve the performance of the production line significantly over policies that use a pure push or a pure pull dispatch policy, or a CONWIP starts policy with pure pull dispatch policy. However, when the PPP control is coupled with a CONWIP starts policy, then for high demand with high variance, the improvement becomes approximately a factor of 4. The unexpected success of a PPP policy with CONWIP is explained using concepts from fluid dynamics that predict that this policy will not work for perishable demand. The prediction is verified through additional simulations.

Multiscale analysis of re-entrant production lines: An equation-free approach
Y. Zou, I.G. Kevrekidis, D. Armbruster

The computer-assisted modeling of re-entrant production lines, and, in particular, simulation scalability, is attracting a lot of attention due to the importance of such lines in semiconductor manufacturing. Re-entrant flows lead to competition for processing capacity among the items produced, which significantly impacts their throughput time (TPT). Such production models naturally exhibit two time scales: a short one, characteristic of single items processed through individual machines, and a longer one, characteristic of the response time of the entire factory. Coarse-grained partial differential equations for the spatio-temporal evolution of a ‘‘phase density’’ were obtained through a kinetic theory approach in Armbruster and Ringhofer [Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling Simul. 3(4) (2005) 782–800.] We take advantage of the time scale separation to directly solve such coarse-grained equations, even when we cannot derive them explicitly, through an equation-free computational approach. Short bursts of appropriately initialized stochastic fine-scale simulation are used to perform coarse projective integration on the phase density. The key step in this process is lifting: the construction of fine-scale, discrete realizations consistent with a given coarse-grained phase density field. We achieve this through computational evaluation of conditional distributions of a ‘‘phase velocity’’ at the limit of large item influxes.

A Model for the Dynamics of large Queuing Networks and Supply Chains
Dieter Armbruster and Christian Ringhofer, ASU, Pierre Degond, Laboratoire CNRS Toulouse, France

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. Based on a simple rule for releasing parts from the buffers into the processors we derive a hyperbolic conservation law for the part density and flux in the supply chain. The conservation law will be asymptotically valid in regimes with a large number of parts in the supply chain. Solutions of this conservation law will in general develop concentrations corresponding to bottlenecks in the supply chain.

Keywords: Supply chains, conservation laws, asymptotics.

Kinetic and fluid model hierarchies for supply chains
D. A., Dan Marthaler, Christian Ringhofer ASU

We present a model hierarchy for queuing networks and supply chains, analogous to the hierarchy leading from the many body problem to the equations of gas dynamics. Various possible mean field models for the interaction of individual parts in the chain are presented. For the case of linearly ordered queues the mean filed models and fluid approximations are verified numerically.

Kinetic and fluid model hierarchies for supply chains supporting policy attributes
Dieter Armbruster ASU, Pierre Degond, Laboratoire CNRS Toulouse, France, and Christian Ringhofer, ASU

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. In a pre vious work, we have derived a hyperbolic conservation law for the part density and flux in the supply chain. In the present paper, we introduce internal variables (named attributes: e.g. the time to due-date) and extend the previously defined model into a kinetic-like model for the evolution of the part in the phase-space (degree-of-completion, attribute). We relate this kinetic model to the hyperbolic one through the moment method and a "monokinetic" (or single-phase) closure assumption. If instead multi-phase closure assumptions are retained, richer dynamics can take place. In a numerical section, we compare the kinetic model (solved by a particle method) and its two-phase approximation and demonstrate that both behave as expected.

Thermalized kinetic and fluid models for re-entrant supply chains
Dieter Armbruster and Christian Ringhofer, ASU

Standard stochastic models for supply chains predict the throughput time (TPT) of a part from a statistical distribution, which is dependent on the Work in Progress (WIP) at the time the part enters the system. So, they try to predict a transient response from data which are sampled in a quasi steady state situation. For re- entrant supply chains this prediction is based on insufficient information, since subsequent arrivals can dramatically change the TPT. This paper extends these standard models, by introducing the concept of a stochastic phase velocity which dynamically updates the TPT estimate. This leads to the concepts of temperature and diffusion in the corresponding kinetic and fluid models for supply chains.

Keywords: Re-entrant supply chains, Traffic flow models, Boltzmann equation, Chapman-Enskog, Fluid limits.

A continuum model for a re-entrant factory
D. A., Dan Marthaler, Christian Ringhofer, Karl Kempf, Tae-Chang Jo

High-volume, multistage continuous production flow through a re-entrant factory is modeled through a conservation law for a continuous-density variable on a continuous-production line augmented by a state equation for the speed of the production along the production line. The resulting nonlinear, nonlocal hyperbolic conservation law allows fast and accurate simulations. Little’s law is built into the model. It is argued that the state equation for a re-entrant factory should be nonlinear. Comparisons of simulations of the partial differential equation (PDE) model and discrete-event simulations are presented. A general analysis of the model shows that for any nonlinear state equation there exist two steady states of production below a critical start rate: A high-volume, high-throughput time state and a low-volume, low-throughput time state. The stability of the low-volume state is proved. Output is controlled by adjusting the start rate to a changed demand rate. Two linear factories and a re-entrant factory, each one modeled by a hyperbolic conservation law, are linked to provide proof of concept for efficient supply chain simulations. Instantaneous density and flux through the supply chain as well as work in progress (WIP) and output as a function of time are presented. Extensions to include multiple product flows and preference rules for products and dispatch rules for re-entrant choices are discussed.

Subject classifications: production/scheduling: approximations; simulations: efficiency; mathematics.

Control and Synchronization in Switched Arrival Systems
D.A. and Bart Rem, Technical University of Eindhoven

A chaotic model of a production flow called the switched arrival system is extended to include switching times and maintenance. The probability distribution of the chaotic return times is calculated. Scheduling maintenance, loss of production due to switching and control of the chaotic dynamics is discussed. A coupling parameter to couple switched arrival systems serially, based on lost production, is identified. Simulations of three parallel and three serial levels were performed. Global synchronization of the switching frequencies between serial levels is achieved. An analytic model allows to predict the self balancing properties of the serial system.

Bucket Brigades when Worker Speeds do not Dominate Each Other Uniformly
D.A. and Esma Gel, Arizona State University

A two worker bucket brigade is studied where one worker has a constant speed over the whole production line and the other is slower over the first portion and faster over the second portion of the line. We analyze the dynamics and throughput of the bucket brigade under two different assumptions: (i) workers can pass each other, and (ii) workers are blocked when an upstream worker runs into a downstream worker. We show that a slight modification of the bucket brigade will always lead to a self organizing production line. The bucket brigade either balances to a fixed point or settles into a period-two orbit. Insights for the management of the bucket brigades for the various scenarios are discussed using results on the throughput performance and self-organization. Extensions to multiple skill levels and more workers are outlined.

Bucket Brigades with Worker Learning
D. A., Esma S. Gel and Junko Murakami, Arizona State University

The dynamics and throughput of a bucket brigade production system is studied when workers' speeds increase due to learning. It is shown that, if the rules of the bucket brigade system allow a re-ordering of its workers then the bucket brigade production system is very robust and will typically rebalance to a self-organizing optimal production arrangement. As workers learn only those parts of the production line that they work on, the stationary velocity distribution for the workers of a stable bucket brigade is non-uniform over the production line. Hence, depending on the initial placement of the workers, there are many different stationary velocity distributions. It is shown that all the stationary distributions lead to the same throughput.

Periodic orbits in a class of re-entrant manufacturing systems
Ivonne Diaz-Rivera AT&T, D. A. and Tom Taylor, ASU

Queue changes associated with each step of a manufacturing system are modeled by constant vector fields (fluid model of a queueing network). Observing these vector fields at fixed events reduces them to a set of piecewise linear maps. It is proved that these maps show only periodic or eventually periodic orbits. An algorithm to determine the period of the orbits is presented. The dependence of the period on the processing rates is shown for a 3(4)-step, 2-machine problem.

Localized Solutions in Parametrically Driven Pattern Formation
Tae-Chang Jo, D. A., ASU

The Mathieu partial differential equation is analyzed as a prototypical model for pattern formation due to parametric resonance. After averaging and scaling it is shown to be a perturbed Nonlinear Schr\"{o}dinger Equation. Adiabatic perturbation theory for solitons is applied to determine which solitons of the NLS survive the perturbation due to damping and parametric forcing. Numerical simulations compare the perturbation results to the dynamics of the Mathieu PDE. Stable and weakly unstable soliton solutions are identified. They are shown to be closely related to oscillons found in parametrically driven sand experiments.

Dynamics of polar reversals in spherical dynamos
Pascal Chossat, Universite de Nice, and D.A.,

Structurally stable heteroclinic cycles (SSHC) are proposed as the mathematical structures that are responsible for the reversals of dipolar magnetic fields in spherical dynamos. The existence of SSHCs involving dipolar magnetic fields generated by convection in a spherical shell for a nonrotating sphere is rigorously proven. The possibility of SSHCs in a rotating shell is proposed and their existence in a low dimensional model of the magnetohydrodynamic equations is numerically confirmed. The resulting magnetic time series shows a dipolar magnetic field, aligned with the rotation axis that intermittently becomes unstable, changes the polar axis, starts rotating, disappears completely and eventually reestablishes itself in its original or opposite direction, chosen randomly.

Does synchronization of networks of chaotic maps lead to control?
Mingqiang Zhu, Dieter Armbruster, Arizona State University, and Ines Katzorke, Department of Physics, University Potsdam,

We consider networks of chaotic maps with different network topologies. In each case, they are coupled in such a way as to generate synchronized chaotic solutions. By using the methods of control of chaos we are controlling a single map into a predetermined trajectory. We analyze the reaction of the network to such a control. Specifically we show that a line of 1-d logistic maps that are unidirectionally coupled can be controlled from the first oscillator whereas a ring of diffusively coupled maps cannot be controlled for more than 5 maps. We show that rings with more elements can be controlled if every third map is controlled. The dependence of unidirectionally coupled maps on noise is studied. The noise level leads to a finite synchronization lengths for which maps can be controlled by a single location. A 2-d lattice is also studied.

Perturbed on-off intermittency
Daniel Marthaler, ASU, D.A., Ying-Cheng Lai ASU, and Eric J. Kostelich, ASU

A basic requirement for on-off intermittency to occur is that the system possesses an invariant subspace. We address how on-off intermittency manifests itself when a perturbation destroys the invariant subspace. In particular, we distinguish between situations where the threshold for measuring the on-off intermittency in numerical or physical experiments is much larger than or is comparable to the size of the perturbation. Our principal result is that as the perturbation parameter increases from zero, a metamorphosis in on-off intermittency occurs in the sense that scaling laws associated with physically measureable quantities change abruptly. A geometric analysis, a random-walk model, and numerical trials support the result.

Noise and O(1) Amplitude Effects on Heteroclinic Cycles
Emily Stone, Utah State University, and D.A.

The dynamics of structurally stable heteroclinic cycles connecting fixed points with one dimensional unstable manifolds under the influence of noise is analyzed. Fokker Planck equations for the evolution of the probability distribution of trajectories near heteroclinc cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained.

Keywords: dynamical systems, heteroclinic cycles, noise-induced dynamics, intermittency.

Noisy Heteroclinic Networks
D.A., Emily Stone, Utah State University, and Vivien Kirk, University of Auckland

The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random switching between the heteroclinic cycles determined by the linear dynamics near one of the saddle points, noise induced stability of a cycle, and intermittent switching between cycles. All three responses are explained by examining the size of the stable and unstable eigenvalues at the equilibria.

Keywords: dynamical systems, heteroclinic cycles, noise-induced dynamics, intermittency.

Information and material flows in complex networks, short survey
Dirk Helbing (ETHZ), Dieter Armbruster, Alexander S. Mikhailov (MPI, Berlin) and Erjen Lefeber (Eindhoven)

In this special issue, an overview of the Thematic Institute (TI) on Information and Material Flows in Complex Systems is given. The TI was carried out within EXYSTENCE, the first EU Network of Excellence in the area of complex systems. Its motivation, research approach and subjects are presented here. Among the various methods used are many-particle and statistical physics, nonlinear dynamics, as well as complex systems, network and control theory. The contributions are relevant for complex systems as diverse as vehicle and data traffic in networks, logistics, production, and material flows in biological systems. The key disciplines involved are socio-, econo-, traffic- and bio-physics, and a new research area that could be called "biologistics".

Keywords: Complex systems; Many-particle systems; Multi-agent models; Network theory; Information flows; Socio- and econo-physics; Traffic physics; Biophysics; Biologistics.

Autonomous Control of Production Networks using a Pheromone Approach,
D. Armbruster, C. de Beer, M. Freitag, T. Jagalski (BIBA Bremen), C. Ringhofer (ASU)

The flow of parts through a production network is usually pre-planned by a central control system. Such central control fails in presence of highly fluctuating and/or unforeseen disturbances. To manage such dynamic networks according to low work-in-progress and short throughput times, an autonomous control approach is proposed. Autonomous control means a decentralized routing of the autonomous parts themselves. The parts’ decisions base on backward propagated information about the throughput times of finished parts for different routes. So, routes with shorter throughput times attract parts to use this route again. This process can be compared to ants leaving pheromones on their way to communicate with following ants. The paper focuses on a mathematical description of such autonomously controlled production networks. A fluid model with limited service rates in a general network topology is derived and compared to a discrete-event simulation model. Whereas the discrete-event simulation of production networks is straightforward, the formulation of the addressed scenario in terms of a fluid model is challenging. Here it is shown, how several problems in a fluid model formulation (e.g. discontinuities) can be handled mathematically. Finally, some simulation results for the pheromone-based control with both the discrete-event simulation model and the fluid model are presented for a time-dependent influx.

Keywords: Production networks; Autonomous control; Pheromones; Discrete-event simulation models; Fluid models.

Structural Properties of third-party logistics networks
D.Armbruster M.P.M. Hendriks, E. Lefeber, J.T. Udding

A third party logistic provider operates a distribution network with minimal control over supply and demand. The operation is characterized by three levels: a strategic level that includes the location and sizing of warehouse, a tactical level that determines the links between customers, warehouses and producers and an operational level that determines the size of the flows through the links at any given time. An algorithm to optimize the operational level is determined for a given network structure. Starting with a fully connected network, optimal operations are determined. A reduced network is found by deleting the least used links until the operational costs increase dramatically. The topological structures of the reduced network as a function of backlog and transportation costs are determined..

Basketball Teams as Strategic Networks
Jennifer H. Fewell, Dieter Armbruster, John Ingraham, Alexander Petersen, James S. Waters

We asked how team dynamics can be captured in relation to function by considering games in the first round of the NBA 2010 play-offs as networks. Defining players as nodes and ball movements as links, we analyzed the network properties of degree centrality, clustering, entropy and flow centrality across teams and positions, to characterize the game from a network perspective and to determine whether we can assess differences in team offensive strategy by their network properties. The compiled network structure across teams reflected a fundamental attribute of basketball strategy. They primarily showed a centralized ball distribution pattern with the point guard in a leadership role. However, individual play-off teams showed variation in their relative involvement of other players/positions in ball distribution, reflected quantitatively by differences in clustering and degree centrality. We also characterized two potential alternate offensive strategies by associated variation in network structure: (1) whether teams consistently moved the ball towards their shooting specialists, measured as “uphill/downhill” flux, and (2) whether they distributed the ball in a way that reduced predictability, measured as team entropy. These network metrics quantified different aspects of team strategy, with no single metric wholly predictive of success. However, in the context of the 2010 play-offs, the values of clustering (connectedness across players) and network entropy (unpredictability of ball movement) had the most consistent association with team advancement. Our analyses demonstrate the utility of network approaches in quantifying team strategy and show that testable hypotheses can be evaluated using this approach. These analyses also highlight the richness of basketball networks as a dataset for exploring the relationships between network structure and dynamics with team organization and effectiveness.

DESIGN OF ROBUST DISTRIBUTION NETWORKS RUN BY THIRD PARTY LOGISTICS SERVICE PROVIDERS
M. P. M. HENDRIKS, D. ARMBRUSTER, M. LAUMANNS, E. LEFEBER and J. T. UDDING

We consider a third party logistics service provider (LSP), who faces the problem of distributing different products from suppliers to consumers having no control on supply and demand. In a third party set-up, the operations of transport and storage are run as a black box for a fixed price. Thus the incentive for an LSP is to reduce its operational costs. The objective of this paper is to find an efficient network topology on a tactical level, which still satisfies the service level agreements on the operational level. We develop an optimization method, which constructs a tactical network topology based on the operational decisions resulting from a given model predictive control (MPC) policy. Experiments suggest that such a topology typically requires only a small fraction of all possible links. As expected, the found topology is sensitive to changes in supply and demand averages. Interestingly, the found topology appears to be robust to changes in second order moments of supply and demand distributions

Analyzing the dynamics of cellular flames
Antonio Palacios, ASU, D.A., Eric J. Kostelich, ASU, and Emily Stone, Utah State University

Video data from experiments on the dynamics of two dimensional flames are analyzed. The Karhunen Loéve (KL-) analysis is used to identify the dominant spatial structures and their temporal evolution for several dynamical regimes of the flames. A data analysis procedure to extract and process the boundaries of flame cells is described. It is shown how certain spatial structures are associated with certain temporal events. The existence of small scale, high frequency, turbulent background motion in almost all regimes is revealed.

Symmetry and the Karhunen Loéve analysis
Nejib Smaoui, Kuwait University, and D.A.

The Karhunen Loéve (K-L) analysis is widely used to generate low dimensional dynamical systems, which have the same low dimensional attractors as some large scale simulations of PDEs. If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K-L eigenmodes instead of symmetrizing the data leads to considerable computational savings, if the K-L analysis is done in the snapshot method. The feasability of the approach is demonstrated with an analysis of Kolmogorov flow.

EVOLUTION OF UNCONTROLLED PROLIFERATION AND THE ANGIOGENIC SWITCH IN CANCER
John Nagy, Dieter Armbruster

The major goal of evolutionary oncology is to explain how malignant traits evolve to become cancer “hallmarks.” One such hallmark—the angiogenic switch—is difficult to explain for the same reason altruism is difficult to explain. An angiogenic clone is vulnerable to “cheater” lineages that shunt energy from angiogenesis to proliferation, allowing the cheater to outcompete cooperative phenotypes in the environment built by the cooperators. Here we show that cell- or clone-level selection is sufficient to explain the angiogenic switch, but not because of direct selection on angiogenesis factor secretion— angiogenic potential evolves only as a pleiotropic afterthought. We study a multiscale mathematical model that includes an energy management system in an evolving angiogenic tumor. The energy management model makes the counterintuitive prediction that ATP concentration in resting cells increases with increasing ATP hydrolysis, as seen in other theoretical and empirical studies. As a result, increasing ATP hydrolysis for angiogenesis can increase proliferative potential, which is the trait directly under selection. Intriguingly, this energy dynamic allows an evolutionary stable angiogenesis strategy, but this strategy is an evolutionary repeller, leading to runaway selection for extreme vascular hypo- or hyperplasia. The former case yields a tumor-on-a- tumor, or hypertumor, as predicted in other studies, and the latter case may explain vascular hyperplasia evident in certain tumor types.

Dynamics of a plant- herbivore model
Yun Kang, Dieter Armbruster and Yang Kuang

We formulate a simple host-parasite type model to study the interaction of certain plants and herbivores. Our two dimensional discrete time model utilizes leaf and herbivore biomass as state variables. The parameter space consists of the growth rate of the host population and a parameter describing the damage inflicted by herbivores. We present insightful bifurcation diagrams in that parameter space. Bistability and a crisis of a strange attractor suggest two control strategies: Reducing the population of the herbivore under some threshold or increasing the growth rate of the plant leaves.

Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions
Yun Kang and Dieter Armbruster

We formulate general plant-herbivore interaction models with monotone plant growth functions (rates). We study the impact of monotone plant growth functions in general plant-herbivore models on their dynamics. Our study shows that all monotone plant growth models generate a unique interior equilibrium and they are uniform persistent under certain range of parameters values. However, if the attacking rate of herbivore is too small or the quantity of plant is not enough, then herbivore goes extinct. Moreover, these models lead to noise sensitive bursting which can be identified as a dynamical mechanism for almost periodic outbreaks of the herbivore infestation. Montone and non-monotone plant growth models are contrasted with respect to bistability and crises of chaotic attractors.

Dynamic simulations of single molecule enzyme networks
Dieter Armbruster, John D. Nagy, E. A. F. van de Rijt, and J. E. Rooda

Along with the growth of technologies allowing accurate visualization of biochemical reactions to the scale of individual molecules has arisen an appreciation of the role of statistical fluctuations in intracellular biochemistry. The stochastic nature of metabolism can no longer be ignored. It can be probed empirically, and theoretical studies have established its importance. Traditional methods for modeling stochastic biochemistry are derived from an elegant and physically satisfying theory developed by Gillespie. However, although Gillespie’s algorithm and its derivatives efficiently model small-scale systems, complex networks are harder to manage on easily available computer systems. Here we present a novel method of simulating stochastic biochemical networks using discrete events simulation techniques borrowed from manufacturing production systems. The method is very general and can be mapped to an arbitrarily complex network. As an illustration, we apply the technique to the glucose phosphorylation steps of the Embden-Meyerhof-Parnas pathway in E. coli. We show that a deterministic version of the discrete event simulation reproduces the behavior of an analogous deterministic differential equation model. The stochastic version of the same model predicts that catastrophic bottlenecks in the system are more likely than one would expect from deterministic theory.

An agent model for the High-End Gamers market
Teun Adriaansen, Dieter Armbruster, Karl Kempf and Hongmin Li

Understanding the driving forces for the markets of their products is a basic necessity for any business. Quantitative models are either ag- gregated over large market segments or restricted to utility models of an individual’s buying decision. While the aggregate models acknowl- edge that customer interactions are important they do not model them and hence have no way to adjust their model to changing business en- vironments. This paper develops crucial methodology to bridge this gap between the individual decisions and the overall market behavior using agent based simulations to model the sales of computer chips in the High- End Gamers market. The simulation environment is dynamic and models the succession of 19 products introduced over a 40 month time horizon which includes the recession of 2008 - 2010. Simulated sales are compared to actual sales data and are used to adjust the parameterization of the agents and their environment. Only two agent parameters are sufficient to obtain a very reasonable fit between simulations and data: The amount of money available for the gaming hobby and a parameter related to the gaming skills of the High-End Gamers.

On the stability of queueing networks and fluid models
Dieter Armbruster, Erjen Lefeber and J.E. Rooda

The stability of the Lu-Kumar queueing network is re-analyzed. It is shown that the associated fluid network is a hybrid dynamical system that has a succession of invariant subspaces leading to global stability. It is explained why large enough stochastic perturbations of the production rates lead to an unstable queuing network while smaller perturbations do not change the stability. The two reasons for the instability are the break- ing of the invariance of the subspaces and a positive Lyapunov exponent. A service rule that stabilizes the system is proposed

A continuous model for supply chains with finite buffers
D. Armbruster, S. Goettlich, M. Herty

An aggregate continuum model for production flows and supply chains with finite buffers is proposed and analyzed. The model extends earlier partial differential equations that represent deterministic coarse grained models of stochastic production systems based on mass conservation. The finite size buffers lead to a discontinuous clearing function describing the throughput as a function of the work in progress. Following previous work on stationary distribution of WIP along the production line, the clearing function becomes dependent on the production stage and decays linearly as a function of the distance from the end of the production line. A transient experiment representing the break down of the last machine in the production line and its subsequent repair is analyzed analytically and numerically. Shock waves and rarefaction waves generated by blocking and re-opening of the production line are determined. It is shown that the time to shutdown of the complete flow line is much shorter than the time to recovery from a shutdown. The former evolves on a transportation time scale whereas the latter evolves on a much longer time scale. Comparisons with discrete event simulations of the same experiment are made.

Biologically inspired mutual synchronization of manufacturing machines
Erjen Lefeber, Herman de Vos, Dieter Armbruster, Alexander S. Mikhailov

A biologically inspired manufacturing control system for synchronous delivery of jobs at a batch machine is developed. This control system is based on a synchronization mechanism of enzymes replacing the role of product molecules with kanbans. This manufacturing control system works well, provided the variability in service times is not too high.


armbruster@asu.edu
updated March 16, 2010