Dieter Armbruster

A MODEL FOR THE DYNAMICS OF LARGE QUEUING NETWORKS AND SUPPLY CHAINS

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, i.e., batches of product or individual product items, 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.

A Continuum Model for a Re-entrant Factory

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. Littles 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.

Kinetic and fluid model hierarchies for supply chains

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 field models and fluid approximations are verified numerically.

Basketball Teams as Strategic Networks

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.

Control of Continuum Models of Production Systems

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 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 versus backlog tracking) are highlighted.

Bucket brigades with worker learning

Abstract 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.

Symmetries and dynamics for 2-D Navier-Stokes flow

Abstract Simulations of forced 2-D Navier-Stokes equations are analyzed. The forcing is spatially periodic and temporally steady. A Karhunen-Loeve analysis is used to identify the structures in phase space that generate the PDE behavior. Their relationship to the invariant subspaces generated by the symmetry group is discussed. It is shown that certain modes that are in the stable eigenspace of the Kolmogorov flow solution play an essential role for the dynamics of the attractor for the 2-D Navier-Stokes equations below a Reynolds number of about 30. In this regime all stable solutions are identified and thier relation to the symmetry structure is elucidated. A new type of gluing bifurcation generated by the symmetry is found and analyzed. A mechanism for the generation of bursting behavior is suggested.

Noisy heteroclinic networks

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.

Bucket brigades revisited: Are they always effective?

Previous work on the dynamics of bucket brigades has focused on systems in which workers can be ordered with respect to their speeds and where this ordering does not change throughout the line. While this assumption is valid in most environments, it may not be satisfied in some. We consider such environments and explore the conditions under which bucket brigades continue to be effective (compared to a traditional static allocation) with respect to self-balancing behavior and throughput performance. A two worker bucket brigade is studied where one worker is faster than the other over some part of the production line and slower over another part of the line. We analyze the dynamics and throughput of the bucket brigade in two environments with passing and blocking. We present the dynamics of the system in each region of the parameter space and provide insights and operating principles for the implementation and management of the bucket brigades under various scenarios.

THERMALIZED KINETIC AND FLUID MODELS FOR REENTRANT SUPPLY CHAINS

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 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 reentrant 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.