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We introduce the optimal in ow control problem for bufler restricted production systems involving a conservation law with discontinuous flux. Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem effciently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.

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... Armed with such a vision, in this paper we consider a continuous model and more precisely an ODE-PDE model, as introduced in [6] and further investigated in Göttlich et al. [7] , Herty et al. [8] , Göttlich et al. [9] . Such models have been solved numerically [10] , also to address optimization issues [7,11] . Similarly, in this paper we will first use numerical methods to solve the model and then formulate and solve the optimization problem. ...

... We prove herein a new property for the approximate solution (11) . Let us consider the generic ODE ...

... By recalling (11) , the approximate solution over the single subinterval for (3) can be written as (omitting subscript): ...

In this paper, we consider supply chains modelled by partial and ordinary differential equations, for densities of parts on suppliers and queues among consecutive arcs, respectively. The considered numerical schemes, whose properties are discussed, foresee the upwind method for goods densities, described by conservation laws, and a Differential Quadrature based explicit formula for queues evolutions. An optimization scheme is also discussed, by considering a cost functional that, according to a pre-defined outflow, weights the queues through variations of processing velocities of suppliers. We get the minimization of the cost functional via a genetic algorithm, which uses mechanisms of selection, crossover and mutation for the processing velocities. An example application is discussed in light of the numerical approaches and the optimization procedure.

... Examples are potentially manifold and address, e.g., thermal treatment and reheating in complex shaped domains (Böhm and Meurer, 2014) or forming processes (Kiefer and Kugi, 2008). Besides considering single process units PDEs can be used to model complete manufacturing flow lines (see, e.g., van den Berg et al. (2008); M.L. Marca and Ringhofer (2010); Göttlich and Schindler (2015)). Under certain assumptions this results in quasilinear hyperbolic PDEs and partial integro differential equations (PIDEs). ...

The control theory of systems governed by partial differential equations has a long history and has reached a certain level of maturity. Based on recent developments, in the following potential research perspectives are discussed that address both methods and applications. The presentation is not intended to be exhaustive and reflects the author's personal view.

... Different numerical approaches are possible for models with conservation laws, see [6,7,8,10,14,23], with emphasis on various optimization problems ( [9,17,24,25]). In our case, the approach considers a numerical scheme in compact form to solve the system of differential equations of our supply system model. ...

In this paper, we discuss a numerical approach for the simulation of a model for supply chains based on both ordinary and partial differential equations. Such methodology foresees differential quadrature rules and a Picard--like recursion. In its former version, it was proposed for the solution of ordinary differential equations and is here extended to the case of partial differential equations. The outcome is a final non--recursive scheme, which uses matrices and vectors, with consequent advantages for the determination of the local error. A test case shows that traditional methods give worst approximations with respect to the proposed formulation.

We develop a numerical method for the solution to linear adjoint equations arising, for example, in optimization problems governed by hyperbolic systems of nonlinear conservation and balance laws in one space dimension. Formally, the solution requires one to numerically solve the hyperbolic system forward in time and a corresponding linear adjoint system backward in time. Numerical results for the control problem constrained by either the Euler equations of gas dynamics or isothermal gas dynamics equations are presented. Both smooth and discontinuous prescribed terminal states are considered.

Our main objective is the modelling and simulation of complex production networks originally introduced in [15, 16] with random breakdowns of individual processors. Similar to [10], the breakdowns of processors are exponentially distributed. The resulting network model is of coupled system of partial and ordinary differential equations with Markovian switching and its solution is a stochastic process. We show our model to fit into the framework of piecewise deterministic processes, which allows for a deterministic interpretation of dynamics between a multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately tracing stochastic events. Numerical results are presented for three exemplary networks, including a comparison with the long-chain model proposed in [10].

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.

We consider a supply network where the flow of parts can be controlled at the vertices of the network. Based on a coarse grid discretization provided in [6] we derive discrete adjoint equations which are subsequently validated by the continuous adjoint calculus. Moreover, we present numerical results concerning the quality of approximations and computing times of the presented approaches.

This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [ ], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [ ].

We analyze the stochastic large time behavior of long supply chains via a traffic flow type random particle model. So items travel on a virtual road from one production stage to the next. Random breakdowns of the processors at each stage are modeled via a Markov process. The result is a conservation law for the expectation of the part density which holds on time scales which are large compared to the mean up and down times of the processors.

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.

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.

We consider models based on conservation laws. For the optimization of such systems, a sensitivity analysis is essential to determine how changes in the decision variables influence the objective function. Here we study the sensitivity with respect to the initial data of objective functions that depend upon the solution of Riemann problems with piecewise linear flux functions. We present representations for the one–sided directional derivatives of the objective functions. The results can be used in the numerical method called Front-Tracking.

A mathematical model describing supply chains on a network is introduced. In particular, conditions on each vertex of the network are specified. Finally, this leads to a system of nonlinear conservation laws coupled to ordinary differential equations. To prove the existence of a solution we make use of the front tracking method. A comparison to another approach is given and numerical results are presented.

We show how to treat supply networks as physical transport problems governed by balance equations and equations for the adaptation of production speeds. Although the nonlinear behavior is different, the linearized set of coupled differential equations is formally related to those of mechanical or electrical oscillator networks. Supply networks possess interesting features due to their complex topology and directed links. We derive analytical conditions for absolute and convective instabilities. The empirically observed "bullwhip effect" in supply chains is explained as a form of convective instability based on resonance effects. Moreover, it is generalized to arbitrary supply networks. Their related eigenvalues are usually complex, depending on the network structure (even without loops). Therefore, their generic behavior is characterized by damped or growing oscillations. We also show that regular distribution networks possess two negative eigenvalues only, but perturbations generate a spectrum of complex eigenvalues.

We present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term. The sensitivity analysis is based on shift-variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations. Based on a first order approximation by shift-variations in L 1 we introduce the concept of shift-differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking-type functionals. In the main part of the paper we show that entropy solutions are generically shift-differentiable at almost all times t > 0 with respect to the control. Hereby we admit shift-variations for the initial data which allows to use the shift-differentiability result repeatedly over time slabs. This is useful for the design of optimization methods with time domain decomposition. Our analysis, especially of the shock sensitivity, combines structural results by using generalized characteristics and an adjoint argument. Our adjoint based shock sensitivity analysis does not require to restrict the richness of the shock structure a priori and admits shock generation points. The analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one-sided Lipschitz condition. As a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking-type functionals.

In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.

In this study we use discontinuous conservation laws to model production networks with finite buffers. A network formulation is characterized by the interaction between several production units, described by discontinuous conservation laws, under the assumption that no parts are lost. Therefore we discuss appropriate Riemann problems and coupling conditions which are derived from a regularized model. To solve the discontinuous problem in an accurate and fast way, we introduce a novel algorithm. Based on different settings, the new method is compared numerically with standard approaches.

We consider a recently proposed model by Armbruster et al. (2011) [2] for supply chains with finite buffers. The continuous model is based on a scalar conservation law and discontinuous flux function. A suitable reformulation of the model is introduced and studied analytically. The latter include waves with infinite negative wave speed. Using wave-front tracking, existence of solutions is obtained for supply chains and sequences of supply chains. Numerical results are presented.

The author studies the initial value problem for the scalar conservation law u t +f(u) x =0 in one spatial dimension. The flow function may be discontinuous with a finite number of jump discontinuities. This paper proves existence of a weak solution, and the proof is constructive, suggesting a numerical method for the problem.

We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to [M. Herty, J.-P. Lebacque and S. Moutari, Netw. Heterog. Media 4, No. 4, 813–826 (2009; Zbl 1187.35137)], we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors.

We propose a rigorous procedure to obtain the adjoint-based gradient representation of cost functionals for the optimal control of discontinuous solutions of conservation laws. Hereby, it is not necessary to introduce adjoint variables for the shock positions. Our approach is based on stability properties of the adjoint equation. We give a complete analysis for the case of convex scalar conservation laws. The adjoint equation is a transport equation with discontinuous coefficients and special reversible solutions must be considered to obtain the correct adjoint-based gradient formula. Reversible solutions of the adjoint transport equation and the required stability properties are analyzed in detail.

In this paper we explicitly construct the entropy solutions for the Lighthill-Whitham- Richards (LWR) tra! c flow model with a flow-density relationship which is piecewise quadratic, concave, but not continuous at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. The existence and uniqueness of entropy solutions for such conservation laws with discontinuous fluxes are not known mathematically. We have used the approach of explicitly construct- ing the entropy solutions to a sequence of approximate problems in which the flow-density relationship is continuous but tends to the discontinuous flux when a small parameter in this sequence tends to zero. The limit of the entropy solutions for this sequence is explicitly constructed and is considered to be the entropy solution associated with the discontinuous flux. We apply this entropy solution construction procedure to solve three representative tra! c flow cases, compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme, and discuss the results from tra! c flow per- spectives.

We introduce a continuous optimal control problem governed by ordinary and partial differential equations for supply chains on networks. We derive a mixed-integer model by discretization of the dynamics of the partial differential equations and by approximations to the cost functional. Finally, we investigate numerically properties of the derived mixed-integer model and present numerical results for a real-world example.

We investigate a model for traffic flow based on the
Lighthill-Whitham-Richards model that consists of a hyperbolic conservation law
with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out
the discontinuity in the flux function over a small distance epsilon << 1 and
then the analytical solution to the corresponding Riemann problem is derived in
the limit as epsilon goes to 0. For certain initial data, the Riemann problem
can give rise to zero waves that propagate with infinite speed but have zero
strength. We propose a Godunov-type numerical scheme that avoids the otherwise
severely restrictive CFL constraint that would arise from waves with infinite
speed by exchanging information between local Riemann problems and thereby
incorporating the effects of zero waves directly into the Riemann solver.
Numerical simulations are provided to illustrate the behaviour of zero waves
and their impact on the solution. The effectiveness of our approach is
demonstrated through a careful convergence study and comparisons to
computations using a third-order WENO scheme.