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We extend the model for supply chains developed in [S. Göttlich, M. Herty and A. Klar, Commun. Math. Sci. 3, No. 4, 545–559 (2005; Zbl 1115.90008)]. The model consists of partial differential equations governing the dynamics on each processor. Furthermore, a modelling of different types of vertices is motivated and discussed. Then, optimization problems are introduced and numerically investigated. A comparison of computing times shows the efficiency of partial differential equations for solving supply chain problems.

Content uploaded by Michael Herty

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... Among the possible applications, vehicular traffic is probably the one more studied; see, for example, [139] and the reference therein. Other applications range over data networks, irrigation channels, gas pipelines, supply chains, and blood circulation; see, for example, [26,62,77,106,108,156]. Therefore, it is natural to consider control problems for such systems, and in particular to consider control functions acting at the level of junctions or nodes [144,160]. ...

... After around a decade, a renewed interest gave rise to many models, see Chitour and Piccoli [70], Coclite et al. [76], Garavello and Piccoli [141,142], Herty et al. [175,177], Holden and Risebro [178], and Lebacque and Khoshyaran [207]. The same models were used for other applications such as telecommunication networks (see D'Apice et al. [108]), gas pipelines networks (see Banda et al. [26] and Colombo and Garavello [77]), and supply chains (see Armbruster et al. [20], D'Apice and Manzo [107], and Göttlich et al. [156]). ...

... Among the possible applications, vehicular traffic is probably the one more studied; see, for example, [139] and the reference therein. Other applications range over data networks, irrigation channels, gas pipelines, supply chains, and blood circulation; see, for example, [26,62,77,106,108,156]. Therefore, it is natural to consider control problems for such systems, and in particular to consider control functions acting at the level of junctions or nodes [144,160]. ...

... After around a decade, a renewed interest gave rise to many models, see Chitour and Piccoli [70], Coclite et al. [76], Garavello and Piccoli [141,142], Herty et al. [175,177], Holden and Risebro [178], and Lebacque and Khoshyaran [207]. The same models were used for other applications such as telecommunication networks (see D'Apice et al. [108]), gas pipelines networks (see Banda et al. [26] and Colombo and Garavello [77]), and supply chains (see Armbruster et al. [20], D'Apice and Manzo [107], and Göttlich et al. [156]). ...

This chapter focuses on control of systems of conservation laws with distributed parameters. Problem with different parameterized fluxes is addressed: in particular, we deal with cases where the control is the maximal speed and look for continuous dependence of the solution on parameters.

... Among the possible applications, vehicular traffic is probably the one more studied; see, for example, [139] and the reference therein. Other applications range over data networks, irrigation channels, gas pipelines, supply chains, and blood circulation; see, for example, [26,62,77,106,108,156]. Therefore, it is natural to consider control problems for such systems, and in particular to consider control functions acting at the level of junctions or nodes [144,160]. ...

... After around a decade, a renewed interest gave rise to many models, see Chitour and Piccoli [70], Coclite et al. [76], Garavello and Piccoli [141,142], Herty et al. [175,177], Holden and Risebro [178], and Lebacque and Khoshyaran [207]. The same models were used for other applications such as telecommunication networks (see D'Apice et al. [108]), gas pipelines networks (see Banda et al. [26] and Colombo and Garavello [77]), and supply chains (see Armbruster et al. [20], D'Apice and Manzo [107], and Göttlich et al. [156]). ...

In this chapter, we introduce Hamilton-Jacobi PDEs. These PDEs are related to conservation laws and their solutions are the anti-derivative (in space) of the Entropy solutions of the corresponding conservation law, given that some assumptions are satisfied.

Conservation and/or balance laws on networks in the recent years have been the subject of intense study, since a wide range of different applications in real life can be covered by such a research.

This chapter focuses on control of systems of conservation laws with boundary data. Problems with one or two boundaries are considered and, in particular, we focus on cases where shocks may be developed by the solution. However, for completeness we briefly discuss in Sect. 2.2 other existing results where singularities are prevented via suitable feedback controls such as in [32].

A vehicle with different (eventually controlled) dynamics from general traffic along a street may reduce the road capacity, thus generating a moving bottleneck , and can be used to act on the traffic flow. The interaction between the controlled vehicle and the surrounding traffic, and the consequent flow reduction at the bottleneck position, can be described either by a conservation law with space dependent flux function [200], or by a strongly coupled PDE-ODE system proposed in [112, 208].

... A nice overview of such models can be found in [256]. Additional models can be seen in [231,[259][260][261][262][263][264]. ...

This review paper is devoted to a brief overview of results and models concerning flows in networks and channels of networks. First of all, we conduct a survey of the literature in several areas of research connected to these flows. Then, we mention certain basic mathematical models of flows in networks that are based on differential equations. We give special attention to several models for flows of substances in channels of networks. For stationary cases of these flows, we present probability distributions connected to the substance in the nodes of the channel for two basic models: the model of a channel with many arms modeled by differential equations and the model of a simple channel with flows of substances modeled by difference equations. The probability distributions obtained contain as specific cases any probability distribution of a discrete random variable that takes values of 0,1,…. We also mention applications of the considered models, such as applications for modeling migration flows. Special attention is given to the connection of the theory of stationary flows in channels of networks and the theory of the growth of random networks.

Nowadays, control strategies are a crucial part of the industrial domain. A challenging aspect in production lines therein is the mismatch between the total desired lots and the total lots of the system output due to the system limitations, the so-called backlog problem. In this paper, optimal control approaches are investigated to address this problem in terms of conservation laws coupled with ordinary differential equations (ODEs) in different interconnection topologies that correspond to dispersing and merging networks. The problem is optimized utilizing open-loop optimal control according to discretize-then-optimize and optimize-then-discretize mechanisms. In addition, model predictive control is designed to solve the problem by using a suitable forward shifting technique to suppress disturbances also taking into account constraints. The numerical results show solvability and associated features for each approach depending on the requirement of the corresponding use case.

This paper focuses on a model for supply chains, based on partial and ordinary differential equations, that model, respectively, densities of parts on suppliers and queues between consecutive arcs. An optimization approach is discussed via a cost functional that, in consideration of a wished outflow, weights queues of materials by variations of processing velocities for suppliers. The minimization of the cost functional is achieved via a genetic algorithm that, as for the processing velocities, considers mechanisms of selection, crossover and mutation. A simulation example is discussed for the optimization procedure.KeywordsGenetic algorithmsSupply chainsSimulations

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. In a previous 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.

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.

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.

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

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.

This report presents a series of lecture notes dealing with the stability and optimization of autonomous supply chains. It begins with an introduction to the terminology and notation. General weak- stability results for autonomous, pull supply chains are then developed. Duality results for "push" supply chains are next presented. The report then examines the ability or inability of autonomous algorithms to return to a steady state, track a time- dependent target, and dissipate small perturbations. A family of strongly stable autonomous algorithms, including "just-in-time" operations, is developed. The procedure for estimating total cost with both autonomous and coordinated system-optimal operations is shown. The report concludes with a presentation of autonomous, strongly stable algorithms for non-linear, multi-commodity, multi-destination networks.

To manage the increasing dynamics within complex production networks, a decen-tralised and autonomous control of material flows is a promising approach. This aims at an easier control of logistic processes in the network, whereby the local decision rules should lead to self-organisation and good logistic performance on the global level. There are two basic questions: (i) How to design the local autonomy to reach the desired global behaviour. (ii) How to design the global structure of the production network to enable local autonomy? For developing and benchmarking such autonomous control methods, dynamic models are essential. Discrete-event simulation models are used for a detailed de-scription of different local decision rules on a micro-level. Continuous fluid models are used to capture the global dynamics on a macro-level and to find a global opti-mum to benchmark the local decision rules. The paper introduces the idea of autonomous logistic processes and presents both a discrete-event simulation (DES) model and a fluid model of a simplified produc-tion network with an autonomously controlled flow of parts based on backward propagated information (pheromone concept).

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions. Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

A hierachy of simplified models for traffic flow on networks is derived from continuous traffic flow models based on partial differential equations. The hierachy contains nonlinear and linear combinatorial models with and without dynamics. Optimization problems are treated for all models and numerical results and algorithms are compared.