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Coupling conditions for supply networks and resulting optimization problems
Abstract
We discuss a continuous model for production lines based on ordinary and partial differential equations and recall coupling conditions for connecting different suppliers of the supply chain. Based on this model we state an optimization problem for distributing flows in supply chains and investigate this numerically. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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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 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.
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.
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).