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A mixed-integer model based on a coupled system of differential equations is presented in order to optimize design and material distribution of production networks. Due to many binary variables arising in this model and in order to guarantee feasible solutions several starting heuristics, which provide incumbents for the branch and cut algorithm, are developed and compared.

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... Here, we combine this model based on partial differential equations with discrete decisions on switching times. The combination of optimization techniques for pde-based problems and integer restrictions on variables has been studied in the field of production in [24,26,29,30,63]. We proceed by transforming the model into a linear mixed integer programming problem. ...
We discuss continuous traffic flow network models including traffic lights. A mathematical model for traffic light settings within a macroscopic continuous traffic flow network is presented, and theoretical properties are investigated. The switching of the traffic light states is modeled as a discrete decision and is subject to optimization. A numerical approach for the optimization of switching points as a function of time based upon the macroscopic traffic flow model is proposed. The numerical discussion relies on an equivalent reformulation of the original problem as well as a mixed-integer discretization of the flow dynamics. The large-scale optimization problem is solved using derived heuristics within the optimization process. Numerical experiments are presented for a single intersection as well as for a road network.
In this work we present a mixed-integer model for the optimal design of production/transportation systems. In contrast to standard design problems, our model is originally based on a coupled system of differential equations capturing the dynamics of manufacturing processes and stocks. The problem is to select an optimal parameter configuration from a predefined set such that respective constraints are fulfilled. We focus on single commodity flows over large time scales as well as highly interconnected networks and propose a suitable start heuristic to ensure feasibility and to speed up the solution procedure. KeywordsProduction systems–Mixed integer models–Heuristics
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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 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.
Strategy and Tactics in Supply Chain Event Management
  • S Göttlich
  • M Herty
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