Continuous Modeling and Optimization Approaches for Manufacturing Systems

This thesis is concerned with two macroscopic models that are based on hyper- bolic partial differential equations (PDE) with discontinuous flux functions. The first model describes the material flow of an entire production line with finite buffers. We consider different solutions of the model, present the novel DFG- method (Discontinuous Flux Godunov), and compare the results with other established numerical methods. Additionally, we investigate a restricted optimization problem with respect to partial differential equations with discontinuous flux functions and consider two different solution approaches that are based on the adjoint method and the mixed integer problem (MIP). Further, we extend the model and its optimization problem to network structures. The second model describe the material flow on conveyor belts with obstacle interactions. We introduce a novel two dimensional model with a discontinuous and a non-local flux function. We consider a finite volume method and the discon- tinuous Galerkin method for solving this model. Finally, we validate the model with real data and present a numerical study with respect to the introduced solution methods

The Transport PDE and Mixed-Integer Linear Programming

Discrete, nonlinear and PDE constrained optimization are mostly considered as different fields of mathematical research. Nevertheless many real-life problems are most naturally modeled as PDE constrained mixed integer nonlinear programs. For example, nonlinear network flow problems where the flow dynamics are governed by a transport equation are of this type. We present four different applications together with the derivation of the associated transport equations and we show how to model these problems in terms of mixed integer linear constraints

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Locating and Protecting Facilities Subject to Random Disruptions and Attacks

Recent events such as the 2011 Tohoku earthquake and tsunami in Japan have revealed the vulnerability of networks such as supply chains to disruptive events. In particular, it has become apparent that the failure of a few elements of an infrastructure system can cause a system-wide disruption. Thus, it is important to learn more about which elements of infrastructure systems are most critical and how to protect an infrastructure system from the effects of a disruption. This dissertation seeks to enhance the understanding of how to design and protect networked infrastructure systems from disruptions by developing new mathematical models and solution techniques and using them to help decision-makers by discovering new decision-making insights. Several gaps exist in the body of knowledge concerning how to design and protect networks that are subject to disruptions. First, there is a lack of insights on how to make equitable decisions related to designing networks subject to disruptions. This is important in public-sector decision-making where it is important to generate solutions that are equitable across multiple stakeholders. Second, there is a lack of models that integrate system design and system protection decisions. These models are needed so that we can understand the benefit of integrating design and protection decisions. Finally, most of the literature makes several key assumptions: 1) protection of infrastructure elements is perfect, 2) an element is either fully protected or fully unprotected, and 3) after a disruption facilities are either completely operational or completely failed. While these may be reasonable assumptions in some contexts, there may exist contexts in which these assumptions are limiting. There are several difficulties with filling these gaps in the literature. This dissertation describes the discovery of mathematical formulations needed to fill these gaps as well as the identification of appropriate solution strategies

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios

MIP presolve techniques for a PDE-based supply chain model

We consider a mixed-integer linear program (MIP) for supply chains that has been derived in [A. Fuumlgenschuh, S. Goumlttlich, M. Herty, A. Klar, and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM J. Sci. Comput. 30(3) (2008), pp. 1490-1507] from a continuous supply chain model based on partial differential equations (PDEs). We develop new presolve techniques where knowledge about the continuous framework is involved. For this purpose, several presolve levels are introduced and compared numerically. The presented methods reduce the size of the MIP in terms of number of variables and constraints, accelerate the solution process of the MIP when using numerical solvers, and finally assure that such solvers are able to find feasible solutions at all, where in some cases they would fail without