# Mixed-integer linear methods for layout-optimization of screening systems in recovered paper production

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## Abstract

The industrial treatment of waste paper in order to regain valuable fibers from which recovered paper can be produced, involves several steps of preparation. One important step is the separation of stickies that are normally attached to the paper. If not properly separated, remaining stickies reduce the quality of the recovered paper or even disrupt the production process. For the mechanical separation process of fibers from stickies a separator screen is used. This machine has one input feed and two output streams, called the accept and the reject. In the accept the fibers are concentrated, whereas the reject has a higher concentration of stickies. The machine can be controlled by setting its reject rate. But even when the reject rate is set properly, after just a single screening step, the accept still has too many stickies, or the reject too many fibers. To get a better separation, several separators have to be assembled into a network. From a mathematical point of view this problem can be seen as a multi-commodity network flow design problem with a nonlinear, controllable distribution function at each node. We present a nonlinear mixed-integer programming model for the simultaneous selection of the network’s topology and the optimal setting of each separator. Numerical results are obtained via different types of linearization of the nonlinearities and the use of mixed-integer linear solvers, and compared with state-of-the-art global optimization software.

## Keywords

Mixed-integer linear programming Nonlinear programming Piecewise linear approximation Global optimization Linear relaxation Topology optimization Network design## Notes

### Acknowledgments

We thank Prof. Dr.-Ing. Samuel Schabel and Dipl.-Ing. Klaus Villforth of the chair of paper technology and mechanical process engineering at Technische Universität Darmstadt for posing the problem and fruitful discussions. We also thank Björn Geiß ler and Antonio Morsi for providing an implementation for ordering triangulations. We thank Stefan Vigerske (GAMS) for his helpful hints about GAMS. The computational results were achieved on computer hardware and software licenses provided by the Zuse Institute Berlin (ZIB). The work of Christine Hayn was partly supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. The submitted version was finished while the third author, Dennis Michaels, was at the Institute for Operations Research at ETH Zürich and financially supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multi-phase Systems” (CRC/Transregio 63 “InPROMPT”). Dennis Michaels thanks the DFG for their financial support

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