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An adaptive discretization algorithm for the design of water usage and treatment networks

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Abstract

In this paper, we consider the design of water usage and treatment systems in industrial plants. In such a system, the demand of water using units as well as environmental regulations for wastewater have to be met. To this end, water treatment units have to be installed and operated to remove contaminants from the water. The objective of the design problem is to simultaneously optimize the network structure and water allocation of the system at minimum total cost. Due to many bilinear mass balance constraints, this water allocation problem is a nonconvex mixed integer nonlinear program (MINLP) where nonlinear solvers have difficulties to find feasible solutions for real world instances. Therefore, we present a problem specific algorithm to iteratively solve this MINLP. In each iteration, this algorithm deals with an interplay of a mixed integer linear program (MILP) and a quadratically constrained program (QCP). First, an MILP approximates the original problem via discretization and provides a suitable network structure. Then, by fixing this structure, the original MINLP turns into a QCP which yields feasible solutions to the original problem. To improve the accuracy of the generated structure, the discretization of the MILP is adapted after each iteration based on the previous MILP solution. In many cases where nonlinear solvers fail, this approach leads to feasible solutions with good solution quality in short running time.

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Authors and Affiliations

  1. Lehrstuhl II für Mathematik, RWTH Aachen University, 52062, Aachen, Germany

    Arie M. C. A. Koster & Sascha Kuhnke

Authors
  1. Arie M. C. A. Koster
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  2. Sascha Kuhnke
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Correspondence to Sascha Kuhnke.

Appendices

Appendix 1: Log information

For the MILPs and QCPs of ADISCM-P with \(m = 5\) and for the MINLPs, Table 4 shows the number of constraints and number of binary and continuous variables. Furthermore, the number of performed iterations of ADISCM-P is stated. For these eight instances, ADISCM-P was able to solve 56% of the QCPs within the desired 1% gap. Another 20% of the QCPs has not been solved within the 1% gap but terminated with an average gap of 17%. For 24% of the QCPs, no feasible solution has been found within the time limit.

Table 4 Size of MILPs, QCPs and MINLPs
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Appendix 2: Detailed computational results

Tables 5 and 6 show detailed results for all 99 test instances of the computational study from Sect. 5.5. In Table 6, the objective values of each instance are divided by the best solution among ADISCM-P and MINLP.

Table 5 Running times in seconds
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Table 6 Relative objective values
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Koster, A.M.C.A., Kuhnke, S. An adaptive discretization algorithm for the design of water usage and treatment networks. Optim Eng 20, 497–542 (2019). https://doi.org/10.1007/s11081-018-9413-6

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  • DOI: https://doi.org/10.1007/s11081-018-9413-6

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