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The Unit Re-Balancing Problem

Author

Listed:
  • Robin Dee

    (Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany)

  • Armin Fügenschuh

    (Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany)

  • George Kaimakamis

    (Faculty of Mathematics and Engineering Sciences, Hellenic Army Academy, Varis-Koropiou Av., 16673 Athens, Greece)

Abstract

We describe the problem of re-balancing a number of units distributed over a geographic area. Each unit consists of a number of components. A value between 0 and 1 describes the current rating of each component. By a piecewise linear function, this value is converted into a nominal status assessment. The lowest of the statuses determines the efficiency of a unit, and the highest status its cost. An unbalanced unit has a gap between these two. To re-balance the units, components can be transferred. The goal is to maximize the efficiency of all units. On a secondary level, the cost for the re-balancing should be minimal. We present a mixed-integer nonlinear programming formulation for this problem, which describes the potential movement of components as a multi-commodity flow. The piecewise linear functions needed to obtain the status values are reformulated using inequalities and binary variables. This results in a mixed-integer linear program, and numerical standard solvers are able to compute proven optimal solutions for instances with up to 100 units. We present numerical solutions for a set of open test instances and a bi-criteria objective function, and discuss the trade-off between cost and efficiency.

Suggested Citation

  • Robin Dee & Armin Fügenschuh & George Kaimakamis, 2021. "The Unit Re-Balancing Problem," Mathematics, MDPI, vol. 9(24), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3205-:d:700212
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    References listed on IDEAS

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    1. Akihiko Takahashi & Yukihiro Tsuzuki, 2017. "Rebalancing static super-replications," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-23, March.
    2. Alex Maynard & Aaron Smallwood & Mark E. Wohar, 2013. "Long Memory Regressors and Predictive Testing: A Two-stage Rebalancing Approach," Econometric Reviews, Taylor & Francis Journals, vol. 32(3), pages 318-360, November.
    3. Schuijbroek, J. & Hampshire, R.C. & van Hoeve, W.-J., 2017. "Inventory rebalancing and vehicle routing in bike sharing systems," European Journal of Operational Research, Elsevier, vol. 257(3), pages 992-1004.
    4. Boyacı, Burak & Zografos, Konstantinos G. & Geroliminis, Nikolas, 2015. "An optimization framework for the development of efficient one-way car-sharing systems," European Journal of Operational Research, Elsevier, vol. 240(3), pages 718-733.
    5. Paul, Nicholas R. & Lunday, Brian J. & Nurre, Sarah G., 2017. "A multiobjective, maximal conditional covering location problem applied to the relocation of hierarchical emergency response facilities," Omega, Elsevier, vol. 66(PA), pages 147-158.
    6. Lakshithe Wagalath, 2014. "Modelling the rebalancing slippage of leveraged exchange-traded funds," Quantitative Finance, Taylor & Francis Journals, vol. 14(9), pages 1503-1511, September.
    7. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, April.
    8. Franco Peschiera & Robert Dell & Johannes Royset & Alain Haït & Nicolas Dupin & Olga Battaïa, 2021. "A novel solution approach with ML-based pseudo-cuts for the Flight and Maintenance Planning problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(3), pages 635-664, September.
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