The Unit Re-Balancing Problem

Robin Dee, Armin Fügenschuh and George Kaimakamis
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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

Mathematics, 2021, vol. 9, issue 24, 1-19

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.

Keywords: re-balancing problem; efficiency; mixed-integer linear programming; bi-criteria optimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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