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.
Highlights
In recent years, many nations tried to re-organize the structure of their public security forces in order to have better readiness and to be more efficient
How do things get un-balanced? Any security force with its units can be found on the hierarchical level of an “organization” in the Living Systems Theory (LST) according to Miller [2]
With our work presented in this paper, we are able to fit nonlinear correlations into a mixed-integer linear programming model via the usage of piecewise linear functions
Summary
Many nations tried to re-organize the structure of their public security forces (army, police, fire departments, first aids, etc.) in order to have better readiness and to be more efficient (see, for example, in [1]). In [15], the authors focus on locating and re-locating a fleet of response units (from army, police, etc.) in a specific transportation network to optimally cover a number of facilities They present the problem as a mixed-integer linear program and develop a heuristic algorithm. With our work presented in this paper, we are able to fit nonlinear correlations (e.g., the efficiency and cost of an army unit depending on various ratings) into a mixed-integer linear programming model via the usage of piecewise linear functions. This offers a new perspective on re-balancing problems.
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