An exact approach for the Minimum-Cost Bounded-Error Calibration Tree problem

Abstract

The Minimum-Cost Bounded-Error Calibration Tree problem (MBCT) is a wireless network optimization problem that arises from the sensors’ need of periodical calibration. The MBCT takes into account two objectives. The first is to minimize the communication distance between the network sensors, while the second is to reduce the maximum post-calibration error in the network. In this paper, we propose a mathematical formulation for the MBCT. Furthermore, we solve the problem using two different implementations of an Augmented $\epsilon$-Constraint Method ($aug\,\epsilon \text {-}CM$) that incorporate the proposed formulation. We also employed the Pareto-fronts obtained by $aug\,\epsilon \text {-}CM$ to evaluate the Node-depth Phylogenetic-based Non-dominated Sorting Artificial Immune System (NPE-NSAIS), the most recent heuristic in the literature for MBCT. Computational experiments showed that $aug\,\epsilon \text {-}CM$ can solve MBCT instances up to 50 nodes. Furthermore, a statistical test demonstrated that the running times of one of the $aug\,\epsilon \text {-}CM$ implementations was significantly smaller than those of the other. Finally, we show that NPE-NSAIS solutions are very close to the Pareto-fronts given by $aug\,\epsilon \text {-}CM$, achieving good results on the evaluated metrics.