Soft Computing

, Volume 23, Issue 9, pp 2959–2967 | Cite as

An effective heuristic for large-scale fault-tolerant k-median problem

  • Igor VasilyevEmail author
  • Anton V. Ushakov
  • Nadezhda Maltugueva
  • Antonio Sforza


We address a general fault-tolerant version of the k-median problem on a network. Unlike the original k-median, the objective is to find k nodes (medians or facilities) of a network, assign each non-median node (customer) to \(r_j\) distinct medians, and each median nodes to \(r_j-1\) other medians so as to minimize the overall assignment cost. The problem can be considered in context of the so-called reliable facility location, where facilities once located may be subject to failures. Hedging against possible disruptions, each customer is assigned to multiple distinct facilities. We propose a fast and effective heuristic rested upon consecutive searching for lower and upper bounds for the optimal value. The procedure for finding lower bounds is based on a Lagrangian relaxation and a specialized effective subgradient algorithm for solving the corresponding dual problem. The information on dual variables is then used by a core heuristic in order to determine a set of primal variables to be fixed. The effectiveness and efficiency of our approach are demonstrated in a computational experiment on large-scale problem instances taken from TSPLIB. We show that the proposed algorithm is able to fast find near-optimal solutions to problem instances with almost 625 million decision variables (on networks with up to 24978 vertices).


k-Median problem Reliable facility location Core selection Fault-tolerant facility location Lagrangian relaxation Disruptions 



This study was funded by Russian Foundation of Basic Research, Project No. 18-07-01037.

Compliance with ethical standards

Conflict of interest

All the authors (I. Vasilyev, A. V. Ushakov, N. Maltugueva and A. Sfroza) declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control TheorySiberian Branch of the Russian Academy of SciencesIrkutskRussia
  2. 2.Department of Electrical Engineering and Information TechnologyUniversity of Naples Federico IINaplesItaly

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