A multifacility location problem on median spaces

AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem

Continuous multifacility ordered median location problems

In this paper we propose a general methodology for solving a broad class of continuous, multifacility location problems, in any dimension and with ℓτ -norms proposing two different methodologies: 1) by a new second order cone mixed integer programming formulation and 2) by formulating a sequence of semidefinite programs that converges to the solution of the problem; each of these relaxed problems solvable with SDP solvers in polynomial time. We apply dimensionality reductions of the problems by sparsity and symmetry in order to be able to solve larger problems. Continuous multifacility location and Ordered median problems and Semidefinite programming and Moment problem.Junta de AndalucíaFondo Europeo de Desarrollo RegionalMinisterio de Ciencia e Innovació

Multifacility location problems on a sphere

A unified approach to multisource location problems on a sphere is presented. Euclidean, squared Euclidean and the great circle distances are considered. An algorithm is formulated and its convergence properties are investigated

A generalized model of equality measures in network location problems

In this paper, the concept of the ordered weighted averaging operator is applied to define a model which unifies and generalizes several inequality measures. For a location x, the value of the new objective function is the ordered weighted average of the absolute deviations from the average distance from the facilities to the location x. Several kinds of networks are studied: cyclic, tree and path networks and, for each of them, the properties of the objective function are analyzed in order to identify a finite dominating set for optimal locations. Polynomial-time algorithms are proposed for these problems, and the corresponding complexity is discussed.Future and Emerging Technologies Unit (European Commission)Ministerio de Educación y Cienci

Minimum 0-Extension Problems on Directed Metrics

For a metric $\mu$ on a finite set $T$, the minimum 0-extension problem 0-Ext$[\mu]$ is defined as follows: Given $V\supseteq T$ and $\ c:{V \choose 2}\rightarrow \mathbf{Q_+}$, minimize $\sum c(xy)\mu(\gamma(x),\gamma(y))$ subject to $\gamma:V\rightarrow T,\ \gamma(t)=t\ (\forall t\in T)$, where the sum is taken over all unordered pairs in $V$. This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. The complexity dichotomy of 0-Ext$[\mu]$ was established by Karzanov and Hirai, which is viewed as a manifestation of the dichotomy theorem for finite-valued CSPs due to Thapper and \v{Z}ivn\'{y}. In this paper, we consider a directed version $\overrightarrow{0}$-Ext$[\mu]$ of the minimum 0-extension problem, where $\mu$ and $c$ are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext$[\mu]$ to $\overrightarrow{0}$-Ext$[\mu]$: If $\mu$ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant ``directed'' edge-length, then $\overrightarrow{0}$-Ext$[\mu]$ is NP-hard. We also show a partial converse: If $\mu$ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then $\overrightarrow{0}$-Ext$[\mu]$ is tractable. We further provide a new NP-hardness condition characteristic of $\overrightarrow{0}$-Ext$[\mu]$, and establish a dichotomy for the case where $\mu$ is a directed metric of a star

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

A branch-and-price approach for the continuous multifacility monotone ordered median problem

Acknowledgements The authors of this research acknowledge financial support by the Spanish Ministerio de Ciencia y Tecnología, Agencia Estatal de Investigación and Fondos Europeos de Desarrollo Regional (FEDER) via project PID2020-114594GB-C21. The authors also acknowledge partial support from project B-FQM-322-UGR20. The first, third and fourth authors also acknowledge partial support from projects FEDER-US-1256951, Junta de Andaluca P18-FR-1422, CEI-3-FQM331, FQM-331, and NetmeetData: Ayudas Fundacin BBVA a equipos de investigacin científica 2019. The first and second authors were par- tially supported by research group SEJ-584 (Junta de Andalucía). The first author was also partially supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/50110 0 011033. The second author was supported by Spanish Ministry of Education and Science grant number PEJ2018-002962-A and the Doctoral Program in Mathematics at the Universidad of Granada. The third author also acknowledges the grant Contratación de Personal Investigador Doctor (Convocatoria 2019) 43 Contratos Capital Humano Línea 2 Paidi 2020, supported by the European Social Fund and Junta de Andalucía.In this paper, we address the Continuous Multifacility Monotone Ordered Median Problem. The goal of this problem is to locate facilities in minimizing a monotone ordered weighted median function of the distances between given demand points and its closest facility. We propose a new branch-and-price procedure for this problem, and three families of matheuristics based on: solving heuristically the pricer problem, aggregating the demand points, and discretizing the decision space. We give detailed discussions of the validity of the exact formulations and also specify the implementation details of all the solution procedures. Besides, we assess their performances in an extensive computational experience that shows the superiority of the branch-and-price approach over the compact formulation in medium-sized instances. To handle larger instances it is advisable to resort to the matheuristics that also report rather good results.Spanish Ministerio de Ciencia y Tecnología, Agencia Estatal de Investigación and Fondos Europeos de Desarrollo Regional (FEDER) via project PID2020-114594GB-C21Partial support from project B-FQM-322-UGR20Partial support from projects FEDER-US-1256951, Junta de Andaluca P18-FR-1422, CEI-3-FQM331, FQM-331, and NetmeetData: Ayudas Fundación BBVA a equipos de investigacin científica 2019Research group SEJ-584 (Junta de Andalucía)Partially supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/50110 0 011033Spanish Ministry of Education and Science grant number PEJ2018-002962-AEuropean Social Fund and Junta de Andalucí

Sensitivity analysis of distance constraints and of multifacility minimax location on tree networks

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent Univ. ,1989.Thesis (Master's) -- Bilkent University, 1989.Includes bibliographical references leaves 77-79.In this thesis, the main concern is to investigate the use of consistency conditions of distance constraints in sensitivity analysis of certain network location problems. The interest is in minimax type of objective functions. A single parametric approach is adopted in the sensitivity analysis for the m-facility minimax location problem on tree networks. Apart from the traditional sensitivity analysis approach, a conceptual framework for imprecision in distance constraints is developed.Doğan, EsraM.S