Mathematical Programming formulations for the efficient solution of the kk-sum approval voting problem

In this paper we address the problem of electing a committee among a set of $m$ candidates and on the basis of the preferences of a set of $n$ voters. We consider the approval voting method in which each voter can approve as many candidates as she/he likes by expressing a preference profile (boolean $m$-vector). In order to elect a committee, a voting rule must be established to `transform' the $n$ voters' profiles into a winning committee. The problem is widely studied in voting theory; for a variety of voting rules the problem was shown to be computationally difficult and approximation algorithms and heuristic techniques were proposed in the literature. In this paper we follow an Ordered Weighted Averaging approach and study the $k$-sum approval voting (optimization) problem in the general case $1 \leq k <n$. For this problem we provide different mathematical programming formulations that allow us to solve it in an exact solution framework. We provide computational results showing that our approach is efficient for medium-size test problems ($n$ up to 200, $m$ up to 60) since in all tested cases it was able to find the exact optimal solution in very short computational times

On the multisource hyperplanes location problem to fitting set of points

In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with an exponential number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions.Comment: 30 pages, 5 Tables, 3 Figure

Constraint relaxation for the Discrete Ordered Median Problem

This paper compares different exact approaches to solve the Discrete Ordered Median Problem (DOMP). In recent years, DOMP has been formulated using set packing constraints giving rise to one of its most promising formulations. The use of this family of constraints, known as strong order constraints (SOC), has been validated in the literature by its theoretical properties and because their linear relaxation provides very good lower bounds. Furthermore, embedded in branch-and-cut or branch-price-and-cut procedures as valid inequalities, they allow one to improve computational aspects of solution methods such as CPU time and use of memory. In spite of that, the above mentioned formulations require to include another family of order constraints, e.g., the weak order constraints (WOC), which leads to coefficient matrices with elements other than {0,1}. In this work, we develop a new approach that does not consider extra families of order constraints and furthermore relaxes SOC -- in a branch-and-cut procedure that does not start with a complete formulation -- to add them iteratively using row generation techniques to certify feasibility and optimality. Exhaustive computational experiments show that it is advisable to use row generation techniques in order to only consider {0,1}-coefficient matrices modeling the DOMP. Moreover, we test how to exploit the problem structure. Implementing an efficient separation of SOC using callbacks improves the solution performance. This allows us to deal with bigger instances than using fixed cuts/constraints pools automatically added by the solver in the branch-and-cut for SOC, concerning both the formulation based on WOC and the row generation procedure

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

Mathematical programming formulations for the efficient solution of the k-sum approval voting problem

In this paper we address the problem of electing a committee among a set of m candidates and on the basis of the preferences of a set of n voters. We consider the approval voting method in which each voter can approve as many candidates as she/he likes by expressing a preference profile (boolean m-vector). In order to elect a committee, a voting rule must be established to ‘transform’ the n voters’ profiles into a winning committee. The problem is widely studied in voting theory; for a variety of voting rules the problem was shown to be computationally difficult and approximation algorithms and heuristic techniques were proposed in the literature. In this paper we follow an Ordered Weighted Averaging approach and study the k-sum approval voting (optimization) problem in the general case 1 ≤ k < n. For this problem we provide different mathematical programming formulations that allow us to solve it in an exact solution framework. We provide computational results showing that our approach is efficient for medium-size test problems (n up to 200, m up to 60) since in all tested cases it was able to find the exact optimal solution in very short computational times.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

A Branch-Price-and-Cut Procedure for the Discrete Ordered Median Problem

International audienceThe Discrete Ordered Median Problem (DOMP) is formulated as a set partitioning problem using an exponential number of variables. Each variable corresponds to a set of demand points allocated to the same facility with the information of the sorting position of their corresponding costs. We develop a column generation approach to solve the continuous relaxation of this model. Then, we apply a branch-price-and-cut algorithm to solve small to large sized instances of DOMP in competitive computational time

Network flow based approaches for the pipelines routing problem in naval design

In this paper we propose a general methodology for the optimal automatic routing of spatial pipelines motivated by a recent collaboration with Ghenova, a leading Naval Engineering company. We provide a minimum cost multicommodity network flow based model for the problem incorporating all the tech- nical requirements for a feasible pipeline routing. A branch-and-cut approach is designed and different matheuristic algorithms are derived for solving efficiently the problem. We report the results of a battery of computational experiments to assess the problem performance as well as a case study of a real-world naval instance provided by our partner company.Ministerio de Ciencia Y Tecnología (MCYT). España PID2020-114594GB-C21European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER) US-1256951Junta de Andalucía P18-FR-1422Junta de Andalucía CEI-3-FQM331Junta de Andalucía B-FQM-322-UGR2