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

Deleplanque, Samuel; Ponce, Diego; Puerto, Justo

doi:10.1287/ijoc.2019.0915

Cited by 18 publications

(8 citation statements)

References 41 publications

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“…The objective function is again the total cost, the cost of satisfying the demand of those customers that do not make use Inequalities (15) and (19) and inequalities in Corollary 1 can be easily adapted as stated in the following remark.…”

Section: Mixed-integer Linear Three-index Formulationmentioning

confidence: 99%

“…Both facts together suggest that a column generation approach is a promising resolution method. Column generation is a popular approach when handling models with a lot variables that have good performance in terms of integrality gap, see [19] for an application to the discrete ordered median problem and [20][21][22][23] for solving different combinatorial problems.…”

Section: Branch and Price Algorithmmentioning

confidence: 99%

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Close-Enough Facility Location

2021

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This paper introduces the concept of close-enough in the context of facility location. It is assumed that customers are willing to move from their homes to close-enough pickup locations. Given that the number of pickup locations is expanding every day, it is assumed that pickup locations can be placed everywhere. Conversely, the set of potential location for opening facilities is discrete as well as the set of customers. Opening facilities and pickup points entails an installation budget and a distribution cost to transport goods from facilities to customers and pickup locations. The (p,t)-Close-Enough Facility Location Problem is the problem of deciding where to locate p facilities among the finite set of candidates, where to locate t pickup points in the plane and how to allocate customers to facilities or to pickup points so that all the demand is satisfied and the total cost is minimized. In this paper, it is proved that the set of initial infinite number of pickup locations is finite in practice. Two mixed-integer linear programming models are proposed for the discrete problem. The models are enhanced with valid inequalities and a branch and price algorithm is designed for the most promising model. The findings of a comprehensive computational study reveal the performance of the different models and the branch and price algorithm and illustrate the value of pickup locations.

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Section: Mixed-integer Linear Three-index Formulationmentioning

confidence: 99%

Section: Branch and Price Algorithmmentioning

confidence: 99%

Close-Enough Facility Location

2021

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“…Branching is applied each time a not integral solution is found until optimality is proved. The reader is referred to the following works and the references therein for further details on the following topics: , for a precise presentation about column generation; Barnhart et al (1996) for a detailed explanation about branchand-price; and to Deleplanque et al (2020), for a recent application of those techniques. A pseudocode of this method is provided in Algorithm 1 and explained in detail in Section 5.…”

Section: The Set Partitioning Formulationmentioning

confidence: 99%

“…Among others, du Merle et al (1999), Pessoa et al (2010), and Sato and Fukumura (2012) have designed procedures to minimize the negative impact of the issue in the convergence of column generation algorithms. See also Sato and Izunaga (2019) or Deleplanque et al (2020) for other recents applications of those techniques.…”

Section: Convergence Of Column Generationmentioning

confidence: 99%

A Branch-and-price procedure for clustering data that are graph connected

Benati¹,

Ponce²,

Puerto³

et al. 2021

Preprint

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This paper studies the Graph-Connected Clique-Partitioning Problem (GCCP), a clustering optimization model in which units are characterized by both individual and relational data. This problem, introduced by Benati et al. ( 2017) under the name of Connected Partitioning Problem, shows that the combination of the two data types improves the clustering quality in comparison with other methodologies. Nevertheless, the resulting optimization problem is difficult to solve; only small-sized instances can be solved exactly, large-sized instances require the application of heuristic algorithms. In this paper we improve the exact and the heuristic algorithms previously proposed. Here, we provide a new Integer Linear Programming (ILP) formulation, that solves larger instances, but at the cost of using an exponential number of variables. In order to limit the number of variables necessary to calculate the optimum, the new ILP formulation is solved implementing a branch-and-price (B&P) algorithm. The resulting pricing problem is itself a new combinatorial model: the Maximum-weighted Graph-Connected Single-Clique problem (MGCSC), that we solve testing various Mixed Integer Linear Programming (MILP) formulations and proposing a new fast "random shrink" heuristic. In this way, we are able to improve the previous algorithms: The B&P method outperforms the computational times of the previous MILP algorithms and the new random shrink heuristic, when applied to GCCP, is both faster and more accurate than the previous heuristic methods.Moreover, the combination of column generation and random shrink is itself a new MILPrelaxed matheuristic that can be applied to large instances too. Its main advantage is that all heuristic local optima are combined together in a restricted MILP, consisting in the application of the exact B&P method but solving heuristically the pricing problem.

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“…Motivated by the recent advances on discrete multifacility location problems with ordered median objectives (Deleplanque et al, 2020;Espejo et al, 2021;Fernández et al, 2014;Labbé et al, 2017;Marín et al, 2020) and the available results on conic optimization (Blanco et al, 2014;Puerto, 2020), we want to address a family of difficult continuous multifacility location problems with ordered median objectives and distances induced by general τ -norms, τ ≥ 1. These problems gather the essential elements of both areas (discrete and continuous) of location analysis making their solution a challenging question.…”

Section: Introductionmentioning

confidence: 99%

A Branch-and-Price approach for the Continuous Multifacility Monotone Ordered Median Problem

Blanco¹,

Ricardo²,

Ponce³

et al. 2021

Preprint

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In this paper, we address the Continuous Multifacility Monotone Ordered Median Problem. This problem minimizes a monotone ordered weighted median function of the distances between given demand points in R d and its closest facility among the p selected, also in a continuous space. We propose a new branch-and-price procedure for this problem, and two mathehuristics. One of them is a decomposition-based procedure and the other an aggregation-based heuristic. 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 performance 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.

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

Cited by 18 publications

References 41 publications

Close-Enough Facility Location

Close-Enough Facility Location

A Branch-and-price procedure for clustering data that are graph connected

A Branch-and-Price approach for the Continuous Multifacility Monotone Ordered Median Problem

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