A column generation approach to capacitated p-median problems

Lorena, Luiz Antônio Nogueira; Senne, Edson Luiz França

doi:10.1016/s0305-0548(03)00039-x

Cited by 135 publications

(62 citation statements)

References 25 publications

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“…This model is very similar to other ILPs arising in CG studies for Capacitated p-Median, see (10) in [18] or (6)- (8) in [7]. The former model is almost exactly the same as (2.1), up to the following notational equivalences: they index the columns with k ∈ [1..m], the decision variables are noted x k and the columns are indicated by vector A k of cost c k -this cost c k is determined by solving n (i.e., |V |) binary knapsack problems, i.e., one for each possible medianm ∈ V .…”

Section: A Set-covering Extended Formulation With An Equality Cardinasupporting

confidence: 69%

“…The p-Median problem is a core topic in customer allocation and distribution system design, and so, it has been tackled with a large variety of algorithms [25], including Column Generation (CG) methods [18,7,1,10,13]. Besides location and distribution applications, p-Median problems can also arise in rather unexpected fields such as cluster analysis (see references in the introduction of [1]), processor scheduling or packet management in networks [14].…”

Section: Problem Introduction and A Basic Boundmentioning

confidence: 99%

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Using dual feasible functions to construct fast lower bounds for routing and location problems

Porumbel

Gonçalves

2015

Discrete Applied Mathematics

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In cutting and packing problems, Dual Feasible Functions (DFFs) represent a well established tool for deriving high quality lower bounds in very short times. A well-known DFF lower bounding approach consists of using DFFs to rapidly generate feasible values for the dual variables of a Column Generation (CG) program (i.e., of an extended formulation commonly associated to CG). This paper presents a method for extending this approach to problems such as Capacitated p-Median, Distance Constrained General Routing and related variants (e.g., Capacitated Arc-Routing). In contrast to classical DFF bounds for cutting and packing, our extended DFF bounds deal with non-equal column costs, i.e., the column costs (primal objective function coefficients) have a more complex structure depending on some sums of distances. The general idea is to use a (reformulated) classical CG model in which a feasible dual solution is expressed as a linear combination of both DFF and non-DFF terms. In fact, one of the proposed approaches (the mixed CG-DFF bound for Capacitated p-Median in Section 2.2) still requires optimizing a restricted CG program, but with fewer variables and easier pricing sub-problems. The most refined bound version is the "DFF warm-started CG" from Section 2.2.2: it takes dual constraints generated while solving the above restricted CG program are re-uses them to warm-start a full CG phase. In the best cases, this can yield a speed-up between 2 and 3 relative to the pure CG. We present numerical experiments on Capacitated p-Median, Capacitated Arc-Routing (with fixed costs) and Distance Constrained Arc Routing; the numerical comparisons with the CG bound concern both the quality and the running time.

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Section: A Set-covering Extended Formulation With An Equality Cardinasupporting

confidence: 69%

Section: Problem Introduction and A Basic Boundmentioning

confidence: 99%

Using dual feasible functions to construct fast lower bounds for routing and location problems

Porumbel

Gonçalves

2015

Discrete Applied Mathematics

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“…Very recently, two heuristic algorithms for the CPMP have been presented: Lorena and Senne [28] followed a column generation approach, finding good solutions on real instances with up to 402 vertices, while Diaz and Fernández [13] attacked an instance with 737 vertices through hybrid scatter search and path relinking.…”

Section: Cardinality Constraintsmentioning

confidence: 99%

“…Both Baldacci et al [3] and Lorena and Senne [28] replaced the location variables y j with the assignment variables x jj , so that each median is forced to be coincident with one of the vertices of its cluster. To be consistent with the terminology explained above, we indicate this problem as the Capacitated P-Concentrator Location Problem (CPCLP).…”

Section: Cardinality Constraintsmentioning

confidence: 99%

A computational evaluation of a general branch-and-price framework for capacitated network location problems

2008

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The purpose of this paper is to illustrate a general framework for network location problems, based on column generation and branch-andprice. In particular we consider capacitated network location problems with single-source constraints. We consider several different network location models, by combining cardinality constraints, fixed set-up costs, concentrator restrictions and regional constraints. Our general branchand-price-based approach can be seen as a natural counterpart of the branch-and-cut-based commercial ILP solvers, with the advantage of exploiting the tightness of the lower bound provided by the set partitioning reformulation of network location problems. Branch-and-price and branch-and-cut are compared through an extensive set of experimental tests.

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“…where X ⊆ L and |X| = p. The capacitated version (see [6,19]) includes a fixed capacity for the facility center located at each point of L that bounds the amount of demand served by it, but usually the problem is uncapacitated and each customer is supplied from its closest facility. Beside this combinatorial formulation, the p-median problem has a formulation in terms of integer programming with matrix D and vector w as data.…”

Section: Algorithm Cnvnsmentioning

confidence: 99%

Parallel Variable Neighborhood Search

Moreno-Pérez

Hansen

Mladenović

2005

Parallel Metaheuristics

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Les textes publiés dans la série des rapports de recherche HEC n'engagent que la responsabilité de leurs auteurs. La publication de ces rapports de recherche bénéficie d'une subvention du Fonds québécois de la recherche sur la nature et les technologies. AbstractVariable Neighborhood Search (VNS) is a recent and effective metaheuristic for solving combinatorial and global optimization problems. It is capable of escaping from the local optima by systematic changes of the neighborhood structures within the search. In this paper several parallelization strategies for VNS have been proposed and compared on the large instances of the p-median problem. RésuméLa Rechercheà Voisinage Variable (RVV) est une métaheuristique récente et efficace pour la résolution de problèmes d'optimisation combinatoire ou globale. Elle permet de quitter un optimum local par des changements systématiques de structures de voisinageà l'intérieur de la recherche. Pour cet article plusieurs stratégies de parallélisation de RVV sont proposées et composées sur de grandes instances de problème de la p-médiane.

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A column generation approach to capacitated p-median problems

Cited by 135 publications

References 25 publications

Using dual feasible functions to construct fast lower bounds for routing and location problems

Using dual feasible functions to construct fast lower bounds for routing and location problems

A computational evaluation of a general branch-and-price framework for capacitated network location problems

Parallel Variable Neighborhood Search

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