A Cut and Branch Approach for the Capacitated p-Median Problem Based on Fenchel Cutting Planes

Boccia, Maurizio; Sforza, Antonio; Sterle, Claudio; Vasilyev, Igor

doi:10.1007/s10852-007-9074-5

Cited by 44 publications

(40 citation statements)

References 16 publications

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“…But it has been demonstrated that the approach based on an exact knapsack separation algorithm in the cutting plane methods outperforms the methods using the column generation on the generalized assignment problem and capacitated p-median problem [4][5][6]. In this paper we propose a new implementation of this approach, where we try to make it more efficient.…”

Section: Which Define Knapsack Polytopes It Is Trivial That P(a B) mentioning

confidence: 99%

“…It has been done by solving a special integer programming problem in [4,5]. Before solving the integer problem, Kaparis and Letchford [13] have suggested the following heuristic rule, which works in over 90 % of cases.…”

Section: Numerical Errorsmentioning

confidence: 99%

“…Previously in [4,5] this problem was solved by MINKNAP algorithm. We suggest solving it by the dynamic programming algorithm, which will be discussed in the next subsection, since it is closely linked with our lifting procedure.…”

Section: Numerical Errorsmentioning

confidence: 99%

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An implementation of exact knapsack separation

2015

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Cutting planes have been used with great success for solving mixed integer programs. In recent decades, many contributions have led to successive improvements in branch-and-cut methods which incorporate cutting planes in branch and bound algorithm. Using advances that have taken place over the years on 0-1 knapsack problem, we investigate an efficient approach for 0-1 programs with knapsack constraints as local structure. Our approach is based on an efficient implementation of knapsack separation problem which consists of the four phases: preprocessing, row generation, controlling numerical errors and sequential lifting. This approach can be used independently to improve formulations with cutting planes generated or incorporated in branch and cut to solve a problem. We show that this approach allows us to efficiently solve large-scale instances of generalized assignment problem, multilevel generalized assignment problem, capacitated p-median problem and capacitated network location problem to optimality.Keywords Knapsack problem · Cutting plane · Exact separation · Generalized assignment problem · Multilevel generalized assignment problem · Capacitated p-median problem · Capacitated network location problem B Igor Vasilyev

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Section: Which Define Knapsack Polytopes It Is Trivial That P(a B) mentioning

confidence: 99%

Section: Numerical Errorsmentioning

confidence: 99%

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An implementation of exact knapsack separation

2015

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“…The CPMP problem is NP-complete according to the proof of [11], so its generalization BMCLP is too. Several other variants of CPMP are treated by [18], [3], [1] and [7]. Dynamic location problems are solved by [2], [8] and [6].…”

Section: Introductionmentioning

confidence: 99%

“…Network problems are appeared in [16], [9], [13] and [19]. For the resolution of the problem and its variants, several exact and approximate approaches are tested: the reference [3] applies a cutting plane algorithm based on the Fenchel cuts, references [18], [5] and [4] have chosen to www.ijacsa.thesai.org use the Branch & Price and branch & Bound methods based on Lagrangian relaxation, a resolution with column generation is applied by [4] and references [6], [15] and [12] used different approaches and techniques.…”

Section: Introductionmentioning

confidence: 99%

Multi-Stage Algorithms for Solving a Generalized Capacitated P-median Location Problem

Amrani¹,

Benadada²

2018

ijacsa

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Abstract-The capacitated p-median location problem is one of the famous problems widely discussed in the literature, but its generalization to a multi-capacity case has not. This generalization, called multi-capacitated location problem, is characterized by allowing facilities to use one of several capacity levels. For this purpose, a predefined list of capacity levels supported by all potential facilities is established. In this paper, we will detail the mathematical formulation and propose a new solving method. We try to construct, indeed, a multi-stage heuristic algorithm that will be called BDF (Biggest Demand First). This new method appears in two approaches: Integrated BDF (IBDF) and Hybridized BDF (HBDF) will be improved by using a local search optimization. A valid lower bound to the optimal solution value is obtained by solving a lagrangian relaxation dual of the exact formulation. Computational results are presented at the end using new instances with higher ratio between the number of customers, facilities and capacity levels or adapted from those of p-median drawn from the literature. The obtained results show that the IBDF is much faster with medium quality solution while HBDF is slower but provides very good solutions close to the optimality.

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Semi-supervised K-Means Clustering via DC Programming Approach

Gruzdeva

Ushakov

2023

Lecture Notes in Computer Science

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A Cut and Branch Approach for the Capacitated p-Median Problem Based on Fenchel Cutting Planes

Cited by 44 publications

References 16 publications

An implementation of exact knapsack separation

An implementation of exact knapsack separation

Multi-Stage Algorithms for Solving a Generalized Capacitated P-median Location Problem

Semi-supervised K-Means Clustering via DC Programming Approach

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