The geometry of optimal partitions in location problems

Mallozzi, Lina; Puerto, Justo

doi:10.1007/s11590-017-1156-3

Cited by 6 publications

(20 citation statements)

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“…Consider problem LL(Q) for a given suitable solution Q ∈ Γ . We point out that for dimensional facilities, we can not directly apply the optimal transport theory as done in [5,16,17], because the characterization of the optimal partition holds when the measure ν has a discrete support. However, we can prove the existence of solution for problem LL(Q) by identifying each dimensional facility with its root point, giving to the measure a discrete support, as the proof of the following theorem shows.…”

Section: Resolution Via Optimal Transport Massmentioning

confidence: 99%

“…By Theorem 2.1 in [17] there exists a solution for problem (5) and it is equivalent to its corresponding Kantorovich relaxed Monge's formulation:…”

Section: Resolution Via Optimal Transport Massmentioning

confidence: 99%

“…We have included several test examples. Some of them were already proposed in [17] and some others are new.…”

Section: End Proceduresmentioning

confidence: 99%

“…Example 1 Our first test illustrates how the approach in this paper applies to one example borrowed from the literature [17]. First, we consider that the demand region Ω, the dimensional facilities P 1 , P 2 , P 3 and all the elements of the lower level problem are the ones given in Example 4.1 in [17].…”

Section: End Proceduresmentioning

confidence: 99%

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On Location-Allocation Problems for Dimensional Facilities

Mallozzi

Puerto

Rodríguez-Madrena

2019

J Optim Theory Appl

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This paper deals with a bilevel approach of the location-allocation problem with dimensional facilities.We present a general model that allows us to consider very general shapes of domains for the dimensional facilities and we prove the existence of optimal solutions under mild, natural assumptions. To achieve these results we borrow tools from optimal transport mass theory that allow us to give explicit solution structure of the considered lower level problem. We also provide a discretization approach that can approximate, up to any degree of accuracy, the optimal solution of the original problem. This discrete approximation can be optimally solved via a mixedinteger linear program. To address very large instance sizes we also provide a GRASP heuristic that performs rather well according to our experimental results. The paper also reports some experiments run on test data.

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Section: Resolution Via Optimal Transport Massmentioning

confidence: 99%

“…By Theorem 2.1 in [17] there exists a solution for problem (5) and it is equivalent to its corresponding Kantorovich relaxed Monge's formulation:…”

Section: Resolution Via Optimal Transport Massmentioning

confidence: 99%

“…We have included several test examples. Some of them were already proposed in [17] and some others are new.…”

Section: End Proceduresmentioning

confidence: 99%

Section: End Proceduresmentioning

confidence: 99%

See 2 more Smart Citations

On Location-Allocation Problems for Dimensional Facilities

Mallozzi

Puerto

Rodríguez-Madrena

2019

J Optim Theory Appl

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show abstract

“…The multipoint or k-order Voronoi diagrams discussed in this paper are one possible way to generalize the classic construction. Some notable generalizations are the cells of more general sets [2,3], the use of non-Euclidean metrics [4,5,6] and the abstract cells that are defined via manifold partitions of the space rather than distance relations [7].…”

Section: Introductionmentioning

confidence: 99%