Optimal capacitated ring trees

Hill, Alessandro; Voß, Stefan

doi:10.1007/s13675-015-0035-x

Cited by 19 publications

(33 citation statements)

References 29 publications

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“…Arguably the most intuitive formulation to ensure circuits that only have one depot (i.e., routes starting from and ending at the same depot) is based on binary variables

z_{i j}^{d}

which have value 1 if arc

(i, j) \in A

is used in the circuit of depot d , and 0 otherwise. Four such similar formulations are given by Albareda‐Sambola et al , Bektaş , Fernández and Rodríguez‐Pereira and by Hill and Voß for variants of multi‐depot problems. Clearly, an arc with an endpoint in a depot

d \in D

cannot be used in the circuit of depot

d' \in D ∖ {d}

or else we could have a path between depots d and

d'

, hence we can define variables

z_{i j}^{d}

for any

d \in D

only for arcs

(i, j) \in A^{c} \cup A^{d}

.…”

Section: Routing Problems With Multiple Depotsmentioning

confidence: 96%

“…To see this, consider by contradiction an inequality (22), assume that d ∈ D and suppose also that there exists i ∈ C such that z d di < x di . Then, we obtain y d = i∈C z d di < i∈C x di = y d , where the first equality is given by constraints (14) and the second equality is given by constraints (1). Thus, constraints (22) and (25) are equivalent.…”

Section: Proposition 3 P Xy ((S3i) L ) Is Contained In the Polyhedrmentioning

confidence: 99%

“…In some computational experiments made, the LP bounds obtained were the same with either set of inequalities. The point of using the equality version of these constraints is that we can eliminate the x variables from the model, and by redefining the

z_{i j}^{d}

variables as 0–1 variables, we obtain a valid formulation for the problem involving only the

z_{i j}^{d}

(and

y_{d}

variables) as in the four previous works by Albareda‐Sambola et al , Bektaş , Fernández and Rodríguez‐Pereira , and Hill and Voß .…”

Section: Routing Problems With Multiple Depotsmentioning

confidence: 99%

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New path elimination constraints for multi‐depot routing problems

2017

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Multi‐depot routing problems arise in distribution logistics where a set of vehicles based at several depots are used to serve a number of clients. Most variants of this problem have the basic requirement that the route of each vehicle starts and ends at the same depot. This article describes new inequalities, namely multi‐cut constraints (MCC), which enforce this requirement in mathematical programming formulations of multi‐depot routing problems. The MCCs are exponential in size, and are equivalent to a compact three‐index formulation for the problem in terms of the associated linear programming relaxations. The article describes how a generalization of the MCCs can be obtained, in a similar manner, by using a stronger version of the three‐index formulation. The connection between the compact and the exponential formulations implies a separation procedure based on max‐flow/min‐cut computations, which has reduced complexity in comparison with a previously known set of constraints described for the same purpose. The new inequalities are used in a branch‐and‐cut algorithm. Computational results are presented for instances with up to 300 clients and 60 depots. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(3), 246–261 2017

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“…Arguably the most intuitive formulation to ensure circuits that only have one depot (i.e., routes starting from and ending at the same depot) is based on binary variables

z_{i j}^{d}

which have value 1 if arc

(i, j) \in A

d \in D

cannot be used in the circuit of depot

d' \in D ∖ {d}

or else we could have a path between depots d and

d'

, hence we can define variables

z_{i j}^{d}

for any

d \in D

only for arcs

(i, j) \in A^{c} \cup A^{d}

.…”

Section: Routing Problems With Multiple Depotsmentioning

confidence: 96%

Section: Proposition 3 P Xy ((S3i) L ) Is Contained In the Polyhedrmentioning

confidence: 99%

z_{i j}^{d}

variables as 0–1 variables, we obtain a valid formulation for the problem involving only the

z_{i j}^{d}

(and

y_{d}

variables) as in the four previous works by Albareda‐Sambola et al , Bektaş , Fernández and Rodríguez‐Pereira , and Hill and Voß .…”

Section: Routing Problems With Multiple Depotsmentioning

confidence: 99%

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New path elimination constraints for multi‐depot routing problems

2017

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“…Capacities are introduced in the capacitated m‐ring star problem (CmRSP), where the number of customers visited by or assigned to a ring is limited as well as the number of possible rings served by the depot. The CmRSP is closely related to the capacitated ring tree problem and is a special case of the recent ring tree facility location problem . In our work, we consider the multi‐depot ring star problem (MDRSP) which generalizes the CmRSP by allowing multiple depots.…”

Section: Introductionmentioning

confidence: 99%

An equi‐model matheuristic for the multi‐depot ring star problem

Hill

Voβ

2016

Networks

Self Cite

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In the multi-depot ring star problem (MDRSP), a set of customers has to be connected to a set of given depots by ring stars. Such a ring star is a cycle graph, also called a ring, with some additional nodes assigned to its nodes by single star edges. Optional Steiner nodes can be used in the network as intermediate nodes on the rings. Depot dependent capacity limits apply to both, the number of customers in each ring star and the number of ring stars connected to a depot. The MDRSP asks for a network such that the sum of the edge costs is minimized. In this article, we present a matheuristic that iteratively refines a solution network in a locally exact fashion. In contrast to existing approaches, we define an equi-model matheuristic. That is a refinement method in which the subproblems are modeled as smaller instances of the global problem. Hence the optimization model that is used to explore the various structural multi-exchange neighborhoods in our algorithm is the MDRSP itself. A first class of neighborhoods considers local subnetworks for optimal improvements. Through an advanced modeling technique, we are able to refine arbitrary subnetworks of suitable size induced by simple node sets. A second class aims at globally restructuring the current network after the application of different contraction techniques. For both purposes, we develop an exact branch & cut algorithm for the MDRSP that efficiently solves the local optimization problems to optimality, if they are chosen reasonably in terms of size and complexity. The efficiency of the approach is shown by computational results improving known upper bounds for instance classes from the literature containing up to 1000 nodes. Ninety-one percent of the known best objective values are improved up to 13% in competitive computational time.

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“…Karasan et al [15] propose a branch-and-cut algorithm for the problem where the backbone network is 2EC and each user node is connected directly to two distinct hub nodes. More recently, Hill and Voß [14] introduce the capacitated ring tree problem where nodes are connected with rings and trees, and rings intersect at a distributor node.…”

Section: Introductionmentioning

confidence: 99%

A branch-and-cut algorithm for two-level survivable network design problems

Rodríguez-Martín

Salazar-González

Yaman

2016

Computers & Operations Research

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This paper approaches the problem of designing a two-level network protected against single-edge failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs, the connection of the hubs through a backbone network (first network level), and the assignment of terminals to hubs and their connection through access networks (second network level). We consider two survivable structures in both network levels. One structure is a two-edge connected network, and the other structure is a ring. There is a limit on the number of nodes in each access network, and there are fixed costs associated with the hubs and the access and backbone links. The aim of the problem is to minimize the total cost. We give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results. Some of the new inequalities can be used also to solve other problems in the literature, like the plant cycle location problem and the hub location routing problem. © 2015 Elsevier Ltd. All rights reserved

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Optimal capacitated ring trees

Cited by 19 publications

References 29 publications

New path elimination constraints for multi‐depot routing problems

New path elimination constraints for multi‐depot routing problems

An equi‐model matheuristic for the multi‐depot ring star problem

A branch-and-cut algorithm for two-level survivable network design problems

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