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The Directed Network Design Problem with Relays Odysseus 2018 Markus - - PowerPoint PPT Presentation

The Directed Network Design Problem with Relays Odysseus 2018 Markus Leitner 1 c 2 Martin Riedler 3 Mario Ruthmair 1 Ivana Ljubi 1 ISOR, University of Vienna, Vienna, Austria 2 ESSEC Business School of Paris, France 3 TU Wien, Vienna, Austria


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The Directed Network Design Problem with Relays

Odysseus 2018 Markus Leitner 1 Ivana Ljubi´ c 2 Martin Riedler 3 Mario Ruthmair 1

1ISOR, University of Vienna, Vienna, Austria 2ESSEC Business School of Paris, France 3TU Wien, Vienna, Austria

June 8, 2018

Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 1 / 36

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SLIDE 2

Network Design with Relays

Models network design problems in transportation and telecommunication. Freight transportation networks: for long haul distance trips, relay points are set along the paths for the exchange of drivers, trucks and trailers. Telecommunication networks: optical signal deteriorates after traversing a certain distance, and has to be re-amplified, i.e., regenerator devices need to be installed. E-mobility networks: batteries of EVs need to be recharged after a certain distance, hence charging stations need to be placed in the network.

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Tesla Supercharger Network (≈1200 stations)

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Network Design with Relays

1 Network Design: Build the network or augment the existing one. 2 Location: Where to place relays, and how many? 3 Routing: How to route each commodity from its source to

destination?

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SLIDE 5

PROBLEM DEFINITION

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Directed Network Design with Relays

Given: directed graph G = (V, A) relay placement costs c: V → Z>0 arc costs w: A → Z≥0 and arc lengths d: A → Z≥0 set K of O-D pairs (commodities) distance limit λmax ∈ Z>0

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Directed Network Design with Relays

Given: directed graph G = (V, A) relay placement costs c: V → Z>0 arc costs w: A → Z≥0 and arc lengths d: A → Z≥0 set K of O-D pairs (commodities) distance limit λmax ∈ Z>0 Goal: install a subset of relays and arcs of minimum cost s.t. there exists a feasible simple path for each O-D pair from K. an O-D path P is feasible if each subpath of P which is longer than λmax contains a relay

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SLIDE 8

Example — Symmetric Instance

λmax = 5, K = {(A, B)} A

5 1

B

5 5 3 1 1 1 1 3 1 Instance Acyclic Solution (cost=5)

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Example — Symmetric Instance

λmax = 5, K = {(A, B)} A

5 1

B

5 5 3 1 1 1 1 3 1 Instance Cyclic Solution (cost=1)

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Previous Work

Undirected NDPR: Cabral et al. (2007): Set-covering formulation (each column is an O-D path, including relays) Heuristics: VNS, Xiao and Konak (2017), tabu search, Lin et al. (2014), GAs, Kulturel-Konak and Konak (2008); Konak (2012) Exact algorithms based on B&P&C (columns are segments between the relays):

◮ Yıldız et al. (2018) ◮ Leitner et al. (2018) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 8 / 36

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SLIDE 11

Previous Work

Undirected NDPR: Cabral et al. (2007): Set-covering formulation (each column is an O-D path, including relays) Heuristics: VNS, Xiao and Konak (2017), tabu search, Lin et al. (2014), GAs, Kulturel-Konak and Konak (2008); Konak (2012) Exact algorithms based on B&P&C (columns are segments between the relays):

◮ Yıldız et al. (2018) ◮ Leitner et al. (2018)

Directed NDPR: Introduced in Li et al. (2012), exact, 2 models:

◮ compact Node-Arc model ◮ Set-Covering model (similar to Cabral et al. (2007)) ⇒ B&P

Heuristic: Li et al. (2017)

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Our contribution:

Directed NDPR: New models based on layered graphs (distance-expanded graphs):

◮ multi-commodity flows ◮ cut-sets

Branch-and-Cut (B&C) algorithms for both models Both B&C significantly outperform the previous state-of-the-art from Li et al. (2012)

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A BASIC FORMULATION

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Node Arc Formulation from Li et al. (2012)

bk

i =

     1 if k = (i, v) −1 if k = (u, i)

  • therwise

(u, v) ∈ K vk

i = distance of node i from the preceeding relay for commodity k.

yi =

  • 1

if relay is installed at node i

  • therwise

i ∈ V xa =

  • 1

if arc a is installed

  • therwise

a ∈ A

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SLIDE 15

Node Arc Formulation from Li et al. (2012)

(NA) min

  • i∈V

ciyi +

  • a∈A

waxa

  • a∈δ+(i)

f k

a −

  • a∈δ−(i)

f k

a = bk i

∀k ∈ K, ∀i ∈ V (1) vk

i + d(i,j)f k (i,j) − λmax(1 − f k (i,j) + yj) ≤ vk j

∀k ∈ K, ∀(i, j) ∈ A (2) vk

i + d(i,j)f k (i,j) ≤ λmax

∀k ∈ K, ∀(i, j) ∈ A (3) f k

a ≤ xa

∀k ∈ K, ∀a ∈ A (4) 0 ≤ vk

i ≤ λmax(1 − yi)

∀k ∈ K, ∀i ∈ V (5) vu,v

u

= 0 ∀(u, v) ∈ K (6) f k

a ∈ {0, 1}

∀k ∈ K, ∀a ∈ A (7) yi ∈ {0, 1} ∀i ∈ V (8) 0 ≤ xa ≤ 1 ∀a ∈ A (9)

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SLIDE 16

MODELS ON LAYERED GRAPHS

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Solution Structure

Set S of commodity sources, set T u of targets of source u

Single-source case:

If S = {u}, there exists an optimal solution which is a Steiner arborescence rooted at u, with leaves from T u. Each O-D path in this tree must be made feasible by installing some relays (when needed).

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Solution Structure

Set S of commodity sources, set T u of targets of source u

Single-source case:

If S = {u}, there exists an optimal solution which is a Steiner arborescence rooted at u, with leaves from T u. Each O-D path in this tree must be made feasible by installing some relays (when needed).

Multiple sources:

An optimal solution is a union of Steiner arborescences rooted at u, with required placement of relays when needed. Steiner arborescence: rooted subtree connecting a given set of terminals.

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SLIDE 19

Example

How to integrate the fact that on some nodes of the Steiner tree relays have to be installed?

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Basic Idea

Create node copies according to feasible distances at which a node can be reached Embed Steiner trees into this network, for each source u

1 2 3 2 1 3

00 10 20 30 11 21 12 02 22 03 13 23 33 04 14 24 34 Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 16 / 36

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SLIDE 21

00 10 20 30 11 21 12 02 22 03 13 23 33 04 14 24 34

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Solution (Single Source)

Solution: Steiner tree rooted at 0, each target reached at some layer.

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Layered Graph Models

Model Name Connectivity Aggregation Type L-CUT cutsets per source B&C L-MCF multi-commodity flow none pseudo-compact B&C

L-CUT: zu

a ∈ {0, 1}

∀u ∈ S, ∀a ∈ Au

L

L-MCF: fuv

a

∈ {0, 1} ∀(u, v) ∈ K, ∀a ∈ Au

L

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Layered Cut Model

(L-CUT) min

  • i∈V

ciyi +

  • a∈A

waxa Ensure connectivity between the source u and a copy of v ∈ T u:

  • a∈δ−(W)

zu

a ≥ 1

∀u ∈ S, ∀v ∈ T u, {vl|vl ∈ V u

L } ⊆ W ⊂ V u L ,

u / ∈ W

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Layered Cut Model

(L-CUT) min

  • i∈V

ciyi +

  • a∈A

waxa Ensure connectivity between the source u and a copy of v ∈ T u:

  • a∈δ−(W)

zu

a ≥ 1

∀u ∈ S, ∀v ∈ T u, {vl|vl ∈ V u

L } ⊆ W ⊂ V u L ,

u / ∈ W Indegree of a node v over all layers is at most one for i ∈ T u, and exactly one for i ∈ T u.

  • il∈V u

L

  • a∈δ−(il),a/

∈Ar

L

zu

a ≤ 1

∀u ∈ S, ∀i / ∈ T u, i = u

  • il∈V u

L

  • a∈δ−(il),a/

∈Ar

L

zu

a = 1

∀u ∈ S, ∀i ∈ T u

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Layered Cut Model (cont.)

Vertical arcs linked to relays:

  • (il,i0)∈Au

L

zu

(il,i0) ≤ yi

∀u ∈ S, ∀i ∈ V Each (i, j) ∈ A can be used in at most one layer

  • (il,jm)∈Au

L

zu

(il,jm) ≤ x(i,j)

∀u ∈ S, ∀(i, j) ∈ A

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Layered MCF Model: No linking with zu

a needed

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Comparing the strength of the two models

Theorem

Formulations L-MCF and L-CUT are equally strong, i.e., the LP-relaxation values of the two models coincide.

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Comparing the strength of the two models

Theorem

Formulations L-MCF and L-CUT are equally strong, i.e., the LP-relaxation values of the two models coincide. Further strengthening is possible for L-CUT There are symmetries induced by the layered graph

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L-CUT: Strengthening Cuts

Flow-balance

In-degree ≤ out-degree for every non-target node in LG:

  • a∈δ−(il)

zu

a ≤

  • a∈δ+(il)

zu

a

∀u ∈ S, ∀il ∈ V u

L , i /

∈ T u ∪ {u}

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L-CUT: Strengthening Cuts

Flow-balance

In-degree ≤ out-degree for every non-target node in LG:

  • a∈δ−(il)

zu

a ≤

  • a∈δ+(il)

zu

a

∀u ∈ S, ∀il ∈ V u

L , i /

∈ T u ∪ {u}

Symmetry Breaking

The same optimal solution may have multiple embeddings in the LG (2 commodities share a subpath, and only one of them uses a relay). Force that in routing path, if relay is installed, it must be used:

  • (il,jm)∈Au

L:l>0∧m>0

zu

(il,jm) ≤ Mu i · (1 − yi)

∀u ∈ S, ∀i ∈ V, i = u Mu

i =

  • min(|T u|, |δ+(i)|)

i / ∈ T u min(|T u| − 1, |δ+(i)|) i ∈ T u

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L-MCF

Symmetry Breaking

  • (il,jm)∈Au

L:l>0∧m>0

fuv

(il,jm) ≤ 1 − yi

∀(u, v) ∈ K, ∀i ∈ V, i = u

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COMPUTATIONAL RESULTS

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Implementation Details

preprocessing

◮ remove “2-cycle arcs” for |δ+(i)| ≤ 1 or |δ−(i)| ≤ 1 vertices ◮ remove unreachable vertices including their outgoing arcs

initial heuristic based on Cabral et al.’s CH1

  • riginal graph cuts to improve convergence speed of the cut model
  • a∈δ−(W)

xa ≥ 1 ∀(u, v) ∈ K, W ⊂ V, u / ∈ W, v ∈ W (10) nested back cuts cost-based branching priorities

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Four Settings

NA: node-arc based model by Li et al. (2012) L-MCF L-CUT-d, dynamic (separation below) L-CUT-s, static (steps 2. and 3. skipped)

Separation

1 separate cut-set inequalities on the original graph 2 separate flow-balance constraints 3 separate two-cycle inequalities 4 if no flow-balance constraints and two-cycle inequalities added,

separate cut-sets on the LG

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Quality of Lower Bounds

Each line is average over 10 instances (from Cabral et al. (2007), EJOR)

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Quality of Lower Bounds

Instances from Konak (2012), EJOR

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Speedup ratio to the NA model

CPU time(NA) / CPU time(algorithm)

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L-MCF vs. L-CUT

Directed Konak instances, type II (inversely correlated distance)

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Conclusion

significantly beats state-of-the-art very strong LP bounds

  • ur algorithms find optimal solutions for instances with:

◮ 160 vertices and more than 7000 arcs

Future Work Network Design with Relays Under Uncertainty (robust or stochastic models?) Applications:

◮ Telecommunications: Quantum-Key-Distribution (QKD), placing of

encryption keys along the network, so as to make sure each O-D path is encrypted according to the Quantum Computing technology.

◮ E-mobility: maximum number of recharging stops, distance limits for

the trips?

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SLIDE 41

Literature I

  • E. A. Cabral, E. Erkut, G. Laporte, and R. A. Patterson. The network

design problem with relays. European Journal of Operational Research, 180(2):834–844, 2007.

  • A. Konak. Network design problem with relays: A genetic algorithm with a

path-based crossover and a set covering formulation. European Journal

  • f Operational Research, 218(3):829–837, 2012.
  • S. Kulturel-Konak and A. Konak. A local search hybrid genetic algorithm

approach to the network design problem with relay stations. In

  • S. Raghavan, B. Golden, and E. Wasil, editors, Telecommunications

Modeling, Policy, and Technology, volume 44 of Operations Research/Computer Science Interfaces, pages 311–324. Springer US, 2008.

  • M. Leitner, I. Ljubi´

c, M. Ruthmair, and M. Riedler. Exact approaches for network design problems with relays. INFORMS Journal on Computing, To appear, 2018.

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Literature II

  • X. Li, Y. P. Aneja, and J. Huo. Using branch-and-price approach to solve

the directed network design problem with relays. Omega, 40(5): 672–679, 2012.

  • X. Li, S. Lin, S. Chen, Y. Aneja, P. Tian, and Y. Cui. An iterated

metaheuristic for the directed network design problem with relays. Computers & Industrial Engineering, 113:35 – 45, 2017. ISSN 0360-8352.

  • S. Lin, X. Li, K. Wei, and C. Yue. A tabu search based metaheuristic for

the network design problem with relays. In Service Systems and Service Management (ICSSSM), 2014 11th International Conference on, pages 1–6, 2014.

  • Y. Xiao and A. Konak. A variable neighborhood search for the network

design problem with relays. Journal of Heuristics, 23(2-3):137–164, 2017.

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Literature III

  • B. Yıldız, O. E. Kara¸

san, and H. Yaman. Branch-and-price approaches for the network design problem with relays. Computers & Operations Research, 92:155–169, 2018.

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