the general routing problem: mathematical formulations, exact … · 2008. 11. 4. · the general...

The General Routing Problem: Mathematical Formulations, Exact Methods, Related

Metaheuristics and Perspectives

Marcos Negreiros, Gilbert LaporteMarcos Negreiros, Gilbert Laportenegreiro@graphvs.com.br gilbert@crt.umontreal.ca

State University of CearState University of Cearáá

Lecture Organization

Introduction to GRPPART I - GRP Symmetric VersionPART II – GRP Asymmetric VersionPART III – System XNÊSGRP Perspectives...

Lecture Organization

Introduction to GRPPART I - GRP Symmetric VersionPART II – GRP Asymmetric VersionPART III – System XNÊSGRP Perspectives...

Introduction to GRP

1. Historic Perspective2. Definition3. Relation to Other Combinatorial Problems4. Reported Applications5. GRP Versions (Solving)

Introduction to GRP

1. Historic Perspectives

1. Introduced by Orloff in 1974

2. Origins in a mixture of a GRAPH Optimization problem in Arcs and Nodes

3. Reported to be a very difficult problem NP-Hard

1. Historic Perspectives1. Introduced by Orloff in 1974

2. A mixture of a GRAPH Optimization problem simultaneously in Arcs and Nodes

3. Reported to be a very difficult problem NP-Hard

4. Orloff (1974/76) = Propose a way to solve it only by unique transformation

1. GRP to Arc Routing or 2. GRP to Node Routing.

5. Lenstra & Kan (1976) = Complexity and prove of NP-Hardness…

Introduction to GRP

2. Definition

• Let G(V,L) be a strongly connected positive weighted graph (cij≥0 | ∀ij∈L), where L={A,E}, is the set of links composed by a set of arcs and a set of edges.

• Let, AR⊆A, ER⊆E the sets of required arcs and edges respectively, or the set of required links LR ={ER, AR}, and VR⊆V, is the sub-set of vertices formed by the required vertices.

• Let LR ≠∅ and VR ≠∅, and v0 is a departure vertex.

• “The GRP wishes to minimize the tour that traverse all the required elements of the sets (LR ,VR) from the departure vertex v0”

1. Definition

Introduction to GRP

3. Relation to Other CP

Generalized TSP

Graphical TSP

Introduction to GRP

4. Reported Applications

1. Garbage Collection

2. Snow Removal

3. Climbing Robots

4. Reported Applications

4. Selective Garbage Collection

5. Cutting

6. Other: Mail delivery, Propane Gas Distribution, Railway application for testing tracks for faults over the course of a particular year,

Inspection of Lines (Utility wire and voltage transformers, water and oil pipes, etc.)

Introduction to GRP

5. GRP Versions (Solving)

We will present different ways to solve the GRP, as most recently reported by the literature.

Accordingly Orloff…

…OR…

By the recent literature...

SymmetricGRP

Asymmetric GRP

GTSP/TSP

Why solving like this?“Because it is necessary to profit the best in the polynomial part of solvable cases of the Arc Routing Problems”

Arc Routing – CPP, Some Classes Non Oriented and Oriented are Polynomial

Node Routing – TSP, All are NP-Hard

The General Routing Problem: Mathematical Formulations, Exact Methods, Related Metaheuristics and Perspectives

PART I – Symmetric GRP

State University of CearState University of Cearáá

Agenda

1. Definition2. Transformation to RPP3. Literature Overview4. Rural Postman Problem5. UGRP/RPP Exact Methods6. UGRP/RPP Heuristic Methods

1. Definition (UGRP)

• Let G(V,E) be a connected positive weighted graph (cij≥0 | ∀ij∈L), where E is the set of edges.

• Let, ER⊆E the sets of required edges, or the set of required links, and VR⊆V, is the sub-set of vertices formed by the required vertices.

• Let ER ≠∅ and VR ≠∅, and v0 is a departure vertex.

• “The GRP wishes to minimize the tour that traverse all the required elements of the sets (ER ,VR) from the departure vertex v0”

1. Definition

Versions of the GRP- Symmetric

- Pure RPP

- Undirected GRP

Agenda

2. Transformation UGRP to URPP• Twin Vertices (Golden & Wong 1981)

Required vertices are expanded to required edges with cost zero,one extremity stay connected with the same adjacent vertices andthe other has just its twin as adjacent.

2. Transformation UGRP to URPP• Twin and Adjacent (Fernández et al 2003)

Required vertices are expanded to required edges with cost zero,where both extremities become connected with the same adjacent vertices from the original vertice.

Agenda

3. Literature Overview

- Symmetric Major Advances- Exact

- Corberán and Sanchis (1994) - RPP- Ghiani and Laporte (2000) - RPP- Corberán, Letchford and Sanchis (2001) - GRP

- Hybrid Exact- Fernández, Garfinkel, Meza and Ortega (2003) - RPP/GRP

2. Lliterature Overview- Symmetric

- Heuristic- Frederickson, Hecht and Kim (1978) > Stacker Crane- Christofides, Campos, Corberán and Mota (1981)

>> Frederickson Heuristic- Pearn and Wu (1995) >> Lagrangean Frederickson Heuristic- Ghiani, Langaná and Musmanno (2006) >> Insertion Heuristic

- Improvement- Hertz, Laporte and Hugo (1999) >> 2Opt- Groves and Vuuren (2005) >> LN, 2Opt/3Opt- Muyldermans, Beullens, Cattrisse, Oudheusden (2005)

>>GLS, Flip, Reverse, Dir-Opt / 2Opt and 3Opt

Agenda

Rural Postman Problem (URPP)

1. Basics

2. Basic Formulation

3. Extended Properties3.1. Corberán and Sanchis (1994-1996)3.2. Ghiani and Laporte (2000)3.3. Corberán, Letchford and Sanchis (2001)3.4. Fernández, Meza, Garfinkel and Ortega (2003)

1. Basics

1. Basics• UCPP

– Eulerian Graph (CPP) ⇒ Eulerian Subgraphs (RPP)

If G is an undirected, connected graph, G is eulerian if and only if all its vertices are of even degree – Euler (1736).

Leonhard Euler

Rural Postman Problem (URPP)• Traversing an Eulerian Graph (Unicursal), Hierholzer (1873), see in

Edmonds and Johnson (1973), End-Pairing Algorithm

• Euler tour Traversal Algorithm

Procedure c-Euler ) ´, ( Γ G - [Bukard & Derigs, 1980]// Generic procedure for tescribe the eulerian path.// Input: G´(V,L’) aumented multigraph from G, GR[vi] array with the vertex vi degree// Output: Γ, K[vi] number of visited edges of vi,01. Begin02. NoAtual:=NoInicial:=NoPartida;03. FimPercurso:=true;04. Гd:={NoAtual}; // Direct Path05. Гd;:=φ; // Inverse Path06. While not FimPercurso do07. begin08. // Cuircuits construction phase09. Repeat10. ProxNo:=L[NoAtual].Prox; // from the list of todes of NoAtual take the next node11. K[NoAtual]:= K[NoAtual] +1;12. Гd:= Гd � {ProxNo}; // keep iin the set the direct path13. Until NoAtual=NoInicial;14.15. // Circuit separation phase16. If | Гd | + |Гi | < |L’| then17. begin18. J:={K[vi] | |K[vi] < GR[vi], �vi � Гd}; // select a vertex not yet fully traversed19. If J≠φthen // in the direct path20. begin21. s:=posiition of J[1] in Гd; // take the position of the vertex in the direct path22. For i:=s+1 to | Гd | do Гi:= Гd[i]; // transfer the direct path to the inverse path23. |Гd|:=s; // set the size of the direct path 24. Continue;25. end26. J:={K[vi] | |K[vi] < GR[vi], �vi � Гi}; // select a veritex not yet traversed27. begin // in the inverse path28. p := posiition dofJ[|J|] in Гi; // take the vertex position in the inverse path29. For i:= | Гi | downto p do Гd:= Гi[i]; // tranfer the inverse path to the direct30. |Гi|:=p; // set the size of the direct path 31. end32. end33. else FimPercurso:=false;34. end;35. For i:= | Гi | downto 1 do Гd:= Гi[i]; // transfer the inverse path to the direct path36. Result:= Гd;37. End; // c-Euler

1. Basics

1. Basics (Graph Simplification Example)

1. Basics

2. Basic FormulationLet, E\ER – is the set of non required edges;Ck- is a k connected component of required edges of G, where (k=1,…,p);VR – set of vertices of vi such that edge (vi, vj) exists in ER;Vk ⊆ VR – the vertex set of Ck, where (k=1,..,p);S⊂V;δ(S) – set of edges of E with one extremity in S and one in V\S.

If S={v}, then we write δ(v) instead of δ(S);xe= xij is the number of additional (deadheading) copies of edge e=(vi,vj) that must be

added to G, to make its required part (GR) eulerian

2. Basic Formulation (Corberán and Sanchis, 1994)

(*) xe is the deadheading times to traverse an edge –The solution is never optimal if an edge is traversed more than twice.

URPP1 is a full dimensional polyhedron.

A number of important facets (induced constraints) of the polyhedron are “naturally” known. 1. Connectivity Inequalities - x(δ(S)) ≥ 2, �S�V, δR(S)=∅2. R-odd cut inequalities - x(δ(S)) ≥ 1, �S: |δR (S)| is odd

1. Basics

Rural Postman Problem (URPP)3. Extended Properties

3.1. Corberán and Sanchis (1994-1998) Similarity with the Graphical TSP

• GTSP is a full dimensional polyhedron

• TSP(G) polytope is a face of GTSP(G), or TSP(G) = GTSP(G) ∩ {xe∈ ℜ|E| : xe =|V|}

• For the GTSP(G) many classes of facets are known:– Connectivity– Path– Path-tree

3. Extended Properties3.2. Ghiani and Laporte (2000)

Facet 1: All but p-1 variables are 0-1

• In an optimal solution, only variables corresponding to shortestspanning tree over connected components need be 0–1–2.

• For any such variable xe , set

all variables of the problem are now 0–1

Facet 2: R-Even

Facet 2: R-Even (generalized by co-circuit inequalities)

3. Extended Properties3.2. Ghiani and Laporte (2000) - First self contained formulation using edge variables only

3. Extended Properties3.3. Corberán, Letchford and Sanchis (2001)

K-Component (K-C)Path BridgeHoneycombInclude these and all others in a cutting-plane framework…

3.3. Corberán and Sanchis (1994)

K-C Component – (K-Connected Component)

Definitions:

Consider a subgraph of G obtained by deleting all non required vertices of G.

1. We call a connected component of this subgraph an R-connected component.2. A set of vertices defining a R-connected component will be called a R-set. An

R-set with only one member will be an isolated vertex.

3.3. Corberán and Sanchis (1994)

K-Component – (K-C Inequalities)

A K-C is a partition {V0, V1, …, VK} of vertex sets of V, with K≥3, such that, V0, V1, …, VK and V0 ∪ VK are clusters of one or more components of GR,

|ER(V0:VR)| ≥ 2 and even, and E(Vi:Vi+1)≠∅, for i=0,…,K-1.

The K-C inequality can be written as:

)1(2)):((||)):(( )2(

},0{},{0

0 −≥=−+− ∑≠≤<≤

KVVExjiVVExK ji

KjiKji

Path Bridge

Definitions:

The Path Bridge Inequalities are generalizations of the K-C inequalities.It is defined by means of a path bridge (PB) configuration…Let, p≥1 and b≥0, positive integers such that p+b≥3 and odd. Let ni≥2, for i=1,…,p also be integers, a PB configuration is a partition of V into vertex sets A, Z and for i=1,…,p, j=1,…,ni with the following properties:i

Path Bridge

Where, (Handle sets)`

(Teeth sets)

1))(())((1

1−++≥+∑∑

pnnpTxHx j

ii δδ

piiii VVHH ∪∪∪= − L1

njjj VVT ∪∪= L1

Honeycomb

Definitions:

The Honeycomb Inequalities are also generalizations of the K-C inequalities, but in different direction and neither class contain the other.The Honeycomb configuration is a partition of V into sets Si , such that:

1. for all i, | δ(Si ) \ δR(Si ) | = ∅ and | δR(Si ) | is even or zero;2. there are at least two values i such that δR(Si ) = ∅;3. there are at least two values i such that δR(Si ) = ∅.together with a set of non-required edges crossing between the Si which form a tree spanning the Si .

A valid honeycomb inequality can be written as:• )1(2 −≥∑

αe is equal to the numberof edges traversed (if any) in the spanning tree to get from one end-vertex of e to the other, except for the edges with one end-vertex in Vi and the other in Vj , i = j, whenthe coefficient is 2 units less.

3. Extended Properties3.4. Fernández, Meza, Garfinkel and Ortega (2003)

Model conception is very different from the others, because considers, the 1-matching blossoons and flow variables.

Dominance Relations1. Ghiani and Laporte (2000)2. Garfinkel and Webb (1999)

3. Extended Properties3.4. Fernández, Meza, Garfinkel and Ortega (2003)Dominance Relations

1. Ghiani and Laporte (2000)Consider the multigraph Gc derived from GTr, by letting the vertices of Gc be the components of GR and the edges of Gc be those edges of ETr that connect components of GT.

Edges of Gc retain their distance values from GT. Let ET* ∗be the edges of any

minimum spanning tree on Gc.

Dominance Relation 1 can be written as:

**\,1,,2 TTrT

e EEexEex ∈≤∈≤

3.4. Fernández, Meza, Garfinkel and Ortega (2003)Dominance Relations

2. Garfinkel and Webb (1999)

2. Within any component of GR, any vertex pair i,j is connected by at most one edge of EP∗

3. Within any component of GR, edge i,j � EP∗ only if |δR(i ) |and |δR(j )| are odd.

4. |δR(P*(i))| � {0,2} if δR(i) is even.

5. |δR(P*(i))| = 1 if |δR(i)| is odd.

3.4. Fernández, Meza, Garfinkel and Ortega (2003)Dominance Relations

2. Garfinkel and Webb (1999)

6. If |δR(i)| is even and there exist j,k, where k � Cj and j≠ k, such that I,j � EP∗ and I,k �EP∗ , then |δR(j)| and |δR(k)| are both odd.

7. At most one R-even vertex in a given component Cj is P∗-incident to one or more edges in another given component Ck, where Cj≠Ck.

8. If {i,j} and {i,k} are in EP∗where k � Cj and Ci ≠ Ck, then {i,j} and {i,k} are the only edges in EP∗ connecting Ci and Cj.

3.4. Fernández, Meza, Garfinkel and Ortega (2003)

FMGO has EA binary variables, m(m−1) continuous variables, and m+VA+m(m−1) constraints.

Agenda

UGRP/RPP Exact Methods

1. Ghiani and Laporte (2000) – Pure Exact (RPP)

2. Corberán, Letchford and Sanchis (2001) – Framework of Cutting Planes (GRP)

3. Fernández, Meza, Garfinkel and Ortega (2003) – Hybrid “Exact” (RPP)

1. Ghiani and Laporte (2000)

1. Ghiani and Laporte (2000) – Instances and Results – RPP Random Planar

1. Ghiani and Laporte (2000) – Instances and Results – RPP Random Planar (Type 3)

1. Ghiani and Laporte (2000) – Evolution of the exact results for URPP

2. Corberán, Letchford and Sanchis (2001) – Computational framework of cutting planes to GRP/URPP

Step 1: Initial Relaxation (LP) for the basic model with specific starting inequalities already added (connectivity and upper bounds xe≤2

Step 2: Inequalities added in each iteration in this sequence: R-odd cut and connectivity, Exact connectivity separation, exact R-odd cut, K-C separation, Honeycomb separation, n-Path Bridge separation, K-C and Honeycomb with interactively merging of adjacent R-Components, and if all fail, do 2-PB.

Step 3: Constraint Management and Cut Pool (Memory management and controlling growth of the LP)

Step 4: Additional Phase (apply the general MST. For GRP if xe ≤2 ∀e∈T and xe≤1 ∀e∉T, it is the optimal tour. This UB dominates the others. Insert this new bound an apply again the separation inequalities. If after all tries the solution is still not integer, then invoke B&B (to avoid large LP, the constraints with slack of 0.01 are deleted from the LP).

2. Corberán, Letchford and Sanchis (2001) RESULTS

Christofides Instances…

Hertz et al Type 1

Hertz et al Type 2

Hertz et al Type 3

Hertz et al GRP

Albaida GRP (116V, 174E)

ALBA35

Madrigueras GRP (196V, 316E)

3. Fernández, Meza, Garfinkel and Ortega (2003)

Hybrid “Exact” Method

Phase 1: Lower Bounding (Cutting Plane)Step 1: Let LP be the LP relaxation of FMGO, with xij relaxed to 0≤ xij ≤1, {i,j}∈EA. Go to step 2;Step 2: Solve LP with optimal solution x*. Go to step 3Step 3: If x* is all integer go to Step 8. Otherwise go to step 4.Step 4: Add Matching and connectivity inequalities violated by x* to LP. Go to step 5.Step 5: If any inequalities were added in Step 4, go to Step 2. Otherwise, search for K-C inequalities

violated by x* . Go to Step 6.Step 6: If any violated K-C inequalities were found in Step 5, add them to LP and go to Step 2.

Otherwise, go to Step 7.Step 7: If the objective function value, z(x*) is not integer, then add cx≥ ⎡z(x*)⎤ to LP, where ⎡z(x*)⎤ to

LP, where ⎡.⎤ is “round up”. Let x* be the optimal solution of LP. Go to step 8.Step 8: If x* is integer, stop. EP given by x* solves the RPP. Otherwise, let Z”=z(x*) be a lower bound

and go to Phase 2.

Phase II – Upper bounding (3-Tree Heuristic)

Let Z*:=∞, where Z* is na upper bound on Z(RPP).For i:=1 to 3 do

Let ET be the tree ETi on GC, and GR∪T=(VR, ER∪T).Find a minimum distance perfect matching EM on the odd (R∪T)-degree vertices of V. (The resulting solution EM∪ET is feasible to the RPP)Apply Eulerian reduction to EM∪ET until no more modification is possible.Let EPi be the resulting solution (Euler Tour) and Z*=min{Z+,Z(EPi)}

EndforEnd.

The 3-Trees

ET1 . Ecand = ED and ce*=ce // Frederickson et al (1979)

ET2 . Ecand = {e | x*e≠0 } . ce

* = 1- x*e, with ties broken by ce. That is , if x*

e’ = x*e”

and ce’< ce”, then e’ is chosen before e” in the MST algorithm

ET2 . Ecand = {e | x*e≠0 } . ce

* = ce, with ties broken by x*e.

Eulerian Reduction (Try to delete edges from EP that do not improve its cost)

Step 1: If EP contains two copies of any edge, and if both can be removed from the tour without disconnecting it, do so.

Step 2: If EP contain edges {i,j} and {i,k}, such that {i,j} and {i,k} can be replaced by {i,k} without disconnecting the tour do so.

3. Fernández, Meza, Garfinkel and Ortega (2003)Instances

3. Fernández, Meza, Garfinkel and Ortega (2003)Results

Agenda

UGRP/RPP Heuristic Methods

1. Constructive2. Improvement3. Meta-Heuristic

1. Constructive

FredericksonInsertionGeneralized

1. Constructive – Frederickson

Step 1: Apply Graph Transformation

Step 2: Apply MST between components of GR and set the selected non-required edges of the optimal tree as required - ET

Step 3: Apply 1-matching for the odd-degree required vertices (formed only by required links), and set the selected edges in the optimal matching paths to EM

Step 4: The Euler Tour is made of ET ∪ EM

1. Frederickson - Step1

Frederickson Step 2

Frederickson Step 3

MST (G1, G2) – ET ={(1,4)}

1-Matching (1, 5) – EM = {(1,4),(4,5)}

Frederickson Final Solution

2. Constructive – Insertion – Ghiani, Laganà, Musmanno (2006)- Definitions

Polygonals – Chains between required vertices of different components. The chains contain just one vertex per component

p.ex.: (9-8-11), (4-3-6), (5-10)

Maximal Polygonal – If it is not included into a distinct polygonal or if it is not a sub-chain of a polygonal.

p.ex.: (9-8-11), (3-6-10-5)

2. Constructive – Insertion – Ghiani, Laganà, Musmanno (2006)- Definitions

Matching– Chains between required vertices (R-odd) of only two different components.

p.ex.: (2-6), (3-6), (6-10), (8-9)

2. Constructive – Insertion

Procedure Insertion(G, ER, C)Begin

Construct a partial solution including two componentsLet {ch, h∈L} be the set of non inserted componentsWhile L≠∅ dobegin

Feasibly insert a component ch , h∈L into the partial solution;Postoptimize the maximal polygonals;Postoptimize the matching;Perform the polygonal-matching joint optimization;L:=L-{h};

end;End; This procedure is O(p n3)

2. Constructive – Insertion – Ghiani, Laganà, Musmanno (2006)

Required Components of G

1. Includes a single component from h

h dCuuh

∈∈

min minarg,...,1

p.ex.: (C1, C2), (C3, C2), (C4, C1)or (C4, C2) or (C4,C3)

C2C1 2. Insertion of a new component

},,{min1 Ehtijijjtit VvPldddD

h∈∈−+=

tiitij

,v,vvijl

and chains theofinsertion with theassociated ][D

and by chain theofon substituti ][D

)( Matching-polygonal ofinsertion with associated ][D

)( polygonal aby link theofon substituti a is ][D

},,2{min2 Ehi

Lkkiij VvVvdD

h∈∈=

}, , ,,{min3 Ehktijijjtikijjkit VvvPlddddddD

h∈∈−+−+=

},, ,,{min4kt

Lkkiikit vvVvvVvddD

h≠∈∈+=

),,,{min minarg 4321* hhhhh DDDDC =

C2C1 2. Insertion of a new component

Steps 3-5 post-optimization calls

3. Polygonals postoptimization

Local search in which a neighbor is obtained by replacing each “internal”vertex vij with another vertex of the same connected component .

p.ex: (3-4) and (6-7) → (2-6-7)

4. Matching postoptimization

2-Opt (Hertz et al) is performed to improve the matching of current partial solution.

5. Joint postoptimization

Removing repeated chains in polygonals and substitution to a maximal.

3. Constructive – Generalized (Negreiros & Laporte)General TREEGeneralized TSP

3. Constructive – Generalized (Negreiros and Laporte)General TREE

Step 1: Identify the GR components – v0 may be taken as a component if and only if it is not part of any required component.

Step 2: Apply MST between components of GR and set the selected non-required edges of the optimal tree as required - ET

Step 3: Apply 1-matching for the odd-degree required vertices (required links) defined by (ER ∪ET), and set the selected edges in the optimal matching paths to EM

Step 4: The Euler Tour is made of ET ∪ EM

GTree - Step1

Components of GR

The rest of steps is the same as Frederickson…

But what is different from Frederickson? Are the results always be always the same?…

GTree – Step 2

Extreme vertices of the components

G1 G2G1 0 Min{sp14,sp34,sp15,sp35}=Min{3,3,5,5}=3

G2 Min{sp41,sp43,sp51,sp53}=Min{3,3,5,5}=3 0

General Tree in Red

GTree – Step 31-Matching for the odd-degree required vertices (formed of required links)

Final eulerian RPP multigraph

3. Constructive – Generalized (Negreiros and Laporte)Generalized TSP

Step 1: Identify the GR components – v0 may be taken as a component if and only if it is not part of any required component

Step 2: Apply GTSP (Zero cost in the required edges) between components of GR(including the departure vertex) and set the selected non-required edges of the optimal tree as required - ET

Step 3: Apply 1-matching for the odd-degree required vertices (required links) defined by (ER ∪ET), and set the selected edges in the optimal matching paths to EM

Step 4: The Euler Tour is made of EGTSP ∪ EM

3. Constructive – Generalized (Negreiros & Laporte)Generalized TSP

GTSP - Step1

Components of GR

3. Constructive – Generalized (Negreiros and Laporte)Generalized TSPGTSP – Step 2

GTSP preparation

GTSP tour and induced required edges

Mark as required the non required edges of

the shortest GTSP tour.

GTSP preparation

GTSP tour and induced required edges

1-matching preparation

1-matching solution

3. Constructive – Generalized (Negreiros and Laporte)Generalized TSPGTSP – Step 4 (Final Euler Graph)

Although using a GTSP – NP-Hard, it is very fast, and computationally efficient once it never “destroy” the support graph.

2. ImprovementLayer NetSlow Improvement 2OptFast Improvement 2Opt/3Opt

2. Improvement - Layer Net (L)Groves and Vuuren (2005)

… …(v2,v3)1

(v3,v2)1

<v5,v4>l

<v4,v5>l

(v2,v3)l+

(v3,v2)l+

(v1,v2)Lr

(v2,v1)Lr

Alg 1: Shortest path through the LN – O(V(L))

Alg 2: Complexity reduction method (low complexity to obtain variation between two different LN solutions)

2. Improvement - Layer Net - Groves and Vuuren (2005)

(v2,v3)1

(v3,v2)1

<v5,v4>l

<v4,v5>l

(v2,v3)l+

(v3,v2)l+

(v1,v2)Lr

(v2,v1)Lr

v0 …

Suppose the final euler tour defined by visiting a set of required edges in the following order: E=(v0, l1, l3, l7, … , l4, v0)

l1 l3 l7 l4…

2. Improvement - Layer Net - Groves and Vuuren (2005)

v0 …

Suppose a change in the previous euler tour order: E=(v0, l1, l4, l7, … , l2 , l5 , l3, v0) – Alg 2considers that internal paths do not change in relative cost and the recalculation may only be needed in short steps because of the propagation in this variation through the LN.

l1 l3l7l4 … l5l2

First gap to the previous

LN vertex cost

Control LN cost change in

vertices label

Second gap to the previous LN

vertex cost

2. Improvement - Slow Improvement 2OptHertz, Laporte and Nanchen-Hugo (1999)The postman postpone

some required tasks

From a solution S, DROP+ADD seeks a better solution S* by successively removing an edge from E’, shortening the solution, and reinserting the edge in E’.

2. Improvement - Slow Improvement 2OptHertz, Laporte and Nanchen-Hugo (1999)

2. Improvement - Fast ImprovementCutting Duplications – Fernández et al (2003)

Step 1: If Ep contains two copies of any edge, and both can be deleted without disconnecting the graph, do so;

Step 2: If Ep contains edges ij and Ik, such that ij and ik can be replaced by ik without disconnecting the graph, do so.

2. Improvement - Fast Improvement2Opt/3Opt – with Layer Net – Groves and Vuuren (2005)

Use of a TSP two-opt and three-opt without modifications, by applying Alg 1 to calculate the initial LN from a constructive solution, and Alg 2 to hold the exchanges and implicit calculation of the improvement. They perform first improvement strategy for both methods.

3. Meta-Heuristic

Guided Local Search - Muyldermans, Beullens, Cattrysse and Oudheusden, 2005

Scatter Search - Benavent, Corberán,Pinãna, Plana and Sanchis, 2006 (Windy Rural Postman)

3. Meta-HeuristicGuided Local Search - Muyldermans, Beullens, Cattrysse and Oudheusden, 2005

Generalized RPP (GRPP)

In a directed GRPP, we consider a strongly connected, directed graph D(V,A) with vertex set V and arc set A. Every arc a ∈A has a nonnegative length ca and a subset of candidate required arcs AR⊆A is given. The arcs in AR are partitioned into K disjoint classes .

The GRPP is the problem of determining a minimum length tour in D, passing through at least one required arc of each class i.

),...,1( KiA iR =

Generalized RPP

D(V,A, AR)

Euler(GRPP) tour

3. Meta-HeuristicGuided Local Search - Muyldermans, Beullens, Cattrysse and Oudheusden, 2005 (Generalized RPP)

Consider a solution S={σ0, σ1,…, σi+1,…, σi+r, σi+r+1,…, σK-1} and name si+1,r= (σi+1,…, σi+r) a sequence of S starting at σi+1 of size r, 1≤r≤K-1, and K = |ER|. The deadheading associated with si+1,r is given by:

∑−+

+=++ =

11,1 ))(),(()(

ijjjri thdsz σσ

Where, σI is an arc of D(V,A, AR), transformed from G(V,E,ER)h(σi), t(σi) is the head and tail of arc σId(h(σi), t(σi)) is the length of the shortest path between h(σi), t(σi)

Improvement Moves - FLIP

Improvement Moves - REVERSE

Improvement Moves – DIROPT (LN)

Neighbor Lists

For every required edge e(v) in the tour, construct a neighbor list where each list contains the NL nearest required edges sorted in ascending order of the distance to e(v).

The distance between e(v) and another required edge e’(v’) is the minimum shortest path distance to reach a vertex on e’ from vertex on e. The size of the list NL is specified as input.

AL is the active list or the list of a required edge taken in the present step.

“The idea of GLS (Voudouris and Tsang, 1996) is to penalize long deadheading paths, which are unlikely to be incorporated in a good tour.

Some deadheading distances are modified and a local search procedure is recalled (using an adapted distance dmod hoping to escape from the local minimum.”

)(pen ))(),(())(),(( 111mod +++ += iiiiii thdthd σσλσσσσ

0Sαλ =

A function α of the average deadheading distance between two consecutive representatives in the local minimum S0 reached at the first call to the GSL

Voudouris and Tsang, 1999 used the GLS for the TSP to

1000 cities with great success

The solution quality for the GRP and URPP is not too

sensitive to α. α=0.3

3. Meta-HeuristicGuided Local Search (Voudouris and Tsang 1999)

Procedure GLS(S,g,λ,[l1,…,lm],[c1,…,cm],M)01. Begin02. k:=0;03. s0 := random or heuristically generated solution in S;04. For i:=1 to M do pi:=0; // set penalties to 005. while StopCriterion do06. begin07. h:=g+λ*Σpi*li;08. sk+1:=LocalSearch(sk,h);09. For i:= 1 to M do utili:=li(sk+1)*cj/(1+pi);10. for each i such that utili is maximum do pi:= pi +1;11. k:=k+112. end;13. s*:=best solution found with respect to cost function g;14. return s*;15. End;

Where,

li = 1 if the solution has property i, and 0 otherwise.

ci, vector of feature costs

cslfsutil

1)(),( 00

GLS iterates until a defined number of iterations jmax at the (j+1)th iteration given solution Sj from the local search, we look for the index m with the largest value for:

⎭⎬⎫

⎩⎨⎧+

= ),(pen 1))(),((

maxarg:1

),...,1( ii

σσσσ

Procedure 2OptLocal Search

Input : Solution Sj, distance matrix d, neighbor lists nbl and active list ALOutput: Solution Sj+1

Step 1 If AL is empty, go to step 4. Otherwise, select the first element e(v) from AL. All edges and vertices in the neighbor lists are called unexamined

Step 2: (Next Neighbor) If all neighbors of e(v) are examined, go to step 3. Select the first unexamined neighbor e’(v’) in the neighbor list of e(v). Let si+1,r be the subsequence to be investigated. Test for an improvement with, e.g., the 2-opt hybrid approach. If the move leads to an improvement, perform it; append the relevant edges or vertices to AL (marking); and go to step 1. If no improvement is found, e’(v’) is examined. Go to Step 2.

Step 3: (Unmark) Delete e(v) from AL, and go to step 1.

Step 4: Return the current solution Sj+1.

Procedure 2Opt-Guided Local Search

Input : GRP Instance, neighbor list size NL, GLS iteration limit jmax, parameter αOutput: Local minimum S* with deadheading cost z*

Step 0: (Preprocessing) Calculate the deadheading matrix d, construct and sort the neighbor lists, construct an initial solution S0 by a heuristic.

Step 1 (First local Search) Truncate the neighbor lists nbl to size NL. Add all required edges and vertices to AL, call 2-Opt_local_Search(S, d, nbl, AL, S0) and calculate λ. Set S*= S0 and z*=z(S0). Initialize the penalty matrix pen (all entries are zero) and set the modified distance matrix dmod := d. Set the GLS iteration counter j:=0;

Step 2: (GLS) If j=jmax, go to Step 3; else do the following. Look in solution Sj for the deadheading path m. Increment the penalties of this path, adapt dmod, and add the adjacent edges (vertices) to AL (Marking). Call 2-Opt_local_Search(Sj, dmod,, nbl, AL, Sj+1). If Sj+1 differs from Sj, evaluate Sj+1 with the original costs d, and if Sj+1improves S*, set z*:=z(Sj+1), S*:=Sj+1. Set j:=j+1 and go to Step 2.

Step 3: (Final Local Search) Add all required edges and vertices to AL, and call 2-Opt_local_Search(S*,d, nbl, AL, S*) with the original distance d, and return S* and z*=z(S*).

UGRP/RPP Heuristic Methods1. Results

General Statistics for the RPP Heuristic Methods BehaviorGAP of the MethodsGroup of

Instances Source Instances V p E Er FRED FRED+2O INS INS+2OPT FGMO MBCO GTSP GTSP+2OPT GMST GMST+2O2.94% 0.72% 4.41% 1.09% 0.26% 0.00% 3.53% 1.33% 3.09% 0.98% Average

7 2 10 4 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% Minimum50 8 184 78 15.79% 5.45% 18.63% 6.61% 6.40% 0.00% 12.75% 7.84% 17.14% 6.61% MaximumChristofides Corberán 24

4.07% 1.45% 4.48% 1.86% 1.31% 0.00% 4.00% 1.84% 3.96% 1.86% Deviation0.40% 4.39% 3.19% 2.85% 1.96% Average

116 2 174 86 0.00% 0.00% 0.00% 0.00% 0.00% Minimum196 43 316 238 5.09% 6.26% 4.69% 5.39% 3.93% MaximumGRP - Vx=0 Fernandez &

Meza 42

1.15% 2.04% 1.63% 1.56% 1.15% Deviation1.01% 0.00% 4.46% 3.20% 3.39% 2.47% Average

122 6 190 105 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% Minimum265 111 594 242 8.42% 0.00% 10.64% 8.33% 9.08% 6.55% MaximumGRP - Vx<>0 Fernandez &

Meza 40

1.89% 0.00% 3.40% 2.73% 2.94% 2.13% Deviation11.51% 0.94% 10.19% 1.19% 3.46% 0.89% 3.28% 1.45% Average

150 4 489 48 2.29% 0.00% 4.43% 0.00% -6.07% -8.03% -6.81% -7.92% Minimum350 61 1717 187 21.61% 6.77% 18.37% 7.39% 9.58% 5.18% 7.42% 4.71% MaximumType A Ghiani &

Langana 34

4.97% 1.66% 3.89% 2.10% 3.12% 2.68% 3.04% 2.54% Deviation13.58% 0.45% 14.34% 1.48% 16.79% 13.81% 16.28% 14.19% Average

50 7 98 18 5.44% 0.00% 0.00% 0.00% 0.16% -4.42% -1.66% -3.18% Minimum250 57 500 146 25.05% 5.43% 28.65% 6.22% 45.41% 43.49% 43.45% 41.29% MaximumType C Ghiani &

Langana 20

5.52% 1.29% 6.36% 1.86% 13.46% 13.36% 13.09% 12.87% Deviation0.24% 5.04% 3.28% 3.63% 2.63% Average

36 5 72 27 0.00% 0.00% 0.00% 0.00% 0.00% Minimum100 23 200 121 6.74% 12.73% 11.61% 11.37% 9.02% MaximumDegree4p Fernandez &

Meza 36

1.16% 3.42% 2.99% 2.83% 2.39% Deviation0.00% 7.27% 4.66% 6.90% 5.24% Average

20 4 37 3 0.00% 0.62% 0.00% 0.00% 0.00% Minimum50 13 203 20 0.00% 22.15% 20.13% 26.61% 26.61% Maximum

Planar Random

Fernandez & Meza 20

36 4 60 24 0.00% 2.70% 1.39% 1.39% 0.00% Minimum100 21 180 113 0.00% 19.05% 16.67% 17.39% 15.38% MaximumGrid Fernandez &

Meza 36

200 8 399 156 0.00% 1.07% 0.99% 0.46% 0.12% Minimum300 32 599 312 0.00% 6.66% 6.22% 2.71% 2.65% MaximumNuevos - RPP Fernandez &

Meza 15

0.00% 1.50% 1.36% 0.74% 0.70% Deviation

(*) Negative gap means new best know value in the group of the instances(**) Missing data means that the results were not reported by the authors.

PART II – Asymmetric GRP

State UniversityState Universityof Cearof Cearáá

Agenda

1. Challenges2. Introduction3. MGRP Mathematical Formulation4. Exact Methods5. Heuristic Methods

Agenda

1. Challenges

1. Knowledge of the pertinent literature; 2. The Model and Cutting Plane Method;3. Development of quality heuristics for the MRPP/MGRP;4. Extensive use of classical optimization methods to

solve Asymmetric Rural Postman/Asymmetric GRP; 5. Few works done in the literature but “very good”!

Agenda

2. Introduction

Versions of the MGRP/MRPP- Asymmetric

- Oriented (Directed)

- Stacker Crane

2. Introduction

Versions of the MGRP/MRPP- Asymmetric

- Pure Mixed

- Mixed GRP

2. IntroductionLiterature- Asymmetric

- Exact- Directed, Christofides, Campos, Corberán and Mota (1986)- Mixed, Laporte (1997) – by Transformation (RPP-GTSP)- MGRP

- Corberán, Romero and Sanchis (2003) >> First MP Formulation

- Blais & Laporte (2003)

- Corberán, Mejía and Sanchis (2005) >> Introduces New Facets

- Corberán, Mota and Sanchis (2006) >> Formulations (one index and two index)

- Heuristics- Stacker Crane, Frederickson, Hecht and Kim (1978)- Mixed RPP, Corberán, Marti, Romero (2000) >> TS

2. Introduction

Versions of the MGRP/MRPP- Windy (mixture of Symmetric and Asymmetric)

- Pure Windy

- WGRP

Agenda

3. MGRP Mathematical FormulationLet,

V – set of vertices;E – set of edges;ER – set of required edges;E\ER – is the set of non required edges;AR – is the set of required arcs;A\AR – is the set of non required arcs;

GR = G(V,E,AR) – the graph obtained by deleting in G all non-required arcs A\AR, in general this graph is not connected;

p – is the number of connected components of GR;

V1∪V2∪…∪Vp = V – the R-sets, corresponding to the p connected components of GR;

Agenda

3. MGRP Mathematical FormulationCi =G(Vi), i=1,..,p, are the sub-graphs of G induced by the R-sets and they will be referred to as R-connected components. Notice that every isolated required vertex is a R-connected component of G.

Let, S1, S2 ⊂ V, S1∩S2=∅, where (S1 : S2) = {(i,j)∈E∪A : i∈S1, j∈S2 or i∈S2, j∈S1};δ(S) = (S : V\S) (Called link cut-set of G defined by S);A+(S) = A(S : V\S);A-(S) = A(V\S : S);E(S) = E(S : V\S).

All the sets (A+, A-, E, δ(S)), referring to required and non-required links follow the above definition.

Given x∈ℜ|E∪A| and given T⊂E∪A, x(T) denotes ∑e∈T xe..

Agenda

3. MGRP Mathematical Formulation (CRS 2003)

The resulting graph may be even and

strongly connected

The balance set conditions may be

satisfiedThe tour may use non required arcs

All the required links are in the solution

MGRP(G) is the convex hull formed by all semi-tours that satisfies (2-8)MGRP(G) is an unbounded polyhedron, Dim(MGRP(G))=|E∪A|-q+1, q - number of edge-connected components of G

Agenda

4. Exact Methods

1. Blais & Laporte Transformation2. RPP to GTSP Results3. Cutting Planes – Corberán, Mejía and Sanchis4. CP Results

1. MGTSP to MRPP

v12 v14 6

i j cij

v12 v24 0

v12 v13 6

v12 v41 3

v12 v42 3

Each pair (vij, vkl) in the transformed problem defines an arc of cost cjk = spjk

Previous Laporte’s idea…

1. MGTSP to MRPP

v12 v14 6

i j cij

v12 v24 0

v12 v13 6

v12 v41 3

v12 v42 3

Each required vertex is relabeled to vii . Each vertex pair (vki , vlj ) in the transformed problem defines an arc of cost cij = spil + clj , where cii=0.

Blais and Laporte extension to GRP…

γ5 v12 v44 5

1. MGTSP to MRPP

The transformed problem is the GTSP (Generalized TSP), and one can use theNoon and Bean (1991) approach to solve exactly the problem, or Renaud and Boctor (1998), Cacchiani, Muritiba, Negreiros and Toth (2008) using efficient heuristics for this approach.

If an ATSP is the transformed problem, it is necessary to include an arc of cost -M between any pair of vertices of the same group (representing edges). After applying Carpaneto, Dell’Amico and Toth, Applegate, Bixby, Chvátal, and Cook. By Concorde exact algorithm platform for the ATSP, or any efficient heuristic.

Blais and Laporte extension to GRP…

4. Exact Methods

1. Blais & Laporte Transformation2. RPP to GTSP/ATSP Results3. Cutting Planes – Corberán, Mejía and Sanchis4. CP Results

4. Exact Methods

2. RPP to GTSP/ATSP Results

4. Exact Methods

1. Blais & Laporte Transformation2. RPP to GTSP Results3. Cutting Planes – Corberán, Mejía and Sanchis 20054. CP Results

4. Exact Methods

3. Cutting Planes – Corberán, Mejía and Sanchis 2003

> Balance Conditions to a mixed graph/subgraph be eulerian (Ford and Fulkerson, 1962) - Unicursal

A+(S) – A-(S) ≤ E(S)

A-(S) – Arcs entering S

A+(S) – Arcs leaving S

E(S) – Edges in SS⊂V

4. Exact Methods

Trivial, connectivity, Balanced-Set, R-odd Cut, Path Bridge and PB02 are all facet-inducing inequalities to MGRP(G).

They introduce a generalization of K-C inequalities as Honeycomb inequalities where it is done in different direction as in UGRP. Honeycomb02 inequalities are also introduced.

Honeycomb Configuration Honeycomb02 Configuration

4. Exact Methods

Cutting-Plane Algorithm

1.R-odd cut and connectivity separation heuristics2.Exact connectivity separation if the heuristic failed3.Exact R-Odd cut separation if he heuristic failed4.Exact balanced-set separation5.If the number of violated inequalities detected so far is ≤10, for each R-set, try the K-C and K-C02 separation heuristic6.If no violated inequalities have been detected so far, try Honeycomb and Honeycomb02 (That can also find K-Cs) 7.If the number of violated inequalities detected so far is ≤10, for each R-set, try the regular Path-Bridge separation heuristic8.If no violated inequalities have been detected so far, try heuristics for K-C, K-C02 , Honeycomb and Honeycomb02 by

interactively merging two R-sets9.Call B&B if no more violated integralities is found and LP is still integral (The ILP to be solved is formed by all inequalities

set in the pool of inequalities evaluated).

4. Exact Methods

1. Blais & Laporte Transformation2. RPP to GTSP Results3. Cutting Planes – Corberán, Mejía and Sanchis4. CP Results

4. Exact Methods

4. CP Results

65/81 instances were solved to optimality using CP

Group of Instances Source Instances V Vr E A Er Ar p

116 0 77 12 7 11 1209 93 164 263 148 99 103

ALBA CMS2005 25

215 1 224 129 0 129 3428 214 224 555 188 555 214

ALDA CMS2005 31

197 1 118 32 13 32 2342 146 298 443 250 443 168

MADRI CMS2005 25

4. Exact Methods

4. CP Results

4. Exact Methods

4. CP Results

4. Exact Methods4. CP Results

Agenda

5. Heuristic MethodsOrientation of Required Links

Sherafat (1988)

xi xjc

xi xj<c,1,∞>

5. Heuristic MethodsOrientation of Non Required Links

xi xjc

xi xj<c,0,∞>

xi xjc

xi xj<c,0,∞>

<c,0,∞>

5. Heuristic Methods• VR to Fixed Cost ARTransformation (Negreiros and Laporte 2008)

Required vertices are expanded to one required arc with cost oneand another with cost zero, one extremity stay connected with the same adjacent vertices and the other has just its twin as adjacent. To the required arc a unit cost flow is set as minimum circulation.

vi vi v|V|+s<0,0,∞>

<1,1,∞>

5. Heuristic Methods

1. TS - Corberán, Martí and Romero (2000)2. GTSP - Negreiros & Laporte (2008)3. GTree – Negreiros & Laporte (2008)4. Results

1. TS - Corberán, Martí and Romero (2000)Constructive_CMR2000_MRPP

Step 1: (Graph Transformation) G → GR(VR,E,A), similar to Christofides et at 1984Step 2: (Connectivity) Define vertices degree costs, set non required arcs cost, orient by MCF

apply shortest spanning tree over the modified graph;Step 3: (Improving Connectivity)

3.1. Assign a direction to the remaining undirected edges;3.2. Construct a directed graph;3.3. Solve over the transformed graph MCF, and accordingly the flow orientation between components include new arcs forming a new graph GΓ;

Step 4: (Obtain a feasible solution)4.1. Build the balance mixed graph from previous GΓ resulting a GB4.2. Convert GB into an even graph by applying 1-matching for the remaining odd degree vertices

Step 5: (Improvement of the solution) If the cost of an arc, or a path, in the augmentation of the graph exceeds the length of a shortest path from i to j, it is advantageous to replace the arc or path by the smallest shortest path.

O(max{O(MCF(G)),O(1-Matching(GB))}

Major Principle: Build an initial MRPP tour by exploring unicursal property of the given mixed graph

1. TS - Corberán, Martí and Romero (2000)

Initial MRPP Graph

Example - Constructive

Step 1: Graph Transformation

Step 2: Initial connection

Step 2: Shortest spanning tree

Step 3: Balanced mixed graphObtained from initial connection

Step 3.1/3.2: Balanced mixed graph

Step 3.3: Final Eulerian, After 1-matching

Step 4: Improvement Phase

Tabu Search CMR2000 Algorithm

Neighborhood

Let <i,j> is a connecting arc and Γ(k,i) and Γ(j,s) paths from k to i and j to s, in the augmentation graph induced by required links (GR).

m(I,j) is a move by deleting the path Γ(Γ(k,i),<i,j>,Γ(j,s)) from current solution andadding the shortest path from Γ(k,s).

Moves may be feasible and infeasible!!

Neighborhood

Let <i,j> is a connecting arc and Γ(k,i) and Γ(j,s) paths from k to i and j to s, in the augmentation graph induced by required links (GR).

m(I,j) is a move by deleting the path Γ(Γ(k,i),<i,j>,Γ(j,s)) from current solution andadding the shortest path from Γ(k,s).

Moves may be feasible and infeasible!!

Intensification

m(i,j) move is executed and is a move and the removed arc (i,j) becomes tabu-active for a number of tabu iterations it can not be added to the solution during this time.

Connecting arcs are randomly selected in the next iterations, its corresponding moveis performed if it is available and does not add any tabu arc.

Diversification

For each tabu iteration a nontabu connecting arc (i,j) is probabilistically selected as in the intensification phase. If move(I,j) is available and does not add a tabu arc, it is Performed. Otherwise, two alternative moves are considered, depending on whether The current solution is connected or not.

(Improving or not the solution) If it is connected the shortest path is calculated and a new graph appears;

If the graph being not connected, it is randomly connected

Long-Term Diversification

It manages the selection of connecting arcs according to their number of occurrences in previous solutions. These frequencies are mapped onto selectionprobabilities: the lower the frequency, the higher the probability to be chosen.

1. TS - Corberán, Martí and Romero (2000)1. Procedure TBSearch_CMR20002. p:=number of R-Connected Components3. Iter:=0;4. While iter<p*Global do5. begin6.iter:=iter+1;7.Long Term Diversification (Generate a Solution)8. Select a connecting arc set9. Apply Constructive_CMR2000_MRPP10. Let X be the solution obtained;11. update the frequencies of the connecting graphs12.If cost(X)<Pt*Cost(BestSolution) Then13. Apply Basic Procedure to X;14. Update, if necessary, BestSolution;15.end;16.End. // TBSearch_CMR2000

3. GTSP Based Method

Procedure ARPP_GTSP(G(V,E,A,ER,AR))Step 1 : Required Components IdentificationStep 2 : Applying GTSPStep 3 : Orienteering and Connecting the R-Components of GStep 4 : Apply the MCF on GT or

If any triangle flow remains ThenGRASP

Random Phase (forcing flow in the triangular structures)Improvement phase (Graph Reduction and GV Reduction)

GTSP - O(p log |p|) if a heuristic procedure is applied, and O(2p) if exact.

It is dominated by MCF step if the number of required edges and selected used links is high, O(|L|2).

4. GTSP Negreiros & LaportePure GTSP with internal costs

GTSP with no internal cost

Paths used by the GTSP solution are oriented and

transformed to a Required Path

3. GTSP Based Method

… …

Layer Net (adapted from GV05 Reduction)…

After the calculations, the LN improvement is applied and the

optimal eulerian tour from the given sequence of visiting the links is

returned

Example

Given Mixed Graph

Example

Extracting and identifying the components and extreme vertices

Example

Applying GTSP in the components, using the extreme vertices of each component as representing each group…

Example

Orienteering the graph…

5. Example

Defining the new required arcs…

5. Example

Applying a MCF in the graph…

5. Example

Continue applying a MCF in the graph until no triangular flows appears…

5. Example

Reduce the graph taking the “best” orientations…

Example

Final circulation…

Example

Final multi-graph…

Example

Final optimal multi-graph…

Evolution

Evolution of the method…

Meta-HeuristicBest Solution

Meta-Heuristic and Best Solution Performance

Iterations1 2 3 4 5 6 7 8 910 12 14 16 18 20 22 24 26 28 30 32 34 36

t1.060

OF value dropped motivated by reduction

and LN calculation

3. GTree – Negreiros & Laporte (2008)Paths used by the GMST solution are transformed

to a Required Path

3. GTree Method

Procedure GTree(G(V,E,A,ER,AR))Step 1 : Required Components IdentificationStep 2 : Apply GMST between the connected components of GStep 3 : Orienteering and Connecting the R-Components of GStep 4 : Apply the MCF on GT or

If any triangle flow remains ThenGRASP

Random Phase (forcing flow in the triangular structures)Improvement phase (Graph Reduction and GV Reduction)

GMST - O(p2).

It is dominated by MCF step if the number of required edges and selected used links is high, O(|L|2).

3. GTree Method

GTree obtain a lower bound to the MGRP after the first call to the MCF method;

Its performance is dependent to the number of required edges induced by the paths it generates and the proper required edges of the MGRP net;

It is easy to control, and surprisingly results may be obtained for the GRP classes of instances.

1. TS - Corberán, Martí and Romero (2000)2. GTSP - Negreiros & Laporte (2008)3. GTree – Negreiros & Laporte (2008)4. Asymmetrical GRP/MRPP Results

Results

5|6 GRASP…

GAP to BestGroup of Instances Source Instances V E A Er Ar p TS-CMR GTSP GTSP+EI

0.86% 1.33% 0.85% Average48 52 10 106 42 3 0.21% 0.34% 0.00% Minimum

125 226 150 318 213 5 1.45% 3.22% 1.68% MaximumANTO-1 CMR2000 10

0.45% 0.85% 0.56% Deviation1.22% 1.84% 1.27% Average

76 60 20 130 42 4 0.00% 0.22% 0.00% Minimum130 192 100 382 230 15 7.17% 5.48% 4.47% MaximumANTO-2 CMR2000 10

7.07% 4.95% 1.86% Deviation1.89% 2.53% 1.55% Average0.00% 0.22% 0.00% Minimum

21.10% 14.61% 6.42% MaximumMethods and the GAP to the best known values for ANTO Instances

2.03% 1.95% 1.13% Avg Dv

Number of GRASP Iterations – 30 (30 greedy random solutions with 6 improvement calls per each set of 5 random

Results

Score Table (5|6)…

SCORETS-CMR GTSP GTSP+EI

ANTO-1 8.00 - 3.00ANTO-2 5.00 3.00 5.00ANTO-3 7.00 1.00 3.00ANTO-4 6.00 - 5.00ANTO-5 3.00 - 7.00ANTO-6 7.00 - 5.00ANTO-7 7.00 - 3.00ANTO-8 2.00 2.00 8.00ANTO-9 5.00 - 5.00ANTO-10 2.00 1.00 8.00

52.00 7.00 52.00

Results

MRPP ANTO Instances

CRS03 TS-CMR GTSP GTSP+EI Gtree Gtree+EI MIN0.00% 0.86% 1.38% 0.96% 1.44% 1.02% 0.72% Average

48 52 10 106 42 3 0.00% 0.21% 0.46% 0.00% 0.38% 0.00% 0.00% Minimum125 226 150 318 213 5 0.00% 1.45% 2.49% 2.04% 3.62% 3.14% 1.53% Maximum

0.00% 0.45% 0.70% 0.69% 1.00% 0.94% 0.51% Deviation0.02% 1.22% 1.96% 1.27% 1.62% 1.07% 0.96% Average

76 60 20 130 42 4 0.00% 0.00% 0.38% 0.00% 0.00% 0.00% 0.00% Minimum130 192 100 382 230 15 0.22% 7.17% 5.73% 3.63% 7.84% 4.30% 3.63% Maximum

49 52 20 107 15 5 0.00% 0.11% 0.56% 0.37% 0.47% 0.19% 0.19% Minimum123 183 100 365 246 6 0.00% 1.81% 3.02% 1.63% 3.53% 1.88% 1.42% Maximum

64 61 10 148 24 6 0.00% 0.00% 0.80% 0.00% 0.48% 0.15% 0.00% Minimum138 181 120 342 205 6 0.00% 2.32% 3.04% 1.91% 4.63% 2.32% 1.91% Maximum

63 58 5 130 12 6 0.00% 0.38% 0.50% 0.00% 0.50% 0.32% 0.00% Minimum139 202 120 341 234 8 0.00% 8.05% 4.66% 2.54% 4.66% 3.39% 2.54% Maximum

57 51 20 145 43 8 0.00% 0.31% 0.15% 0.15% 0.38% 0.38% 0.15% Minimum138 222 120 396 181 8 0.00% 1.93% 5.38% 3.07% 4.30% 1.84% 1.84% Maximum

46 47 15 109 19 9 0.00% 0.49% 0.98% 0.23% 0.70% 0.59% 0.23% Minimum152 214 120 332 238 9 0.00% 9.95% 10.22% 4.57% 7.80% 3.51% 3.25% Maximum

46 20 10 150 38 10 0.00% 0.62% 1.47% 0.99% 0.76% 0.39% 0.39% Minimum147 192 224 373 261 10 0.00% 7.30% 9.96% 5.97% 9.29% 4.87% 4.87% Maximum

36 15 93 5 7 3 0.00% 0.00% 1.29% 1.05% 0.19% 0.00% 0.00% Minimum158 211 327 120 217 13 0.00% 7.39% 8.02% 6.42% 13.37% 5.15% 5.15% Maximum

45 25 5 68 6 8 0.00% 1.07% 1.22% 0.58% 0.99% 0.76% 0.58% Minimum164 192 316 333 205 16 0.00% 21.10% 11.04% 11.04% 12.92% 10.42% 6.82% Maximum

0.00% 7.07% 3.36% 3.60% 3.77% 3.31% 2.37% Deviation0.00% 1.89% 2.44% 1.69% 2.46% 1.54% 1.29% Average0.00% 0.00% 0.15% 0.00% 0.00% 0.00% 0.00% Minimum0.22% 21.10% 11.04% 11.04% 13.37% 10.42% 6.82% Maximum0.01% 2.03% 1.75% 1.27% 2.12% 1.21% 1.00% Avg Deviation

Group of Instances Source Instances V E Er pA Ar

ANTO-4 CMR2000 10

ANTO-1 CMR2000 10

ANTO-2 CMR2000 10

ANTO-7 CMR2000 10

ANTO-8 CMR2000 10

ANTO-5 CMR2000 10

ANTO-6 CMR2000 10

ANTO-3 CMR2000

ANTO-9 CMR2000 10

GAP to Best

Methods and the GAP to the Best Known for ANTO Instances

ANTO-10 CMR2000

Results

Score Tables

SCOREInstance TS-CMR GTSP GTSP+EI GTree GTree+EIANTO1 5.00 1.00 3.00 1.00 4.00 ANTO2 4.00 - 3.00 3.00 4.00 ANTO3 5.00 - 1.00 - 5.00 ANTO4 6.00 - 4.00 - 1.00 ANTO5 3.00 - 7.00 - -ANTO6 4.00 1.00 5.00 1.00 2.00 ANTO7 6.00 - 1.00 - 3.00 ANTO8 2.00 1.00 2.00 - 7.00 ANTO9 3.00 1.00 2.00 2.00 7.00 ANTO10 2.00 2.00 5.00 2.00 3.00

40.00 6.00 33.00 9.00 36.00

Results

Score Tables min of Generalized

SCORETS-CMR MIN

ANTO-1 7.00 6.00 ANTO-2 5.00 6.00 ANTO-3 8.00 6.00 ANTO-4 6.00 5.00 ANTO-5 3.00 7.00 ANTO-6 5.00 7.00 ANTO-7 8.00 4.00 ANTO-8 3.00 8.00 ANTO-9 6.00 8.00

ANTO-10 2.00 8.00

53.00 65.00

Results

GRP/MRPP Instances

CP GTSP GTSP+EI Gtree Gtree+EI MIN0.00% 6.08% 3.63% 13.51% 7.51% 2.59% Average

116 0 77 12 7 11 1 0.00% 0.38% 0.00% 0.05% 0.00% 0.00% Minimum209 93 164 263 148 99 103 0.00% 14.26% 10.90% 76.23% 41.71% 10.08% Maximum

0.00% 4.35% 3.27% 18.22% 10.00% 2.82% Deviation0.00% 10.44% 6.57% 13.65% 8.20% 6.17% Average

215 1 224 129 0 129 3 0.00% 2.44% 1.38% 2.71% 1.49% 1.38% Minimum428 214 224 555 188 555 214 0.00% 21.31% 16.02% 42.83% 30.79% 16.02% Maximum

197 1 118 32 13 32 2 0.00% 0.45% 0.28% 0.00% 0.00% 0.00% Minimum342 146 298 443 250 443 168 0.00% 17.94% 12.94% 29.11% 21.80% 16.02% Maximum

502 2 311 305 92 305 3 0.00% 0.47% 0.31% 0.35% 0.06% 0.06% Minimum643 143 1021 1200 759 1200 245 0.00% 15.73% 11.34% 18.39% 9.83% 9.07% Maximum

1000 1 611 532 171 532 2 0.00% 0.43% 0.00% 0.40% 0.00% 0.00% Minimum1000 292 2033 2031 1515 2031 480 6.04% 6.84% 3.70% 10.53% 3.93% 3.70% Maximum

357 0 261 210 86 210 1 0.00% 0.19% 0.11% 0.87% 0.85% 0.11% Minimum498 0 987 976 739 976 102 0.00% 11.74% 9.14% 18.22% 10.47% 9.00% Maximum

708 0 521 470 159 470 1 0.00% 0.22% 0.16% 0.31% 0.17% 0.16% Minimum999 0 1969 1984 1481 1984 188 0.00% 17.15% 10.02% 21.03% 13.95% 10.02% Maximum

0.00% 5.18% 3.26% 6.07% 3.61% 3.05% DeviationAverage 0.14% 6.85% 4.43% 9.41% 5.44% 4.00%

Instances 153 Minimum 0.00% 0.19% 0.00% 0.00% 0.00% 0.00%Maximum 6.04% 21.31% 16.02% 76.23% 41.71% 16.02%

Group of Instances Source Instances V Vr E A Er Ar

ALBA CMS2005 25

ALDA CMS2005 31

GD WEB 18

GB CMS2006 18

GAP to Best Known Solution

MRPP-RB WEB 18

MRPP-RD WEB 18

MADRI CMS2005

Results

Score Table

SCOREGTSP GTSP+EI GTree Gtree+EI

ALBA 0 14 1 12ALDA 0 22 0 9MADRI 0 11 1 14GB 0 3 0 15GD 0 6 0 12MRPP-RB 1 8 0 10MRPP-RD 0 5 0 13

1 69 2 85

Results

Performance (p × Time (s) ) of GTree+EI and GTSP+EI for instances ALBA (left) and ALDA (right).

Results

Some new results appear in comparison to cutting plane method proposed by Corberán et al (2005) in two sets of instances, ANTO (1 instance) and GD (8 instances).

These results reveal that there is still room for evaluating new facets and properties to describe better the MGRP.

PART III – System XNÊS

System XNÊS

An Academic Software Project developed by the team of young OR researches from State University of Ceará, Federal University of Espirito Santo, and lately by GRAPHVS and CIRRELT/Université de Montreal

Including in the CD

Exact CPP Versions – XCPP Version (FULL)RPP Versions - XRPP Version (DEMO)

System XNÊS

An Academic Software Project developed by the team of young OR researches from State University of Ceará, Federal University of Espirito Santo, and lately by GRAPHVS and CIRRELT/Université de Montreal

It is a MVI (interactive visual modelling) graph based platform.

Graphs may be drawled as simple graphs.

Including in the CD

Exact CPP Versions – XCPP Version (FULL)RPP Versions - XRPP Version (DEMO)

System XNÊS

General Interface

General Menu

Speed Buttons Menu

Graph visualization area

System XNÊS

Speed ButtonsOpen New Graph File

Open existing Graph File

Save Graph

Show textual Solution

System XNÊS

Speed Buttons

Reset any operation

Move vertex

Mark and demark Origin

Add and Exclude Vertex

Move GraphDelete Graph

System XNÊS

Speed Buttons

Include/Exclude label in Links

Add and Exclude Edges

and Arcs

See /Hide Vertices Labels

Links costs Editor (Metrics)

All Link attributes Editor

System XNÊS

Speed Buttons

Restart euler tour animation

Include/exclude a picture

Change Link Status to Required to non-

Required

Expand/Retract Required Vertex

Print current Graph

System XNÊS

Speed Buttons

Execute related RPP

Stop euler tour animation Close the

system

Show multigraph solution view

Show Graph description

System XNÊS

Asymmetric RPP/GRP Solver

Test bed for algorithm design

System XNÊS

Symmetric RPP/GRP Solver

Solution dateil text editor

Multigraph Euler Tour visualization

System XNÊS

Windy RPP/GRP Solver

PART IV – GRP Perspectives

GRP Perspectives1. Still room for Metaheuristics in Symmetric and Asymmetric;

2. New facets and more effective cuts may appear for the MGRP;

3. The new WGRP proposed by Corberán, Plana, Sanchis (2008);

4. WGRP introduce new Zig-Zag inequalities for CP;

5. Bounds as good as TSP in the Asymmetrical Instances;

6. More effective solution procedures and new related problems (m-GRP, Generalized GRP).

Acknowledgments1. CNPq Pos-Doctoral GRANT 202457/2006-0 ;

2. CIRRELT / Université de Montreal;

3. GRAPHVS Ltda and its Research Team;

4. State University of Ceará (UECE);

5. Augusto Wagner Palhano, MSc;

6. Prof Francisco José Negreiros Gomes, DSc (in memoria).

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