A tabu search algorithm for the min max k-chinese postman problem

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A tabu searchalgorithm for the min- max k-Chinese postman problem Dino Ahr, Gerhard Reinelt* Computers & Operations Research 33 (2006) 3403–3422 報告人: 陳政謙

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Outline • The min-maxk-Chinese postman problem • Merging and separating edges • Improving tours • The tabu search algorithm • Computational results

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k-Chinese postman tour •For a given graph G=(V, E), k-Chinese postman tour is k closed walks(tours) which each tour starts and ends at the depot node and each edge e ∈ E is covered by at least one tour. v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 1 2 24 27 1 8 7 7 1 26 210

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k-Chinese postman tour •For a given graph G=(V, E), k-Chinese postman tour is k closed walks(tours) which each tour starts and ends at the depot node and each edge e ∈ E is covered by at least one tour. v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 1 2 24 27 1 8 7 7 1 26 210 depot node = v1

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k-Chinese postman tour •For a given graph G=(V, E), k-Chinese postman tour is k closed walks(tours) which each tour starts and ends at the depot node and each edge e ∈ E is covered by at least one tour. v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 1 2 24 27 1 8 7 7 1 26 210 depot node = v1 k = 2

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k-Chinese postman tour •For a given graph G=(V, E), k-Chinese postman tour is k closed walks(tours) which each tour starts and ends at the depot node and each edge e ∈ E is covered by at least one tour. v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 1 2 24 27 1 8 7 7 1 26 210 depot node = v1 k = 2

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7 1 1 k-Chinese postman tour •For a given graph G=(V, E), k-Chinese postman tour is k closed walks(tours) which each tour starts and ends at the depot node and each edge e ∈ E is covered by at least one tour. v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 depot node = v1 k = 2

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The min-max k-Chinesepostman problem • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C2: C1: • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C1) = 107 C2: C1: • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 78 w(C1) = 107 C2: C1: • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 78 w(C1) = 107 C2: C1: The maximum weight is 107 • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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The min-max k-Chinesepostman problem 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 94 w(C1) = 93 C2: C1: The maximum weight is 94 • The aim for the min-max k-Chinese postman problem (MM k-CPP) is to minimize the length of the longest tour of k- Chinese postman tour.

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Merging and separatingedges • Two Algorithms – SeparateWalkFromTour – MergeWalkWithTour

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Algorithm: SeparateWalkFromTour • Input:The tour Ci and the walk H to be separated. • Output: The tour represents a feasible tour formed from the remaining of edges of Ci after removing edges from H and possibly additional edges needed to re-establish feasibility.

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Algorithm: SeparateWalkFromTour • Step1:Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2:

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: u = v4 • Step1: Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: u = v4 • Step1: Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: u = v4 • Step1: Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: v = v8 H: u = v4 • Step1: Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: v1 is not an internal node of H. H: u = v4 v = v8 • Step1: Let u and v be the endnodes of H (u = v is possible). Check whether the depot node v1 is an internal node of H.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: • Step2: Remove all edges of H from Ci to yield Ĉi. Check whether the depot node v1 is contained in Ĉi. u = v4 v = v8

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: u = v4 v = v8 • Step2: Remove all edges of H from Ci to yield Ĉi. Check whether the depot node v1 is contained in Ĉi.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: u = v4 v = v8 • Step2: Remove all edges of H from Ci to yield Ĉi. Check whether the depot node v1 is contained in Ĉi.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C2: H:Ĉ1: u = v4 v = v8 • Step2: Remove all edges of H from Ci to yield Ĉi. Check whether the depot node v1 is contained in Ĉi.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C2: H:Ĉ1: v1 is contained in Ĉ1. u = v4 v = v8 • Step2: Remove all edges of H from Ci to yield Ĉi. Check whether the depot node v1 is contained in Ĉi.

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C2: H:Ĉ1: • Step3: We have to consider two cases when we connect u and v: – If v1 is an internal node of H and v1 is not contained in Ĉi , then we let evolves from Ĉi by connecting u and v with SP(u, v1), SP(v1, v). – Otherwise we let evolves from Ĉi by connecting u and v with SP(u, v). u = v4 v = v8

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Algorithm: SeparateWalkFromTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C2: H:Ĉ1: • Step3: We have to consider two cases when we connect u and v: – If v1 is an internal node of H and v1 is not contained in Ĉi , then we let evolves from Ĉi by connecting u and v with SP(u, v1), SP(v1, v). – Otherwise we let evolves from Ĉi by connecting u and v with SP(u, v). u = v4 v = v8

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Algorithm: MergeWalkWithTour • Input:The tour Ci and the walk H to be merged. • Output: The tour represents a feasible tour formed from edges contained in Ci and H and possibly additional edges.

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Algorithm: MergeWalkWithTour • Step1:Remove those edges from the beginning and the end of H which also occur in Ci to yield the result Ĥ. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2:

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: H: • Step1: Remove those edges from the beginning and the end of H which also occur in Ci to yield the result Ĥ.

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: • Step1: Remove those edges from the beginning and the end of H which also occur in Ci to yield the result Ĥ.

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: • Step2: Let u and v be the endnodes of Ĥ. Determine the node t on Ci which minimizes w(SP(u, t)) + w(SP(v, t)). Ĥ:

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 • Step2: Let u and v be the endnodes of Ĥ. Determine the node t on Ci which minimizes w(SP(u, t)) + w(SP(v, t)).

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 v = v8 • Step2: Let u and v be the endnodes of Ĥ. Determine the node t on Ci which minimizes w(SP(u, t)) + w(SP(v, t)).

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 v = v8 t = v4 • Step2: Let u and v be the endnodes of Ĥ. Determine the node t on Ci which minimizes w(SP(u, t)) + w(SP(v, t)).

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 v = v8 t = v4 • Step3: The result evolves from splicing SP(t, u), Ĥ, SP(v, t) into Ci at node t.

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• Step3: Theresult evolves from splicing SP(t, u), Ĥ, SP(v, t) into Ci at node t. Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 v = v8 t = v4

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Algorithm: MergeWalkWithTour 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 224 27 1 8 7 26 210 C1: C2: Ĥ: u = v4 v = v8 t = v4 • Step3: The result evolves from splicing SP(t, u), Ĥ, SP(v, t) into Ci at node t.

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Required edge andredundant edge • Øi(e) denote the frequency of e which occurs in Ci. • Ø(e) = Øi(e) • An edge e is called required for Ci if Øi(e) ≥ 1 and Ø(e) = Øi(e). • An edge e is called redundant for Ci if Øi(e) ≥ 1 and Ø(e) > Øi(e). C1: C2: 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210

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Required edge andredundant edge • Øi(e) denote the frequency of e which occurs in Ci. • Ø(e) = Øi(e) • An edge e is called required for Ci if Øi(e) ≥ 1 and Ø(e) = Øi(e). • An edge e is called redundant for Ci if Øi(e) ≥ 1 and Ø(e) > Øi(e). (v1, v7) is a required edge for C1. C1: C2: 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210

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Required edge andredundant edge • Øi(e) denote the frequency of e which occurs in Ci. • Ø(e) = Øi(e) • An edge e is called required for Ci if Øi(e) ≥ 1 and Ø(e) = Øi(e). • An edge e is called redundant for Ci if Øi(e) ≥ 1 and Ø(e) > Øi(e). (v4, v6) is a redundant edge for C1. C1: C2: 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210

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Improving tours • Threeimprovement procedures – RemoveReplicateEdgesKeepingParity – RemoveEvenRedundantEdges – ShortenRequiredEdgeConnections

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RemoveReplicateEdgesKeepingParity • For edgee which occurs n ≥ 3 times in Ci, we can remove n – 2 times (n – 1 times) e if n is even(odd). 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2:

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RemoveEvenRedundantEdges • If edgee is a redundant edge with even frequency, e will be removed completely from Ci if the remaining of Ci remains connected and still contains the depot node. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2:

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 P:C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 P:C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v3 P:C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step2: Traversethe tour Ci further on to build up walk Q until a redundant edge (vl, vj) entering vj is found. If such a walk Q = {vj, ..., vl, vj} is found, reverse the orientation of Q in Ci and continue with step 1 starting again at vh 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v3 P:C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step2: Traversethe tour Ci further on to build up walk Q until a redundant edge (vl, vj) entering vj is found. If such a walk Q = {vj, ..., vl, vj} is found, reverse the orientation of Q in Ci and continue with step 1 starting again at vh 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v3 Q:P:C1: C2: C1: {v1, v2, v3, v5, v6, v4, v6, v8, v7, v1, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step2: Traversethe tour Ci further on to build up walk Q until a redundant edge (vl, vj) entering vj is found. If such a walk Q = {vj, ..., vl, vj} is found, reverse the orientation of Q in Ci and continue with step 1 starting again at vh 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v3 C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1} Q:P:

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 P:C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step1: Constructthe longest possible walk P (while traversing Ci) which consists of redundant edges. Let vh be the origin and vj be the terminus of walk P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v1 P:C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections • Step2: Traversethe tour Ci further on to build up walk Q until a redundant edge (vl, vj) entering vj is found. If such a walk Q = {vj, ..., vl, vj} is found, reverse the orientation of Q in Ci and continue with step 1 starting again at vh 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v1 P:C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1}

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ShortenRequiredEdgeConnections Step3: Replace Pby the shortest path SP(vh, vj) if w(SP(vh, vj)) < w(P). Continue with step 1 with the edge following the last edge of P. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 vh = v1 vj = v1 Q:P:C1: C2: C1: {v1, v2, v3, v1, v7, v8, v6, v4, v6, v5, v3, v4, v7, v1}

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What is Tabusearch? • Tabu search is created by Fred W. Glover in 1986. • Tabu search is used for optimization problem. • Neighborhood is a solution set which improves the current solution. • If a potential solution has been previously visited within a certain short-term period or if it has violated a rule, it is marked as "tabu" (forbidden) so that the algorithm does not consider that possibility repeatedly.

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Neighborhoods • Three Neighborhoodstructures – TwoEdgeExchange – RequiredEdgeWalkExchange – SingleRequiredEdgeExchange

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The flow ofexchange The longest tour Ci another tour Cj

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The flow ofexchange The longest tour Ci Walk H another tour Cj Separate

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The flow ofexchange The longest tour Ci Walk H Apply improvement procedure another tour Cj Separate

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The flow ofexchange The longest tour Ci Walk H Apply improvement procedure another tour Cj Separate Merge

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The flow ofexchange The longest tour Ci Walk H Apply improvement procedure another tour Cj Apply improvement procedure Separate Merge

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TwoEdgeExchange • The TwoEdgeExchangeneighborhood considers two consecutive edges e and f in the longest tour Ci. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 94 w(C1) = 93 C2: C1:

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RequiredEdgeWalkExchange • P iscomposed of redundant edges. • e is a required edge. • The RequiredEdgeWalkExchange neighborhood considers walks H = {P1, e, P2} in the longest tour Ci. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 94 w(C1) = 93 C2: C1:

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SingleRequiredEdgeExchange • The SingleRequiredEdgeExchangeneighborhood considers required edges e in the longest tour Ci. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 94 w(C1) = 93 C2: C1:

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SingleRequiredEdgeExchange • The SingleRequiredEdgeExchangeneighborhood considers required edges e in the longest tour Ci. 7 1 1 v2 v1 v5 v3 v6 v4 v7 v8 13 38 8 2 24 27 1 8 7 26 210 w(C2) = 94 w(C1) = 93 C2: C1:

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The tabu searchalgorithm • Input: A k-postman tour, the maximum number of iterations without improvement called maxNOfItsWithoutImprovement, a tabu tenure called tabuTenure and a flag called improvementProcedure. • Output: A possibly improved k-postman tour

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The tabu searchalgorithm A k-postman tour

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The tabu searchalgorithm Compute a list of neighborhood solutions N in decreasing order of improvement value. A k-postman tour

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The tabu searchalgorithm Compute a list of neighborhood solutions N in decreasing order of improvement value. Select the first solution of the list as current solution if it is non-tabu or tabu but better than the best solution. (if no such solution then the algorithm terminates) A k-postman tour

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The tabu searchalgorithm Compute a list of neighborhood solutions N in decreasing order of improvement value. Select the first solution of the list as current solution if it is non-tabu or tabu but better than the best solution. (if no such solution then the algorithm terminates) A k-postman tour If current solution value < best solution value then best solution = current solution.

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The tabu searchalgorithm Compute a list of neighborhood solutions N in decreasing order of improvement value. Select the first solution of the list as current solution if it is non-tabu or tabu but better than the best solution. (if no such solution then the algorithm terminates) If the number of iterations < the max number of iterations then continue this algorithm, or the algorithm terminates. A k-postman tour If current solution value < best solution value then best solution = current solution.

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Computational results • Parameters –tabuTenure = 20 – maxNOfIterationsWithoutImprovement = 100

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Computational results • Twoimprovement index – Let x be the value of the best heuristic solution from [2] – Let y be the best solution value obtained by the tabu search algorithm for all possible configurations of neighborhood structures and improvement procedures. – Let z be the best lower bound obtained by the lower bounds SPT-LB and CPP/k-LB presented in [2]. – Average impr.(%) = (x - y) * 100 / x – Average gap(%) = (y - z) * 100 / y

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Computational results

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A tabu search algorithm for the min max k-chinese postman problem

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