Leticia Vargas under the supervision of

Transcription

1 A Dynamic Programming Operator for Meta-heuristics to Solve Vehicle Routing Problems with Optional Visits Leticia Vargas under the supervision of Nicolas Jozefowiez Sandra U. Ngueveu juin 0 /

2 Overview Introduction Solution approach Selector operator Metaheuristic Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

3 Introduction Thesis focus: Vehicle Routing Problems with Optional Visits (VRPOV) Overall goal Construct an optimal tour through a subset of vertices subject to a set of constraints. Dierence with VRP? Vehicle routing problems require that all vertices of the network be visited. Example of VRPOV tour Example of VRP tour /

4 Introduction VRPOV More dicult to solve than classical VRP. select customers to serve segment them into routes in the case of multi-vehicle problems order them within the route(s) /

5 Introduction Families of VRPOV VRPOV Covering Problems Prot Problems /

6 Introduction Families of VRPOV VRPOV Covering Problems Prot Problems Some customers can be visited, while others must lie close enough to a visited customer. Model situations where unavailable roads or scarce resources prevent from visiting a customer. /

7 Introduction Families of VRPOV VRPOV Covering Problems Prot Problems Some customers can be visited, while others must lie close enough to a visited customer. Model situations where unavailable roads or scarce resources prevent from visiting a customer. There is a level of attractiveness (prot) associated with each customer. Model situations where one has to select which are the most protable/urgent/interesting customers to visit under given constraints. /

8 Introduction Examples of VRPOV Covering Problems Covering Tour Problem (CTP) Covering Salesman Problem Ring Star Problem Generalized TSP m-ctp Cumulative m-ctp (CCTP) /

9 Introduction Examples of VRPOV Covering Problems Covering Tour Problem (CTP) Covering Salesman Problem Ring Star Problem Generalized TSP m-ctp Cumulative m-ctp (CCTP) Prot Problems Traveling Salesman Problems with Prots Orienteering Problem (OP) Protable Tour Problem Prize-Collecting TSP Team Orienteering Problem Capacitated Protable Tour Problem VRP with Private Fleet and Common Carrier Prize Collecting VRP /

10 Introduction Applications of covering problems Application Publication location of post boxes Labbé and Laporte () bimodal distribution systems Current and Schilling () health care delivery Hodgson et al. () design of tours in entertaining industry ReVelle and Laporte () humanitarian logistics Naji-Azimi et al. (0) design of urban patrolling services Oliveira et al. (0) /

11 Introduction Applications of prot problems Application Publication deliver heating fuel where the customer's Golden et al. () tank level sets the urgency of service route customer visits to maximise sales Ramesh and Brown () recruit athletes for college teams Butt and Cavalier () outsource unprotable customers select the suppliers auditors visit in order to maximise recovered inventory claims design personalised tourist routes to maximise the value of the attractions Chu (00) Bolduc et al. (00) Ilhan et al. (00) Vansteenwegen et al. (0) select customers for waste collection Aksen et al. (0) /

12 Introduction Combinatorial Optimisation Problem (COP) A COP P = ( Ψ, f, S) can be dened by a set of variables X = {x,..., x n }; variable domains D,..., D n ; set of constraints Ψ; an objective function f to be optimized; the set of all possible feasible solutions (search space) S = {s = {(x, v ),..., (x n, v n )} v i D i, s satises all the constraints}. To solve a COP one has to nd the globally optimal feasible solution s S, that is f (s ) f (s) s S. /

13 Introduction Thesis Main goal Develop a heuristic methodology that addresses the solution of VRPOV in a unied manner. Tasks completed to achieve the goal Implement a general-purpose operator which can eciently select the optimal set of customers to visit by solving a sub-problem using dynamic programming. Implement a generic meta-heuristic which embeds the operator. Apply the methodology proposed to solve several VRPOV: CTP, m-ctp, Cumulative CTP, and OP. 0 /

14 Introduction Solution approach Selector operator Metaheuristic Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

15 Solution approach General principle Route rstcluster second approach (Beasley, ) Route-rst Construction of the giant tour Travelling Salesman Problem /

16 Solution approach General principle Route rstcluster second approach (Beasley, ) Route-rst Construction of the giant tour Cluster-second Splitting of the giant tour Travelling Salesman Problem Shortest Path Problem /

17 Solution approach Route rstcluster second approach It was not adequately recognized (Laporte and Semet, 00). Many successful meta-heuristics have been implemented using it (Prins et al., 0). /

18 Solution approach Route rstcluster second approach It was not adequately recognized (Laporte and Semet, 00). Many successful meta-heuristics have been implemented using it (Prins et al., 0). These procedures explore only the space of giant tours, but evaluate them using a splitting operator. Reason for their success: smaller solution space is searched (space of giant tours vs space of VRP solutions). /

19 Solution approach Route rstcluster second approach It was not adequately recognized (Laporte and Semet, 00). Many successful meta-heuristics have been implemented using it (Prins et al., 0). These procedures explore only the space of giant tours, but evaluate them using a splitting operator. Reason for their success: smaller solution space is searched (space of giant tours vs space of VRP solutions). Nowadays, it is known as Order rstsplit second. /

20 Solution approach How to solve each stage? ) ordering phase handled by a generic meta-heuristic. giant tour (σ) σ 0 σ σ σ σ σ /

21 Solution approach How to solve each stage? ) ordering phase handled by a generic meta-heuristic. giant tour (σ) σ 0 σ σ σ σ σ ) splitting phase solved by the new Selector operator. visited nodes,, σ 0 σ σ σ σ σ skipped nodes, /

22 Introduction Solution approach Selector operator Metaheuristic Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

23 Solution approach Selector: Principle Formulates the problem of selecting the best customers as giant tour (σ) RCESPP optimal subsequence Goal of Resource Constrained Elementary Shortest Path Problem Find the elementary least cost path from a source node to a target node, such that overall resource usage is within given bounds. /

24 Solution approach Selector: Principle Formulates the problem of selecting the best customers as giant tour (σ) RCESPP optimal subsequence Problem is NP-Hard (Dror, ). The standard approach to solve it is dynamic programming. It can be solved in pseudo-polynomial time (Feillet et al., 00). Selector adapts the algorithm of Desrochers () (labelling-reaching algorithm). Selector includes no additional side restrictions like adjacency, for example. /

25 Solution approach Dynamic programming Recurrence Relation Iterative Process Sequence of Partial Solutions (steps) Local Decisions (state) Dene: X k, state at step k f(k, x), partial optimum for x X k c(y, x), cost of transition y x Algorithm: f(0, x) = cost(initial state) for k = 0,..., n do f(k +, x) = min y Xk {f(k, y) + c(y, x)}; store y x return X n Problem solved must exhibit:. Bellman's principle of optimality.. Space of subproblems must be polynomial in the input size. /

26 Solution approach Similar operators Split (Prins, 00): Finds optimal set of routes in CVRP. Solves a Shortest Path Problem, polynomial algorithm. /

27 Solution approach Similar operators Split (Prins, 00): Finds optimal set of routes in CVRP. Solves a Shortest Path Problem, polynomial algorithm. Bouly et al. (00): Selects m routes that satisfy tour length constraint in TOP. Uses a PERT/CPM method, polynomial algorithm. Restricts route must be formed by adjacent customers. /

28 Solution approach Similar operators Split (Prins, 00): Finds optimal set of routes in CVRP. Solves a Shortest Path Problem, polynomial algorithm. Bouly et al. (00): Selects m routes that satisfy tour length constraint in TOP. Uses a PERT/CPM method, polynomial algorithm. Restricts route must be formed by adjacent customers. Vidal et al. (0): Finds optimal set of routes in routing problems with prots. Solves a RCESPP, pseudo-polynomial algorithm. /

29 Solution approach How it works: auxiliary graph /

30 Solution approach How it works: auxiliary graph /

31 Solution approach How it works: auxiliary graph /

32 Solution approach How it works: auxiliary graph /

33 Solution approach How it works: auxiliary graph /

34 Solution approach How it works: extension of labels Purpose: store partial results and keep accounting of used resources. /

35 Solution approach How it works: extension of labels Purpose: store partial results and keep accounting of used resources. Every node receives several labels. /

36 Solution approach How it works: extension of labels Purpose: store partial results and keep accounting of used resources. Every node receives several labels. A label represents a partial path from the depot σ 0 up to node σ ν. λ = [ζ, ν, R] ζ = travel cost (Euclidean distance) ν = index of last visited vertex R = resource consumption /

37 Solution approach How it works: extension of labels Purpose: store partial results and keep accounting of used resources. Every node receives several labels. A label represents a partial path from the depot σ 0 up to node σ ν. λ = [ζ, ν, R] ζ = travel cost (Euclidean distance) ν = index of last visited vertex R = resource consumption Prune redundant labels with the aid of dominance rules. /

38 Solution approach How it works: extension of labels Purpose: store partial results and keep accounting of used resources. Every node receives several labels. A label represents a partial path from the depot σ 0 up to node σ ν. λ = [ζ, ν, R] ζ = travel cost (Euclidean distance) ν = index of last visited vertex R = resource consumption Prune redundant labels with the aid of dominance rules. Prevent the creation of labels by verifying the feasibility of the partial path (Feillet et al., 00) /

39 Solution approach Extend λ[0, 0, 0]: extension may visit or skip 0 /

40 Solution approach Extend λ[0, 0, 0]: extension may visit or skip 0 /

41 Solution approach Extend λ[0, 0, 0]: extension may visit or skip 0 /

42 Solution approach Extend λ[0, 0, 0]: extension may visit or skip 0 nodes are skipped when redundant or when its visitation violates a restriction /

43 Solution approach Extend λ[0, 0, 0]: extension may visit or skip 0 nodes are skipped when redundant or when its visitation violates a restriction /

44 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction /

45 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction /

46 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction /

47 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction search in that trayectory is abandoned when ζ(λ) > Bound best /

48 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction search in that trayectory is abandoned when ζ(λ) > Bound best /

49 Solution approach Extend λ[0, 0, 0]: extension may visit or skip nodes are skipped when redundant or when its visitation violates a restriction search in that trayectory is abandoned when ζ(λ) > Bound best /

50 Solution approach Extend skipping dened sequences of nodes 0 Label extended is the one of lowest cost. Extension starts at ν +, where ν is the last visited node. Possible extensions = n ν. /

51 Solution approach Extend skipping dened sequences of nodes 0 0 Label extended is the one of lowest cost. Extension starts at ν +, where ν is the last visited node. Possible extensions = n ν. /

52 Solution approach Extend skipping dened sequences of nodes Label extended is the one of lowest cost. Extension starts at ν +, where ν is the last visited node. Possible extensions = n ν. /

53 Solution approach Extend skipping dened sequences of nodes Label extended is the one of lowest cost. Extension starts at ν +, where ν is the last visited node. Possible extensions = n ν. /

54 Introduction Solution approach Selector operator Metaheuristic Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

55 Solution approach ALNS: Adaptive Large Neighborhood Search (Ropke & Pisinger, 00) /

56 Solution approach ALNS: Adaptive Large Neighborhood Search (Ropke & Pisinger, 00) Pool of sub-heuristics chosen based on their past performance. /

57 Solution approach ALNS: Adaptive Large Neighborhood Search (Ropke & Pisinger, 00) Pool of sub-heuristics chosen based on their past performance. Three destruction sub-heuristics Shaw Removal, Shaw () Worst Removal, Ropke & Pisinger (00) Random Removal /

58 Solution approach ALNS: Adaptive Large Neighborhood Search (Ropke & Pisinger, 00) Pool of sub-heuristics chosen based on their past performance. Three destruction sub-heuristics Shaw Removal, Shaw () Worst Removal, Ropke & Pisinger (00) Random Removal Three construction sub-heuristics Best Greedy First Greedy Regret-k /

59 Solution approach ALNS: Adaptive Large Neighborhood Search (Ropke & Pisinger, 00) Pool of sub-heuristics chosen based on their past performance. Three destruction sub-heuristics Shaw Removal, Shaw () Worst Removal, Ropke & Pisinger (00) Random Removal Three construction sub-heuristics Best Greedy First Greedy Regret-k An outer meta-heuristic guides the search: simulated annealing, which allows to accept a worse solution. /

60 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Selector(σ new) yes UB < αub best no decide acceptance of solution update scores /

61 Solution approach Outline of general algorithm -opt(σ 0) /

62 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) /

63 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) /

64 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Search_UB is a reduced version of Selector /

65 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Search_UB is a reduced version of Selector UB < αub best α {.00,.0,.} /

66 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Search_UB is a reduced version of Selector Selector(σ new) yes UB < αub best α {.00,.0,.} /

67 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Search_UB is a reduced version of Selector Selector(σ new) yes UB < αub best α {.00,.0,.} no decide acceptance of solution update scores /

68 Solution approach Outline of general algorithm select_dh(p r) σnew DH(σ current) -opt(σ 0) select_ch(p i) σ new CH(σnew) UB Search_UB(σ new) Search_UB is a reduced version of Selector Selector(σ new) yes UB < αub best α {.00,.0,.} no decide acceptance of solution update scores /

69 Introduction Solution approach Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

70 Application to the Covering Tour Problem CTP Covering Tour Problem (Gendreau et al., ) depot V: nodes that can be visited /

71 Application to the Covering Tour Problem CTP Covering Tour Problem (Gendreau et al., ) depot V: nodes that can be visited W: nodes to cover c: covering distance /

72 Application to the Covering Tour Problem CTP Covering Tour Problem (Gendreau et al., ) depot V: nodes that can be visited W: nodes to cover c: covering distance /

73 Application to the Covering Tour Problem Former solution approaches Exact methods Branch and cut algorithm Gendreau et al. () V 00 Approximate methods GENIUS and PRIMAL Gendreau et al. () V 00 GRASP Motta et al. (00) V W 00 Three scatter search methods Baldacci et al. (00) V 00 0 /

74 Application to the Covering Tour Problem Implementation. Label denition. Dominance rule. Feasibility checking. Look ahead mechanism. Extension of a label /

75 Application to the Covering Tour Problem Implementation. Label denition: λ = [ζ, ν, ω] ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W /

76 Application to the Covering Tour Problem Implementation. Label denition: λ = [ζ, ν, ω] ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W ω[i] indicates either: - w i W is covered or - the number of nodes of V that can still be visited to cover it /

77 Application to the Covering Tour Problem Implementation. Label denition: λ = [ζ, ν, ω] ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W ω[i] indicates either: - w i W is covered or - the number of nodes of V that can still be visited to cover it ω updated at every label extension /

78 Application to the Covering Tour Problem Implementation. Label denition: λ = [ζ, ν, ω] ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W ω[i] indicates either: - w i W is covered or - the number of nodes of V that can still be visited to cover it ω updated at every label extension. Feasibility checking: makes certain a node can be skipped. /

79 Application to the Covering Tour Problem Implementation. Label denition: λ = [ζ, ν, ω] ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W ω[i] indicates either: - w i W is covered or - the number of nodes of V that can still be visited to cover it ω updated at every label extension. Feasibility checking: makes certain a node can be skipped. - Let Ω be the set of nodes w not covered by λ - For each w i Ω feasible labels yield ω[i] > 0 /

80 Application to the Covering Tour Problem Implementation. Dominance rule: A label λ = [ζ, ν, ω ] dominates a label λ = [ζ, ν, ω ] with λ λ if and only if: /

81 Application to the Covering Tour Problem Implementation. Dominance rule: A label λ = [ζ, ν, ω ] dominates a label λ = [ζ, ν, ω ] with λ λ if and only if: i. ν = ν /

82 Application to the Covering Tour Problem Implementation. Dominance rule: A label λ = [ζ, ν, ω ] dominates a label λ = [ζ, ν, ω ] with λ λ if and only if: i. ν = ν ii. ζ ζ /

83 Application to the Covering Tour Problem Implementation. Dominance rule: A label λ = [ζ, ν, ω ] dominates a label λ = [ζ, ν, ω ] with λ λ if and only if: i. ν = ν ii. ζ ζ iii. Ω Ω Ω = set of nodes w already covered /

84 Application to the Covering Tour Problem Implementation. Look ahead mechanism: - notices ahead of time that a node is the only one that remains to cover a set - allows to reduce the number of labels created σ i σ i σ j σ k w w w Ω j = {w } Ω k = {w, w, w } sentinel w /

85 Application to the Covering Tour Problem Implementation. Extension of a label: at every step it veries node is non-redundant /

86 Application to the Covering Tour Problem Implementation. Extension of a label: at every step it veries node is non-redundant ζ(λ j ) < UB best /

87 Application to the Covering Tour Problem Implementation. Extension of a label: at every step it veries node is non-redundant ζ(λ j ) < UB best label is non-dominated /

88 Application to the Covering Tour Problem Implementation. Extension of a label: at every step it veries node is non-redundant ζ(λ j ) < UB best label is non-dominated These tests seek to control label proliferation. /

89 Application to the Covering Tour Problem Performance enhancements. Computation of a lower bound. Bidirectional search with bounding /

90 Application to the Covering Tour Problem Computation of a lower bound Let λ = [ζ, σ i, ω], /

91 Application to the Covering Tour Problem Computation of a lower bound Let λ = [ζ, σ i, ω], h(λ) be LB on cost incurred visiting only nodes in ϱ = (σ i+,..., σ n ), /

92 Application to the Covering Tour Problem Computation of a lower bound Let λ = [ζ, σ i, ω], h(λ) be LB on cost incurred visiting only nodes in ϱ = (σ i+,..., σ n ), µ(λ) = ζ + h(λ) be the total estimated cost. /

93 Application to the Covering Tour Problem Computation of a lower bound Let λ = [ζ, σ i, ω], h(λ) be LB on cost incurred visiting only nodes in ϱ = (σ i+,..., σ n ), µ(λ) = ζ + h(λ) be the total estimated cost. h(λ) is found solving a Fractional Knapsack Problem. /

94 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } /

95 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } prot of object = number of nodes w j W the object covers /

96 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } prot of object = number of nodes w j W the object covers weight of object = min in-going arc weight + min out-going arc weight (arcs in giant tour) /

97 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } prot of object = number of nodes w j W the object covers weight of object = min in-going arc weight + min out-going arc weight (arcs in giant tour) capacity of knapsack = Ω where Ω = set of nodes not covered yet /

98 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } prot of object = number of nodes w j W the object covers weight of object = min in-going arc weight + min out-going arc weight (arcs in giant tour) capacity of knapsack = Ω where Ω = set of nodes not covered yet Problem is solved with a greedy approach (Dantzig, ) /

99 Application to the Covering Tour Problem Computation of a lower bound µ(λ) = ζ + h(λ) Fractional Knapsack Problem yields h(λ) objects: set ϱ = {σ i+,..., σ n } prot of object = number of nodes w j W the object covers weight of object = min in-going arc weight + min out-going arc weight (arcs in giant tour) capacity of knapsack = Ω where Ω = set of nodes not covered yet Problem is solved with a greedy approach (Dantzig, ) If µ(λ) UB best, the label is pruned or not extended when retrieved. /

100 Application to the Covering Tour Problem Bidirectional search with bounding (Pohl, ) σ 0 σ σ σ σ σ σ + 0 forward search backward search λ λ λ λ λ λ queues in increasing order of µ(λ) It performs two search processes simultaneously. /

101 Application to the Covering Tour Problem Bidirectional search with bounding (Pohl, ) σ 0 σ σ σ σ σ σ + 0 forward search backward search λ λ λ λ λ λ queues in increasing order of µ(λ) It performs two search processes simultaneously. Shorter paths produce less labels. /

102 Application to the Covering Tour Problem Bidirectional search with bounding (Pohl, ) σ 0 σ σ σ σ σ σ + 0 forward search backward search λ λ λ λ λ λ queues in increasing order of µ(λ) It performs two search processes simultaneously. Shorter paths produce less labels. In each iteration, search having the best cost is the one extended. /

103 Application to the Covering Tour Problem Bidirectional search with bounding (Pohl, ) σ 0 σ σ σ σ σ σ + 0 forward search backward search λ λ λ λ λ λ queues in increasing order of µ(λ) It performs two search processes simultaneously. Shorter paths produce less labels. In each iteration, search having the best cost is the one extended. Labels of opposite search meet when ν forward = ν backward. /

104 Application to the Covering Tour Problem Bidirectional search with bounding (Pohl, ) σ 0 σ σ σ σ σ σ + 0 forward search backward search λ λ λ λ λ λ queues in increasing order of µ(λ) It performs two search processes simultaneously. Shorter paths produce less labels. In each iteration, search having the best cost is the one extended. Labels of opposite search meet when ν forward = ν backward. Termination criterion: µ(λ ) + µ(λ ) UB best and λ λ form a complete solution. /

105 Application to the Covering Tour Problem Computational experiments GB, Intel Core i-0 CPU@.0Ghz, Linux OS bits algorithm coded in C++ ALNS parameters tuned with irace package (López-Ibañez et al., 0) instances derived from TSPLIB costs treated as integer values instances solved with dierent seeds: 0 times number of iterations: 0,000 θ percentage of deviation from the best-known solution running time in seconds T = 0 /

106 * algorithm of Gendreau et al. () / Application to the Covering Tour Problem Results: small and medium-sized instances B&C Selector-ALNS Instance V W Best Based on t(s) θ(%) t(s) Gap(%) kroa kroa kroa krob krob krob kroa kroa kroa kroa krob krob krob krob

107 Application to the Covering Tour Problem Results: large-sized instances,000 ITERATIONS Instance V W MONODIRECTIONAL BIDIRECTIONAL w/bnd Based on θ (%) t (s) θ (%) t (s) lin_ rd_ pcb_ d_ att_ ali_ rat_ θ = average deviation from best value found /

108 Introduction Solution approach Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

109 Application to the Multi-Vehicle Covering Tour Problem m-ctp Multi-Vehicle Covering Tour Problem depot V: nodes that can be visited W: nodes to cover c: covering distance /

110 Application to the Multi-Vehicle Covering Tour Problem m-ctp Multi-Vehicle Covering Tour Problem depot V: nodes that can be visited W: nodes to cover c: covering distance /

111 Application to the Multi-Vehicle Covering Tour Problem m-ctp Multi-Vehicle Covering Tour Problem depot V: nodes that can be visited W: nodes to cover c: covering distance m: number of vehicles p: number of nodes/route q: route length /

112 Application to the Multi-Vehicle Covering Tour Problem Former solution approaches Exact methods Branch and cut algorithm Hà et al. (0) V 00 Branch and price algorithm Lopes et al. (0) V 00 Branch and price algorithm Jozefowiez (0) V 0 Column generation + heuristics Murakami (0) V 00 Approximate methods Classical: savings, sweep, clustering Hachicha et al. (000) V 00 Multi-start heuristic Naji-Azimi et al. (0) V 0 Two-phase hybrid meta-heuristic Hà et al. (0) V 00 Variable neighborhood search Kammoun et al. (0) V 00 /

113 Application to the Multi-Vehicle Covering Tour Problem Methods used as reference Two-phase hybrid meta-heuristic Hà et al. (0) Solves Set Covering Problem to form a set that covers all w W Solves resulting CVRP with Evolutionary Local Search (Prins, 00) Does local search over obtained routes Uses four neighborhood structures: relocation, swapping, merging, and replacing /

114 Application to the Multi-Vehicle Covering Tour Problem Methods used as reference Two-phase hybrid meta-heuristic Hà et al. (0) Solves Set Covering Problem to form a set that covers all w W Solves resulting CVRP with Evolutionary Local Search (Prins, 00) Does local search over obtained routes Uses four neighborhood structures: relocation, swapping, merging, and replacing Variable Neighborhood Search Kammoun et al. (0) Implements Variable Neighborhood Descent, nd-best Uses two neighborhood structures: insertion and swapping /

115 Application to the Multi-Vehicle Covering Tour Problem Implementation. Cost of a label is the best total route cost. m-ctp implementation similar to CTP implementation Main modication: cost function CTP: simple addition m-ctp: modied version of the Split operator (Prins, 00) c() c b() 0 d() b d a() e() a trip trip 0 trip e 0. Giant tour T=(a,b,c,d,e). Optimal splitting, cost 0 ab() cd() a(0) b(0) c(0) d(0) e(0) 0 bc() bcd(0) de(). Auxiliary graph H of possible trips for Q=0 (shortest path in boldtype) /

116 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 /

117 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] /

118 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] /

119 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] /

120 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] /

121 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] /

122 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 /

123 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ /

124 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ V [] /

125 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ V [] /

126 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ V [] V [] /

127 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ V [] V [] /

128 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ σ V [] V [] /

129 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ σ V [] V [] V [] /

130 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ σ V [] V [] V [] /

131 Application to the Multi-Vehicle Covering Tour Problem Implementation: cost of a label demand = i V p = and q = + n = (Split) 0 V [] V [] V [] σ 0 σ σ σ σ V [] V [] V [] V [] /

132 Application to the Multi-Vehicle Covering Tour Problem Implementation. Label denition remains almost the same: λ = [ζ, ν, ω A]. ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W A = vectors dened by the Split algorithm /

133 Application to the Multi-Vehicle Covering Tour Problem Implementation. Label denition remains almost the same: λ = [ζ, ν, ω A]. ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W A = vectors dened by the Split algorithm. m is a decision variable. /

134 Application to the Multi-Vehicle Covering Tour Problem Implementation. Label denition remains almost the same: λ = [ζ, ν, ω A]. ζ = travel cost (Euclidean distance) ν = index of last visited vertex ω = vector indexed by the nodes of W A = vectors dened by the Split algorithm. m is a decision variable.. All customers have the same demand. /

135 Application to the Multi-Vehicle Covering Tour Problem Implementation. Dominance rule adds a fourth test. A label λ = [ζ, ν, ω A] dominates a label λ = [ζ, ν, ω A] with λ λ if and only if: i. ν = ν ii. ζ ζ iii. Ω Ω iv Q Q Ω = set of nodes w already covered Q = remaining capacity of vehicle 0 /

136 Application to the Multi-Vehicle Covering Tour Problem Results: medium-sized instances Best values found for 00-node instances when p constant and q = + Instance V W p α m-selector-alns Hà et al. Kammoun et al. Based on θ (%) t (s) Opt t (s) θ (%) t (s) θ (%) t (s) kroa krob kroa krob /

137 Application to the Multi-Vehicle Covering Tour Problem Results: medium-sized instances Best values found when q ϕ + ϑ and p = + Instance V W ϑ α m-selector-alns Jozefowiez Based on t (s) θ (%) Opt t (s) kroa kroa krob krob kroa kroa krob krob ϕ = max v j V\{v {c 0,j} 0 } c 0,j is the cost of reaching vertex v j from the depot /

138 Introduction Solution approach Application to the Covering Tour Problem Application to the Multi-Vehicle Covering Tour Problem Conclusions and Perspectives /

139 Conclusions and Perspectives Contributions The (m)selector operator and its performance improvement features /

140 Conclusions and Perspectives Contributions The (m)selector operator and its performance improvement features The unied heuristic methodology that incorporates the operators to solve VRPOV /

141 Conclusions and Perspectives Contributions The (m)selector operator and its performance improvement features The unied heuristic methodology that incorporates the operators to solve VRPOV Novel ways proposed to solve the problems treated: CTP, m-ctp, CCTP, and OP /

142 Conclusions and Perspectives Future paths Solve other VRPOV /

143 Conclusions and Perspectives Future paths Solve other VRPOV Enhance or change the ALNS /

144 Conclusions and Perspectives Future paths Solve other VRPOV Enhance or change the ALNS Implement graph sparsication /

145 Conclusions and Perspectives Future paths Solve other VRPOV Enhance or change the ALNS Implement graph sparsication Adapt Selector to treat multi-objective VRPOV /

146 Conclusions and Perspectives Publications L. Vargas, N. Jozefowiez, and S.U. Ngueveu. A Selector operator-based adaptive large neighborhood search for the covering tour problem. In Proceedings of the th Learning and Intelligent OptimizatioN Conference, LION, pages 0-, Springer, New York, 0. L. Vargas, N. Jozefowiez, and S.U. Ngueveu. A dynamic programming operator for tour location problems applied to the covering tour problem. Submitted to Journal of Heuristics, 0. Has undergone rst round of revisions. L. Vargas, N. Jozefowiez, and S.U. Ngueveu. A dynamic programming-based meta-heuristic for the m-covering tour problem. Submitted to Computers & Operations Research, 0. L. Vargas, N. Jozefowiez, and S.U. Ngueveu. Solving the orienteering problem using a dynamic programming operator for tour location problems. Technical report. LAAS-CNRS, 0. /

147 /

148 A Dynamic Programming Operator for Meta-heuristics to Solve Vehicle Routing Problems with Optional Visits Leticia Vargas under the supervision of Nicolas Jozefowiez Sandra U. Ngueveu juin 0 /

149 0 Orienteering Problem number inside indicates prot collected 0 /

150 0 Orienteering Problem number inside indicates prot collected L max 0 /

151 Implementation. Label denition: λ = [π, ν, ζ, µ] π = prot collected ν = last visited vertex ζ = travel cost (Euclidean distance) µ = upper bound on complete path prot /

152 Implementation. Label denition: λ = [π, ν, ζ, µ] π = prot collected ν = last visited vertex ζ = travel cost (Euclidean distance) µ = upper bound on complete path prot. A label λ = [π, ν, ζ, µ ] dominates a label λ = [π, ν, ζ, µ ] with λ λ if and only if: /

153 Implementation. Label denition: λ = [π, ν, ζ, µ] π = prot collected ν = last visited vertex ζ = travel cost (Euclidean distance) µ = upper bound on complete path prot. A label λ = [π, ν, ζ, µ ] dominates a label λ = [π, ν, ζ, µ ] with λ λ if and only if: i. ν = ν /

154 Implementation. Label denition: λ = [π, ν, ζ, µ] π = prot collected ν = last visited vertex ζ = travel cost (Euclidean distance) µ = upper bound on complete path prot. A label λ = [π, ν, ζ, µ ] dominates a label λ = [π, ν, ζ, µ ] with λ λ if and only if: i. ν = ν ii. ζ ζ /

155 Implementation. Label denition: λ = [π, ν, ζ, µ] π = prot collected ν = last visited vertex ζ = travel cost (Euclidean distance) µ = upper bound on complete path prot. A label λ = [π, ν, ζ, µ ] dominates a label λ = [π, ν, ζ, µ ] with λ λ if and only if: i. ν = ν ii. ζ ζ iii. π π /

156 0 Performance enhancements. Restrict the number of labels extended. Computation of an upper bound /

157 0 Performance enhancements. Restrict the number of labels extended Search extends only a predened number of the most promising labels. Search nishes when this number is reached. It guarantees computations are restricted by a value. Side eect: search obtains a non-optimal solution.. Computation of an upper bound /

158 Results instance optimum optimum time GLS gap time GRASP gap time Selector gap gap time Fischetti Campos F C F C tsi tsi eil tsi eil rd kroa kroa pr lin rd Average /