Node Edge Arc Routing Problems (NEARP)
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1 Node Edge Arc Routing Problems (NEARP) Nicolas Briot Coconut-LIRMM janvier 0 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
2 What is NEARP? Arc and node routing problem consists of finding a set of routes covering arcs, edges and/or vertices of a graph, which meet conditions. Prins, C., Bouchenoua, S. (00). A memetic algorithm solving the VRP, the CARP and general routing problems with nodes, edges and arcs. In Recent advances in memetic algorithms (pp. 6-8). Springer Berlin Heidelberg. Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
3 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
4 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
5 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
6 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} 0 Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
7 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, demand q; 0; ; 0; 0; 0 ; Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
8 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, c ij c ji, demand q;cost c 0;0 ;0 ;6 0; 0;0 0 ;00 Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
9 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, c ij c ji, q =0 demand q;cost c 0;0 ;0 ;6 q =0 0; q =0 0;0 ;00 0 Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
10 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, c ij c ji, k number of vehicles, q =0 demand q;cost c 0;0 ;0 ;6 q =0 0; q =0 0;0 ;00 0 Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
11 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, c ij c ji, k number of vehicles, W vehicles capacity, q =0 demand q;cost c 0;0 ;0 ;6 q =0 0; q =0 0;0 ;00 0 Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
12 The Capacited General Windy Routing Problem (CGWRP) Let G = (V, E) a graph with : V R = {,, } E R = {(, ), (, ), (, )} q ij q ji, c ij c ji, k number of vehicles, W vehicles capacity, demand q;cost c 0;0 ;00 Find a least-cost set of k routes which satisfied all demands (edges and vertices) and not exceed W. q =0 ;0 ;6 q =0 0; 0;0 q =0 0 NP-hard. Branch and Cut algorithm (Corberan et al. 0). Baldacci, R., Bartolini, E., Laporte, G. (00). Some applications of the generalized vehicle routing problem. Journal of the Operational Research Society, 6(7), Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
13 The Capacited General Windy Routing Problem with ed (CGWRP-ed). Let G = (V, E) an undirected graph with : V R V E R E q ij q ji q ij = q ji, c ij c ji, k number of vehicles, W vehicles capacity, demand q;cost c 0;0 0;00 Find a least-cost set of k routes which satisfied all demands (edges and vertices) and not exceed W. q =0 ;0 ;6 q =0 0; 0 0;0 q =0 0 NP-hard. no specific algorithm. Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
14 Capacited Arc Routing Problem (Golden and Wong 98) Let G = (V, E) an undirected graph with : V R V V R = E R E q ij q ji q ij = q ji, c ij c ji, k number of vehicles, W vehicles capacity, demand q;cost c 0;0 0;00 Find a least-cost set of k routes which satisfied all demands (edges) and not exceed W. NP-hard (even <. approx). Branch and cut algorithm (belenguer and benavent 99). Several lower bound and heuristics (Pearn 989, 99) have been proposed. /-approximation for CARP (Wohlk 008). ;0 ;6 0; 0 0;0 0 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
15 Rural Postman Problem (Orloff 97) Let G = (V, E) an undirected graph with : V R V V R = E R E q ij q ji q ij = q ji, demand q;cost c ;0 ;6 ; c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, ;0 0 Find the shortest route which visit all required edges. NP-hard. Branch and Bound (Frederickson 979,Christofides 986). On mixed graph (Stacker Crane Problem) : 9/ approx (Fredericks et al. 978). Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 6 /
16 Chinese Postman Problem (CPP) (Kwan 96) Let G = (V, E) an undirected graph with : V R V V R = E R E E R = E q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, demand q;cost c ;0 ;6 ; ;0 Find the shortest total travelling route length whereby every edge is visited at least one time. Polynomial (undirected : Matching-based algorithm (Edmonds and Jonhson 97), directed : Flow algorithm (Orloff 97,..). Mixed : NP-hard, Branch and Cut (Grotschel and Win 99). Heuristic / approx (Frederickson 979). Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 7 /
17 Eulerian circuit (Koningsberg bridge problem : Euler 7) Let G = (V, E) an undirected graph with : V R V V R = E R EE R = E q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, demand q;cost c Find a route that visits all edges exactly once. Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 8 /
18 Eulerian circuit (Koningsberg bridge problem : Euler 7) A connected graph has an Euler circuit iff every vertex has an even degree.(tijms 00) It is possible to find an Euler path iff 0 or vertices have an odd degree.(tijms 00) Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 9 /
19 Eulerian circuit (Koningsberg bridge problem : Euler 7) Let G = (V, E) an undirected graph with : V R V V R = E R EE R = E q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, demand q;cost c Find a route that visits all edges exactly once. Polynomial. Undirected : end-pairing algorithm (Edmonds and Johnson 97). Directed : Spanning arborescence algorithm (De Bruijn 9) Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 0 /
20 No problem. (Briot 0) Let G = (V, E) an undirected graph with : V R V V R = E R EE R = q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, Find a route that visits no edges. Constant. Undirected : constant. Directed : constant. Mixed : constant Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
21 Hamiltonian circuit (Hamilton 97). Let G = (V, E) an undirected graph with : V R V V R = V R = V E R EE R = EE R = q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, demand q;cost c Find a circuit or cycle that visits each vertex exactly once (except for one). If E and degree(v) + degree(w) E (v, w V, (v, w) / E) then there is a Hamilton circuit. NP-complete : inclusion-exclusion principe (A. Bjorklund 0) with O(.67 n ). In bipartite graph : O(. n ). If the maximum degree of graph is three O(. n ) ( Iwama,et al. (007)). Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
22 Traveling Salesman Problem (Menger 90) Let G = (V, E) an undirected graph with : V R V V R = V R = V E R EE R = EE R = q ij q ji q ij = q ji, c ij c ji c ij = c ji, k = number of vehicles, W vehicles capacity, demand q;cost c ;0 ; 0 ;00 ; Find the shortest possible route that visits each vertex exactly once (except the origin vertex). NP-hard. Exact algorithm : Concorde (Hahsler, Michael ; Hornik, Kurt (007)). Heuristic : Lin, Shen and Kernighan (97). 7/9 approx for Max-TSP (Paluch 009). Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
23 Vehicle Routing Problem (Dantzig and Ramsey 99) Let G = (V, E) an undirected graph with : V R V V R = V R = V E R EE R = EE R = q ij q ji q ij = q ji, c ij c ji c ij = c ji, demand q;cost c ;0 ; 0 ; k number of vehicles, W vehicles capacity, ;00 0 Find at most k tours that visit all vertex that have a total minimal length. NP-hard. Branch and Bound. With CP (De Backer 000). Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
24 Capacited Vehicle Routing Problem Let G = (V, E) an undirected graph with : V R V V R = V R = V demand q;cost c q =0 E R EE R = EE R = q = ;0 q =0 q ij q ji q ij = q ji, c ij c ji c ij = c ji, ; 0 ; k number of vehicles, W vehicles capacity, q =0 ;00 q = 0 Find k tours that visit all vertex that have a total minimal length and not exceed W. NP-hard. Integer Linear Programming. Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
25 Bin Packing problem Let G = (V, E) an undirected graph with : V R V V R = V R = V q =0 E R EE R = EE R = q = q =0 q ij q ji q ij = q ji, c ij c ji c ij = c ji, k number of vehicles, W vehicles capacity, q =0 q = In the bin packing problem, objects of different volumes must be packed into a finite number of bins or containers each of volume W in a way that minimizes the number of bins used NP-hard. Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 6 /
26 Asymetric Capacited Vehicle Routing Problem (ACVRP) Let G = (V, E) an undirected graph with : q =0 V R V V R = V R = V demand q;cost c ;0 E R EE R = EE R = ;0 ;6 q =0 q ij q ji q ij = q ji, ; c ij c ji, k number of vehicles, ;0 0 W vehicles capacity, q =0 ;00 Find k tours that visit all vertex that have a total minimal length and not exceed W. NP-hard.? Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 7 /
27 The Capacited General Windy Routing Problem with ed (CGWRP-ed). q =0 Let G = (V, E) an undirected graph with : demand q;cost c 0;0 V R V E R E q ij q ji q ij = q ji, ;0 ;6 0; 0 q =0 c ij c ji, k number of vehicles, W vehicles capacity, q =0 0;0 0;00 0 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 8 /
28 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 9 /
29 Many problems on Arc Routing Problem Mail delivery, Road sweeping, Road gritting, School bus, Garbage collection, Meter reading,... Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 0 /
30 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
31 Many problems on Node Routing Problem VRP CVRP Open VRP VRP with time-window VRP Heterogeneous fleet VRP Multi-Depot... Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
32 Nicolas Briot (Coconut-LIRMM) Node Edge Arc Routing Problems (NEARP) janvier 0 /
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