The RPP is to determine in a graphG(V , A, E)a least-cost route traversing a subset R⊆A∪Eof required arcs and edges at least once. Its applications arise in garbage collection, mail delivery, network maintenance, snow removal and meter reading in scarcely populated areas.
Let G1(V1, A1, E1), . . . , Gp(Vp, Ap, Ep) be the p connected components of graphG(V , R)induced by the required arcs and edges (see Figures (7.27) and (7.28)).
The RPP solution can be obtained in two steps.
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Figure 7.27 A mixed graphG(V , A, E).
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Figure 7.28 Connected components induced by the required arcs and edges of graphGin Figure 7.27.
Step 1. Determine a least-cost set of arcsA(a)and edgesE(a)such that the multigraph G(a)=(V , (R∩A)∪A(a), (R∩E)∪E(a))is Eulerian (see Figure 7.29).
Step 2. Determine an Eulerian tour inG(a).
The first step is NP-hard even for directed and undirected graphs, ifp > 1. For p=1, the RPP can be reduced to a CPP. The second step can be solved in polynomial time with the end-pairing procedure. In what follows, the first stage of two constructive heuristics is illustrated for directed and undirected graphs.
Directed rural postman problem. A heuristic solution to the directed RPP can be obtained through the ‘balance-and-connect’ heuristic.
Step 1. Using the procedure employed for the directed CPP, construct a directed symmetric graphG(a)(V , R∪A(a)), by adding toG(V , R)a suitable set of least-
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Figure 7.29 Least-cost Eulerian multigraph associated with graphG(V , R)of Figure 7.27.
cost paths between nonsymmetric vertices. IfG(a)(V , R∪A(a))is connected, STOP,G(a)=G(a)is Eulerian.
Step 2. Letp(1< p⩽p) be the number of connected components ofG(a)(V , R∪ A(a)). Construct an auxiliary undirected graphG(c)=(V(c), E(c)), in which there is a vertexh∈V(c)for each connected component ofG(a), and, between each pair of verticesh, k∈V(c),h=k, there is an edge(h, k)∈E(c). With edge(h, k)is associated a costghkequal to
ghk= min
i∈Vh,j∈Vk{wij +wj i},
wherewij andwj i are the costs of the least-cost paths from vertexi to vertex j and from vertexj to vertexi inG, respectively. Compute the minimum-cost tree T(c)=(V(c), E(c)T )spanning the vertices of graph G(c). Construct a sym- metric, connected and directed graph G(a)(V , R∪A(a)∪A(a)) by adding to G(a)(V , R∪A(a))the set of arcs A(a) belonging to the least-cost paths cor- responding to the edgesET(c)of the treeT(c).
Step 3. Apply, when possible, the shortcuts method (see the Christofides algorithm for the STSP) in order to reduce the solution cost.
The ‘balance and connect’ algorithm is applied to problem represented in Fig- ure 7.30. The directed graph G(V , R)has five connected components of required arcs. At the end of Step 1 (see Figure 7.31),A(a)is formed by arcs (2,1) (the least- cost path from vertex 1 to vertex 2), (3,4) (the least-cost path from vertex 3 to vertex 4), (5,6) (the least-cost path from vertex 5 to vertex 6), (8,9) and (9,11) (the least-cost path from vertex 8 to vertex 11), and (10,7) (the least-cost path from vertex 10 to vertex 7). At Step 2,V(c)= {1,2,3}, and the least-cost paths from vertex 1 to vertex 4 (arc (1,4) and vice versa (arc (4,1)), and from vertex 6 to vertex 7 (arc (6,7)) and vice versa (arcs (7,2), (2,3) and (3,6)) are added to the partial solution (see Figure 7.32).
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Figure 7.30 Directed graphG(V , A).
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Figure 7.31 Directed graphG(a)(V , R∪A(a))obtained at the end of Step 1 of the ‘balance and connect’ algorithm.
Finally, using the end-pairing procedure, the following circuit of cost 379 is obtained:
{(0,2), (2,1), (1,4), (4,5), (5,6), (6,7), (7,8), (8,9), (9,11),
(11,10), (10,7), (7,2), (2,3), (3,6), (6,3), (3,4), (4,1), (1,0)}. It can be shown that in this case the ‘balance and connect’ solution is optimal.
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Figure 7.32 Symmetric and connected directed graphG(a)(V , R∪A(a)∪A(a)) obtained at the end of Step 2 of the ‘balance and connect’ algorithm.
Undirected rural postman problem. A heuristic solution to the undirected RPP can be obtained through the Frederickson procedure.
Step 1. Using the matching procedure illustrated for the undirected CPP, construct an even graphG(a)(V , R∪E(a)). IfG(a)(V , R∪E(a))is also connected,STOP, G(a)=G(a)is Eulerian.
Step 2. Letp(1< p⩽p) be the number of connected components ofG(a)(V , R∪ E(a)). Construct an auxiliary undirected graphG(c)(V(c), E(c)), in which there is a vertexh∈V(c)for each connected component ofG(a), and between each couple of verticesh, k∈V(c),h=k, there is an edge(h, k)∈E(c). With each edge(h, k) is associated a costghkequal to
ghk= min
i∈Vh,j∈Vk{wij},
wherewij is the cost of the least-cost path between verticesiandj inG. Com- pute a minimum-cost treeT(c)=(V(c), ET(c))spanning the vertices of graphG(c). Construct an even and connected graphG(a)(V , R∪E(a)∪E(a))by adding to R∪E(a) the set of edges E(a) (each of which taken twice) belonging to the least-cost chains corresponding to the edgesET(c)of treeT(c).
Step 3. Apply, if possible, the shortcuts method (see the Christofides algorithm for the STSP) in order to reduce solution cost.
Tracon distributes newspapers and milk door-to-door all over Wales. In the same road subnetwork as in the Welles problem, customers are uniformly distributed along some roads (represented by continuous lines in Figure 7.33). The entire demand of the
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Figure 7.33 GraphG(V , E)associated with the Tracon problem (costs are in kilometres).
subnetwork can be served by a single vehicle. By applying the Frederickson heuris- tic, the even and connected multigraphG(a)(V , R∪E(a))shown in Figure 7.34 is obtained. Finally, by using the end-pairing procedure, the following cycle is gener- ated:
{(0,9), (9,12), (12,11), (11,10), (10,8), (8,7), (7,6),
(6,5), (5,4), (4,3), (3,1), (1,3), (3,2), (2,5), (5,6), (6,0) (total cost is 31.3 km). It is worth noting that edge (5,6) is traversed twice without being served. It can be shown that the Frederickson solution is optimal.