Comput Manag Sci (2012) 9:531–542 DOI 10.1007/s10287-012-0154-2 ORIGINAL PAPER
Network design for time-constrained delivery using subgraphs Hui Chen · Ann M. Campbell · Barrett W. Thomas
Received: 23 June 2011 / Accepted: 11 September 2012 / Published online: 27 September 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract Delivery companies are offering an increasing number of time-definite services. Yet, little research has been done that explores the design of delivery networks that can support these types of services. In this paper, we explore such design problems for networks with a specified number of edges B > n − 1, where n is the number of nodes in the problem. We present a two-phase heuristic solution approach that first constructs a network and then improves the network via local search. For the improvement phase, we extend neighborhood structures that have proven effective for tree-structured solutions and also identify a new search neighborhood that takes advantage of specific features of subgraph solutions. We present a computational analysis of our solution approach as well as managerial insights. Keywords
Logistics · Heuristics · Time Definite
1 Introduction As the global economy has weakened, companies looking for supply-chain savings have been shifting away from time-definite services, such as the 10 a.m. delivery guarElectronic supplementary material The online version of this article (doi:10.1007/s10287-012-0154-2) contains supplementary material, which is available to authorized users. H. Chen Delta Airlines, Atlanta, GA, USA e-mail:
[email protected] A. M. Campbell · B. W. Thomas (B) Department of Management Sciences, University of Iowa, Iowa City, IA, USA e-mail:
[email protected] A. M. Campbell e-mail:
[email protected]
123
532
H. Chen et al.
antees made famous by small-package carriers such as FedEx, to less expensive daydefinite services (Maurits de Jonge 2008). By 2010, day-definite deliveries represented 12.1 % of all parcels (under 70 kg) delivered (Datamonitor 2010). While time-definite services are most closely associated with air freight, day-definite services most often use strictly ground transportation such as is offered by UPS (United Parcel Service 2011) or by rail/truck or ship/truck intermodal services (Thuermer 2008). While daydefinite services are growing in popularity, there is little academic literature that addresses how a delivery company should design their network to accommodate time constraints. We propose the minimum sum violation subgraph problem (SVS) to address the design of a time-constrained delivery network. Given a set of delivery time constraints and a graph, we seek a subgraph such that the sum of the deadline violations for all point-to-point travel is minimized. Formally, given a simple, connected, and undirected graph G = (V, E), a flow between nodes vi and v j in V defined by ri j ≥ 0, a service time commitment between nodes vi and v j in V given by Hi j ≥ 0, and an edge budget B, the problem is to identify a spanning subgraph W = (V, E W ) with E W ⊆ E and |E W | ≤ B that minimizes
{max{ri j (dW (vi , v j ) − Hi j ), 0}}.
(1)
vi ∈V v j ∈V
For two nodes vi and v j , the value dW (vi , v j ) represents the service time or time required to travel between vi and v j in a given subgraph W . Because there may be more than one path connecting two nodes in a subgraph, the service time for a pair of nodes is based on the length of the shortest path between the two nodes in the subgraph. If B = n − 1, where n = |V |, the solution will be a spanning tree. For this case, the problems reduce to the tree-structured network design problems discussed in Chen et al. (2008, 2009), and Chen (2008). In this study, we consider only problems with B > n − 1 and ri j = r . We note that we ignore the flow considerations as a result of the authors’ experience with tree-structured problems where the solution methods performed equally well when flow was included and when it was not. See Chen et al. (2008) for details. It is important to consider when B is greater than n − 1 for several reasons. For one, this case is more typical in practice. Second, it can be helpful for practitioners to understand the value, in terms of improved service, of adding additional edges/connections to the network. Because of the costs of hiring a driver and leasing a truck, each edge represents a significant investment. If we can develop efficient solution methods, managers can more easily experiment with different B values and better evaluate the tradeoffs between these investments and their impact on customer service. Third, we want to develop an understanding of the structure of the resulting subgraphs. With B greater than n − 1, there will be cycles, so we want to develop insights on where these cycles should be to create the most impact on customer service. As the analogous tree-structured network design problem addressed in Chen et al. (2008) is shown to be N P-Complete, the same result holds for the SVS. Further, Chen et al. (2008) demonstrates computationally that exact solutions to even the treestructured problem are limited to very small problem sizes. We focus this paper on
123
Network design for time-constrained Delivery using subgraphs
533
heuristic solution methods and introduce a two-phase solution method. The first phase constructs an initial solution and the second phase uses local search to improve the solution. For the improvement phase, we extend neighborhood structures that have proven effective for tree-structured solutions and identify a new search neighborhood that takes advantage of specific features of subgraph solutions. To the best of our knowledge, this is the first paper to study the question of subgraphstructured network design in the presence of time constraints. In the literature, there are three related minimum spanning subgraph design problems. The first is the Shortest Total Path Length Spanning Subgraph Problem (see Johnson et al. (1978); Plesník (1984), and Wu et al. (2002)), the second is the Minimum Subgraph with Diameter Constraint Problem (see Plesník (1981); Caccetta et al. (2001), and Rosenberg (2005)), and and the third is a network (or graph) augmentation problem called the Minimum Augmentation with Diameter Constraint Problem (see Li et al. (1992) and Rosenberg (2005)). Related tree-structured problems are discussed at length in Chen et al. (2008, 2009), and Chen (2008). Section 2 discusses our two-phase solution approach, including the new and modified local search neighborhoods used within this approach. Section 3 presents the results of our computational study. We examine the quality of our two-phase approach and try to understand the structure of solutions for SVS. Section 4 summarizes our managerial insights for subgraphs. 2 Solution methodology for SVS While the N P-Completeness of the SVS does not eliminate the possibility of solving it exactly, the complexity result coupled with an inability to solve even nine customer tree-structured problems (see Chen et al. (2008)) suggest heuristic approaches are necessary. Consequently, in this section, we present a heuristic solution approach for the SVS. Our approach proceeds in two phases. The first phase constructs an initial subgraph, and the second phase applies local search to improve the initial subgraph. We first describe two methods that can be used to generate initial solutions. We then present the post-construction local search approach and the neighborhoods utilized within this approach. 2.1 Initial subgraph generation In this section, we detail two heuristics for generating initial subgraphs. The first method builds on the authors’ previous work. As discussed in the introduction, the addition of edges to a delivery network can be an expensive proposition, and delivery companies consequently want as sparse of a graph as possible. If the company has a limited edge budget, it should make sense computationally to build the subgraph from a tree. This was one of the key motivations for our study as it would show the solution for the SVT is a good starting point for solving the SVS. Our Edge Addition Heuristic (EAH) begins by creating a tree. For this, we use the method proposed in Chen et al. (2008) for solving minimum sum violation tree problems. To these tree solutions, we iteratively add the required number of edges to construct a subgraph.
123
534
H. Chen et al.
For these additions, we use a best-improving criterion. At each iteration i, we iterate through all of the edges in e ∈ E \ E Wi , where Wi is the current subgraph and add to Wi the edge that best improves Eq. 1. We also consider an alternative construction heuristic. The second construction heuristic is inspired by Caccetta et al. (2001). Caccetta et al. (2001) describe four heuristic algorithms for the MSD problem that start from the original graph G and iteratively remove edges. Similarly, our Edge Deletion Heuristic (EDH) iteratively deletes a selected edge from the solution until the solution satisfies the edge number budget constraint |E W | ≤ B. Given the graph G as the initial solution W0 = G, at each iteration i, EDH checks all the edges e ∈ E Wi−1 and generates Wi by deleting the edge from Wi−1 that least increases Eq. 1. 2.2 Post-construction improvement In this section, we discuss the local search heuristic designed to improve the initial solutions returned by the subgraph construction heuristics. As in Chen et al. (2008), we use a form of the variable neighborhood descent (VND) method. A VND is a local search method that employs a sequence of search neighborhoods in order to escape local minima in a search. In our method, four of these neighborhoods are designed to exploit the structure of SVS solutions. These neighborhoods are Move-one-hub (MOH), Add-one-hub (AOH), Merge-two-hubs (MTH), and Link-two-hubs (LTH). The fifth neighborhood is the Edge 1-Switch neighborhood. The first three neighborhoods were originally designed to exploit structure in tree-structured problems. These neighborhoods were introduced Chen et al. (2008) and discussion of adaptations for subgraphs can be found in Chen et al. (2012). Also in Chen et al. (2012), we discuss a general network neighborhood, the Edge 1-Switch neighborhood. A discussion of LTH can be found in the subsequent subsection. In our implementation, we begin with an initial solution generated using one of the methods described above. We then iterate through the structure-based neighborhoods in the order MOH, AOH, MTH, LTH. We continue to iterate through the structurebased neighborhoods until a cycle through all of them yields no improvement. We then employ Edge 1-Switch until it yields no improvement. Once Edge 1-Switch converges, we return again to the structure-based neighborhoods. We pass through the structure-based and Edge 1-Switch neighborhoods until neither yields an improving solution. 2.3 Link-two-hubs (LTH) neighborhood This section introduces the LTH neighborhood, which is designed to exploit solution structure specific to subgraphs. In the rest of the paper, we define a hub node as any node whose degree is two or higher and a leaf node as a node whose degree is one in a particular solution. In our preliminary computational tests, the cycles in the subgraph-structured solutions tend to connect high degree nodes. Consequently, we propose a new neighborhood that rearranges the connections between hub nodes in a subgraph. To rearrange
123
Network design for time-constrained Delivery using subgraphs
535
Fig. 1 Sample link-two-hubs move
the connections, we delete an edge connecting two hub nodes in the current subgraph and select the edge (that is not yet in the subgraph) between two hub nodes that best improves the sum of violations as in Fig. 1. Algorithm 1 provides pseudocode for our implementation of the LTH neighborhood. The set V represents all the hub nodes in W . The set E V represents the set of edges in G connecting two hub nodes in V . The V represents the set of edges in W connecting two hub nodes in V . The algorithm EW ensures that the subgraph remains connected. For LTH, the neighborhood for each hub edge is searched, and the best move from each hub edge neighborhood is made. Thus, one iteration of LTH Algorithm may consist of multiple local search moves. Our experiments in Sect. 3 will verify the quality of this new neighborhood for solving the SVS.
3 Computational results for SVS This section discusses both the results of experiments designed to determine the best method for generating SVS solutions as well as the results of an analysis that seeks to identify general structural properties of SVS solutions. In this research, we focus our computational testing on instances US4, US6, US8, US10, US11, US12, EU4, EU6, EU8, EU10, EU11, and EU12 first proposed in Chen et al. (2008). A detailed description of all instances can be found therein. All of the instances are derived from a data set of the 150 largest cities in the U.S. and 147 E.U. cities (Daskin 1997).
123
536
H. Chen et al. Deletion Then Local Search
Tree-based Solution Then Local Search
5000 4500
Violations
4000 3500 3000 2500 2000 1500 1000 500 0
0
1
2
3
4
5
6
7
8
9
10
# of edges more than (n-1)
Fig. 2 The solution values associated with the addition of arcs to subgraphs for EU12
The datasets differ in the rules used to determine the time deadlines between cities and the number of connections between cities (sparsity/density). We focus on subgraphs with edge budgets of n − 1 plus three and six edges. That is, our subgraphs will have either three or six more edges than a tree. We focus on these limited edge budgets because preliminary tests indicate that larger edge budgets have minimal effect on solution values. For example, Fig. 2 shows the after local-search solution values for both the EDH- and EAH-initialized subgraphs for the set EU12. In both cases, the greatest improvements in solution value occur within the addition of the first three edges beyond the tree solution. After the addition of the sixth edge, the EDH-initialized solution does not change at all while the tree-initialized solution changes only minimally. This pattern is representative of results for all of our datasets. A manager can do a similar study with his or her data to help understand the dataset specific tradeoffs between service and number of edges. The tests are run using an Intel Pentium D CPU 3.20 GHz, 1024 KB cache size, and 2 GB RAM. Runtimes reported are in terms of seconds. 3.1 Results In our tests, we seek to answer four questions: 1. 2. 3. 4.
What method provides the best initial solutions for the local search? Does a particular initialization method lead to improved final solution values? Does a particular initialization method lead to improved runtime? Does the Link Hub neighborhood offer value?
Figure 3 summarizes the numerical results of the two-phase heuristic for the three additional edge instances. For brevity, the results with three additional edges are presented here, and the results with six additional edges as well as detailed data are included in Sect. 2 of Chen et al. (2012). Results for six additional edges all yielded similar conclusions. Figure 3 presents percentage differences for each initializing
123
Network design for time-constrained Delivery using subgraphs
(a)
Difference in Solution Values for Initial Solu-
(b)
537
Difference in Runtimes for Initial Solution
tion
(c)
Difference in Solution Values for Improved Solution
(d)
Difference in Runtimes for Improved Solution
Fig. 3 A comparison of EAH and EDH in terms of solution values and runtimes
method, labeled EAH and EDH. As noted previously, exact solutions of SVS problems are not obtainable, and the value of a solution can be measured only relative to the heuristics proposed in this paper. Thus, for all solution values and runtimes, a percentage difference between the EAH solutions and the EDH solutions are computed as (EDH Value − EAH Value) . EAH Value
(2)
Thus, positive percentages reflect that EDH has a higher cost or runtime than EAH, and negative percentages reflect that EAH has a higher cost or runtime than EDH. Figure 3a shows that the tree-based initial solution (EAH) values are always better than those generated through EDH. This improved performance is likely due to the greedy nature of EDH and the fact that EAH modifies high-quality tree-structured solutions. However, Fig. 3b shows that the EAH initial solutions require more runtime in seven of 12 cases, while EDH requires more than an order of magnitude more runtime for the other five datasets. Despite the initial solution values, Fig. 3c show that EDH-initialized solutions lead to better solutions after the application of the local search phase. For three edges, the post-construction improved and EDH-initialized solutions are better than the postconstruction improved and EAH-initialized solutions in 11 of the 12 instances. The largest percentage differences come from the datasets with the least restrictive time
123
538
H. Chen et al.
Fig. 4 Comparison of initial solution number of hubs for three edge budgets
commitments but these are also the instances with the lowest initial solution values. For some of these instances, a reduction in violation of 100, for example, can translate to a large percentage difference. The reason for the success of the EDH-initialized solutions is likely found in the number of hubs before improvement. Figure 4 presents the percentage differences of the number of hubs associated with each of the initial solutions for the three edge budget. The six edge results yield similar conclusions. The percentage differences are computed analogously to Eq. (2). A positive percentage indicates that the EDHinitialized solution has more hubs. The Figure shows that, for the three edge budgets, the initial solution generated by EDH has more hubs than EAH in 10 of 12 cases. The advantage of this greater number of hubs is that it offers more opportunities for the structure-based neighborhoods to find improved solutions. We note that there is an exception to this conclusion. For datasets EU10 and EU11, the solutions initialized by EAH heuristic have more hubs with both edge budgets. As shown in Fig. 3d, the relatively better performance of EDH-initialized solutions does come at the expense of runtime. The total runtime for the two-phase heuristic initialized with EAH is less in 11 of 12 three-edge instances. The reason for the increased computation time associated with the EDH-initialized solutions, particularly in the local-search phase, is that the search finds more improving solutions when working with EDH-initialized solutions. It appears that the better initial solutions associated with EAH may lead to premature convergence where the EDH solutions provide more opportunities for improvement. The fourth question of interest is the value of the subgraph specific structurebased neighborhood, the LTH neighborhood. Figure 5 presents the after-improvement solution values for both initialization methods when the local search does and does not include the LTH neighborhood and three additional edges are used. The percentage differences are computed analogously to Eq. (2). In this case, a negative value indicates that the solution using the LTH neighborhood is better than the solution found without it. The improvement with the LTH neighborhood is better in nine of 12 instances for initialization with both EDH and EAH, and many of the improvements for the EDHinitialized solutions are sizable. In six instances, the improvement is greater than 10 %.
123
Network design for time-constrained Delivery using subgraphs
(a) Difference in Solution Values when using LTH with EAH-initialized Solutions
539
(b) Difference in Runtimes when using LTH with EDH-initialized Solutions
Fig. 5 The value of the link hub neighborhood for three additional edges
On the other hand, the performance of the search initialized with the EAH is not as impressive. The improvement is greater than 10 % in only two instances.
3.2 Insights into the structure of good subgraph solutions This section seeks to identify general properties of SVS solutions and understand how they are different from tree-based solutions. For example, we want to understand how the hubs change (if they do) and where the cycles tend to be. Based on the results of the previous section, we focus on solutions that were initialized with EDH and then improved using the local search heuristic that incorporates all of the structured neighborhoods and the Edge 1-switch neighborhood. All tree solutions are generated using the methodology described in Chen et al. (2008). For each dataset and for the tree, three additional edge, and six additional edge solutions, we measure the number of hubs of each degree and the number of “hub edges.” As previously discussed, we define a hub to be a node of degree two or higher. A “hub edge” is an edge that links together two hubs. Our analysis is based on the use of tree-based solutions as a basis for comparison. For brevity, the results for the three additional edges are included here and the results for the six additional edges are included in Sect. 3 of Chen et al. (2012). Both sets of results yield similar conclusions. Table 1 presents the number of hubs and “hub edges” in the solution graphs associated with each dataset. For each graph structure and each dataset, Table 1 reports the number of hubs of degree greater than or equal to two, three, and four. These measures are labeled “d ≥ 2,” “d ≥ 3,” and “d ≥ 4,” respectively. For each graph structure and each dataset, the table also presents the number of hubs edges that connect two hubs whose degree is greater than or equal to two, three, and four. These measures are also labeled “d ≥ 2,”“d ≥ 3,” and “d ≥ 4,” respectively. We begin our discussion with the solutions to the US-based problems. As Table 1 shows, there are more hubs in three-edge solutions than in the tree solutions. One idea we wanted to verify computationally was whether the additional edges present in the subgraph are being used to connect leaf nodes, thus turning them into additional hubs
123
540
H. Chen et al.
Table 1 The number of hubs and hub edges in the tree and graphs with three additional edges Problem Tree Hubs
Three additional edges Hubs
Tree Hub edges
Three additional edges Hub edges
d≥2 d≥3 d≥4 d≥2 d≥3 d≥4
d≥2 d≥3 d≥4 d≥2 d≥3 d≥4
US4
23
17
13
27
18
16
22
16
10
29
18
15
US6
26
20
14
32
19
18
25
19
11
34
19
18
US8
19
13
12
20
14
14
18
12
11
22
15
15
US10
24
18
16
25
22
19
23
16
14
27
23
17
US11
20
18
17
27
24
20
19
17
16
29
24
18
US12
22
18
17
25
22
20
21
16
15
27
23
20
EU4
23
22
17
21
19
16
22
21
15
23
20
16
EU6
21
17
17
24
18
18
20
15
15
26
19
19
EU8
21
18
15
31
27
22
20
17
14
33
28
20
EU10
21
17
16
28
24
21
20
16
15
30
26
20
EU11
22
17
15
22
17
15
21
16
14
21
16
14
EU12
25
18
15
34
25
14
24
16
14
36
24
12
of degree 2. Looking at the results, we see that the number of hubs of degree three or higher increases for all but US6, and all US datasets have an increase in the number of hubs of degree four or higher. An examination of the hub edges suggests that the increased degree of the hubs is the result of increased number of connections between hubs rather than between leaf nodes. For all but US6, the number of edges between hubs of degree three or higher increases in comparing the tree to three additional edge solutions. For US6, the number stays the same. Further, for all graphs, the number of edges between hubs of degree four or higher increases. The minimum increase is two for US11, while the rest are three or higher with a maximum of seven. The increase in the number of high degree hubs indicates that the extra edges in a subgraph are being used to create high degree “super” hubs that are connected to one another. Thus, the cycles created by the increase in edges from the tree to the subgraph solutions are connecting these super hubs. With the emergence of super hubs, it is useful to compare the nodes that are hubs in the tree-structured solutions and those that are hubs in the subgraph-structured solutions. If the hubs remain the same between the two solution structures, one could conclude that a simple heuristic for an increased edge budget would be to connect existing hubs. However, such a heuristic would miss important differences that exist in the nodes chosen for hubs in the tree- and subgraph-structured solutions. As an example, consider the US12 solutions. The tree-structured solution has 17 hubs of degree greater than or equal to four and the three-edge solution has 20 hubs of degree greater than or equal to four. The three-edge solution maintains all but two of the treestructured solution hubs of degree greater than or equal to four. Of the subgraph’s other five hubs of degree four or higher, however, none is even a hub in the tree-structured solution. Similar conclusions exist for the other datasets as well.
123
Network design for time-constrained Delivery using subgraphs
541
The results for the EU-based datasets are not quite as generalizable as those for the US-based datasets. One possibility is that there are many nearly identical solutions to the EU-based sets, perhaps as a result of the more compact nature of the geography. In our experiments, the number of hubs still tends to increase for the EU-based sets as more edges are added. In general, it is also true that the hubs in the solutions with different numbers of edges have some differences.Except for EU4 and EU11, the total number of hubs increases, the three degree hubs increase, and the number of degree four or greater hubs increase for the EU-based datasets from the tree solutions to the subgraphs with three additional edges. This indicates that the additional edges are again being used to connect higher degree hubs rather than just leaf nodes. 4 Conclusions In this paper, we examine the SVS and highlight the differences from the version of the problem restricted to trees. We conclude by emphasizing the managerial insights based on the computational study: – By carefully adding connections to the network, a small number of added connections can significantly improve customer service. The exact tradeoffs between the number of edges and impact on service can be verified by implementing the tools in this paper. – The modification of the network structure to include a larger number of connections than a tree has a particularly large impact percentage-wise on instances with the least restrictive commitments. – Iteratively adding edges to a tree-based solution can provide a lower objective value (before improvement) than using a deletion-based approach. This may yield a reasonable solution when time is limited. – To create the best final solution, though, it is better to start from a deletion-based initial solution and give the local search heuristic time to converge. – The added edges in a subgraph are often used to increase the connections between hubs, resulting in “super” hubs. This structure supports the idea of a “hub and spoke” type of network design with certain nodes emerging as these hubs. In practice, this translates to certain nodes having a much larger amount of flow through them than others. These nodes would need to be equipped to handle this additional traffic, in terms of both equipment and staff. – When comparing the nodes of high degree in a subgraph network with those in a tree network or network with fewer edges, there may be some nodes in common but there also may be some significant differences.Thus, it is important to solve instances of the problem for subgraphs separately rather than simply adding edges between existing hubs. An area of future work is the consideration of a broader set of instance sizes. In the work presented in this paper, we found minimal value to edge budgets greater than six. Yet, all of our experiments are conducted on datasets of 150 or 147 nodes, and experiments on datasets with greater and smaller numbers of nodes might offer different conclusions about the size of the edge budgets.
123
542
H. Chen et al.
Another consideration for future research is to consider an objective that minimizes maximum lateness. Chen (2008) found that algorithms similar to those in this paper were inappropriate for the min-max objective. In particular, the objective function can be improved only by changing arcs that fall on the maximum path, so a different solution approach is required. Acknowledgments useful comments.
We would like to thank an Editor-in-Chief and two anonymous reviewers for their
References Caccetta L, Hartman IBA, Huang J (2001) Concerning the number of edges in a graph with diameter constraints. J Comb Math Comb Comput 36:161–173 Chen H (2008) Network-design for time-constrained delivery. PhD thesis, The University of Iowa, Tippie College of Business Chen H, Campbell AM, Thomas BW (2008) Network-design for time-constrained delivery. Naval Res Logist 55(6):493–515 Chen H, Campbell AM, Thomas BW, Tamir A (2009) Minimax flow tree problems. Networks 54(3): 117–129 Chen H, Campbell A, Thomas B (2012) Online supplementary materials for network design for timeconstrained delivery using subgraphs. Online at Computational Management Science website Daskin MS (1997) 150 U.S. largest cities. http://users.iems.northwestern.edu/msdaskin/ Datamonitor (2010) Express logistics in the united states. http://www.fastmr.com/prod/66006_logistics_ in_the_united_states.aspx Johnson DS, Lenstra JK, Kan AHGR (1978) The complexity of the network design problem. Networks 8:279–285 Li CL, Mccormick ST, Simchi-Levi D (1992) On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum-diameter edge addition problems. Oper Res Lett 11:303–308 Maurits de Jonge J (2008) No break in the storm. Traffic World Plesník J (1981) The complexity of designing a network with minimum diameter. Networks 11:77–85 Plesník J (1984) On the sum of all distances in a graph or digraph. J Graph Theory 8:1–21 Rosenberg E (2005) Hierarchical topological network design. IEEE/ACM Trans Netw 13:1402–1409 Thuermer KE (2008) Tweaking the supply chain to optimize value and minimize cost. World Trade 21(10):24 United Parcel Service (2011) U.S. ground maps for calculating service time and cost. http://www.ups.com/ maps?loc=en_US&WT.svl=SubNav, accessed on June 23, 2011 Wu BY, Chao KM, Tang CY (2002) Light graphs with small routing cost. Networks 39:130–138
123
