Transportation Research Part B 93 (2016) 750–761

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Transportation Research Part B

journal homepage: www.elsevier.com/locate/trb

Finding potential hub locations for liner shipping

Zhuo Sun, Jianfeng Zheng ∗

Transportation Management College, Dalian Maritime University, Dalian 116026, China

a r t i c l e i n f o

Article history:

Received 1 February 2015

Revised 15 February 2016

Accepted 7 March 2016

Available online 4 April 2016

Keywords:

Hub location problem

Concave cost network flow

Arctic routes

a b s t r a c t

The current models for hub location problems are unable to find potential hub locations

in uncharted areas that currently have no port. To explore hub locations that are not se-

lected from the present ports, this study proposes a two-stage method to address this gap

in knowledge. A concave cost multicommodity network flow model is solved in the first

stage to obtain container traffic in waterways. Then, in the second stage, a hubbing proba-

bility is evaluated for each node to indicate potential hub locations. A case study including

emerging Arctic routes is provided to demonstrate this method.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

At the strategic decision level, the liner shipping network design problem has attracted a great deal of interest from many

researchers ( Meng et al., 2014 ). The selection of hub port locations is a key problem that significantly impacts decisions in

liner shipping network design because large container ships (or megaships) are usually deployed to serve certain hubs

and major ports, and small ships often serve some feeder ports. Hence, the location of hub ports has a definite impact

on the routing of ships. There have been many studies on the hub location problem in conventional freight transportation,

where hubs are selected from the present nodes using two- or four-dimensional decision variables ( Alumur and Kara, 2008 ).

Unfortunately, the process for designing a liner shipping network cannot follow the construction protocols found in urban

and aviation transportation networks for three major reasons. First, the physical links of a liner shipping network (i.e.,

waterways) are flexible to a certain extent, and there are few limits on the topology aspects. Second, the vehicles used in

liner shipping, i.e., container ships, can be very large, unlike the vehicles used in urban transportation. The more containers

one ship can carry, the lower the transportation cost for each container. Thus, carriers benefit from economies of scale in

liner shipping. Economies of scale related to ship size have been successively addressed by McLellan (1997), Gilman (1999),

Cullinane and Khanna (1999, 20 0 0 ), and Imai et al. (2006) . Third, a container ship often calls at several ports, leading to a

complicated liner shipping network, unlike that in aviation transportation.

The current models for the hub location problem face challenges when handling a liner shipping network. Priori network

structures are used, and hub locations are selected from existing ports, which significantly limits the results. For example,

some potential locations may be excluded from the candidates. Let us consider an extreme case depicted in Fig. 1 , in which

three ports are located at the three vertices of a triangle. Each pair of ports has a symmetric container demand. The cur-

rent models will find the potential hubs at location 1, 2 or 3. However, if we extend the network topology and consider

economies of scale in transportation costs, the traffic paths will bend, and location 4 emerges as a potential hub, as shown

in Fig. 1 . Obviously, location 4 is a previously uncharted area with no port available and no traffic passing through.

∗ Corresponding author. Tel.: + 86 18840921746. E-mail address: jfzheng@dlmu.edu.cn (J. Zheng).

http://dx.doi.org/10.1016/j.trb.2016.03.005

0191-2615/© 2016 Elsevier Ltd. All rights reserved.

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Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 751

3

Port

Demand

1

2

3

Port

Traffic

Potential hub

1

2

4

Fig. 1. A potential hub can be located in an uncharted area.

The aforementioned concept is inspired by the actual planning processes that are performed when a liner shipping com-

pany wants to change its long-term shipping network. A real-world example is the case of emerging Arctic routes; the

retreating Arctic ice cap has exposed two Arctic travel routes. The sailing distances from Asia to North Europe and to the

East Coast of America can be shortened by more than 30% ( Somanathan et al., 2009 ; Liu and Kronbak, 2010 ). To utilize

these Arctic routes, a liner shipping company would first have to assess the possible traffic passing through the new Arctic

routes and the other existing shipping routes. Then, potential hubs would be set up at major traffic intersections, with a

high predicted volume of traffic and many connections. Based on the hubs and traffic, the ship route services would be

designed. This paper focuses on finding potential hub locations for liner shipping companies, where liner shipping network

design (i.e., the design of ship routes) is not considered.

In view of the above discussion, the objective of this study is to imitate practical planning processes using a two-stage

method. In the first stage, a concave cost multicommodity network flow model is proposed, in which the link costs are

concave, reflecting the economies of scale in liner shipping. This NP-hard problem can then be efficiently solved using an

improved branch-and-bound algorithm. In the second stage, each node in the resulting flow pattern network is evaluated

using a hubbing probability. We show that this problem is equivalent to finding the stationary probabilities for a transition

matrix.

The rest of this paper is organized as follows. In the next section, a number of studies related to the hub location problem

are reviewed. In Section 3 , a concave cost multicommodity network flow model is presented and solved. In Section 4 , the

hubbing probability is defined and evaluated. A case study is presented in Section 5 , and the Arctic problem is revisited.

Our conclusions are given in Section 6 .

2. Literature review

The hub location problem has been studied for decades. Alumur and Kara (2008) published a comprehensive review of

the problem, and early studies discussed airline and telecommunication networks ( O’kelly, 1987 ). Classical single allocation

p -hub median models ( Campbell, 1992; Ernst and Krishnamoorthy, 1996; Ebery, 2001 ) were widely applied, in which p hubs

must be selected from the present nodes, and each non-hub node is allocated to a single hub. The total transportation costs

are minimized when the inter-hub costs are discounted by a factor. Later studies extended the single-allocation policy to

allow non-hub nodes to connect to multiple hubs ( Campbell, 1994; O’Kelly et al., 1996 ). To factor in the cost of opening a

hub, the number of hubs was introduced as a decision variable, and the corresponding models became much more complex

( O’Kelly, 1992 ).

Illia and Wynter (2005) proposed a non-linear mixed integer program model for identifying freight hubs. The number of

hubs in their model cannot exceed two, and the routes in the network are predefined. For maritime transportation networks,

this problem is always coupled with the network design problem. In addition to the hub-and-spoke structure, the multi-

port-calling structure has been widely studied, where container vessels sail “directly” to visit all of the necessary ports

on their itineraries. As indicated in certain studies, for imbalanced demand, multi-port-calling networks are superior to

conventional hub-and-spoke networks in terms of the container management cost ( Imai et al. 2009 ). Later, Meng and Wang

(2011) further combined multi-port-calling and hub-and-spoke models to allow for direct container shipment between any

two ports. Gelareh et al. (2010) and Gelareh and Pisinger (2011) investigated the hub location problem together with liner

shipping network design and ship fleet deployment using a pre-determined discount factor to reflect economies of scale

in ship size. Recently, Zheng et al. (2014, 2015 ) proposed the multi-stage decomposition method to split the liner shipping

network design problem into a hub location problem and liner ship route design problem, both of which can be solved

independently. In addition, there have been many studies on network design, ship fleet deployment, ship scheduling and

container routing for the shipping industry; please refer to the review papers, i.e., Ronen (1983, 1993 ), Christiansen et al.

(2004, 2013 ) and Meng et al. (2014) .

Methods from previous studies for finding potential hub locations are not practical for large-scale realistic problems due

to the following two reasons. (a) The explicitly predefined network structures used in previous studies significantly confine

any improvement in the solutions, and (b) the complexity of the model (for a large-scale network, there are thousands of

752 Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761

Waterway link

Ordinary node

Port node1

5

2

3

Access/Egress link

+

+

Super deficit node

Super surplus node

-

-

Dummy link4

6

Fig. 2. A spatial maritime network with empty container repositioning.

integer decision variables and countless path permutations) makes the computation intractable. Only candidate nodes or

routes are considered as inputs, which further limits the identification of practical solutions.

To address the shortcomings in previous works, this study makes use of a different method. The major purpose of this

paper is to broaden the potential hub selection range and to render the processes for finding potential hub locations more

practical.

3. The concave cost multicommodity network flow model

In this section, a strategic-level concave cost multicommodity network flow model is introduced. A spatial maritime net-

work is constructed, and we assume that the containers (including empty containers) are transported directly on it. The

objective function is concave, representing the economies of scale observed in liner shipping. An improved branch-and-

bound algorithm is devised to solve the model. Our model is similar to the two container assignment models proposed by

Bell et al. (2011, 2013 ). There are three major differences. First, in Bell’s models, containers are assigned in a fixed ship-

ping service network; in our model, containers are assigned in a spatial maritime network. They are therefore on different

decision levels. Our model acts at a higher decision level, as we aim to find the potential hub locations, which can be

regarded as the predetermined candidate hub ports in the hub location problem or liner shipping network design for con-

tainer transshipment operations. Second, the objective function of Bell’s models is linear, and ours is concave. Third, Bell’s

models separately formulate full and empty container flow to address different demand patterns, while our model modifies

the network structure to adapt different demand patterns (see Section 3.1 ). This will make the algorithm implementation

much more efficient.

3.1. Network construction with empty container repositioning

The network used to assign containers is constructed according to the spatial information of the physical network. We

will discuss the construction of a real network in the case study. Fig. 2 shows the key concepts. We assume that the ocean

area is covered by a navigable body of water, i.e., waterways. We create waterway links accordingly. At the intersections of

waterway links, ordinary nodes are created. Each port node is associated with an ordinary node and an access/egress link.

The laden container demands are originated and terminated at port nodes.

The laden container demand at each port node is not always balanced, which generates empty container demands. As

shown in Fig. 2 , the + and − signs at the port nodes represent the surplus and deficit of empty containers, respectively.The empty container repositioning is recognized as the Hitchcock problem ( Hitchcock, 1941 ). To address it, we extend the

network by adding an artificial super surplus node and an artificial super deficit node. Dummy links with zero costs and

corresponding capacities (the surplus/deficit volume of the port node) are installed between these nodes. As a result, the

empty container’s demand originates at the super surplus node and travels to the super deficit node.

To better understand the role of dummy nodes and links, here we present an example based on Fig. 2 . Suppose there are

two weekly laden container demands: from port 3 to 1, volume 5 and from 4 to 2, volume 3. At the end of each week, port

1 has 5 surplus empty containers, port 2 has 3 surplus empty containers, port 3 has 5 deficit empty containers and port 4

has 3 deficit empty containers. Carriers need to balance empty containers at each port, which requires transporting empty

containers from surplus ports to deficit ports and making the surplus/deficit number zero. By using the dummy nodes and

links we can transform the empty container demands to a normal OD-type demand. The new empty container demand is

from 5 to 6, volume 8. The dummy links have zero cost and the following capacities: link 5–1, capacity 5; link 5–2, capacity

3; link 3–6, capacity 5; link 4–6, capacity 3.

3.2. Assumptions

To simplify the problem and present the basic idea behind the proposed multicommodity network flow model, the fol-

lowing assumptions are made:

Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 753

• To easily model economies of scale, we assume that the liner shipping company does not own any of the ships. All of the

slots for transporting containers are chartered from the market. The containers are handled on the access/egress links

and transported on the waterway links. • Only three types of costs are considered: the chartering cost, voyage cost and handling cost. The charter cost and voy-

age cost are considered for the waterway links, and the handling cost is considered for the access/egress links. For the

dummy links, the cost is zero. • The container demand is fixed and given in advance. • Different types of containers are considered. The containers are assumed to be interchangeable for capacity occupation.

For compact formulation, each OD (origin–destination) pair demand is distinguished by a unique container type. • Each link in the spatial maritime network has a capacity limit. For most of the waterway links, this capacity is fairly large

and can be treated as infinity. For the access/egress links and some of the waterway links representing channels and

straits, the capacity is set to their throughput. For dummy links, the capacity is set to the corresponding surplus/deficit

container volume.

3.3. Notation and model

Some mathematical notations have to be defined to facilitate formulation of the problem.

Parameters

V Set of nodes P Set of port nodes, including the super-surplus/deficit node, P ⊂ V A Set of links

Ā Set of access/egress links, Ā ⊂ A K Set of container types O Set of origin ports, O ⊆ P D Set of destination ports, D ⊆ P ηk Capacity sharing factor for a k -type container h k

i j Handling cost for a k -type container at an access/egress link ( i, j )

d k i

Net flow of a k -type container at node i

u i j Capacity of link ( i, j )

q k i

Volume of the k -type container demand at node i

Decision variable

x k i j

Flow of k -type containers at link ( i, j )

To formulate our problem, let c ij ( · ) and v ij ( · ) denote the chartering cost function and the voyage cost function forlink ( i, j ), respectively. The chartering cost consists of the operating costs, periodic maintenance costs and capital costs,

which correspond to the charter rate in the market. The voyage cost consists of the fuel consumption and port costs. Note

that each port node is associated with an ordinary node and an access/egress link in our constructed spatial maritime net-

work, as illustrated in Fig. 2 . When transporting laden (empty) containers from the origin (surplus) ports to the destination

(deficit) ports, the container transshipment operations at any intermediate port can be formulated through its associated ac-

cess/egress link, which will incur an extra cost for relaying laden and empty containers. Hence, the container transshipment

operations do not occur in our constructed spatial maritime network. In other words, any OD flow will not pass through a

third port node. When the container demand is fixed and given in advance, the ports to be visited are fixed. Therefore, the

port costs can be treated as a constant. Thus, we only consider fuel consumption in the voyage cost function. Due to scale

economies related to ship size, both the chartering cost function and the voyage cost function have a property of concavity.

The detailed formulations of these two cost functions are regressed using the real data, as shown in Section 5 .

The concave cost multicommodity network flow model can be expressed as follows:

min ∑

( i, j ) ∈ A c i j

( ∑ k ∈ K

ηk x k i j

) +

∑ ( i, j ) ∈ A

v i j

( ∑ k ∈ K

ηk x k i j

) +

∑ ( i, j ) ∈ ̄A

∑ k ∈ K

h k i j x k i j (1)

subject to ∑ j∈ V

x k i j −∑ j∈ V

x k ji = d k i ∀ i ∈ V, k ∈ K (2)∑ k ∈ K

ηk x k i j ≤ u i j ∀ ( i, j ) ∈ A (3)

x k i j ≥ 0 ∀ ( i, j ) ∈ A (4)

754 Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761

d k i =

⎧ ⎨ ⎩

q k r , i f i = r ∈ O ;−q k s , i f i = s ∈ D ;0 , otherwise.

(5)

The objective function ( 1 ) is the sum of the chartering, voyage and handling costs for all types of containers being

transported between ports. Because any OD flow will not pass through a third port node, the sum of the handling costs is

constant and can be removed from the objective function.

Constraint ( 2 ) guarantees flow conservation at each node. Constraint ( 3 ) handles the capacity limit for each link. Con-

straint ( 4 ) forces the decision variables to be non-negative. The origin and destination constraints are given in ( 5 ).

3.4. Solution method

The concave cost multicommodity network flow model ( 1 )–( 5 ) proposed in this study is known as an NP-hard problem

(Zangwill, 1968) . Several exact and heuristic methods have been developed over the years to solve this problem. Guisewite

and Pardalos (1990) conducted a comprehensive review on related studies. Here, we introduce an improved branch-and-

bound method, originally proposed by Soland (1974) , in which the feasible region is not branched.

For the network flow problem, if the classic branch-and-bound method is applied, the feasible region is broken up into

several pieces, and the variables are bounded accordingly. This results in link splitting and non-zero initial flow, which will

make the problem much more difficult to solve. In Soland’s branch-and-bound method, the feasible region is branched, but

it does not need to be bounded. The linearized problem becomes a linear assignment problem at each stage, and a shortest

path algorithm can be used to solve it. In our case, this technique can save immense computational effort. For a large-scale

network, we further propose a heuristic to speed up the branching process.

Note that the decision variables are bundled in the two concave functions of objective ( 1 ). We set the unified flow vari-

able w i j = ∑

k ∈ K ηk x k i j and the objective function f i j ( w i j ) = c i j ( w i j ) + v i j ( w i j ) . At branch n of the branch-and-bound method,for link ( i, j ), there is an interval [ l n

i j , u n

i j ] in the feasible region of w ij (see Fig. 3 ). However, w ij is not bounded by the interval.

The linearized function is defined by f̄ n i j ( l n

i j ) = f i j ( l n i j ) and f̄ n i j ( u n i j ) = f i j ( u n i j ) . Suppose the optimal solution of the linearized

problem on link ( i, j ) at branch n is w n ∗i j

. Then, the upper and lower bounds of the problem are defined as

UB = min { ∑

( i, j ) ∈ A f i j

(w n ∗i j

): n = 1 , 2 , . . . , N

} (6)

LB = min { ∑

( i, j ) ∈ A f̄ n i j

(w n ∗i j

): n = 1 , 2 , . . . , N

} (7)

where N is the total number of branches.

If the optimal solution w n ∗i j

of the linearized problem lies inside the interval [ l n i j , u n

i j ] , the difference between the concave

function and the linearized function δn i j

is positive; otherwise, δn i j

is negative. Therefore, if LB < UB , there must be at least

one link that makes δn i j

positive.

At the subsequent two branches, N + 1 and N + 2 , the feasible region of w ij is further branched into two intervals:[ l n

i j , w n ∗

i j ] and [ w n ∗

i j , l n

i j ] . The linearized function is also branched into two linear functions f̄ N+1

i j and f̄ N+2

i j , respectively (see

Fig. 3 ).

Fig. 3. Branch n for the branch-and-bound method at link ( i, j ).

Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 755

Let w UB ∗

and w LB ∗

be the optimal solutions found at UB and LB , respectively. The Soland’s branch-and-bound method

applied to the concave cost multicommodity network flow model can then be described as follows:

Step A: Set N = 1 . Find UB and LB . Go to step B. Step B: If LB ≥ UB, w UB ∗ is the optimal solution. Otherwise, go to step C Step C: Find the link ( i, j ) that has the maximum δij for w

LB ∗ . Branch at w LB ∗

i j and create two new branches, N + 1 and

N + 2 . Set N = N + 2 . Update UB and LB . Go to step B.

Although Soland’s branch-and-bound method can significantly reduce the computational effort compared to the classic

method, it still cannot run in a reasonable time for a large-scale network. To speed up the convergence, we change the

branch scheme by selecting several links simultaneously in step C. Let δmax i j

be the maximum δij for all links for w LB ∗ . We

add a scaling factor α ∈ (0, 1] to control the link selection process. Now, step C is as follows: Step C: Find δmax

i j for w LB

∗. Find a set of links Ā in which each link ( i, j ) satisfies δi j ≥ α · δmax i j . Branch at w LB

∗i j

for each

link ( i, j ) ∈ Ā and create two new branches N + 1 and N + 2 . Set N = N + 2 . Update UB and LB . Go to step B. If we phase the branch-and-bound method in terms of a binary tree, the heuristic proposed above prunes the binary

tree at each stage by branching many links together. For the network flow, links in the same path carrying a flow are likely

branched together, rerouting the flow at each step of the method.

4. Evaluation of hubbing probabilities

The result of the concave cost multicommodity network flow model shows the best container traffic flow pattern of

“economies of scale” in the spatial maritime network. The incoming and outgoing volumes of a node indicate the through-

put, and the number of incoming and outgoing flow-carrying links indicates the connectivity in the network. To find a po-

tential hub, the above two characteristics must be considered simultaneously. Here, we propose a network analysis method

( Wasserman and Faust, 1994 ) to evaluate the hubbing probabilities of the nodes in the resulting flow pattern network (see

Fig. 4 ).

The hubbing probability for node i is defined as:

p i = ∑

j∈ V, t j > 0

πi j p j t j

+ ∑

j∈ V, t j > 0

p j

| V | (8)

where π ij is the sum of unified traffic flow between i and j , πi j = π ji = ∑

k ∈ K ηk ( x k i j + x k ji ) , or the link weight of the resultingnetwork. t j is the throughput for node j , which is equal to the sum of all incoming and outgoing flows, t j =

∑ i ∈ V π ji . We

use 1/| V | to address a dangling node with no throughput. This equation implies that the hub probability of a node depends

on the probability of the surrounding nodes. If a node shares relatively high traffic flow with another node, these two nodes

have strong ties. We quantify this relationship by transferring a proportion of the probability from one node to the other

node. The probability of a dangling node is distributed to all other nodes.

The vector form of Eq. (8) can be rewritten as

p = Mp (9)where p is the column vector of the hubbing probabilities and the matrix M is defined by

M i j = {

πi j t j

, i f t j > 0 ;1 | V | , otherwise.

(10)

Note that ∑

i M i j = 1 . The matrix M is recognized as the transition matrix or Markov matrix, which has an eigenvalueequal to 1. The vector p is the stationary probability vector, which is the eigenvector of the matrix associated with the

eigenvalue 1.

ijπi

j jtjp

ip

Fig. 4. Evaluation of the hubbing probability for the resulting flow pattern network.

756 Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761

Access/Egress link Port Node

Ordinary NodeWaterway link

Arctic waterways

Fig. 5. Construction of a real-world spatial maritime network in GIS.

To evaluate all nodes in a large-scale network and obtain a meaningful result, we propose two additional processes. First,

we propose a filtering process in which small flows are ignored because small flows will spread the probabilities through-

out the network and make it difficult to identify real hubs. Second, we propose spatial grouping in which the hubbing

probabilities are aggregated within a certain distance to more clearly demonstrate the results.

Let W denote the minimum traffic volume that will be considered. Let r ij denote the spatial distance between node i and

j , and let R denote the grouping distance. The hubbing probability evaluation algorithm is described below:

Step A: Collect all flows that satisfy π ij ≥ W for each ( i, j ) ∈ A . Calculate t i for each i ∈ V . Create the transition matrix M .Go to step B.

Step B: Use the power method to find the eigenvector p associated with the eigenvalue 1 of the matrix M . Go to step C.

Step C: (1) Find the largest available p i in p . If p i = 0 , end. (2) Aggregate the hubbing probability for node i by p i =∑ j∈ V, r i j ≤R p j . Set p j = 0 for each node j that satisfies r ij ≤ R . Go to (1).

The hubbing probability evaluation method proposed above is a type of eigenvector centrality analysis, which was pop-

ularized by the PageRank algorithm in Google ( Page et al., 1999; Langville and Meyer, 2011 ), where billions of webpages

are ranked for the search engine. There are two major differences between the webpage network and the flow pattern net-

work described here. One is the link weight. The webpage network uses the number of links between pages as the network

weight, which is discrete. In this paper, the flow pattern network adopts the traffic volume as the link weight, which is

continuous. The link weight can be very small in the case of low traffic volume. Therefore, we use a filtering process. The

second difference is the spatial information. As a network node, a webpage does not contain spatial information, i.e., co-

ordinates. However, in our method, the spatial information is important. Therefore, we apply a spatial grouping process to

make the results more meaningful.

5. Case study

To test our method, we employed the real data in our model. We not only determine the current operational waterways,

we also examine the Arctic waterways that are expected to open in future decades. Our data and model are organized

and developed under an open-source software framework, MicroCity ( http://microcity.github.io ). We will release the add-on

modules for this paper later on that site.

A real-world spatial maritime network is first constructed in GIS (see Fig. 5 ). The major waterways are created based on

geographical data retrieved from an empirical trading route analysis ( Rodrigue et al., 2009 ) and a historical ship movement

analysis ( Kaluza et al., 2010 ). The Arctic waterways, presented as dotted lines in the figure, are obtained from the ArcticData

portal ( http://www.arcticdata.is/ ). The port nodes (red dots) and OD demands are provided by a leading shipping company.

We use a computer program to topologize the digitalized waterways and to create the necessary nodes and links for the

network. The statistics of input data can be found in Table 1 . As for multi-type containers adopted in practice, we mainly

consider TEU (Twenty-foot Equivalent Unit) and FEU (Forty-foot Equivalent Unit). For parameter ηk in constraints ( 3 ), oneFEU can be regarded as two TEUs.

http://microcity.github.iohttp://www.arcticdata.is/

Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 757

Table 1

Statistics of input data.

Without Arctic waterways With Arctic waterways

Number of waterway links 550 648

Number of access and egress links 334 334

Number of ordinary nodes 243 282

Number of port nodes 167 167

Number of demand OD pairs 7247 7247

Demand volume 287,155 TEU 287,155 TEU

Empty container demand volume 96,471 TEU 96,471 TEU

Fig. 6. Economies of scale for charter rates (left) and fuel consumption (right).

Table 2

Statistics of model output.

Without Arctic waterways With Arctic waterways

Running time of Soland B&B 17 h Memory depletion

Running time of proposed B&B 4623 s 5647 s

Running time of hub evaluation 3 s 7 s

Number of traveled links 746 794

Container traveled distance 3.50 ×10 9 TEU km 3.37 ×10 9 TEU km Number of potential hub locations 13 12

The chartering cost function of objective ( 1 ) in the network flow model is set to 1 . 4 4294 4 · x · ( 1 + x ) −0 . 757464 (dollarunit), which is regressed and converted from the container ship time charter rates published in the market ( UNCTAD, 2014 )

(see left side of Fig. 6 ). The voyage cost function is set to 1 . 375249 · x · ( 1 + x ) −0 . 31489 (dollar unit), which is regressed andconverted from the container ship fuel consumption data ( https://www.ihs.com/ ) (see right side of Fig. 6 ).

Our program is coded with MicroCity Script and run in JIT (just-in-time compilation) mode on a 3-GHz Quad Core PC

with 8GB of RAM. Executing the exact Soland’s branch-and-bound method can deplete the memory; hence, we use the

proposed branch-and-bound heuristic with the parameter α = 0 . 7 . The running time is approximately 1 h. Then, we evalu-ate the hubbing probabilities for the parameters W = 2500 and R = 2500 , which are the empirical values provided by theshipping company. The statistics of the algorithm running times and some model outputs can be found in Table 2.

Fig. 7 shows the results obtained from the proposed method for the current navigational conditions, which do not contain

the Arctic waterways. The blue lines indicate the eastbound container traffic flow, and the magenta lines indicate the west-

bound container traffic flow. The width of a line represents the traffic volume. The red dots indicate potential hub locations,

and the size of a dot represents the hubbing probability. The container flows are concentrated along one major route across

Asia, Europe and North America. Hubs are likely located near traffic concentrating places, and the region with the largest

hubbing probability is located near Shanghai. Other major potential hub locations are observed near Singapore, Mumbai,

Haifa, the Gibraltar Strait, Wilmington, the Panama Canal and San Francisco, as shown in Fig. 7 . Additionally, small hubbing

probabilities are found near Melbourne and Helsinki. The shipping company confirmed these regions as hub locations. Some

of these locations already have operational hubs, some are under construction, and others are under consideration.

We further test our method using the same parameters while considering the Arctic Ocean. Fig. 8 shows that approx-

imately one-third of the westbound South Asia container traffic and almost all westbound Asia–Europe container traffic

are rerouted along the Arctic waterways north of Russia, the so-called Northern Sea Route (NSR). A portion of the Asia–

North America traffic is rerouted north of Canada along the Northwest Passage (NWP). The largest hubbing probability is

https://www.ihs.com/

758 Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761

8500035000500

0.05

0.2

0.1

Container Flow

Hubbing Probability

DirectionEastboundWestbound

Fig. 7. The container traffic flow pattern and potential hub locations for the current navigational conditions.

8500035000500

0.05

0.2

0.1

Container Flow

Hubbing Probability

DirectionEastboundWestbound

Fig. 8. The container traffic flow pattern and potential hub locations with consideration for the Arctic Ocean.

still located near Shanghai, and the Gibraltar Strait, which is identified in the previous results, is replaced by Amsterdam.

The probabilities near Singapore and Mumbai become smaller and move south slightly. The location at Haifa is moved to

the Persian Gulf because the traffic from the Arabian Sea to the Mediterranean Sea decreases substantially. The hubbing

probabilities for the three North American locations of Wilmington, the Panama Canal and San Francisco do not change

significantly. A new hub location can be found at the Bering Sea, which is the entry of the Arctic routes. Interestingly, in the

middle of the two Arctic traffic routes, there are two small hubbing probabilities, indicating that the middle of the Arctic

routes is slightly more important than other parts. Thus, it would be profitable for a shipping company to construct hubs at

these locations.

Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 759

6. Conclusions

This study proposed a two-stage method for identifying potential hub locations for liner shipping. Some locations are in

uncharted areas and do not yet have traffic. In the first stage, a concave cost multicommodity network flow model is built to

reflect the economies of scale for shipping containers. We employ Soland’s branch-and-bound method to achieve an optimal

solution as well as a heuristic improvement to increase the efficiency. In the second stage, the resulting traffic flow pattern is

converted to a weighted network. We define the hubbing probability for each node and then build the transition matrix for

the network. The solution is the eigenvector of the transition matrix. We further test our proposed method using real data

for a shipping company. The results show that the method can effectively predict hub locations for current circumstances.

We then apply the method to a new setting that includes Arctic waterways. The results indicate changes in the old hub

locations and provide new potential hub locations.

The method presented here is superior to the existing models in three respects. First, there is freedom in the network

structure. Most of the existing models adopt priori network structures in the modeling processes, where spokes are allocated

to hubs or hubs are directly connected. Our method does not have this limitation. Instead, we focus on the traffic flow

pattern. Second, the existing models must determine the number of hubs, possible nodes and predefined routes in advance,

which restricts the solutions. In our method, we do not need to explicitly define the hubs to solve the model. Third, our

method exhibits improved computation efficiency. Most of the existing models are mixed integer programming problems,

which are very difficult to solve when the number of integer variables is large. With our method, it is easy to implement

an efficient algorithm in light of network flow and network analysis algorithms to achieve a much shorter execution time.

This practical method can be useful for strategic planning in liner shipping companies.

Acknowledgments

This research is supported by the NOL Fellowship Program of Singapore ( R-264-0 0 0-244-720 ), the National Basic Re-

search Program of China ( 2012CB725400 ), the National Natural Science Foundation of China ( 61304179 , 71501021 , 71372088 ,

71431001 , 71202108 , 71302085 ), National Social Science Fund of China ( 13&ZD170 ), Program for Changjiang Scholars and

Innovative Research Team in University ( IRT13048 ) and the Fundamental Research Funds for the Central Universities

( 3132015064 ). The authors also thank Hung Song Goh and K.H. Eddie Ng from APL Co. Pte Ltd. for their valuable comments

on this study. We are indebted to the two anonymous reviewers for their helpful comments and suggestions.

Appendix

Based on the potential hub locations obtained by solving our proposed two-stage method, one can further determine

hub location and feeder allocation by using the conventional hub-and-spoke network design models (see Alumur and Kara,

2008 ). Here we provide the formulation of a hub-and-spoke network design model with single allocation.

Let P 1 denote the set of potential hub locations. Let c in v est p represent the investment cost to establish hub at port p ( ∀ p∈ P 1 ). Let q ij be the container demand which to be delivered from port i to port j , including the empty container demandwith respect to the super-surplus/deficit node.

The decision variables are as follows:

z im : A binary variable which takes value 1 if port i ∈ P is allocated to the hub port m ∈ P 1 , and 0 otherwise. z ii equals 1if port i is selected to be a hub port, and 0 otherwise.

Z ijmn : Fraction of the container demand from port i to port j that is routed via the first hub port m and then hub port n .

x ij : The container flow on arc ( i, j ) ∈ A .

By mainly considering the investment cost, the chartering cost and the voyage cost, the hub-and-spoke network design

model with single allocation can be given as follows:

min ∑ i ∈ P

∑ j∈ P

f i j (x i j

)+

∑ m ∈ P 1

(c in v est m × z mm

)(A1)

subject to ∑

m ∈ P 1 z im = 1 ∀ i ∈ P (A2)

Z i jmn ≤ z mm ∀ i, j ∈ P, ∀ m, n ∈ P 1 (A3)

Z i jmn ≤ Z nn ∀ i, j ∈ P, ∀ m, n ∈ P 1 (A4)

http://dx.doi.org/10.13039/501100001809

760 Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761

∑ m ∈ P 1

∑ n ∈ P 1

Z i jmn = 1 ∀ i, j ∈ P and q i j > 0 (A5) ∑

m ∈ P 1 Z i jmn = z jn ∀ i, j ∈ P, ∀ n ∈ P 1 (A6)

∑ n ∈ P 1

Z i jmn = z jm ∀ i, j ∈ P, ∀ m ∈ P 1 (A7)

x i j = ∑ n ∈ P

∑ m ∈ P 1

(q im Z im jn

)+

∑ n ∈ P

∑ m ∈ P

(q mn Z mni j

)+

∑ n ∈ P

∑ m ∈ P 1

(q n j Z n jmi

) ∀ i, j ∈ P (A8) x i j ≤ u i j ∀ i, j ∈ P (A9)

0 ≤ Z i jmn ≤ 1 ∀ i, j ∈ P, ∀ m, n ∈ P 1 (A10)

z im ∈ { 0 , 1 } ∀ i ∈ P, ∀ m ∈ P 1 (A11)

x i j ≥ 0 ∀ i, j ∈ P (A12) The objective function ( A1 ) aims to minimize the total cost, where the chartering cost and the voyage cost on arc ( i, j )

is denoted as function f ij ( · ). Constraints ( A2 ) state the requirement of single allocation. Constraints ( A3 ) and ( A4 ) meanthat the container flow for each OD pair should be routed via hub port. Constraints ( A5 ) ensure that the container flow for

each OD pair should be delivered. Constraints ( A6 ) and ( A7 ) show the relationship between Z ijmn and z im or z jn . Constraints

( A8 ) describe the relationship between the arc flow and the path flow. Constraints ( A9 ) are capacity constraints. Finally,

constraints ( A10 )–( A12 ) define the domain of the decision variables.

References

Alumur, S. , Kara, B.Y. , 2008. Network hub location problems: the state of the art. European Journal of Operational Research 190 (1), 1–21 . Bell, M.G.H. , Liu, X. , Angeloudis, P. , Fonzone, A. , Hosseinloo, S.H. , 2011. A frequency-based maritime container assignment model. Transportation Research

Part B 45 (8), 1152–1161 . Bell, M.G.H. , Liu, X. , Rioult, J. , Angeloudis, P. , 2013. A cost-based maritime container assignment model. Transportation Research Part B 58, 58–70 .

Campbell, J.F. , 1992. Location and allocation for distribution systems with transshipments and transportion economies of scale. Annals of Operations Re-search 40 (1), 77–99 .

Campbell, J.F. , 1994. Integer programming formulations of discrete hub location problems. European Journal of Operational Research 72, 387–405 .

Christiansen, M. , Fagerholt, K. , Nygreen, B. , Ronen, D. , 2013. Ship routing and scheduling in the new millennium. European Journal of Operational Research228 (3), 467–483 .

Christiansen, M. , Fagerholt, K. , Ronen, D. , 2004. Ship routing and scheduling: status and perspectives. Transportation Science 38 (1), 1–18 . Cullinane, K. , Khanna, M. , 1999. Economies of scale in large container ships. Journal of Transport Economics and Policy 33 (2), 185–208 .

Cullinane, K. , Khanna, M. , 20 0 0. Economies of scale in large containerships: optimal size and geographical implications. Journal of Transport Geography 8,181–295 .

Ebery, J. , 2001. Solving large single allocation p-hub problems with two or three hubs. European Journal of Operational Research 128, 447–458 .

Ernst, A.T. , Krishnamoorthy, M. , 1996. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science 4 (3), 139–154 . Gelareh, S. , Nickel, S. , Pisinger, D. , 2010. Liner shipping hub network design in a competitive environment. Transportation Research Part E 46 (6), 991–1004 .

Gelareh, S. , Pisinger, D. , 2011. Fleet deployment, network design and hub location of liner shipping companies. Transportation Research Part E 47 (6),947–964 .

Gilman, S. , 1999. The size economies and network efficiency of large containerships. Maritime Economics and Logistics 1 (1), 39–59 . Guisewite, G.M. , Pardalos, P.M. , 1990. Minimum concave-cost network flow problems: applications, complexity, and algorithms. Annals of Operations Re-

search 25 (1), 75–99 .

Hitchcock, F.L. , 1941. The distribution of a product from several sources to numerous localities. MIT Journal of Mathematics and Physics 20, 224–230 . Illia, R. , Wynter, L. , 2005. Optimal location of intermodal freight hubs. Transportation Research Part B 39 (5), 453–477 .

Imai, A. , Nishimura, E. , Papadimitriou, S. , Liu, M. , 2006. The economic viability of container mega-ships. Transportation Research Part E 42 (1), 21–41 . Imai, A. , Shintani, K. , Papadimitriou, S. , 2009. Multi-port vs. Hub-and-Spoke port calls by containerships. Transportation Research Part E 45 (5), 740–757 .

Kaluza, P. , Kölzsch, A. , Gastner, M.T. , Blasius, B. , 2010. The complex network of global cargo ship movements. Journal of The Royal Society Interface 12, 1–11 .Langville, A.N. , Meyer, C.D. , 2011. Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press .

Liu, M. , Kronbak, J. , 2010. The potential economic viability of using the Northern Sea Route (NSR) as an alternative route between Asia and Europe. Journal

of Transport Geography 18 (3), 434–4 4 4 . McLellan, R.G. , 1997. Bigger vessels: How big is too big? Maritime Policy and Management 24 (2), 193–211 .

Meng, Q. , Wang, S. , 2011. Liner shipping service network design with empty container repositioning. Transportation Research Part E 47 (5), 695–708 . Meng, Q. , Wang, S. , Andersson, H. , Thun, K. , 2014. Containership routing and scheduling in liner shipping: overview and future research directions. Trans-

portation Science 48 (2), 265–280 . O’Kelly, M.E. , 1987. A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research 32 (3), 393–404 .

O’Kelly, M.E. , 1992. Hub facility location with fixed costs. Papers in Regional Science 71 (3), 293–306 .

O’Kelly, M.E. , Bryan, D. , Skorin-Kapov, D. , Skorin-Kapov, J. , 1996. Hub network design with single and multiple allocation: a computational study. LocationScience 4 (3), 125–138 .

Page, L. , Brin, S. , Motwani, R. , Winograd, T. , 1999. The PageRank Citation Ranking: Bringing Order to the Web. Technical Report. Stanford InfoLab . Rodrigue, J.-P. , Comtois, C. , Slack, B. , 2009. The Geography of Transport Systems. Taylor & Francis .

Ronen, D. , 1983. Cargo ships routing and scheduling: Survey of models and problems. European Journal of Operational Research 12, 119–126 . Ronen, D. , 1993. Ship scheduling: The last decade. European Journal of Operational Research 71, 325–333 .

http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0001http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0001http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0001http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0002http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0003http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0003http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0003http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0003http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0003http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0004http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0004http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0055http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0055http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0005http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0005http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0005http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0005http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0005http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0006http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0006http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0006http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0006http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0007http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0007http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0007http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0008http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0008http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0008http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0009http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0009http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0010http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0010http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0010http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0013http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0013http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0013http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0013http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0014http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0014http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0014http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0015http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0015http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0016http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0016http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0016http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0017http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0017http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0018http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0018http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0018http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0019http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0019http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0019http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0019http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0019http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0020http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0020http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0020http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0020http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0021http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0021http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0021http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0021http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0021http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0022http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0022http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0022http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0023http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0023http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0023http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0024http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0024http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0025http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0025http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0025http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0026http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0026http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0026http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0026http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0026http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0027http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0027http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0028http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0028http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0029http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0029http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0029http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0029http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0029http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0030http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0030http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0030http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0030http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0030http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0031http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0031http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0031http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0031http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0032http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0032http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0033http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0033

Z. Sun, J. Zheng / Transportation Research Part B 93 (2016) 750–761 761

Soland, R.M. , 1974. Optimal facility location with concave costs. Operations Research 22 (2), 373–382 . Somanathan, S. , Flynn, P. , Szymanski, J. , 2009. The Northwest passage: a simulation. Transportation Research Part A 43 (2), 127–135 .

UNCTAD, 2014. Review of maritime transportation 2014. In: Paper presented at the United Nations Conference on Trade and Development. New York andGeneva http://www.unctad.org/en/docs/rmt2014 _ en.pdf .

Wasserman, S. , Faust, K. , 1994. Social Network Analysis: Methods and Applications. Cambridge University Press, New York . Zangwill, W.I. , 1968. Minimum concave cost flows in certain networks. Management Science 14 (7), 429–450 .

Zheng, J. , Meng, Q. , Sun, Z. , 2014. Impact analysis of maritime cabotage legislations on liner hub-and-spoke shipping network design. European Journal of

Operational Research 234 (3), 874–884 . Zheng, J. , Meng, Q. , Sun, Z. , 2015. Liner hub-and-spoke shipping network design. Transportation Research Part E 75, 32–48 .

http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0034http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0034http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0035http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0035http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0035http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0035http://www.unctad.org/en/docs/rmt2014_en.pdfhttp://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0037http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0037http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0037http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0038http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0038http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0039http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0039http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0039http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0039http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0040http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0040http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0040http://refhub.elsevier.com/S0191-2615(16)00042-4/sbref0040

Finding potential hub locations for liner shipping1 Introduction2 Literature review3 The concave cost multicommodity network flow model3.1 Network construction with empty container repositioning3.2 Assumptions3.3 Notation and model3.4 Solution method

4 Evaluation of hubbing probabilities5 Case study6 Conclusions Acknowledgments Appendix References