Accepted Manuscript An Intermodal Hub Location Problem for Container Distribution in Indonesia Hamid Mokhtar, A.A.N. Perwira Redi, Mohan Krishnamoorthy, Andreas T. Ernst PII: S0305-0548(18)30239-9 DOI: https://doi.org/10.1016/j.cor.2018.08.012 Reference: CAOR 4548 To appear in: Computers and Operations Research Received date: 26 November 2017 Revised date: 22 August 2018 Accepted date: 29 August 2018 Please cite this article as: Hamid Mokhtar, A.A.N. Perwira Redi, Mohan Krishnamoorthy, Andreas T. Ernst, An Intermodal Hub Location Problem for Container Distribution in Indonesia, Com- puters and Operations Research (2018), doi: https://doi.org/10.1016/j.cor.2018.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Accepted Manuscript
An Intermodal Hub Location Problem for Container Distribution inIndonesia
Hamid Mokhtar, A.A.N. Perwira Redi, Mohan Krishnamoorthy,Andreas T. Ernst
Received date: 26 November 2017Revised date: 22 August 2018Accepted date: 29 August 2018
Please cite this article as: Hamid Mokhtar, A.A.N. Perwira Redi, Mohan Krishnamoorthy,Andreas T. Ernst, An Intermodal Hub Location Problem for Container Distribution in Indonesia, Com-puters and Operations Research (2018), doi: https://doi.org/10.1016/j.cor.2018.08.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a serviceto our customers we are providing this early version of the manuscript. The manuscript will undergocopyediting, typesetting, and review of the resulting proof before it is published in its final form. Pleasenote that during the production process errors may be discovered which could affect the content, andall legal disclaimers that apply to the journal pertain.
An Intermodal Hub Location Problem for Container Distribution in
Indonesia
Hamid Mokhtar∗1, A. A. N. Perwira Redi†2,3, Mohan Krishnamoorthy‡1,2, and Andreas T.Ernst§3
1Faculty of Engineering, Architecture and IT, The University of Queensland, St Lucia QLD 4072, Australia2Department of Mechanical & Aerospace Engineering, Monash University, Clayton VIC 3800, Australia
3School of Mathematical Sciences, Monash University, Clayton VIC 3800, Australia
Abstract
In this paper, we extend traditional hub location models for an intermodal network design on asparse network structure. While traditional hub location problems have been employed for developingnetwork designs for many specific applications, their general assumptions – such as full connectivity,uniform transfer mode, and direct connections between access nodes and hubs – restrict their directapplicability to real-world logistics problems in several ways. In many network design contexts, theusage of versatile transfer modes and hubs is required due to different pricing of modes and topologicalconsiderations. In this paper, we extend the traditional hub location problem by incorporatingthree transfer modes and two kinds of hubs. As an important additional modification, we do notassume that the underlying network is fully connected, or that hubs and access nodes are directlyconnected. The context for our modelling is intermodal container movements in an archipelago.We develop and formulate an intermodal hub location problem. We show that this problem is NP-hard. Furthermore, a dataset for intermodal hub location problem is provided, based on a real-worldcontainer distribution problem in Indonesia. This dataset involves three modes of transport and asparse network structure. We perform computational experiments and analyse our computationalresults. Our model provides insights for decision making and determining pricing policies for thedesired levels of network flow.
telecommunication systems are examples where hub network designs are required. In such network
design problems, the transfer modes may represent different transport vehicle types that carry goods
between nodes and hubs and between hubs. Hubs are equipped with special facilities that enable the
interchange of commodity between such transport modes. Each pair of nodes in the network (also
referred to as access nodes) have a flow demand for which a route or path in the network must be
chosen. The routes take advantage of hubs for cost effective transfers between hubs. Therefore, the
employment of hubs reduces network operational and installation costs through delivering economy of
scale (see Campbell and O’Kelly (2012)), and utilising the different transport modes. A hub location
problem (also called hub-and-spoke network design) determines the location of the hubs and designs a
set of routes through hubs to fulfil the origin-destination flow demands at minimum cost. In traditional
hub location problems, the network design is restricted to one transfer mode, and a few essential
assumptions. A common assumption in many of these networks is that hubs are fully connected. It
is also assumed that each access node can only be directly connected to one or many hubs, but that
it must be directly connected to a hub. In this context, for a given positive integer p, a hub location
problem in which the number of hubs is fixed to p is called the p-hub median problem. Then, we either
get the uncapacitated single allocation p-hub median problem (USApHMP), in which each access-node
must be allocated to exactly one hub, or the uncapacitated multiple allocation p-hub median problem
(UMApHMP) if access nodes can be allocated to multiple hubs.
Variants of the hub location problem (HLP) have been well-studied in the literature. Following seminal
works of O’Kelly (1986, 1987), a few hub median problems were introduced and formulated by Campbell
(1994, 1996). HLPs have been studied in several contexts including telecommunications, parcel delivery
systems, airline hub design and transportation networks (see Klincewicz (1998); Sen et al. (2015, 2016);
Ernst and Krishnamoorthy (1996, 1998a,b); Cetiner et al. (2010); Jaillet et al. (1996); Powell and
Sheffi (1983)). There have been many well-known complex network design problems which have been
analysed and implemented using a hub location model/framework. Ernst and Krishnamoorthy (1996,
1998a,b) analysed the Australia Post delivery system using UMApHMP and USApHMP. O’Kelly (1987)
implemented USApHMP in order to analyse the 25-city airline passengers network, known as Civil
Aeronautics Board (CAB) dataset. Tan and Kara (2007) analysed Turkey’s Cargo Delivery network of
81 cities through the implementation of a hub location model. Gelareh et al. (2010) used hub network
design to address a liner shipping problem. There are a few surveys which review early and recent
works on HLP including classification, modelling, real-life case studies, and solution methodologies
(see Farahani et al. (2013); Campbell et al. (2002); Alumur and Kara (2008)).
Although traditional hub location problems have been used for modelling and solving hub-and-spoke
problems, it is not comprehensive to model and analyse all real world problems, in general. These
problems often arise when hubs are not fully connected, an access node is not necessarily directly
connected to a hub, or any route can interchange flow between multiple transfer modes. In order
to overcome such requirements, research works have been carried out in which some of the underlying
assumptions have been relaxed. The resulting models, while retaining the essential characteristics of hub
location models, have become increasingly general and applicable. For instance, a common assumption
in HLP is that flow demands must be routed only through hubs. However, it is not rational to exclude
cheaper direct links between access nodes in the model for some practical applications. Aykin (1995)
and Jaillet et al. (1996) considered hub network design in which direct connections are possible while
Kartal et al. (2017) considered multi-node routes for each vehicle. Another assumption, which has been
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relaxed in previous works, is around the full connectivity of hub networks (see Campbell (2009); Labbe
and Yaman (2008); Akgun and Tansel (2018)). The traditional HLP models are not directly applicable
for delivering network designs in which the network of hubs is sparse, or when several transfer modes are
possible. Therefore, while the traditional HLP has been helpful in modelling and solving hub network
designs, there is room for more general hub location problems to be developed and used. Furthermore,
since existing datasets cannot be fully utilised for these generalisations of the HLP, new datasets are
also needed to allow evaluation and comparison of alternative formulations and algorithms.
The most popular benchmark datasets in the HLP literature are the Civil Aviation Board (CAB)
dataset, the Australia Post (AP) dataset, and the Turkish Cargo Delivery (Turkey) datasets. Most
tests of HLP models have been carried out on these datasets. In such tests, the transfer mode is
assumed to be the same throughout the whole network design. However, different modes of flow are
involved in many transportation problems in practice. In such problems, there are multiple transport
modes available and often interchange between modes is allowed but occurs at some cost.
For example, logistics companies usually operate various transportation modes such as air, ground, and
sea. In the literature, intermodal network designs address the combinations of modes that logistics
companies need to employ in order to be efficient and effective. Some of these intermodal network
design models have employed a hub location problem perspective. Note that in multimodal hub location
problems, while several modes are involved, the transfer mode for each route is uniform between its
origin and destination (see for example Alumur et al. (2012)), which is different with the intermodal
case. In many studies on intermodal network design, the graph model is partitioned into two subgraphs,
each for one mode of transfer, and possibly a set of arcs to connect the two subgraphs. Arnold et al.
(2004) gave an integer linear programming model for a problem with two modes (rail and road), where
transfer costs using two modes is minimised and the number of hubs is fixed. In this work, hubs are not
utilised to bring about a change of mode. They argued that intermodal network design is a worthwhile
substitute for unimodal network flow, in which a smaller ratio of rail to road costs results in a larger
portion of traffic flow to be routed through rail links. Ishfaq and Sox (2010), generalised UMApHMP
to model an intermodal hub location problem. In their model, hubs are fully connected in three modes,
but direct paths between pairs of nodes is allowed only in one mode. Ishfaq and Sox (2011) studied a
p-hub median model in which two copies of hubs are considered to facilitate mode changes between two
modes, and the network of hubs is fully connected. In their work, a service time at hubs is incorporated,
although costs of mode changes are not considered. Ishfaq (2012) considered UMApHMP for intermodal
networks, with a focus on incorporating delay at hubs, and developed a nonlinear ILP for the problem.
In their experiments on a dataset (that was generated with 15 nodes, and sea-road modes), they found
that there is a trade-off between the number of inland ports and operational costs. Limbourg and
Jourquin (2009) also extended the p-hub median model for the intermodal hub location problem with
single allocation for determining the locations for European transfer terminals in a rail-road network.
An intermodal hub location model is implemented on the CAB dataset by Dukkanci and Kara (2017).
Ghane-Ezabadi and Vergara (2016) proposed a model for intermodal network design, in which a set
of routes for each flow demand is calculated in advance. As a result, the proposed model is relatively
large, even when their implementation was on a sparse demand matrix.
In most of the intermodal hub location studies, it is assumed that demand nodes can access any hub
node. Furthermore, it is assumed that all hubs are homogeneous in their operations and interchange
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capabilities. However, we may have different types of hubs when there are more than two transfer
modes. Ishfaq and Sox (2011) generated a dataset from CAB dataset to examine their intermodal
network design model. They consider two modes and defined a cost ratio to prioritise one over the
other (in terms of costs).
Here, we present a dataset for the intermodal hub location problem that is drawn from a real-world
case study. It involves a sparse network, three different transport modes, and two types of hubs. We
introduce and study an intermodal hub location problem, with the archipelago of Indonesia as the
context. Our study provides a strategic solution for flow through the integrated use of two or more
modes of transportation for delivering goods.
The problem of designing an optimal Indonesia Container Distribution (ICD) network is a hub location
problem that cannot be modelled by traditional hub location models. The major Indonesian cities are
located in six main ‘areas’ (islands) and are connected through a sparse network of road, rail, and sea
links. Trucks, trains and ships are three major large volume transfer modes. Each route may use any
combination of these three modes, with hubs facilitating the intermodal transfer. Because of the sparse
nature of the ICD network, the assumption that hubs are fully connected is not valid. Also there may
not be a direct link from any access node to a hub node. Furthermore, direct connections of access
nodes in ICD may sometimes result in a cheaper route design. Thus, considering these characteristics,
the traditional HLP is not an appropriate model to accommodate models for ICD. Therefore, we need
to develop and evaluate a new kind of hub model that draws on elements from traditional hub models.
Existing hub location models in the literature do not offer adequate mathematical formulations that
take into account sparse networks and the specific intermodal requirements of the ICD example. Thus,
apart from introducing ICD, a new dataset to the HLP literature, through this paper, we also contribute
to the HLP literature by providing a new formulation and model that looks at container flows via a
mix of intermodal container hubs in a sparse network.
This paper extends the existing research in the field of hub location problems by utilising intermodal
flow transfers on sparse networks. We also provide a new dataset and introduce a new modelling
approach for the intermodal hub location problem. In Section 2, we describe the ‘Indonesian Container
Distribution’ (ICD) dataset, in which three transfer modes are included on a sparse network. The
network has unique characteristics based on the topology of Indonesia and includes both existing and
potential new infrastructure. In Section 3, we introduce the intermodal p-hub median problem, and
we prove that this problem is NP-hard. Our computational results are provided in Section 4. We then
analyse the results in great detail in Section 5.
2 The ICD Dataset
The movement of goods between islands in the archipelago of Indonesia relies heavily on maritime
transport. The volume of Indonesian container traffic has been increasing steadily. The prediction is
that the total volume of container traffic in 2020 will be four times the volume that was experienced
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in 2009. Further, the expectation is that this volume will double again in the period 2020–20301.
The future development of Indonesian transportation infrastructure is moving strongly towards the
utilisation of an intermodal transport strategy2. Thus it is vitally important to model and analyse
the network infrastructure and configurations at the strategic level before making capital investment
decisions. Given the large capacity and the cost-effectiveness of sea-based container transportation, and
the throughput effectiveness of equipment at ports, there is a need to improve and augment the efficiency
of inland transportations modes substantially. Due to the flexibility (and the inexpensive nature) of
road transportation in Indonesia, trucks have been a major vehicle for the transfer of containers to
and from seaports. This has the effect of creating large amounts of traffic at the seaports and road
networks that surround the seaports. Since seaports are mostly located in medium to large cities, road
congestion has become a major issue in these cities in Indonesia. This has, not surprisingly, impacted
many aspects of livability. The trucks queue up at ports, clog the surrounding road networks and
cause significant traffic problems in the cities. Thus, even though most of the ports in Indonesia are
modern and efficient, the large vehicular traffic around the ports means that these modern ports are
under-utilised. It leaves some of the high-capacity and expensive ship loading/unloading equipment
idle for some of the time.
As opposed to trucks, trains are able to transfer large volume of containers without causing significant
congestion problems in the cities that these ports are located in. Apart from ensuring that container
cargo are quickly moved away from the port, trains are also able to provide the modern Indonesian ports
with substantially increased throughput that means that the ship loading/unloading capital investment
is more usefully and productively utilised. Therefore, there are good reasons to look for ways to use
trains and inland container depots much more than they currently are in Indonesia. The motivation is
that through the use of trains and inland container depots, along with new rail links that need to be
constructed, the result will be: (a) be greater port capital efficiency, and (b) along with incentives and
taxes for greater utilisation of rail transportation, there will be a reduction in congestion in port cities.
The Network Structure
We first develop and describe the ICD dataset, based on real data from the Indonesian container
transportation network. The demand pattern is asymmetric, with some areas having higher levels of
production than consumption due to the presence of industry, while other areas have a net inflow of
containers. The network is rather sparse, particularly the availability of rail and maritime links is
limited for obvious reasons. We also look at utilisation levels and future development of infrastructure
to facilitate efficient intermodal transportation.
The ICD dataset represents the Indonesian container distribution network. It is useful to consider that
Indonesia is an archipelago country with a total area of 1,913,579 square kilometres. It consists of
approximately 17, 500 islands in 34 provinces3. The ICD dataset contains an origin-destination flow
demand matrix, operational and fixed costs, a set of potential hub locations, and a set of links with three
1Australian Aid-Indonesia Infrastructure Initiatives (IndII). Academic Paper to Support National Port Master PlanDecree: Creating an Efficient, Competitive and Responsive Port System for Indonesia, Technical Report, 2012
2Masterplan Percepatan dan Perluasan Pembangunan Ekonomi Indonesia 2011-2025, Kementerian koordinator BidangPerekonomian Republik Indonesia, 2011
3Badan Pusat Statistik Indonesia (BPS), 2017. https://www.bps.go.id/. Accessed: 2018-08-18.
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transfer modes in the network. The network consists of 225 nodes, which represent 195 demand nodes
and 30 potential hubs. The demand nodes correspond to 195 administrative districts in 34 provinces
of Indonesia. The potential hubs are selected in accordance with the strategy identified in the Master
Plan for Acceleration and Expansion of Indonesias Economic Development4. The potential hubs include
five inland terminals, which are hubs that enable transfers between road and rail modes of transport.
Inland terminals play a vital role in the network. They offer intermodal transportation opportunities,
increase the usage of rail transport and cause a substantial increase in container throughput into the
seaports by train. Through this, they also cause a reduction in road traffic around seaports. The rest
of the potential hubs are selected based on a visionary plan to connect the major islands, and the
West-East corridors (Maritime Highway Initiative and Pendulum Nusantara program) for container
shipping5. Reduced ICD datasets have also been created that have the same flow volumes (but with
nodes aggregated) to produce solvable instances. Illustrations for the distribution of demand nodes
with 225 nodes and 53 nodes are presented in Figures 1 and 2.
Figure 1: Illustration of demand nodes in the ICD dataset with 225 nodes
The Demand Matrix
The traffic flow demand matrix is based primarily on a survey of Indonesian origin-destination trans-
port6, in which the pattern of goods delivery for origin-destination pairs in Indonesia is estimated. The
survey data is broken up by three zones (main islands, provinces, and districts) and 33 commodity
types. The ICD dataset is based on the data at the provincial level, and uses 11 out of 33 commodities
to estimate the volume of containerised good transport in Indonesia. In order to map this demand data
at provincial level down to a higher spatial resolution, the population density data from the Indonesian
National Statistical Bureau3 is used as a proxy for the relative level of demand in different cities.
4Kementerian Koordinator Bidang Perekonomian.Masterplan Percepatan dan Perluasan Pembangunan Ekonomi In-donesia 2011-2025. Kementerian Koordinator Bidang Perekonomian, 2011.
5Indonesia Ministry of National Development Planning: National Medium Term Development Plan of Indonesia20152019 (RPJMN 20152019). Kementerian Perencanaan Pembangunan Nasional/Badan Perencanaan PembangunanNasional (Kementrian PPPN/BAPPENAS), Indonesia, 2015
6Indonesia Ministry of Transportation: Survei Asal Tujuan Transportasi Nasional Barang, 2016. URL attn-barang.dephub.go.id/data. Accessed: 2018-08-18
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Figure 2: Illustration of demand nodes in the ICD dataset with 53 nodes
Figure 3: Demand distribution in the ICD dataset
The Distance Matrix and Costs
The third element of the ICD dataset is the utilisation of three transport modes in the network, namely,
sea, road and rail links. The weight of rail and sea links in the dataset is based on the distances from
publicly available information provided by Indonesian Ministry of Transportation7 and PT Kereta Api
Logistics (KALOG) 8. Meanwhile, the lengths of road links are produced by using the Google Maps
API9 as a primary reference. The weight of rail, sea, and road links are measured in travel distance. It
is simply because a distance matrix can be easily transformed into cost and travel time. Any ferry link
is considered as a road link to reserve sea transportation as a high volume aggregation and discounted
transfer mode.
The distance matrix in the ICD dataset is asymmetric. In other words, we may not have dij = dji
7Kementrian Perhubungan Republik Indonesia (Kemenhub), 2017. URL www.dephub.go.id. Accessed: 2018-04-18.8PT Kereta Api Logistik (KALOG), 2017. URL http://www.kalogistics.co.id/. Accessed: 2018-08-18.9Google maps, 2017. URL developers.google.com/maps/documentation/distance-matrix/. Accessed: 2018-08-18.
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Figure 4: Demand distribution in the ICD dataset
in general, where dij is the length of arc (i, j). In practice, the ICD network is sparse. However, by
using the Google Maps API to produce the road distance matrix, we obtained the shortest path length
making the road network appear to be fully connected. Therefore, we use the triangular inequality to
filter the arcs which are not physical links in the network. A direct link from i to j is considered in the
dataset if and only if |dij − (dik + dkj)| ≥ ε for all nodes k distinct from both i and j, for some ε > 0.
At a strategic level, logistics planning in the sparse Indonesian archipelago network with intermodal
transportation requires the consideration of rail and inland container depots. This is likely to result
in a cost structure in which rail and sea transportation are prioritised over road-based transportation.
Since rail-based transportation benefits from high throughput, such a design leads to more efficient
ports and less road traffic congestion issues. Recall that economy of scale in the traditional HLP
occurs through the same transportation method. As opposed to the traditional HLP, the application
of economy of scale in ICD is through different transportation methods, with lower per container costs
for rail and ship transport. The marginal costs of road, sea, and rail transport modes depend on many
factors. Therefore, we collect the price of container distribution from relevant sources to select the most
appropriate cost factors for each mode (Prasetyo and Hadi, 2013; Limbourg and Jourquin, 2009). The
cost factors and distance matrix are used to calculate the cost of flow.
The original dataset on 225 nodes is a large instance and hub location models are unable to solve
problems of this size. Therefore, we generate smaller instances from the original dataset. The smaller
instances are produced on 53, 66, 73, 115, 147, 158 and 195 nodes. In generating a smaller instance
with n nodes, a subset of n nodes from 225 nodes is selected. In order to maintain the same pattern and
distribution of the original problem, each province is represented by at least one node. Therefore, the
smallest instance has 53 nodes and consists of 34 demand nodes (each of which represents a province)
and 19 potential hubs. The flow demand of each node is the aggregation of flow demands of the district
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they represent. Nodes in the instances with 66 to 195 nodes represent districts with highest density
and flow demands. A summary of features of these ICD instances is given in Table 1.
CAB Sym 25 Sym Sym 1 complete USA passenger airline networkAP Asym. 200 Asym. Sym. 1 complete Australia Post delivery systemTurkey ∼Sym. 81 Sym. Sym. 1 complete Turkey cargo delivery systemICD Asym. 225 Asym. Asym. 3 sparse Indonesian container distribution
Table 3: A comparison of a few hub location problem datasets
Any oriented path in the network connecting a pair of nodes can consist of arcs in A with different
modes. However, any two adjacent arcs in a path must be either in the same mode, or be incident to a
hub node. A hub is a node which facilitates a transfer mode interchange. A hub can be selected from
a given subset H ⊆ N of nodes. In other words H is a subset of nodes which represents the set of
potential hubs in the network. Changing the mode for each unit of flow incurs some cost. Furthermore,
the establishment of a node as a hub is only possible if a fixed cost is incurred. The problem of locating
a set of hubs among n nodes, and routing each flow demand with minimum total cost is called the
intermodal hub location problem (IHLP). Figure 9 illustrates an intermodal network for 4 islands with
three different transfer modes. Figure 9 also contains potential inland terminals and rail links. As we
discuss in Sections 4 and 5, we may include potential inland terminals and rail links in our dataset to
study the impact of network expansions.
In the IHLP, the number of located hubs, including seaports and inland terminals is to be decided in
such a way that the total cost is minimised. Inland terminals enable transfer between road and rail
while seaports provide transfer to ships. For a given pair of integers (q, p), where q ≤ p, the intermodal
multiple allocation hub location problem is one in which the total number of hubs is fixed to p and
the minimum number of inland terminals is set to q. We denote this problem as the (q, p)-intermodal
hub median problem ((q, p)-IHMP). As specified earlier, this problem is important for a few practical
reasons. At a strategic level, we use this model to understand the infrastructure budget that needs
to be allowed for capital and infrastructure works that are required to facilitate efficient container
flows in the network. We use this model to also understand the impact that various permutations of
inland terminals and seaports has on both traffic congestion as well as infrastructure costs. We expect
that there is a trade-off that needs to be struck between reduced congestion and costs of establishing
inland terminals. We expect to use this model to ascertain what the optimal balance is through the
provisioning of appropriate incentives that encourage rail transfers. We formulate and discuss this
particular problem in this paper from here on in, noting that there are many other interesting variants
of this problem that may also be explored. However, for the remainder of this paper we focus specifically
on the (q, p)-IHMP.
Mathematical formulation for (q, p)-IHMP
Let δ+i,m denote the subset of arcs with transfer mode m ∈M whose tails are i, and δ+i is the subset of
arcs whose tails are i, where M = {1, 2, 3}. In other words, δ+i,m = {e ∈ Am : e = (i, j) for some j ∈ N},for i ∈ N andm ∈M , and δ+i = ∪m∈Mδ+i,m. Analogously, define δ−i,m = {e ∈ Am : e = (j, i) for some j ∈N} and δ−i = ∪m∈Mδ−i,m. Denote by Ghkl the cost of changing transfer mode from k to l at hub h, for
each unit of flow and k, l ∈ M . Denote by Fh the cost of establishing node h as hub. We denote the
subset of inland terminal potential hubs by D, where D ⊂ H.
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roadsea linkrail link
seaport
inland terminal
demand node
potential rail link
potential inland terminal
Figure 9: A depiction of an intermodal network for 4 islands and three transfer modes
Let the binary decision variable zh = 1 if node h is chosen as hub, and zh = 0 otherwise, for each
h ∈ H. We assume that there is no flow demand if either of the terminals is a potential hub. Let yijebe the fraction of flow request Wij which uses arc e, for all i, j ∈ N \H, i 6= j, and e ∈ A. Then the
cost of using arc e, when e ∈ Am, for m ∈ M , is Ce = αmde, where αm is the cost factor for transfer
in mode m ∈ M . By xijhkl we denote the fraction of flow Wij whose transfer mode changes from k
to l at hub h. Note that at each hub node, the total of flow fractions which can either pass through
the hub without change of transfer mode, or change from one transfer mode to another one is at most
the number of connection requests, which is (|N | − |H|)2. Since there are 3 transfer modes, and so,
9 possible mode changes or transfers without mode change, the total of flow fractions which can be
transferred through each hub is bounded above by Γ = 9(|N | − |H|)2. We use this upper bound in
conjunction with the binary variables zh below. We present an integer linear programming formulation
of the problem below:
(q, p)-IHMP: min∑
h∈HFh zh +
∑
i∈N
∑
j∈N
∑
e∈ACe Wij yije +
∑
i,j∈N
∑
h∈H
∑
k,l∈MGhklWij xijhkl (1)
s.t.∑
h∈Hzh = p, (2)
∑
h∈Dzh ≥ q, (3)
∑
e∈δ+i
yije −∑
e∈δ−i
yije = 1 ∀i, j ∈ N \H (4)
∑
e∈δ+j
yije −∑
e∈δ−j
yije = −1 ∀i, j ∈ N \H (5)
∑
e∈δ+h
yije −∑
e∈δ−h
yije = 0 ∀i, j ∈ N \H,h ∈ N,h 6= i, j (6)
∑
e∈δ+h,m
yije −∑
e∈δ−h,m
yije =∑
k 6=mxijhkm −
∑
l 6=mxijhml ∀m ∈M,h ∈ H, i, j ∈ N (7)
∑
i,j∈N
∑
k,l∈Mxijhkl ≤ Γ zh, ∀h ∈ H (8)
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zh ∈ {0, 1}, xijhkl ≥ 0, yije ≥ 0 ∀i, j ∈ N,h ∈ H, k, l ∈M, e ∈ A. (9)
Equation (2) fixes the number of located hubs to p, and equation (3) ensures that the required minimum
number of inland terminals are located. The equations (4)-(6) are flow conservation constraints for flow
between demand nodes. Equations (4) and (5) ensure that the total flow for any flow demand is
sourced and terminated at corresponding terminals, and the set of constraints (6) ensures that inflow
and outflow of a demand at any intermediate node are equal. Through the set of constrains (7), xijhklcaptures the exact amount of flow portion for demand (i, j) which changes transfer mode from k to l
at hub h, for all i, j ∈ N, k, l ∈ M , and h ∈ H. The right term of constraints (7) gives the increased
fraction of flow Wij in transfer mode m at hub h. In (7), for a fixed (i, j) and h, there are |M | equations
and |M |(|M | − 1) variables xijhkl (we already excluded xijhmm for m ∈ M). In any optimal solution,
at most |M |(|M | − 1)/2 of these variables are positive. When |M | = 3, the set of equations in (7) has
a unique solution for the variables xijhkl for fixed yije for e ∈ A. Thus xijhkl captures the fraction of
Wij with mode change from k to l at hub h in any feasible solution. From xijhkl we can assess the
congestion at links incident to hubs in different modes, and congestion at facilities for transfer mode
changes. The set of constraints (8) guarantees that all mode changes only occur at the selected hubs.
This constraint can be disaggregated to create a tighter but larger formulation. Finally, the objective
function reflects the total cost of fulfilling flow demands. The cost of locating hubs, the cost of flow
through links, and the cost of transfer mode changing in all paths are respectively considered in the
first, second and third terms of (1).
Note that (1)-(9) provides a rigorous and straightforward mathematical formulation for (q, p)-IHMP,
which differs from the existing formulations in the literature (for example, see Ishfaq and Sox (2011);
Arnold et al. (2004)) in several ways. Our formulation allows three distinct transfer modes and the
transfer mode changes at hubs, and relaxes the full-connectivity assumption for the hub network.
Our formulation is partly based on the original hub location formulations, and partly based on the
choice of variables xijhkl, which bring about these advantages. But the number of variables in (q, p)-
IHMP remains in the same order as in existing formulations for the hub location problems. When the
underlying network is sparse, the number of arcs can be considered to be of linear order in the number
of nodes, that is |A| = O(n). The number of origin destination pairs is at most O(n2). In formulation
(1)-(9), the number of binary variables is |H|, the number of real variables is O(n3), and the number of
constraints is O(n3). Interestingly, for any feasible set of p hubs by which a path for every flow demand
exists in the network, there is an optimal solution in which the real variables yije and xijhkl are either
0 or 1. In fact, if there are two fractional flow paths between two nodes (i, j), say P1 and P2, then we
can always choose the cheaper path since there is no capacity on links/nodes. If the total cost of one
of them is larger, then the fraction of flow through that path is zero. If P1 and P2 have the same cost,
then either of them can be chosen to have zero flow. Hence, since the coefficients of all variables in the
objective function are positive, there is an optimal solution in which all variables have integer values.
Theorem 3.1. (q, p)-IHMP is NP-hard.
Proof. To prove this claim, we show that a given instance of UMApHMP can be reduced to an instance
of (q, p)-IHMP. Since the p-hub median problems are NP-hard (Love et al., 1988), this reduction proves
that (q, p)-IHMP is NP-hard. Suppose we are given an instance of symmetric UMApHMP, denoted by
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P1, on the set of nodes {1, 2, . . . , n} to locate p hubs, in which Wij is the flow request for pair (i, j),
and dij is the distance between i and j. Let N = {1, 2, . . . , n} × {0, 1}, and let A be the set of arcs
{((i, 0), (j, 1)), ((i, 1), (j, 0)), ((i, 1), (j, 1)) : i, j = 1, 2, . . . , n, i 6= j}. Now we construct a network on the
set of nodes N and the set of arcs A. Any arc ((i, 0), (j, 1)) or ((i, 1), (j, 0)) is considered to be in mode
2, and any arc ((i, 1), (j, 1)) is considered to be in mode 3, where i, j = 1, 2, . . . , n. This network does
not contain any arc in mode 1. Then we construct P2, an instance of (q, p)-IHMP, in which q = 0 and
the set of potential hubs H is the set of nodes {(i, 1) : i = 1, 2, . . . , n} (see Figure 10 for an illustration).
We set the flow demands for ((i, oi), (j, oj)) to be Wij if oi = oj = 0, and zero otherwise. Moreover, we
set the distance of (i, oi) and (j, oj) to be dij , for oi, oj ∈ {0, 1}, and (oi, oj) 6= (0, 0). We set mode cost
factors α2 to be the collection/distribution factor, and α3 to be the transfer cost between hubs in P1.
We set the cost of change modes and installation fixed costs to zero in P2.
(i, 1)
(i, 0)
(j, 1)
(j, 0)
Figure 10: Construction of a (q, p)-IHMP instance from a UMApHMP instance
We show that any optimal solution of P1 corresponds to an optimal solution of P2 with the same
optimal value, and vice versa. For a given feasible solution of P1, suppose {kt : t = 1, 2, . . . , p} is the
set of located hubs, and (ksij)s is the ordered visited hubs on the route from i to j. This is equivalent
to a solution of P2, in which {(kt, 1) : t = 1, 2, . . . , p} is the set of located hubs. Moreover, there is a
route from (i, 0) to (j, 0) which visits hubs ((ksij , 1))s in order. It is obvious that the cost of the two
path in P1 and P2 are the same.
On the other hand, clearly any set of p hubs in P2 gives rise to a set of p hubs in P1. Suppose
Pij = (i, 0), (k1, o1), . . . , (kt, ot), (j, 0) is the shortest path in P2 between (i, 0) and (j, 0). Note that
ol = 1 for l = 1, 2, . . . , t. Otherwise the path (i, 0), (k1, 1), . . . , (kt, 1), (j, 0) is less expensive than Pij .
Then i, k1, . . . , kt, j is an equivalent path of Pij in P1 with the same cost. Therefore, any optimal
solution of P1 corresponds to an optimal solution of P2 with the same cost, and vice versa. This
completes the proof.
The (q, p)-IHMP is an interesting extension of the traditional hub location problem for intermodal
versions. In (q, p)-IHMP, (a) hubs serve consolidations/unconsolidations of goods and facilitate transfer
mode changes, and (b) two types of hubs are distinguished so that any requirement for prioritising the
installation of one type of hub can be accommodated. A similar discussion holds for the IHLP, in which
the optimal number of hubs is determined by optimal solutions. A formulation for IHLP can be obtained
from the (q, p)-IHMP formulation by dropping (2) and (3). However, IHLP and (q, p)-IHMP are only
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two of many possible extensions of the traditional hub location problems in the intermodal networks.
Other extensions, including the capacitated hub location problem (see Ernst and Krishnamoorthy
(1999)), single allocation problem (see Ernst and Krishnamoorthy (1996)), and hub center problem
(see Ernst et al. (2009)) could be explored. We leave these investigations for the future.
In the following we use a commercial solver to solve instances of (q, p)-IHMP and analyse the solutions
thus produced. Due to computational restrictions, we are not able to solve the very large problem
instances. Because of the inherent complex nature of (q, p)-IHMP we can borrow from the literature
on traditional hub location problems, and employ many effective approaches for solving the problem.
These would include heuristics (see Ernst and Krishnamoorthy (1998b)) and Benders decomposition
(see Mokhtar et al. (a,b,c)). Future research should explore these approaches for solving larger instances
of (q, p)-IHMP efficiently. In our current work, we wished to motivate our research, indicate why new
models were required, present a new dataset, provide a formulation for the new problem, provide initial
computational results to simply demonstrate what sorts of analyses could be carried out, and present
our computational analysis. In the next sections, we present our computational results using CPLEX
and the analyse these results.
4 Computational Results
In this section we present extensive computational results of our experiments on the ICD dataset and
intermodal hub location problems 11. We consider different scenarios in which the cost factors, or the
network structure in the ICD dataset is modified. We then analyse our computational results using
different scenarios in Section 5. We show how inland terminals, congestion taxes and rail subsidies can
actually improve intermodal network flow and reduce road congestion in large cities. In Table 4, we
first summarise the notations that we will use throughout this section.
notation description
time the computational time in secondsObj the optimal value of objective function divided by 1.0e13πs number of 100,000 containers routed through sea linksπd number of 100,000 containers routed through road links incident to seaports with rail terminalsπl number of 100,000 routed through rail links incident to seaports with rail terminalsRR% percentage of containers routed through road links incident to seaports with rail terminalsd<>l number of 100,000 containers interchanged modes between road and rail at a seaport with rail terminald<>s number of 100,000 containers interchanged modes between sea and road at a seaport with rail terminall<>s number of 100,000 containers interchanged modes between sea and rail at a seaport with rail terminal
Table 4: Notations used in computational results
All computations were performed on a computer with 8 cores of 2.5 GHz processors and 32 Gb
memory, with 64-bit Linux RedHat operating system. All methods were coded in C++ using the
Concert Technology CPLEX 12.6. To improve computational efficiency, we set a few parameters in
CPLEX empirically, based on limited numerical experiments with small instances. We set the rule for
selecting the branching variable to be based on pseudo-shadow prices, and the solution algorithm at
root nodes to be the dual simplex method. We also switched off the usage of ‘heuristic in nodes’. The
time limit for all computations was fixed to 7200 seconds. If within the specified time limit, a method
is able to find the optimal solution, the corresponding CPU time is presented in seconds (sec). If the
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Figure 11: An optimal solution for the instance with n = 73, p = 12, q = 4 (road links are not shown)
method only finds a feasible non-optimal solution, the gap between the best solution is presented in
the ‘time’ column.
We present our computational results for (q, p)-IHMP for the ICD dataset in Tables 6 and 7. Since
the instances on 53 nodes are not representative of distribution for demand nodes in the high demand
area Java, we have only present results on instances with more than 60 nodes. The largest instance
we could solve optimally with our settings was the 73-node instance. An optimal location of hubs and
inland terminals for an instance with 73 nodes is illustrated in Figure 11. In fact, we experimented
our program on instances with 97 nodes or more with the 3-hour time limit. In most cases, no feasible
solution was found, and in a few cases, the quality of solutions were not satisfactory for presentation
here. Over the last two decades the hub location research community has made significant progress in
solving larger instances, in part due to the challenge provided by larger realistic data sets such as the
Australia Post and Turkey datasets. This paper is in part providing a similar challenge to push the
boundaries of what problems can be solved in the hub location area. Also, there are many ways to
extend the computational experiments and analyse solutions. We believe that further research, newer
algorithms and more experiments will be needed to solve these larger problem instances more effectively.
In the archipelago of Indonesia, there are five main islands which can only be connected thorough sea
links. Therefore, p ≥ 5 and q ranges between 0 and p − 5. Note that for p larger than the number
of potential seaports, at least p − k inland terminals are already included, where k is the number of
seaports. Therefore, problems with q = 0, 1, . . . , p − k are equivalent. Thus, we only consider one of
the equivalent problems in our computational experiment.
Congestion along roads that lead to seaports is a major issue in Indonesia7. Thus, we analyse the
number of containers transferred to seaports through road links, either for shipping, loading trains, or
bypassing seaports. Tables 6 and 7 respectively present our computational results for instances with 66
and 73 nodes respectively. In the first scenario, the original pricing of all modes are used (see Section 2).
Table 5: Cost factors of different modes in different scenarios
In scenarios ICD-66%-discnt and ICD-99%-discnt, respectively, the pricing of train transportation is
discounted by 66% and 99%, respectively (see Table 5). Of course ICD-99%-discnt is not a realistic
scenario. So, we use it only for the purpose of analysis. The corresponding results are presented in the
columns ICD-66% and ICD-99% in Tables 6 and 7. Also we assess the effect of increasing the number
of rail links too. We add more rail links to the existing rail network to the 10%, and 25% highest
demand nodes (explained later). Since the rail network in Indonesia is very sparse, these scenarios are
helpful in the analysis of the impact of the rail network expansion on throughput and congestion. The
corresponding results are presented in the columns ICD+10% and ICD+25% in Tables 6 and 7 respectively
(where we were able to solve the instances). In total, our computational experiments are performed on
680 instances. All computation results are available in an online repository 11.
As shown in Tables 6 and 7, the computational time for (q, p)-IHMP increases with the number of
nodes because of the sharp increase in the number of variables in the model. Generally, there is an
increase in the computational effort as p decreases and n, q are fixed, or when q increases and n, p are
fixed (due to more intense competitions for the location of hubs). Also, for smaller differences of p
and q (that is p − q), the computational effort surges because of the high competition for resolving
potential hub locations. As the number of p-hub combinations increase, the search space grows and the
computational effort increases. For fixed n and increasing p, the fixed costs increase and the network
flow costs decreases (since flow costs dominate the fixed costs) and the objective value decreases as p
increases. For fixed (n, p), the feasible set of the problem shrinks as q grows, and the objective value
increases. Intuitively, with larger q, there is a tougher competition between seaports (and also inland
terminals) to be chosen, and hence the computational time increases in general. In most cases, the
computational effort is dominated by these trade-offs.
In Tables 6 and 7, the number of containers routed through road links incident to some seaport which
have rail terminals, indicated by πd, has a decreasing trend as q grows and p or p−q are fixed in general.
This is due to the presence of more inland terminals and deviation of transportations from seaports to
inland terminals. It is interesting that very few inland (rail only) hubs are selected by the model. The
first time that this happens is for p = 12, as can be seen by the fact that the cost does not increase
when the use of an inland terminal is forced (q = 1). d<>s, πd and πl demonstrate the usage of road
for sea transfer, the road congestion and usage of trains respectively. d<>s and πd decrease while πlgrows as q increases and p− q is fixed in general (see Figures 14-18).
When the number of inland hubs is small, seaports are used not only for shipping through sea links,
but also for change of transfer modes between road and rail. It can be observed from Tables 6 and
7 that for larger numbers of inland hubs, that is q ≥ 3 and p ≤ 11, there is a significant drop in the
volume of πd, πl, and d<>l. This is justified by the smaller number of containers that change mode
to sea using roads and more usage of inland terminals to change mode between land modes instead of
using seaport facilities. This observation can also be a signal to the significant importance of inland
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terminals in (q, p)-IHMP. When the number of inland terminals is small (generally when q is small),
a larger number of transfer mode interchanges to sea are delivered by trucks, and seaports play the
role of inland terminals to facilitate the interchange between land modes, especially for routes between
nodes in the same island. There are cases in which a few seaports in the optimal solutions are solely
used to facilitate d<>l interchanges. Note that in many instances with q ≤ 2, inland hubs are mostly
used for transportation between two parts of large demand nodes rather than facilitating transfer of
containers to seaports. For q ≥ 3, there are a few inland terminals which have direct rail links to some
seaports. Thus, such inland terminals support a cheaper transfer of containers to seaports. As a result,
road congestion at seaports is also avoided. This leads us to a conclusion that the installation of inland
terminals can help easing the traffic congestions at seaport roads, and a more comprehensive utilisation
of infrastructure at seaports for sea shipments (as a result of capacity-reduction inducing queuing) at
seaports for shipments through sea links.
For the scenarios with discounted rail costs, we make similar observations to those in the above para-
graphs. In both scenarios ICD-66%-discnt and ICD-99%-discnt, a larger number of inland terminals
– up to the case when q ≤ p − 6 – causes reductions in congestion at seaport roads and the number
of interchange of modes at seaports. Due to less expensive train transportation, the transportation
through seaport by roads decreases from over 46% to about 34% in ICD-66%-discnt on average. Using
the ICD-99%-discnt scenario, we obtain a lower bound on the number of containers routed through
road, and interchanges at seaports in this given network structure.
In our experiments, we further included more rail links to analyse how a denser rail network could
help the network flow in this problem since the number of rail links in Indonesia is quite small. We
considered demand nodes with the highest flow amounts in the network which contribute to 10% of
total flow in the problem. An inland terminal for each of such demand nodes is added. Then we
added a tree of rail links to connect potential hubs. We repeated the same procedure for demand nodes
with highest flow amounts contributing to 25% of total containers. We denote these two scenarios by
ICD+10% and ICD+25% respectively.
In these two experiments, the total rail and road transportations and d<>s at seaports reduces, while
the portion of rail transportation and l<>s increases. In scenarios with more rail links or discounted
train transportation, the usage of trains is enhanced. However, a larger portion of demands in the
scenarios with new rail links are fulfilled by trains which reduced the usage of trucks at seaports in
general.
There are many interesting variations of IHLP and, in particular (q, p)-IHMP. We leave the exploration
of these variants for subsequent work on this topic. One interesting variant of (q, p)-IHMP is one in
which the number of hubs is not fixed. We get a different intermodal hub location problem when we
consider this option.
In this free-hubs variant of (q, p)-IHMP, the optimal number of hubs is left to be determined by the
optimisation based on fixed costs for hubs. We also experiment with this version of the intermodal hub
location problem, and present the computational results in Table 8. The number of tested instances for
this variation is smaller since the argument on experimentation for different combinations of p and q is
not valid. In this table, computational results on instances with 66 and 73 nodes are presented to be
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ICD ICD-66% ICD-99% ICD+10% ICD+25%
p q time Obj πs πd πl RR% d<>l d<>s l<>s time Obj πs πd πl RR% d<>l d<>s l<>s time Obj πs πd πl RR% d<>l d<>s l<>s time Obj πs πd πl RR% d<>l d<>s l<>s time Obj πs πd πl RR% d<>l d<>s l<>s
Table 8: Intermodal hub location problem on ICD where the numbers of hubs are not fixed
compared with the results on (q, p)-IHMP in Tables 6 and 7. Nonetheless, computational experiments
on large instances for either variations may encounter memory limitation issues. As shown in Table 8,
the optimal number of seaports is 14, and the optimal number of inland terminals is 5 in most cases.
When there is a discount on train transportation costs, or when new inland terminals are added to the
network structure, the total network cost decreases. However, the road congestion at seaport roads
increases with decreased rail cost. This somewhat counter-intuitive result is due to optimal solution
making greater use of rail transportation but needing to move all of the containers by truck to the port
where the mode change occurs. Hence the share of container movements by rail at the port increases
at the same time as the road movements to the port increases. The same trend is also observed for
the number of intermodal changes at seaports. In comparison with (q, p)-IHMP, the small number of
inland terminals causes higher loads on seaport roads.
5 Analysis of Solutions
Our experiments with the various different scenarios provide us insights into decision making based
on pricing options on different transfer modes and based on investments that are made to network
infrastructure. For instance, it is inferred from our experiments that seaports are less congested when
either more inland terminals (and associated rail links) are installed, or when a discount factor is applied
for rail transportation. It is what we would expect, intuitively. However, the model and the results
demonstrate that this is indeed the case.
As shown in Figures 12 and 13, when n and p are fixed, the traffic congestion on seaport roads decreases
as q grows in all scenarios for q ≤ 5. In other words, by requiring the problem to include more inland
terminals among p hubs, trains are more involved in transferring containers to seaports. When train
costs are well-discounted or new rail links are added to the network, simultaneously the number of
containers shipped through seaports declines and the usage of trains becomes more attractive. So the
flow on road links in the network is deviated towards inland terminals. In scenarios ICD-66%-discnt
and ICD-99%-discnt, the road congestion on seaport roads and d<>s decrease as q grows. However,
once q becomes too large so that the number of seaports drops to 5, the usage of trains is not cost
effective, even if more rail links for high demand nodes are added in the network. Note that, as depicted
in Figures 12 and 13, the operational costs in different scenarios increases in all scenarios by at most
2%.
In order to understand the impact of inland terminals and rail transportation, we analyse congestion
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20%
30%
40%
50%
60%
1 2 3 4 5 6 7 8
ICD ICD-99%-dscnt ICD-66%-dscnt ICD10xtra
(a) Road usage percentage
40
52
64
76
88
100
112
1 2 3 4 5 6 7 8
x 10
0000
ICD ICD-99%-dscnt ICD10xtra ICD-66%dscnt
(b) Road Load
110
120
130
140
150
160
170
2 3 4 5 6 7 8
Trill
ions
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(c) Total costs
Figure 12: Comparison of road and rail usage at seaports as q increases for n = 66 and p = 15
20%
30%
40%
50%
60%
0 1 2 3 4 5
ICD ICD-99%-dscnt ICD-66%-dscnt ICD10xtra
(a) Road usage percentage
6.3E+06
7.5E+06
8.7E+06
9.9E+06
1.1E+07
0 1 2 3 4 5q
ICD ICD-99%-dscnt ICD-66%dscnt
(b) Road Load
1.2E+14
1.3E+14
1.4E+14
1.5E+14
1.6E+14
0 1 2 3 4 5
ICD ICD-99%-dscnt ICD-66%-dscnt
(c) Total costs
Figure 13: Comparison of road and rail usage at seaports as q increases for n = 73 and p = 14
and also, costs for the case when the total number of hubs p and the minimum number of inland
terminals, q, are incremented together. This way, we can observe the impact when the problem itself is
not forced to sacrifice a seaport for an inland terminal. Hence, we can observe the impact of adding an
inland terminal to the set of hubs. As shown in Figures 14 and 15, the proportion of road congestion at
seaport links is considerably decreased and the total operational and fixed costs decreases by at most
5%. Figures 14-18 indicate that the number of containers transferred by trucks to seaports for transfer
through sea links is considerably decreased. There is an increase in road transfer to seaports (mostly
for l<>d) in most scenarios when the number of inland terminals reaches to 2 because there is a better
opportunity to substitute road transfer by rail transfer in a few islands, but the proportion of road
usage remains declining as q grows. Besides installing more inland terminals, the facilities of rail usage
(by new rail links, or discounted rails) promotes the ratio of the rail usage to the road usage. As shown
in Figures 16 and 17, there is a shift in the number of containers shipped to seaports by road to rail as
q grows. Furthermore, the total operations of modal interchange at seaports is significantly deviated
from seaports as shown in Figure 18.
In the following, we compare the fixed costs and congestion costs in seaports. The congestion cost
is associated with the opportunity cost of delays on roads. It is proportional to the number of road
users and reciprocal of the average speed on roads when the capacity of roads is fixed. Since container
transport by trucks increases the number of road users and decreases the average speed, we expect
that the deviation of container transport to rail improves the congestion cost. There are many sophis-
ticated models to incorporate variable traffic flow, network flow equilibrium conditions, and speed-flow
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20%
30%
40%
50%
60%
0 1 2 3 4 5 6 7 8
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(a) Road usage percentage
45
65
85
105
125
145
0 1 2 3 4 5 6 7 8
x 10
0000
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(b) Road load
1.1E+14
1.2E+14
1.3E+14
1.4E+14
1.5E+14
1.6E+14
0 1 2 3 4 5 6 7 8
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(c) Total costs
Figure 14: Comparison of road and rail usage at seaports as q increases for n = 66 and p− q = 9
20%
40%
60%
0 1 2 3 4 5
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(a) Road usage percentage
5.00E+06
6.00E+06
7.00E+06
8.00E+06
9.00E+06
1.00E+07
1.10E+07
1.20E+07
1.30E+07
0 1 2 3 4 5
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(b) Road load
1.2E+14
1.3E+14
1.4E+14
1.5E+14
1.6E+14
0 1 2 3 4 5
ICD ICD-99%-dscnt ICD+10% ICD-66%-dscnt
(c) Total costs
Figure 15: Comparison of road and rail usage at seaports as q increases for n = 73 and p− q = 10
5
8
11
0 1 2 3 4 5
Mill
ions
Truck Train
(a) ICD
5
8
11
14
0 1 2 3 4 5
Mill
ions
Truck Train
(b) ICD-66%-dscnt
5
8
11
14
0 1 2 3 4 5 6 7 8
Mill
ions
Truck Train
(c) ICD+25%
Figure 16: Comparison of road and rail usage at seaports in different scenarios as q increases for n = 66and p− q = 8
relationship to come up with optimum congestion costs (Yang and Huang, 1998; Verhoef, 1999). To
measure the cost of congestion in seaports in our study, we use an economic model for congested roads,
which is simple to interpret and requires few parameters. Evans (1992) proposed congestion costs as
follows:
W = β
(1
1− QCap
− 1
)+
βQ
Cap× 1
(1− QCap)2
,
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0.00E+00
3.00E+06
6.00E+06
9.00E+06
1.20E+07
1.50E+07
1.80E+07
0 1 2 3 4 5
Truck Train
(a) ICD
0.00E+00
3.00E+06
6.00E+06
9.00E+06
1.20E+07
1.50E+07
1.80E+07
2.10E+07
2.40E+07
0 1 2 3 4 5
Truck Train
(b) ICD-66%-dscnt
Figure 17: Comparison of road and rail usage at seaports in different scenarios as q increases for n = 73and p− q = 10
where β is a parameter that reflects the monetary value of the equation, and Q and Cap are the total
number of containers using roads and the overall capacity of the road system, respectively. We also
associate cost factors Crr and Crs for the number of containers which change transfer modes between
road and rail (denote it by Xrr), and road to ship (denoted by Xrs), respectively, in order to measure
avoidable usage of seaport infrastructures by using inland terminals. All together, we use the congestion
cost function W +CrrXrr+CrsXrs. Since queues at seaport facilities are not desired, we set Crs � CrrAs shown in Figure 19, with larger number of inland terminals, the total cost of congestion and fixed
costs is minimum for q = 4. In other words, when the intermodal hub location problem parameters
are set to have at least 4 inland terminals, the cost of installations and road congestion at large cities
(in other words, seaports), is minimum. In this way the modelling can give some guidance as to the
level of investment in rail infrastructure that can be justified by the reduction in congestion, though
the practical difficulty is that the congestion costs are carried by all road users, while the infrastructure
has to be paid for by the state or transport companies.
0
20
40
60
80
100
120
140
160
0 5 0 5 0 5
ICD ICD10xtra ICD3
x 10
0000
d<>l d<>s l<>s
ICD + 10% ICD - 66%
q:
(a) n = 66 and p− q = 8
0
20
40
60
80
100
120
140
160
180
0 5 0 5 0 5
ICD ICD10xtra ICD3
x 10
0000
d<>l d<>s l<>s
ICD + 10% ICD - 66%
q:
(b) n = 73 and p− q = 10
Figure 18: Comparison of interchanges at seaports for q = 0, 5
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50
100
150
200
250
300
350
400
0 1 2 3 4 5
x 10
0000
000
Fixed Costs (inland terminals) Congstion Costs Total Costs
(a) n = 66
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5
x 10
0000
000
Fixed Costs (inland terminals) Congestion Costs Total Costs
(b) n = 73
Figure 19: Trend of congestion costs and fixed costs for new inland hubs
6 Conclusions
In this paper, we present a network design framework as an extension of the traditional hub location
problem. Here, a few important restrictions of the traditional hub location problem are relaxed in order
to model the intermodal network design problem for container transportation. We relax the assumption
that the network is complete. We assume that the cost of interchanges between modes is non-trivial.
The (q, p)-IHMP presented and formulated in this paper is such a network design problem. It includes
3 transport modes and 2 types of hubs. Network flow costs and fixed costs are also incorporated.
We then proved that (q, p)-IHMP is equivalent to the traditional hub location problem in terms of
computational complexity. Furthermore, a dataset based on a real-world problem for intermodal hub
location problems is provided. This includes a sparse network structure, 3 types of links, 2 types of
hubs, cost factors, and flow demands.
An analysis on our computational results on (q, p)-IHMP provides an insight into some of the most
influential factors for decision making in this case study. It is shown that by investing in the estab-
lishment of inland container hubs, the total congestion cost and fixed cost is reduced. Furthermore,
it is shown that the introduction of discounted pricing on rail transportation can influence network
flow, throughput and can lead to congestion reduction at roads around seaports. Through these capital
investment instruments, a larger volume of containers are transferred to seaports through the rail net-
work. This causes shorter queues at ports for loading ships, and a better utilisation of port facilities.
However, not all seaports automatically see reduced congestion due to increased rail facilities, as the
rail station can attract additional truck traffic to transfer containers to rail. Hence careful analysis
using the type of hub location models proposed here is required. In general, a better pricing policy
results in a more efficient usage of port facilities at seaports, and also causes far less road congestion
at the corresponding cities.
The intermodal hub location problem is an important extension of the general hub location problem.
Also, the sparsity of the underlying network reflects a more practical framework for further research.
There is room to extend the intermodal hub location problem to other interesting variations. A varia-
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tion of the problem that considers specific allocation strategies and capacities on arcs and nodes will be
particularly interesting and practically relevant. In addition, the ICD dataset provided in this paper
can be used in the literature of intermodal hub location problems (and hub location problems, in gen-
eral) to design more complex and interesting problems. The new problem also creates a new challenge
as solving the full sized instance, or even medium to large sized instances in the ICD dataset is beyond
the capability of standard integer linear programming solvers (in a reasonably short time) using the
formulation presented here. We are confident that several alternative and more sophisticated mathe-
matical models are possible for the problem that we have presented. We also believe that it is possible
to develop a more tailored, powerful and specialised solution method (or methods) for the problem that
we presented. Indeed, given the potential practical importance of the model, further research is needed
to develop such methods for this problem. The current literature on solutions to hub location problems
suggests that decomposition methods and/or heuristic methods could well be employed as promising
directions. Furthermore, different transfer modes and types of hubs can be exploited for interesting
experimentations and solution analyses with the aim of advising strategic investment decisions and
tactical flow management arrangements. We also believe that future approaches should incorporate
more sophisticated congestion models.
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