1 Introduction

The transportation industry is rapidly changing due to technological advances and the constant need to find faster and cheaper ways to transport freight across the globe. Intermodal freight transport is a system for transporting goods, particularly over longer distances and across international borders, which has played a significant role in the freight transportation industry. An intermodal freight transportation system includes ocean and coastal routes, inland waterways, railways, roads, and airways. Intermodal transportation is the shipment of commodities from one point to another using combinations of at least two different transport modes (e.g., truck to train to barge to ocean-going vessel) (Bektaş and Crainic 2007).

The volume of freight transport has grown rapidly in the last four decades. In 1971, the domestic freight market totalled just 134 billion tonne-kilometres, while by 2010 it has expanded to more than 220 billion tonne-kilometres (Department for Transport 2012). Increasing freight transportation brings with it concerns about air quality and climate change. Freight transportation is largely driven by fossil fuel combustion, mostly diesel fuel, resulting in emissions of greenhouse gases (GHG), such as Carbon dioxide (CO2), Nitrogen oxide (NOx), Sulfur oxide (SOx), particulate matters and air toxics.

GHG emissions not only are harmful to the health of humans, but also have harmful impacts on environment, including the increased drought, more heavy downpours and flooding, greater sea level rise and harm to water resources, agriculture, wildlife and ecosystems. Global emissions of CO2 as the primary GHG increased by 3 % in 2011, reaching an all-time high of 34 billion tonnes in 2011 (Oliver et al. 2012). In the UK, the total GHG emissions from transport have increased by 13 % between 1990 and 2009. The proportion of total UK GHG emissions from transport have increased from 18 % in 1990 to 27 % in 2009 (Department for Transport 2011). In 2009, the freight transport within the UK is estimated to account for 21 % of domestic transport GHG emissions and 5 % of all UK domestic GHG emissions (Department for Transport 2011).

Traditional logistics models have concentrated on minimizing operational transportation costs. However, the consideration of the wider objectives and issues especially related to GHG emissions leads to new models and technologies. In this paper, an intermodal transportation problem is described that includes consideration of GHG emissions, in which CO2 emissions are explicitly modelled. In our model, the objective is to minimize the total costs in an intermodal system, including the capital costs, operational costs, intermodal transfer cost and the GHG emission cost, such that a number of commodities are shipped from their origin to destination. The decisions to be made comprise: (i) the selection of routes and transport modes and (ii) the flow distribution through the selected route and mode. The resulting green service network design model is a non-linear mixed integer program. We propose a linearization to transform the model into an integer linear programming formulation which is then solved by off-the-shelf optimization software.

The key contributions of this paper can be summarized as follows: (i) We present a model which, to the best of our knowledge, is the first to explicitly include intermodal transfer cost in its objective when modeling a green intermodal transportation system. (ii) we present the results of a hypothetical but realistic numerical study based on data collected from the UK.

The rest of the paper is organized as follows: In the following section, a brief literature review is presented. Section 3 discusses a way of estimating emissions and presents a non-linear service network design model with intermodal transfer cost, which is then linearized. In Section 4, a hypothetical case study from the UK is provided and results of computational experiment and analyses are presented. In Section 5, we present a bi-criteria analysis for the multimode multicommodity network design problem. The paper concludes in Section 6.

2 A Brief Review of the Literature

An informative overview on intermodal transport is given in Bektaş and Crainic (2007) and Crainic and Kim (2007). A general description of current issues and challenges related to the large-scale implementation of intermodal transportation systems in the United States and Europe is presented by Zografos and Regan (2004). A large body of mathematical solutions, mostly operational research models and methods, have been applied to generate and evaluate the transportation network. Macharis and Bontekoning (2004) and Janic and Bontekoning (2002) present the opportunities for operational research in the intermodal transportation research application field. They have concluded that modeling intermodal freight transport is more complex than modeling uni-modal systems. They also define the problems and review the associated mathematical models that are currently in use in this field. However, their overview covers publications only up to 2002. In the past 10 years, a significant number of papers on this topic have appeared. For example, Meng and Wang (2011) propose a mathematical formulation to design an intermodal hub-and-spoke network for multi-type container transportation, which is suitable when there are multiple stakeholders, such as the network planner, carriers, hub operators, and intermodal operators. Arnold et al. (2004) present a systematic approach to deal with the problem of optimally locating the rail/road terminals for freight transport in relation to the cost criterion. The problem is solved using a heuristic approach involving the solution of a shortest path problem for each commodity. Yamada et al. (2009) describe a bi-level programming model for strategic, discrete freight transportation network design, in which a local search method is suggested to find near-optimal actions to maximise the freight-related benefit-cost ratio depending on their impact on freight and passenger flows. The model is applied to an actual interregional intermodal freight transport network in Philippines. The above three models however do not consider the external costs associated with the logistics and transportation network.

According to Crainic and Laporte (1997), decision makers in transportation are faced with planning problems of three different time horizons, namely strategic, tactical and operational levels of planning. Strategic planning (long term) involves the highest level of management. Decisions at this planning level affect the design of the physical infrastructure network, see Steenbrink (1974), Magnanti and Wong (1984), Crainic and Rousseau (1986) and surveys in Campbell et al. (2002) and Ebery et al. (2000). Tactical planning (medium term) helps improving the performance of the whole intermodal transportation system by ensuring an efficient allocation of existing resources over a medium term horizon, see Crainic and Kim (2007) and Crainic and Laporte (1997). Operational planning (short term) is performed by local management, at which level, day-to-day decisions are made in a highly dynamic environment where the time factor plays an important role, see Crainic and Laporte (1997), Christiansen et al. (2007) and Crainic and Kim (2007).

The service network design problem is a key tactical problem in intermodal transport, which is also the case in our paper. Service network design formulations are generally used to determine the routes on which service will be offered, as well as the frequency of the schedule of each route (Crainic 2000). The performance of the service network design model is evaluated by the tradeoffs between total operating costs and service quality. In solving freight transportation network design problems, much effort has been dedicated to the variant of the problem where there are no limitations on the transportation capacity and significant results have been obtained. Balakrishnan et al. (1989) have presented a dual-ascent procedure for large-scale uncapacitated network design that very quickly achieves lower bounds within 1–4 % of optimality. Later research has focused on the capacitated service network design problem to capture more realistic settings. Holmberg and Yuan (2000) formulate a general model for capacitated multi-commodity network design and propose its Lagrangian heuristic algorithm. Multicommodity flow formulations have been used for an integrated freight origin-destination problem (Holguín-Veras and Patil 2008) and for the hub location problem (Correia et al. 2013). For a recent and up-to-date review of service network design problem, we refer the reader to Crainic and Kim (2007) and Crainic and Laporte (1997).

Service network design models are extensively used to solve a wide range of tactical planning and operations problems in transportation, logistic and telecommunications systems. For example, Ben-Ayed et al. (1992) describe a formulation of the highway network design problem as a bi-level linear program for optimizing the investment in the inter-regional highway network. They then apply this model to the Tunisian network using actual data. Kuby et al. (2001) present a spatial decision support system for network design problems in which different kinds of projects can be built in stages over time and apply this model to the Chinese railway network. They also introduce some easy-to-implement innovations to reduce the size of the problem to be solved by branch-and-bound.

Winebrake et al. (2008b) present a geospatial intermodal freight transport model to help analyse the cost, time-of-delivery, energy, and environmental impacts of intermodal freight transport. Three case studies are also applied to exercise the model. However, they use single criterion objective functions (such as minimizing cost, or time or CO2), which provide the extreme values across the different criteria. We break away from this research by modeling the emissions and transfers as external costs in our model. Moreover, Winebrake et al. (2008b) only consider a single origin-destination commodity. In contrast, we test several commodities in our model for a more systematic analysis.

Bauer et al. (2009) are one of the first to explicitly consider GHG emissions as a primary objective and propose a linear cost, multi-commodity, capacitated network design formulation to minimize the amount of GHG emissions of intermodal transportation activities. They also apply this model in a real-life rail freight service network design and present the tradeoffs between conflicting objectives of minimizing time-related and environmental costs. Our model differs from Bauer et al. (2009) in the objective function. We minimize the total transportation costs, including the operating cost, fixed cost, GHG emission cost and intermodal transfer cost.

One other often ignored aspect when modeling intermodal transportation problems is the terminals where classification operations are carried out. Whilst there exists research on intermodal terminals with respect to, e.g., location and assignment (Vidović et al. 2011) and optimal pricing and space allocation (Holguín-Veras and Jara-Díaz 2006), an explicit consideration within the context of the service network design needs more attention. As pointed out by Bektaş and Crainic (2007), terminals are “perhaps the most critical components of the entire intermodal transportation chain”, and “the efficiency of the latter highly depends on the speed and reliability of the operations performed in the former”. It is therefore important to capture terminal operations in a modeling framework. However, it is difficult to explicitly model speed and reliability of terminal operations at a tactical level, as these measures require a treatment at an operational level. To overcome the difficulty, in this model we capture this phenomenon through the transfer cost.

3 Problem Description and Mathematical Modeling

3.1 Estimating Emissions

There are several approaches for estimating the GHG emissions including an energy-based approach and an activity-based approach. For a review of several vehicle emission models, we refer the reader to Demir et al. (2011). Most of the models presented therein are of microscopic nature. An application of an emission function suitable for traffic on road networks has been presented by Szeto et al. (2013) arising in a road network design problem. For our modeling approach which is at a more strategic/tactical level of planning, we have opted to use an activity-based function by McKinnon and Piecyk (2010) to estimate CO2 emissions in intermodal transportation. This approach has also been used elsewhere, e.g., Treitl et al. (2012) use it to estimate the total transport emissions in a petrochemical distribution network in Europe, and Park et al. (2012) use it to calculate CO2 emissions in a road network for trucks and railway in an intermodal freight transportation network in Korea.

According to McKinnon and Piecyk (2010), the total cost of CO2 emissions of a vehicle carrying a load of l (in tonnes) over a distance of d (in km) calculated by Eq. 1 below,

$$ l\times d\times e,$$
(1)

where e is the average CO2 emission factor (g/tonne-km). To convert CO2 emissions into monetary units, we adopt the figures provided by the World Bank (The World Bank 2012). In particular we use $100 per ton (= £71.6 per tonne).

The rationale for adopting this CO2 emissions function is in its ease of use. First, it has the advantages of including variables to measure the total freight weight as well as the corresponding distance transported, while avoiding the elements that are hard to measure or calculate, such as different vehicle and fuel types for each mode of transport. Second, it is applicable to different transportation modes. For a given mode of transportation, the total CO2 emissions are dependent on the shipping distance and the weight of the commodities.

3.2 Problem Description

The problem is defined on a network, represented by a set of nodes N, a set of links A connecting these nodes, a set of transportation modes M and a set of commodities K. The traffic flows between the nodes can be expressed in terms of an origin-destination flow matrix. Each vehicle of a given mode has a load capacity when running on a particular link. The total transportation cost that results from a vehicle moving over a link includes fixed and variable costs. The variable cost per weight unit of a commodity includes the carrier’s fuel costs, crew costs, overhead costs and administration costs. We assume that the unit variable cost is constant over time and only depends on the link and mode. The fixed cost consists primarily of operator wages and handling costs incurred in moving commodities on and off the vehicles. We assume that each vehicle of the same mode of transport incurs the same fixed cost. Intermodal transfer cost arises from transferring freight from one transportation mode to another in an intermodal terminal (e.g., ports or rail yards). In our modeling, we assume that the transfer cost does not depend on the combinations of nodes involved in the transfer. While this might be seen as a strong assumption, there are two reasons why we do so. First, the literature on intermodal transportation modeling states that the internal handling costs only depend on the load, e.g., Janic (2007) and Winebrake et al. (2008b). The only case where different combinations of modes for a transfer result in different values is external costs (of emissions) although they do not vary significantly (Winebrake et al. 2008b). The second reason is that an explicit consideration of combination of modes in a transfer will require a different model, possibly with more variables, to represent the possible combinations. However, the number of such combinations might be large. For example, in an instance with |M| = 4 there are six possible combinations, whereas for |M| = 10 there could be up to 45 different combinations. Given the figures obtained from the literature, and to simplify the representation, the unit handling cost at terminals is assumed to be independent of the transportation modes involved in the transfer. An alternative model which distinguishes between transfer cost would be worth exploring in future research.

The problem considered in this paper is to design the network by assigning a number of vehicles to links and shipping each commodity from its origin to destination by respecting constraints on the link capacities, flows and requirements for demands. The overall objective is to minimize the total cost, including variable cost, fixed cost, emission cost and transfer cost.

We present three examples in Figs. 1, 2 and 3 showing how transfers occur at an intermodal node. In Fig. 1, 20 units of a commodity are transported by truck. When they reach an intermodal node, the transportation mode is changed to ship, so there is one transfer occurring. In Fig. 2, 50 units of a commodity are transported to the intermodal node by truck where they are split into two parts, 20 units are shipped by ship and 30 units are shipped by rail. In this case, there are two transfers. In Fig. 3, the commodity is brought into the terminal by ship and truck. One transfer occurs at the node. The transportation mode for the 20 units is then changed to truck.

Fig. 1
figure 1

A single transfer from one mode to another

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Fig. 2
figure 2

Two transfers from one mode to two others

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Fig. 3
figure 3

A single transfer from two modes to one

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3.3 Mathematical Modeling

To formulate the green service network design problem described in the previous section, we define the notation as shown in Table 1.

Table 1 Mathematical notation and the explanation in our model
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Using the notation in Table 1, a mathematical model for the problem can be written as follows:

$$ \text{Minimize } \sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}c_{ij}^{m}x_{ij}^{km}$$
(2)
$$ +\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}f_{ij}^{m}y_{ij}^{m}$$
(3)
$$ +\sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}d_{ij}^{m}p^{m}x_{ij}^{km}$$
(4)
$$ +\frac{1}{2}w\sum\limits_{i\in N}\sum\limits_{k \in K}\left(\sum\limits_{m \in M}\left|\sum\limits_{j \in N_{j}^{+}}x_{ij}^{km}-\sum\limits_{j\in N_{j}^{-}}x_{ji}^{km}\right|-{h_{i}^{k}}\right)$$
(5)

subject to

$$ \sum\limits_{j \in N_{j}^{+}}\sum\limits_{m\in M}x_{ij}^{km}-\sum\limits_{j \in N_{j}^{-}}\sum\limits_{m\in M}x_{ji}^{km}={b_{i}^{k}} \qquad \forall i\in N, \forall k\in K$$
(6)
$$ \sum\limits_{k\in K}x_{ij}^{km}\leq u_{ij}^{m}y_{ij}^{m} \qquad \forall (i,j)\in A, \forall m\in M$$
(7)
$$ x_{ij}^{km}\geq 0 \qquad \forall(i,j)\in A, \forall k\in K, \forall m\in M$$
(8)
$$ y_{ij}^{m} \in\left\{0,1,2...\right\} \qquad \forall (i,j)\in A, \quad \forall m\in M.$$
(9)

In this model,

$$ {b_{i}^{k}}=\left\{ \begin{array}{c} r^{k},\ i=o(k) \\ -r^{k},\ i=d(k)\\ \quad 0,\ \text{otherwise},\\ \end{array} \right.$$
(10)

and

$$ {h_{i}^{k}}=\left\{ \begin{array}{c} r^{k},\ i=o(k) \text{ or} \ i=d(k)\\ \quad 0,\ \text{otherwise}.\\ \end{array} \right.$$
(11)

The model (2)–(11) presented above is a non-linear, multi-commodity multi-modal service network design formulation. The objective function measures the total transportation costs. Components (2), (3), (4) and (5) are variable cost, fixed cost, the emission cost and the intermodal transfer cost, respectively.

A bundle of flow conservation constraints are shown by Eq. 6, which also expresses the demand requirements. In this case, each commodity has only one origin and one destination. Constraint set (7) models the capacity constraints. In particular, they guarantee that the total flow on arc (i, j) using mode m ∈ M must not exceed the product of the capacity of each vehicle and the number of vehicles used of mode m ∈ M. If arc (i, j) is not chosen in the shipping network or mode m ∈ M is not used on arc (i, j), the flow on arc (i, j) has to be 0 (i.e., \(y_{ij}^{m}=0\)). Constraint sets (8) and (9) are to ensure the nonnegativity of the decision variables \(x_{ij}^{km}\) describing the flow of each commodity k ∈ K and the integrality of variables \(y_{ij}^{m}\) indicating the number of vehicles of mode m ∈ M on each link.

The model is an extension of the well-known capacitated multicommodity network design problem (Crainic 2000), which it itself is challenging to solve. The non-linear nature of the model due to the objective function makes it even more difficult. It is beyond the scope of this paper to present a bespoke optimization algorithm for this model. However, we will make use of standard linearization methods in the literature to convert it into a linear model. This is shown in the next section.

3.4 Linearization

The second part of the objective function shown by component (5) is non-linear, due to the absolute value used to model transfers. To linearize, we use a variable \(z_{i}^{km}\) defined for each i ∈ N, k ∈ K and m ∈ M. More specifically, \(z_{i}^{km}\) shows the transferred amount of commodity k ∈ K by mode m ∈ M at node i ∈ N if there is a transfer at this node.

Proposition 1

The component \(\left |\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j\in N_{j}^{-}}x_{ji}^{km}\right |\) for ∀i ∈ N, ∀k ∈ K, ∀m ∈ M can be linearized using the following constraints:

$$ \sum\limits_{j \in N_{j}^{+}}x_{ij}^{km}-\sum\limits_{j \in N_{j}^{-}}x_{ji}^{km}\leq z_{i}^{km}, \forall i \in N, \forall k \in K, \forall m\in M,$$
(12)
$$ \sum\limits_{j \in N_{j}^{-}}x_{ji}^{km}-\sum\limits_{j \in N_{j}^{+}}x_{ij}^{km}\leq z_{i}^{km}, \forall i \in N, \forall k \in K, \forall m\in M,$$
(13)

where \(z_{i}^{km}=\left |\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j\in N_{j}^{-}}x_{ji}^{km}\right |\).

Proof

First, notice that \(\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km}=-\left (\sum _{j \in N_{j}^{-}}x_{ji}^{km}-\sum _{j \in N_{j}^{+}}x_{ij}^{km}\right )\). Namely, if \(\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km}\geq 0\), then \(\sum _{j \in N_{j}^{-}}x_{ji}^{km}-\sum _{j \in N_{j}^{+}}x_{ij}^{km}\leq 0\). If \(\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km}<0\), then \(\sum _{j \in N_{j}^{-}}x_{ji}^{km}-\sum _{j \in N_{j}^{+}}x_{ij}^{km}>0\).

Now we consider two cases. If \(\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km}\geq 0\), then constraint (12) and the minimizing objective function (2)–(5) will guarantee that \(z_{i}^{km}=\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km}\). In the other case where \(\sum _{j \in N_{j}^{+}}x_{ij}^{km}-\sum _{j \in N_{j}^{-}}x_{ji}^{km} <0\), then constraint (13) along with the minimizing objective function (2)–(5) will guarantee \(z_{i}^{km}=\sum _{j \in N_{j}^{-}}x_{ji}^{km}-\sum _{j \in N_{j}^{+}}x_{ij}^{km}\).

With the new variable \(z_{i}^{km}\), the formulation can be rewritten as:

$$ \hbox{Minimize} \sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}c_{ij}^{m}x_{ij}^{km}$$
(14)
$$ +\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}f_{ij}^{m}y_{ij}^{m}$$
(15)
$$ +\sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}d_{ij}^{m}p_{m}x_{ij}^{km}$$
(16)
$$ +\frac{1}{2}w\sum\limits_{i\in N}\sum\limits_{k \in K}\left(\sum\limits_{m \in M}z_{i}^{km}-{h_{i}^{k}}\right)$$
(17)

subject to (6)–(9), (12), (13).

The resulting model is now a linear mixed integer program, which can be solved using any available optimization software. In the following section, we present results of computational experiments using this linearized formulation on a case example on data collected from the UK.

4 Computational Experiments

4.1 Description of the Case Study

To show an application and guide the further development of our intermodal model, this section presents a hypothetical but realistic UK intermodal transportation case example. The network consists of 11 nodes, of which nine are important ports in the UK, namely Edinburgh, Newcastle, Liverpool, Milford Haven, Bristol, Felixstowe, London, Folkestone and Southampton. Two of the nodes are important inland cities, Birmingham and Manchester. A geographical representation of the nodes in the network is shown in Fig. 4. There are three transportation modes assumed to be running in the network: truck, rail and ship. For Birmingham and Manchester, only rail and truck options are available, while for other nodes all transportation modes are available. The service network design case example comprises 292 possible directed arcs between 11 nodes. We consider actual route distance instead of straight-line distance to make the computational results more realistic. For road and rail journeys, the distance is provided by Travel Footprint Limited (travelfootprint.org), where road distances come from Google Maps using best route road algorithms and will apply to all road vehicles, while for rail, distance is calculated according to main rail line distances using the most common rail interchanges, see Lane (2006). For sea journeys, distances between pairs of ports are provided by sea-distance.com. There are 30 different commodities to be transported in the network. We randomly generate the origin, destination and demand requirements for each commodity, which is shown in Table 2. The data for capacity, variable cost, transfer cost and CO2 emissions factor of each mode are found from the literature, see Department for Transport (2005), Andersson et al. (2011), Faulkner (2004), Winebrake et al. (2008a) and McKinnon and Piecyk (2010), summarized in Table 3. As can be seen from this table, the capacities, variable cost and emissions tend to vary from one mode to another, although the transfer cost is in this instance the same across all modes (Winebrake et al. 2008b). As for the unit cost of emissions, we use the aforementioned value of £71.6 per tonne (The World Bank 2012).

Fig. 4
figure 4

11 UK cities and ports in the network

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Table 2 Origins, destinations and demands for 30 commodities
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Table 3 Parameters used in the case study
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Since there are three modes used in our case example, there are three corresponding fixed costs. Estimates of fixed costs in the literature vary significantly. Therefore, we look at a range of scenarios and investigate how these scenarios influence the solution. Fixed cost is one possible mean by which government policy can influence the transportation of goods. We denote fixed cost by f t , f r and f s for truck, rail and ship. The relationships between them can be written as:

$$ f_{r}=\alpha f_{t},$$
(18)
$$ f_{s}=\beta f_{t},$$
(19)

where α and β are positive integers.

To see the impact of fixed cost on the optimal structure and send the correct economic signals, we test a number of scenarios by changing the values of α, β and f t in our case example. When the value of f t is fixed, the value of α and β are combinations of 1, 3 and 5. For each f t , this results in nine different combinations. We assume that the fixed costs for truck are £50, £100 and £150, respectively. The number of instances and corresponding fixed costs given different f r and f s are shown in Table 4. In total there are 27 combinations altogether.

Table 4 Fixed costs(£) for instances with different α and β
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4.2 Computational Results and Analysis

Based on the data set introduced in Section 4.1, the computational testing for our model is performed using the CPLEX Interactive Optimizer 12.4.0.0 on a Lenovo ThinkPad T410 laptop computer with Intel Core i5 CPU and 4G RAM. For each instance, the resulting integer linear programming formulation has 8760 continuous and 292 integer variables. The computational time required to solve the integer model to optimality was 1.89 seconds for each instance.

4.2.1 The Effect of Emission and Transfer Costs

In this section, we provide results of nine experiments when f t =£50 to show the effect of incorporating transfer and emission costs into the model. For this purpose, we first solve the model without the emission cost (4) and transfer cost (5), and denote this by F. The total cost generated by F is the total operational transportation cost, including variable cost and fixed cost. Using the solutions obtained, we calculate the resulting emission cost G and transfer cost T, as a consequence of solving F. Then we solve the model with objective (4) denoted F(G), with objective (5) denoted F(T) and with both, denoted F(G+T). Finally, the percentage savings obtained by F(G), F(T) and F(G+T) over the solutions provided by F+G, F+T and F+G+T are calculated. The results of these experiments are represented in Tables 5, 6 and 7, along with the averages calculated across the nine instances.

Table 5 Computational results of models with and without emission cost
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Table 6 Computational results of models with and without transfer cost
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Table 7 Computational results of models with and without emission and transfer costs
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As shown in Table 5, F generated at least £6545 emission cost for each instance. Considering GHG emissions in the objective function significantly decreases the emission cost, with an average saving of 20.66 %. Since operational cost accounts for most of the total cost, the total savings are between 0.48 % and 1.11 %, which is around £4300–£8000. In Table 6, it can be seen that F generated around £3400 transfer cost for each instance. F(T) yields an average savings of 51.97 % on the transfer cost. The significance of the results shown in both tables is that solutions of similar total cost to F+G and F+T can be obtained with our models, but those are with significantly less emission and transfer costs. Table 7 shows the comparison of results between F+G+T and F(G+T). In this case, emission cost in F(G+T) is reduced by between 6.93 % and 11.59 % compared to that generated by F. Transfer cost is also reduced by up to 54.11 %. The average savings in the total cost obtained by using F(G+T) is 0.72 % on average. However, it is noteworthy that these solutions again exhibit an average savings of 9.98 % on emission cost and 24.75 % on transfer cost.

4.2.2 Intermodal vs. Unimodal Transportation

In our next set of experiments, we seek to compare unimodal with intermodal transportation. Since Birmingham and Manchester are inland, transportation by ship is not available for these nodes. As a result we only test truck only and rail only models for unimodal transportation. We test scenarios for unimodal and intermodal transportation models with f t = f r = f s = £50, f t = f r = f s = £100 and f t = f r = f s = £150, to delete the effects of different fixed cost when comparing the result. Computational results are summarized in Table 8. The columns of the table are self-explanatory.

Table 8 Comparison of the results of the unimodal and intermodal transportation
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As the results in Table 8 indicate, when the fixed cost is £50, the total cost for truck only, rail only and intermodal transportation are £112513, £98778 and £87253, respectively. In this case, intermodal transportation is 22.4 % less costly than truck only and 11.7 % less costly than rail only transportation. When the fixed cost increases to £100 and £150, the total savings that intermodal transportation affords is up to 41.2 %.

Variable cost of the three truck only scenarios changes from £75730 to £75775, which is a 0.06 % increase. For rail only scenarios, variable cost stays the same. Unlike unimodal results, variable cost of intermodal scenarios changes from £75327 to £77705, which is an increase of 3.1 % when fixed cost increases from £50 to £100. A larger range of change indicates that when fixed cost is less than £100, it has a greater effect on variable cost in intermodal transportation than that in unimodal transportation. When fixed cost increases from £100 to £150, variable cost of unimodal and intermodal models does not change. In this case, it is possible that variable cost reaches its maximal value and will not be affected by the change of fixed cost.

Since the emission factor of truck is the highest, followed by ship and rail, the emission cost for truck only transportation is nearly three times costly than that of rail only and intermodal transportation. We notice that there is only a slight change for emission cost in truck only transportation by adjusting fixed cost, while for rail only transportation, emission cost stays the same. In intermodal transportation, emission cost decreases from £5920 to £5020, a 15.2 % decrease. When fixed cost keeps increasing, emission cost does not change. Reducing emission cost can be obtained by adjusting fixed cost when fixed cost is less than £100.

Transfer cost is only occurred in intermodal transportation. It is the smallest part of the total cost. When fixed cost increases from £50 to £150, the transfer cost increases slightly from £2806 to £2840. More details on this are provided in the next section where intermodal transportation results are discussed.

In summary, even if transfer costs are incurred in intermodal transportation, the total cost can be reduced by 11.3 %–41.2 % as opposed to unimodal transportation. By adjusting the fixed costs, variable and emission costs can be reduced in intermodal transportation, while they rarely change in unimodal transportation.

4.2.3 Intermodal Transportation Results

This section presents results obtained with intermodal transportation. The results shown in Figs. 5, 6 and 7 show the total cost, including variable cost, fixed cost, emission cost and intermodal transfer cost for nine intermodal instances when f t = £50, £100 and £150, respectively. From these figures, we note that the variable cost, which is around £75000, takes up the largest proportion of the total cost and it does not change greatly across the 27 instances. Intermodal transfer cost takes up the least proportion. For each f t , the total fixed cost increases when f r or f s is increasing. Hence, the total cost is also increasing. More details about the total cost and its components are presented in Table 9, which shows costs that are normalized to 1 against base case scenarios of f t = f r = f s , for all 27 instances.

Fig. 5
figure 5

Computational results for 9 intermodal instances when f t =50

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Fig. 6
figure 6

Computational results for 9 intermodal instances when f t =100

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Fig. 7
figure 7

Computational results for 9 intermodal instances when f t =150

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Table 9 Normalized costs for intermodal instances
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It can be seen from Table 9 that when f t = £50, the normalized costs for the nine corresponding instances is betwen 1.000 and 1.078. For each f t , the total fixed cost is increasing as f r or f s is increasing. It is interesting that the variable cost stays relatively stable even with the changing fixed costs, where the range of difference is from 0.1 % to 2.3 %. For most instances, the emission cost slightly increases as f r or f s increases. The emission cost increases up to 11.6 %. We have mentioned in Section 3.1 that emission cost is measured by the product of distance, amount of load and emission factors. We also know that emission factor for truck is the highest, followed by ship and rail. So when f r or f s is increasing, total truck tonne-miles increase, which results in an increase of the emission cost. Since rail and ship have lower emission factors than truck, reducing f r and f s helps reducing emission cost in intermodal transportation. Transfer cost also does not change significantly across the instances. The transfer cost is between 1.65 % and 3.1 % of the total cost, implying that transfers are not an insignificant component of the intermodal tansportation chain, at least as far as the costs are concerned. Consolidation of loads or incentivizing certain low emission forms of transportation can be efficiently achieved in intermodal transportation.

Tables 10, 11 and 12 present the total tonnage shipped and the capacity utilization of three transport modes when f t = £50, £100 and £150 with the commodities shown in Table 2. In each table, either f r or f s is increases as one moves from instance 1 to instance 9. We note in this case that more trucks are used and a greater total tonnage of commodities are transported by truck when f r or f s is increasing. Except for six instances in total, in which no trucks are used for shipping, the capacity utilization of truck stays above 91 %. In eight out of these 22 instances, the capacity utilization reaches 100 % for truck. By adjusting f t , we could avoid the situation that there are no trucks used in the transportation network and at the same time, get a relatively high capacity utilization. For rail, when f r increases, both total tonnage and total number of vehicles decreases, while the capacity utilization increases. One reason might be that it is more cost effective to travel further to consolidate onto fewer vehicles when f r is high. When f r is low, using more vehicles on direct routes could be more attractive. The capacity utilization of rail across the 27 instances is greater than 77 % overall, and is greater than 84 % in 17 instances. When f t increases, the total tonnage transported by rail is increasing. By changing f t or f r , we might control the total tonnage shipped on rail. Since ships have large capacity, the average capacity utilization of ship is much less than that of truck and rail, with values around 20 %. Due to its large capacity, the number of ships used and total tonnage of commodities transported also do not change as much as trucks and ships.

Table 10 Comparison of the capacity utilization for 3 modes when f t = £50
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Table 11 Comparison of the capacity utilization for 3 modes when f t = £100
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Table 12 Comparison of the capacity utilization for 3 modes when f t = £150
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As part of the computational experiments, we have conducted further tests where variable cost and capacity of rail are changed and demands are increased. In the case that the variable cost and capacity of rail are changed to £0.017 per ton-mile and 3360 tonnes (Forkenbrock 2001), the results with the intermodal model suggested a unimodal transportation plan with rail being the only mode of transport used and ship very rarely. The capacity utilization for rail is 18 % to 32 %. In the event that demands are increased 10-fold, the results do not significantly change although the capacity utilization of rail and ship increases. In particular, rail has a much higher capacity utilization of more than 96 % and ship being more used with capacity utilization of between 76 % and 88 %. Computational results for this case can be found in Table 14, 15 and 16 presented in the Appendix.

5 Bi-criteria Analysis

In this paper, the multimode multicommodity network design problem has so far been treated as a single-objective optimization problem. This was made possible by aggregating the two objective functions, one relevant to operational activities, and the other to emissions, by attaching suitable cost coefficients to each. In this section, we show how the model can be used to produce non-dominated solutions with regard to the two objectives, if they are treated as incommensurable. This analysis is particularly relevant if costs are not a prevailing factor, or are not readily available, as might be the case for emission cost. In multi-objective optimization, a non-dominated, or Pareto-optimal, solution is where no objective can be improved without worsening at lease one other objective (Coello and Lamont 2007). The set of Pareto-optimal solutions form the Pareto-frontier. Decision-makers usually select a particular Pareto-optimal solution based on their preferences on the objectives (Ghoseiri et al. 2004). One may refer to Palacio et al. (2013) for an application of a bicriteria optimization model to a location problem arising in maritime logistics.

In order to find the trade-offs between minimizing total transportation cost and the amount of CO2 emissions in the multicommodity multimodal service network design problem, we use the notation already shown in Table 1 and define the two objective functions below:

$$\begin{array}{@{}rcl@{}} (OBJ1) \qquad f_{1}(x,y,z) & = &\sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}c_{ij}^{m}x_{ij}^{km} \end{array}$$
(20)
$$\begin{array}{@{}rcl@{}}& +&\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}f_{ij}^{m}y_{ij}^{m} \end{array}$$
(21)
$$\begin{array}{@{}rcl@{}}& +&\frac{1}{2}w\sum\limits_{i\in N}\sum\limits_{k \in K}\left(\sum\limits_{m \in M}z_{i}^{km}-{h_{i}^{k}}\right) \end{array}$$
(22)
$$\begin{array}{@{}rcl@{}}(OBJ2) \qquad f_{2}(x) & = &\sum\limits_{k\in K}\sum\limits_{(i,j)\in A}\sum\limits_{m\in M}d_{ij}^{m}e^{m}x_{ij}^{km}. \end{array}$$
(23)

In the above, f 1(x, y, z) measures the total transportation cost, including variable cost, fixed cost and transfer cost, the details of which are given in Section 3.2. On the other hand, f 2(x) measures the total CO2 emissions for all activities. Let S denote the feasible set of solutions defined by the equality and inequality constraints (6)–(9), Eqs. 12 and 13 as described in Section 3.3.

A number of methods have been suggested to generate the Pareto frontier in multi-objective programming, such as goal programming, bound objective function formulations and genetic algorithms. In this paper, we use the 𝜖-constraint method, which is easy to implement and effective. In this method, one of the objectives is optimized, subject to other objective converted into a constraint by imposing appropriate upper bounds on its value. The 𝜖-constraint method can be implemented as follows (Mavrotas 2009, Engau and Wiecek 2007). We first solve the following single-objective optimization problem:

$$\begin{array}{@{}rcl@{}} \text{Minimize} & \qquad f_{2}(x) \end{array}$$
(24)
$$\begin{array}{@{}rcl@{}} \text{subject to} & \qquad x \in S. \end{array}$$
(25)

Let an optimal solution vector of (24)–(25) be denoted by \((x_{2}^{*})\). The minimized amount of CO2 emissions is then fixed as \(f_{2}(x_{2}^{*}) = \epsilon \). Following this step, we solve the following single-objective optimization model:

$$\begin{array}{@{}rcl@{}} \text{Minimize} & \qquad f_{1}(x,y,z) \end{array}$$
(26)
$$\begin{array}{@{}rcl@{}} \text{subject to} & \qquad f_{2}(x) \leq \epsilon \end{array}$$
(27)
$$\begin{array}{@{}rcl@{}} & \qquad x,y,z \in S. \end{array}$$
(28)

Let \((x_{1}^{*},y_{1}^{*},z_{1}^{*})\) denote an optimal solution to Eqs. 26–28. Furthermore, let q be the number of Pareto-optimal solutions that a decision-maker wishes to produce. Then, the value of 𝜖 is to be increased by ς at every iteration where:

$$ \varsigma =(f_{2}(x_{1}^{*})-f_{2}(x_{2}^{*}))/(q-1).$$
(29)

The method iterates by increasing the value of 𝜖 in this way (see Table 13), where each iteration generates another optimal solution. By repeatedly relaxing the upper bound on f 2, and resolving f 1 each time, solutions can be obtain to construct the Pareto-frontier.

Table 13 Value of 𝜖 in the 𝜖-constraint method
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We now present a numerical example to illustrate the application of the 𝜖-constraint method on the multimode multicommodity network design problem. The method is implemented in C and each sub-problem is solved to optimality by ILOG CPLEX Interactive Optimizer 12.4. Therefore, the solutions generated are Pareto-optimal. The instances is based on the data presented in Section 4.1 with the fixed-cost scenario where f t = £50, f r = £150 and f s = £250.

Assuming q=45, the stepsize is calculated as ς=0.7 tonne. The resulting set of Pareto-optimal solutions are shown in Fig. 8.

Fig. 8
figure 8

Pareto optimal curve of the bi-criteria model

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To select the most preferred solution from this set of non-dominated solutions, we have also implemented a normalized distance method, for which the two objectives are normalized as follows:

$$\begin{array}{*{20}l} \bar{f_{1}}(x,y,z)&=\frac{f_{1}(x,y,z)-f_{1}\left(x_{1}^{*},y_{1}^{*},z_{1}^{*}\right)}{f_{1}\left(x_{2}^{*},y_{2}^{*},z_{2}^{*}\right) -f_{1}\left(x_{1}^{*},y_{1}^{*},z_{1}^{*}\right)}\in[0,1], \end{array}$$
(30)
$$\begin{array}{*{20}l} \bar{f_{2}}(x)&=\frac{f_{2}(x)-f_{2}\left(x_{2}^{*}\right)}{f_{2}(x_{1}^{*})-f_{2}\left(x_{2}^{*}\right)}\in[0,1], \end{array}$$
(31)

where \(\bar {f_{1}}\) and \(\bar {f_{2}}\) are the normalized value of OBJ1 and OBJ2, respectively. Assuming an ideal point (0,0), as shown in Fig. 9, the solution that is closest can be considered as the most preferred solution.

Fig. 9
figure 9

Normalized Pareto optimal curve of the bi-criteria model

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As seen in Fig. 9, the normalized solution that yields the minimum distance from the ideal point is \(\bar {f_{1}}=0.31\) and \(\bar {f_{2}}=0.43\). Mapping this on the original functions, the values of the corresponding solutions are OBJ1 =£85869.2 and OBJ2 =83.1 tonne.

6 Conclusions

In this paper, we have described an intermodal freight transportation model, which extends the traditional service network design models by taking GHG emission cost into account. The model includes a nonlinear expression for transfer cost at intermodal nodes, but one which can be linearized as shown in the paper. We have also shown how the problem can be formulated as a bi-objective optimization model. The computational results suggest that the proposed model can provide cost-efficient and emission-efficient ways for transporting commodities. This makes it interesting for practical applications.

One important detail to remember is that the adjusted fixed cost is used for the purpose of sending the correct economic signals. We believe that it provides an opportunity for considerable cost reduction while adjusting fixed costs. By changing the value of the fixed costs for the three modes of transport, the tradeoffs between emissions and transfer costs can be analyzed.