Intermodal terminal location problems

Research on IMTLP has been receiving growing attention by both policy makers and academics (Bontekoning et al. 2004). The research work began with the mathematical formulation of the problem by Arnold et al. (2001). Arnold et al. (2001) focussed on locating intermodal terminals to serve the regional containerised transport markets, where the use of intermodal transport requires the use of exactly two IMTs along the intermodal transport chain.

The chain consists of three leg; the pre-haul leg, the main haul, and the post haul legs as illustrated in Figure 2.2. The pre-leg involves local pickup of cargo by trucks from various cargo origins to the nearest IMT, and the post haul leg consists of local distribution of cargo to various destinations and is also done by trucks due to its flexibility and accessibility to customer facilities. The main haul is done by high capacity mode such as rail.

Figure 2.2 : Regional intermodal transport chain

This intermodal transport market as noted in Section 1.4 involves moving freight over long distances and its competitiveness depends on the economies of distance (lower unit cost per kilometer) and the economies of scale (through the use of rail) relative to road alone mode such as trucks. The model formulated by Arnold et al. (2001) can be stated as follows:

(Arnold et al. 2001): Min Λ = ∑ ∑ ∑ ∑ 𝑐_{𝑖𝑠𝑡𝑗}𝑊_{𝑖𝑠𝑡𝑗} origin nodes and 𝒟 the set of destinations nodes; 𝒯 is the set of all feasible IMT locations on the network; 𝑞_𝑖𝑗 represent the quantity of cargo to be transported from origin 𝑖 to destination 𝑗; 𝑈_𝑖𝑗 the quantity of 𝑞_𝑖𝑗 to be transported unimodally or by trucks from origin 𝑖 to destination 𝑗 with unit cost 𝑐_𝑖𝑗; 𝑏_𝑠 the maximum handling capacity of IMT 𝑠 , and 𝑓_𝑠 the fixed or setup cost of IMT 𝑠. The objective function is composed of three parts; the first part captures the weighted cost of all intermodal transport flows; the second part represents the weighted cost of road alone transport, and the third part comprises the total fixed associated with all opened IMTs. Constraints (2.13) and (2.14) ensure that no cargo can be transported through an IMT, unless it is opened. Constraint (2.15) reveals the existence of competition between road lone and intermodal transport modes and stipulates that, for each origin-destination, the sum of all cargo flows transported by road alone and by regional intermodal transport should equal the demand associated with this origin/destination-pair. Constraint (2.16) enforces capacity limit on each opened IMT. Constraints (2.17) ensure that an IMT should either be opened or closed.

Constraint (2.18) ensures that no demand is transported using only one IMT and also ensure that only non-negative amounts are transported.

The model formulated by Arnold et al. (2001) was associated with a large number of decision variables and constraints, making it difficult to efficiently solve for even small problem instances. This limitation motivated a new formulation of the problem by Arnold et

al. (2004). Their reformulated model is similar to the multicommodity fixed-charge network design problem (MCNDP), where IMTs are considered as arcs instead of vertices in a graph.

The reformulation resulted in a significant reduction in the number of decision variables and constraints especially in sparse networks (Arnold et al. 2004). The generalisation of the intermodal location problem to include non-linear and concave cost functions to capture economies of scale effects on intermodal usage was developed by Racunica and Wynter (2005).

The non-linearity and concavity of the transport cost function means that a linearization procedure is required to solve the problem. The algorithms they proposed includes a linearization algorithm for reducing the non-linear problem to linear and two heuristics for solving large instances of the problem.

Similar work by Rahimi et al. (2008) comprises a location-allocation model for locating hubs to promote the use of rail through the use of hub-and-spoke networks using a concave cost function to capture economies of scale resulting from freight consolidation at hub terminals. Limbourg and Jourquin (2009) cast the intermodal location problem as linked p-hub median problem and multi-modal assignment problem where the demand can be assigned over all the transport modes, with the option of using trans-shipment facilities. The model works by repeatedly solving the p-median problem for each update in trans-shipment costs, which in turn is based on previous estimated flow at each terminal until the relative difference in trans-shipment costs between two iterations is smaller than a pre-defined threshold.

Ishfaq and Sox (2011) employed the multiple-allocation p-hub median modelling approach to formulate a new mathematical model for IMTLP. Their proposed model includes important transport mode attributes such as transport cost, modal connectivity costs, and fixed location costs under service time requirement. Large instances of the problem were solved using a tabu search metaheuristics algorithm with the quality of the solution measured against lower bounds from a Lagrangian relaxation. Ishfaq and Sox (2012) extended their previous work (Ishfaq and Sox 2011) with the integration of a queuing system to model hub operations and investigated the effects of limited hub resources on the design of intermodal logistics networks under service time requirements. The features of the model were illustrated using a 25-city road-rail intermodal logistics network. Sorensen et al. (2012) proposed meta-heuristic algorithms; the greedy randomised adaptive search procedure (GRASP) and the attribute based hill climber (ABHC) for solving the model proposed in Arnold et al. (2001).

To capture the joint effects of CO2 emissions and economies of scale on intermodal terminal location, Zhang et al. (2013) proposed as bi-level programming, where a genetic algorithm was used at the upper level to search for the optimal terminal network configurations, while the lower level performs multi-commodity flow assignment over a multimodal network.

A similar work by Qu et al. (2016) was conducted but their objective focussed on the effects of greenhouse gas emissions and intermodal transfers on intermodal network design. The resulting non-linear model was linearized and solved for a hypothetical case study of eleven candidate locations in the United Kingdom. Recently, Ghane-Ezabadi and Vergara (2016) proposed a new mathematical formulation and decomposition based solution algorithm for designing intermodal networks. The novelty in their approach was the use of composite variable in representing a complete route for a load from origin to destination, thereby allowing exact solution algorithms to be developed for solving relatively large instances of the problem.

The computational efficiency of their proposed decomposition-based algorithm was illustrated through numerical examples, where they show that it could solve for instances of up to 150 nodes in reasonable amount of computational time (few seconds).

The first model for locating city or IMEX intermodal terminals proposed by Teye et al.

(2015) was also based on a MILP with Lagrangian heuristics for solving it. They observed that the MILP formulation leads to all-or-nothing (AON) assignment of demand between competing modes for each origin-destination pair, resulting in unintuitive results during forecasting and policy testing. A summary of work on locating intermodal terminals or hubs are presented in Table 2.3. The next section discusses the gaps identified in the literature and how this research intends to fill the gaps.

Table 2.3: Summary of previous research on IMTLP

Reference Objective Modelling Solution Comments

Arnold et al. (2001) Total cost Mixed integer Heuristic Here the competition is between modes of minimisation programming transport and intermodal transport involves

the use of exactly two terminals

Peeters and Total cost Mixed integer Heuristic Improved formulated of Arnold et al. (2001) Thomas (2004) minimisation programming as a multicommodity fixed-charge network

design problem (MCNDP), to reduce the number of decision variables

Racunica and Total cost Mixed integer Linearization The formulation includes non-linear and Wynter (2005) minimisation programming and heuristics concave cost functions to capture economies

of scale effects on intermodal usage Rahimi et al. (2008) Total cost Single facility Exact Location-allocation model for locating hubs

minimisation location (6 nodes) terminals to promote the use of rail using a concave cost function to capture economies of scale resulting from freight consolidation at hubs Limbourg and Total cost Multiple Exact solution Linked p-hub median problem and multi-modal Jourquin (2009) minimisation allocation for hubs only assignment problem for locating intermodal

phub median locations terminals

Ishfaq and Sox (2011) Total cost Single allocation Tabu search Employed a multiple-allocation p-hub median minimisation p-hub median modelling approach to formulate a new mathematical

model for the intermodal terminal location problem Ishfaq and Sox (2012) Total cost Nonlinear Tabu search Extension of Ishfaq and Sox, (2011) with the

minimisation mixed integer integration of a queuing system to model hub programming operations and investigated the effects of limited hub

resources on the design of intermodal logistics networks under service time requirements Sorensen, Vanovermeire Total cost Mixed integer GRASP and Two heuristic algorithms for solving the intermodal and Busschaert (2012) minimisation programming ABHC terminal location problem

Zhang et al. (2013) CO2 emissions Mixed integer Genetic A as bi-level programming, where the upper level cost minimisation programming algorithm determines the optimal terminal network

configurations, while the lower level performs multi-commodity flow assignment over a multimodal network

Qu et al. (2014) Greenhouse gas Mixed integer Exact Focussed on the effects of greenhouse gas emissions cost programming (11 nodes) emissions and intermodal transfers on

minimisation intermodal network design

Teye et al. (2015) Total cost Mixed integer Lagrangian Intermodal terminals location with and transport

minimisation programming heuristic mode choice

Ghane-Ezabadi Total cost Integer Decomposition Simultaneously determines the location of hubs, and Vergara (2016) minimisation programming approach routes for loads and their transport modes

using composite variables

Current research Welfare Nonlinear Exact and The method strategically places IMTs at locations maximisation mixed integer heuristic where shippers’ or users’ welfares are maximised

Programming