A frequency-based maritime container assignment model

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Abstract

This paper transfers the classic frequency-based transit assignment method of Spiess and Florian to containers demonstrating its promise as the basis for a global maritime container assignment model. In this model, containers are carried by shipping lines operating strings (or port rotations) with given service frequencies. An origin–destination matrix of full containers is assigned to these strings to minimize sailing time plus container dwell time at the origin port and any intermediate transhipment ports. This necessitated two significant model extensions. The first involves the repositioning of empty containers so that a net outflow of full containers from any port is balanced by a net inflow of empty containers, and vice versa. As with full containers, empty containers are repositioned to minimize the sum of sailing and dwell time, with a facility to discount the dwell time of empty containers in recognition of the absence of inventory. The second involves the inclusion of an upper limit to the maximum number of container moves per unit time at any port. The dual variable for this constraint provides a shadow price, or surcharge, for loading or unloading a container at a congested port. Insight into the interpretation of the dual variables is given by proposition and proof. Model behaviour is illustrated by a simple numerical example. The paper concludes by considering the next steps toward realising a container assignment model that can, amongst other things, support the assessment of supply chain vulnerability to maritime disruptions.

Highlights

► The frequency-based transit assignment method is transferred to containers. ► Full containers are assigned to routes to minimize sailing plus dwell time. ► Empty containers are repositioned to minimize sailing plus discounted dwell time. ► The number of container moves per unit time at any port is limited to port capacity. ► Dual variables define surcharges for loading or unloading a container at a congested port.

Introduction

A model capable of representing container flows at a global level has for some time been a holy grail. Such a tool, if it existed, would be useful to shipping lines, terminal operating companies, shippers, port authorities, national and regional planning authorities, marine insurance companies, and others. The Container World project,1 which attempted to develop such a model, took an agent-based simulation approach. Every ship, port, service, shipping line, trucking company and rail operator was represented as a separate agent. A global network was constructed using the actual strings (port rotations) operated by shipping lines at the time of the study (sourced from the MDS-Transmodal Containership Databank). Containers were moved from their origins, via vessels, trucks and trains, to their destinations with each agent involved in this process operating according to its own set of rules. Although there is still interest in this approach (see Newton, 2008, in the context of WORLDNET), Container World proved to be too data intensive in an industry where companies in competition with each other are reluctant to share data.

In the absence of access to confidential information from individual shipping lines needed to create a microscopic simulation model, a macroscopic approach looks more feasible. Perrin et al. (2008) describe a macroscopic container assignment model. A network is created where the nodes represent origins, destinations, ports and maritime waypoints while maritime links represent connections operated by shipping lines and land links represent connections between origins or destinations and ports. An origin–destination matrix of container flows in TEUs2 between countries, obtained from UNCTAD3 and Eurostat4 sources, is input. For each origin–destination pair a set of routes is generated by a k-best path algorithm based on all the available ports for the countries considered.5 After creating sets of routes, a logit route choice model is applied. Currently this is a simple multinomial logit model, but it is planned to replace this by a path-size multinomial logit model to allow for the correlations engendered by overlapping paths. After the assignment of container flows to paths, flow is aggregated for each string operated by shipping lines. Port choice is modelled indirectly by picking a route from the choice set. To calibrate the model, shadow prices are sought for port calls which, after the assignment, reproduce as closely as possible port throughput data.

There are a number of international, national and regional freight flow models, reviewed in de Jong et al. (2004), which look more generally at freight rather than container flows. Tavasszy (2006) identified three decades of freight flow model development. The first attempt in Europe to explain freight flows in the early 1970s (Chisholm and O’ Sullivan, 1973) made use of the gravity model, more usually used for trip distribution. Subsequently Input/Output (I/O) and Land Use-Transport Interaction (LUTI) models were added to explain the generation as well as the distribution of freight flows. As behavioural modelling became popular for passenger transport in the 1970s, the first mode choice models for freight were proposed. In the 1980s there was increased interest in applying general equilibrium principles to freight networks, explaining simultaneously the generation, distribution, mode split and assignment of freight flows (Harker and Friesz, 1986a, Harker and Friesz, 1986b). These models were extended in the 1990s by introducing commodity differentiation (Crainic et al., 1990), improved probabilistic choice models, and inventory considerations.

The conventional four step (generation, distribution, modal split and assignment) approach, originally developed for passenger transport, remains the basis of most freight flow models. Usually all steps are handled at the aggregate (zonal) level, starting with multi-regional or regionalised national input–output models (see Marzano and Papola, 2008) leading to production–consumption matrices. Value-to-weight transformations and vehicle load factors allow production–consumption matrices to be assigned to transport networks (road, rail and air). The models for freight transport in Norway (NEMO) and Sweden (SAMGODS), described in de Jong and Ben-Akiva (2007), are typical of this approach as they contain the conventional four steps (with mode choice and assignment being handled simultaneously in a multimodal assignment) as well as value-to-weight transformations and vehicle load factors, with all steps handled at the zonal level. WSP, 2002a, WSP, 2002b provide useful early reviews of such models. EUNET2.0, described in WSP (2005), is an equivalent model for the UK.

The current decade has seen interest in agent-based simulation and the application of game theory. It was noted by de Jong and Ben-Akiva (2007) that four step models lack important logistical elements, like the determination of shipment size or the use of consolidation and distribution centres, although transhipments between modes may be included indirectly by the use of a multi-modal network. Consequently they specify a logistics model that takes as inputs commodity flows from production to consumption zones generated by a conventional four step approach. The logistics model then disaggregates these flows to firm-to-firm flows. After this disaggregation, the logistical decisions (shipment size, use of consolidation and distribution centres, mode/vehicle/vessel type, and loading unit) are simulated.

This paper focuses specifically on the maritime transport of containers and builds on the observation that the maritime container assignment problem shares a greater affinity with transit assignment than with traffic assignment conventionally used in the four step approach, because containers are generally carried by shipping lines which operate services on strings (or port rotations). To date it is believed that this paper constitutes the first attempt to adapt the classic Spiess and Florian (1989) frequency-based transit assignment method to maritime container transport. This method assigns an origin–destination matrix of passenger trips to a transit (usually a bus) network to minimise expected travel time. Where bus stops are shared by a number of bus routes, it assumes that passengers choose the first bus on an attractive route to arrive at the stop. Buses are assumed to arrive randomly, so the division of traffic across the attractive routes leaving a stop is in proportion to their frequency. At a bus stop a route is labelled as attractive for a given destination if by excluding it from the choice set the expected travel time is increased. Spiess and Florian showed that this assignment problem can be solved as a linear program.

This paper transfers the Spiess and Florian method into the domain of maritime container transport by replacing passengers by containers and routes by strings. The method needs to be extended by firstly including the repositioning of empty containers and secondly by introducing constraints to represent the maximum number of containers a port can handle per unit time. As the resulting container assignment model is also a linear program, we look carefully at the interpretation of the dual variables, showing by proposition and proof that they provide useful information about the solution. The assumptions behind the proposed container assignment model are discussed in greater detail in the next section.

Section snippets

Assumptions and methodology

To demonstrate more clearly the properties of the proposed container assignment method, we assume that there is only one type of container, that containers are interchangeable, and that containers are routed to minimise an objective function measured in time units. The objective function consists of the sum of the time containers spend at sea and their expected dwell time in ports. The extension to containers of multiple types leads to a multi-class formulation of the problem, increasing its

Container assignment model

The container assignment model can be expressed as the following linear program:P0:minx,waA(xa+f+xae)ca+w++f+δw+es.t.:aAi+xasf-aAi-xasf=bisffor alliI,sDaAi+xae-aAi-xae=-biefor alliIxasfwisffafor allaAi-,isI,sDxaewiefafor allaAi-,iIkiaAi-(xa+f+xae)+aAi+(xa+f+xae)for alliIxasf0for allaA,sDxae0for allaAbisf=-trsfifi=rOt+sfifi=sD0otherwisebie=t+if-ti+fifi=rOori=sD0otherwiseNote that + denotes summation by the respective subscript, for example xa+f=sDxasf. Note

Numerical example

Consider the network depicted in Fig. 1. Per day 1000 full containers are shipped from Port A to Port F and empty containers return from Port F to Port A. The black link connects A to F directly with a sailing time of 6 days. By contrast, the red7 link takes 2 days to sail between A and C and the blue link takes 3 days to sail between C and F. Alternatively, the green link takes 1 day to sail between C

Conclusions

This paper presents a promising first step toward a global maritime container assignment model which represents the effects of sailing time, service frequency and port capacity on the pattern of full and empty container flows and therefore on port choice. In order to develop this into a global maritime container flow model, the following would be required:

  • 1.

    A global maritime network with sailing times and frequencies. This could be constructed from published timetables for container liners.

  • 2.

    Global

Acknowledgements

The authors are grateful to Dr Jonathan Carter, Mike Garrat, Phil Gridrod and colleagues in PORTeC for discussions leading to a better understanding of the determinants of container flows.

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