A rolling horizon heuristic for the stochastic cargo mix problem

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Highlights

  • Uncertain demands and bookings in liner shipping call for stochastic approaches.

  • Novel multistage stochastic formulation for the Liner shipping cargo-mix problem.

  • Efficient & effective two-phase matheuristic based on a rolling horizon approach.

  • Realistic size instances with over 32,000 scenarios can be solved within minutes.

Abstract

This paper presents the stochastic cargo-mix problem, which aims at analysing the cargo composition needed for a liner vessel to maximise its revenue on a given service. The unreliability with respect to the demand forecast is included by considering the cargo-flows as being stochastic instead of deterministic. We also take into account accepted bookings, draft, stability and capacity constraints. A compact formulation of the problem is shown to be too complex to solve industrially sized instances. Instead, a rolling horizon matheuristic is presented, and the computational results show that it can achieve high-quality results in reasonable time.

Introduction

Aside from a few years of financial crisis, the liner shipping industry has had a continuous growth. The growing demand has resulted in a fierce competition to deliver the best product concerning efficiency, reliability, and most importantly cost. As a result, shipping rates are historically low, making it vital for the carriers to utilise their vessels as efficiently as possible. In recent years, carriers have been building bigger and bigger vessels to follow demand trends, but also to achieve economies of scale.

While academic focus on vessel intake maximisation is relatively new, it is nothing new in the shipping industry. Container vessels are delivered with a theoretical nominal capacity. Only if the weight distribution is perfect, the full nominal capacity can be reached, which hardly ever happens. With the increasing size of the vessels, a small decrease in utilisation results in hundreds of containers having to be dropped. Stowage coordinators are responsible for planning the cargo and finding a load configuration (stowage plan) that suits the cargo to be loaded at the current port, while also making sure the vessel can be utilised to its maximum in future ports. The unreliability with respect to the demand forecast in the industry further complicates this problem. For the shippers, there is no fee for booking shipments, and they will only pay for a container transport once it has been undertaken by the liner. Thus, a booking does not mean the containers will ever arrive in time. Therefore, it is complicated for the stowage coordinators to make a good plan as the high unreliability is not likely to change without radically changing the cost structure.

The focus of our work is the analysis of vessels’ cargo mix (the cargo mix problem) and, in particular finding a cargo composition needed for a vessel to maximise its revenue on a given service. We include the unreliability of the demand and model confirmed bookings. We focus on out-of-region demands (e.g. from Asia to Europe), where we consider load and discharge of cargo in the current region and only discharge in the destination region. This corresponds to optimising the revenue from an inter-regional leg while still considering the revenue obtained by shipping containers within the region. Delgado (2013) shows that a cargo-mix analysis based on simple capacity constraints overestimates the revenue generated by operating a vessel. Therefore, it is important to include additional features and limits to estimate the capacity and revenue correctly. Additionally, we enforce that the stowage plan adheres to a block stowage strategy used within the industry. This corresponds to a logical partitioning of the vessel into blocks, and by enforcing the block strategy each block is only allowed to stow containers that have the same discharge port. This way of stowing containers is aimed at improving operations at ports since it makes it possible to perform e.g. dual cycling (where load and discharge operations are no longer sequential). Examples of block stowage are found in the stowage planning literature in e.g. Ambrosino et al., 2015a, Ambrosino et al., 2015b.

The proposed model can have multiple applications, e.g., driving rate prices and improving fleet composition. However, we see greater potential in using the model as an analytic tool. We envision its true potential in the analysis of different what-if scenarios. To allow the execution of multiple analyses, fast solution methods are sought. We will thus put a lower priority with respect to optimality and prioritise an adaptable, simple and fast solution method.

In Christensen and Pacino (2017) a deterministic version of the cargo mix problem is described, and multiple matheuristics based on the same idea are compared. The problem studied in Christensen and Pacino (2017) assumes perfect information and is aimed at identifying an optimal cargo-mix for a specific vessel. Such cargo-mix can be used to e.g. evaluate which vessel is better suited for a specific service. In this paper, we present a more operational version of the cargo-mix problem, where the current out-of-region configuration of a vessel is studied. Essentially, we aim at utilising the vessel as best as possible before the long-haul journey.

The contributions of this paper are threefold. First, we present a more operational version of the cargo-mix problem that is able to analyse what-if scenarios and present a novel multi-stage stochastic formulation. Second, among the major container liner shipping problems studied in the literature (e.g. stowage planning, service network design, cargo flow optimisation and empty repositioning), this is, to the best of the authors’ knowledge, the first work that includes stochastic cargo forecast elements. Third, an efficient 2-phase framework for the proposed multi-stage stochastic program has been implemented and tested. The method is inspired by the multi-phase approach from Christensen and Pacino (2017), which has been extended and adapted to address this problem. Moreover, a rolling horizon heuristic has been used for the second phase in order to overcome the problem intractability introduced by the stochastic program.

The remainder of the paper is organised as follows. Section 2 presents background knowledge of vessel architecture and the industry as a whole. In Section 3 relevant existing literature is reviewed. Section 4 gives a detailed description of the problem and presents a compact formulation of the problem. Section 5 describes the matheuristic approach. Section 6 describes the data generated and used for this study, and Section 7 presents the results for the matheuristic and compares it with the result of the compact model. At last, Section 8 contains the final remarks and conclusions.

Section snippets

Background

Liner shipping is the service of long-haul transportation of goods by using ocean-going vessels. The vessels used are of high capacity and operate on a fixed route with published schedules. The high capacity of the vessels helps to keep costs down, and the liner shipping industry is the cheapest and most energy-efficient form of international transportation.

The cargo to be transported is packed in standardised containers, which are then loaded on vessels. Most containers carried on liner

Literature review

The existing literature on the liner shipping cargo mix problem is limited. The problem was formally introduced in the PhD thesis of Delgado (2013). Here a mixed integer programming model is presented, and the multi-port version is shown not to be scalable. To achieve scalability, the problem is decomposed in a similar way as to what is suggested in earlier stowage planning work (see Pacino et al. (2011)). The work of Christensen and Pacino (2017) extends the liner shipping cargo mix problem by

The Stochastic Cargo Mix Problem with Block Stowage (SCMPBS)

Given a vessel, a string of ports, an initial configuration of the vessel, a list of accepted bookings and a distribution of the cargo flow (i.e. origin-destination demand matrix), the Stochastic Cargo Mix Problem with Block Stowage (SCMPBS) aims at optimising the expected revenue of the cargo loaded. A high degree of accuracy is imposed wrt. stability constraints to ensure the vessel is seaworthy. Furthermore, the loading must comply with the block stowage requirement, ensuring that all

Solution method

The number of scenarios in the stochastic model in Section 4 grows exponentially with the number of ports considered. As the number of scenarios quickly increases, even more so does the number of variables. Therefore, it is not expected that the model can solve more than the smallest toy example.

Besides the number of variables, the main contributor to the intractability of the stochastic model is the binary indicator variables enforcing the block stowage. To overcome this, we will describe a

Data

Most liner shipping routes reflect the nature of an international trade, e.g. the Europe to Far East services. Here, containers are exported from the Far East and imported to Europe. Vessels seldom dock at ports between the Suez canal and the Singapore Strait. Due to the length of this leg, it is the most important as regards revenue optimisation. To reflect this, we consider 5 Europe-Far East/Far East-Europe services operated by our industry collaborator. These services will be the basis of

Computational results

Three different solution methods have been tested and compared.

  • Compact - SCMPBS Mixed Integer Programming model

    The mixed integer programming model as presented in Section 4, given a time limit of 5 h.

  • RHH - SCMPBS Rolling Horizon Heuristic

    The rolling horizon heuristic as described in Section 5. All the subproblems are solved sequentially and not in parallel.

  • ModelHeu - SCMPBS Mathematical modelling based heuristic

    This is a version of the heuristic where the model (33), (34) is used instead of the

Conclusion

In this paper, the Stochastic Cargo Mix Problem is studied. The problem aims to find the optimal cargo composition needed for a vessel to maximise its revenue on a given service. A solver for this problem can be a valuable analysis tool for the industry and can be used to perform various kinds of what-if analyses. For this sort of analysis, a fast solver is needed.

The results show that the mathematical model can only solve the smallest of the considered instances. Instead, a rolling horizon

Acknowledgements

This work has been funded by the Danish Innovationsfonden under the GREENSHIP Project (1313-00005B-GREENSHIP).

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