Analyzing sailing network performance in liner shipping

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Analyzing sailing network performance in liner shipping

Niek Rake

Groningen, August 3, 2010

Master’s Thesis Econometrics, Operations Research and Actuarial Studies Specialization Operations Research

Supervisors:

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Analyzing sailing network performance in liner shipping

Master’s Thesis Econometrics, Operations Research and

Actuarial Studies

Groningen, August 3, 2010

Author: Niek Rake

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niekrake@gmail.com

Student Econometrics, Operations Research and Actuarial Studies Specialization Operations Research

University of Groningen

Supervisors: Prof. dr. B. Goldengorin (University of Groningen)

A.J. Dijkstra MSc (TBA)

Abstract: This thesis focuses on the performance of sailing networks in liner

shipping. The planning problem behind a sailing network is the ship routing and scheduling problem, that is in the class of vehicle rout-ing problems. A MINLP model for the ship routrout-ing and schedulrout-ing problem is proposed. The solvability of that model is analyzed, a two-phase greedy heuristic is presented that can generate sailing net-works, and it is investigated to what extent that model is useful in analyzing the performance of sailing networks.

Analyzing analytically the performance of a realistic size sailing net-work turns out to be impracticable. Therefore, a simulation model is developed that is able to analyze the performance of a sailing network in a realistic setting. The performance of a case study sailing network in the Mediterranean Sea is analyzed with the simulation model, and improvements for the performance of that sailing network are pro-posed.

Keywords: Sailing network, ship routing and scheduling problem, vehicle routing

problem, MINLP, two-phase greedy heuristic, simulation.

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Preface

This master’s thesis is my final project for the study Econometrics, Operations Research and Actuarial studies at the University of Groningen. It is a result of an internship at consultancy company TBA in Delft.

I would like to thank everyone who helped me during this final project. In particular, I would like to thank my supervisors Boris Goldengorin (University of Groningen) and Age Dijkstra (TBA) for their useful comments and suggestions.

As this thesis marks the end of my study period, I would also like to thank my family, friends and fellow students for a great time during my study period, and especially my parents for their unconditional support and for giving me the opportunity to study.

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Introduction

1.1 Sailing networks in liner shipping

The world economy is highly dependent on international trade. This dependence increases due to the continuous growth of the world population and of its standard or living. The major transportation mode of international trade is ocean shipping. About 90% of the international trade, measured in tons, takes place by sea (2000). Maritime container transport is a main

category of ocean shipping. In 2001, around 60% of the non-bulk cargo1in ocean shipping was

transported in a container, and this percentage increases. Figure 1.1 shows that the amount of

cargo, measured2 in TEU, transported by container ships has increased considerably over the

last years. Most of the containers are transported by liner shipping. In this thesis, shipping

Figure 1.1: World container trade and annual growth. Source: Clarkson Research, September 2006.

refers to transporting containers by ships. A liner shipping company is like a bus service, it operates container ships according to definite time schedules, routes (sailing lines) and tariffs. An example of a sailing line of a liner shipping company is depicted in Figure 1.2. A large

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Bulk cargo are raw materials such as sand, oil, coal, iron ore and grains. Non-bulk cargo is all other cargo, mainly final and semi-final products such as textiles, computers and a miscellany of manufacturing output.

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TEU (Twenty feet Equivalent Unit) is an unit of cargo capacity that refers to the volume of a twenty feet (± 6 meter) long shipping container. The most common size of a container is 2 TEU (12m × 2.4m × 2.6m).

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Figure 1.2: Sailing line of a liner shipping company. Source: www.hanjin.com.

liner shipping company operates on a network of dozens of sailing lines, called the sailing network, that has a worldwide coverage.

Liner shipping companies face several planning problems. The essential planning problem for a liner shipper company is the routing and scheduling. In the routing and scheduling, the sailing network is designed by creating the sailing lines. That is, the sequence of port visits for each sailing line is determined, and the assignment of ships to the sailing lines is decided upon. This network design is an important aspect to the performance and quality of a sailing network. Relevant performance and quality measures of a sailing network are, amongst other things, reliability, utilization of stowage capacity, and total costs. Reliability indicates to what extent the containers are delivered on time to the customers, and is thus an important measure for the customer satisfaction. Utilization of stowage capacity gives information about the suitability of the ships in terms of their capacity.

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1.2 Overview of thesis

This thesis proceeds as follows. In Chapter 2, the research description is given. Amongst other things, the research objective is stated and a literature review about sailing networks and ship routing and scheduling is given. In Chapter 3, we formulate the mathematical model of the network design problem faced by a liner shipping company. This network design problem is related to the well known vehicle routing problem. Similarities and differences between those two problems are discussed. In Chapter 4, the solvability of the formulated mathematical model is investigated, and we present a heuristic to generate sailing networks. Chapter 5 describes the sailing network of the case study as well as the data collection needed for this case study. In Chapter 6, a method is presented to analyze the performance of a sailing network. This is done by forming a simulation model that uses a sailing network as input, and generates a number of performance measures as output. Chapter 6 also presents the outcomes of the performed simulations, and characteristics of the sailing network are used to propose improvements in the sailing network of the case study. Finally, Chapter 7 provides the conclusions of this thesis, and gives some recommendations for further research. The Glossary at page 63 of this theses contains an explanatory list of abbreviations and subject-matter terms used in this thesis.

1.3 Case study

In this thesis, a case study about a liner shipping company in the Mediterranean Sea is used as an application model for the proposed heuristic to generate sailing networks, and its performance is analyzed with the simulation model. The case study involves a network of about 60 ports and 125 ships, and is, in the matter of its characteristics and size, representative for the ship routing and scheduling problem faced by many liner shipping companies.

1.4 TBA

The research is conducted at TBA in Delft. TBA is a leading international consultancy company that delivers decision support and operation optimization services. TBA’s main specialization is the analysis of logistic processes in container terminals. Complex logistic processes at container terminals are simulated in order to improve or optimize the processes. TBA also analyzes logistic processes outside the world of container terminals, and another area of TBA’s (future) interest is sailing networks of liner shipping companies and their corresponding routing and scheduling problems.

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Chapter 2

Research description

In this chapter, the research description is given. Section 2.1 states the main characteristics and terminology in liner shipping that are necessary for understanding the research questions. Here we also give an informal description of the network design problem in liner shipping, and we observe difficulties in the network design problem. Then, in Section 2.2, the research questions and the methodology used in this research to answer that research questions are explained. Section 2.3 provides a literature review about sailing networks and the correspond-ing routcorrespond-ing and schedulcorrespond-ing. This chapter ends with the discussion of the key performance measures used in this research to analyze the performance of sailing networks.

2.1 Context of research

Shipping is one of the world’s most international industries. The global shipping industry, as it is nowadays, began with the introduction of steam in the 1850s and 1860s. The introduction of steam and the industrial revolution yielded the first independently owned and operated freight lines, offering scheduled services over regular routes. Other essential developments in the shipping industry are the introduction of containers and bulk shipping in the 1960s, to speed up the flow of cargo and to manage the escalating volume of world trade.

Cargo that is transported by shipping activities is, according to Stopford (1988), divided into bulk cargo and non-bulk cargo. Before the introduction of containers, non-bulk cargo was transported in various forms of packaging. Handling these different packaging forms was very time consuming and labor intensive. Therefore, vessels spent most of their time in ports for loading and unloading, which led to congestion in ports. The introduction of the “containerization” is often described as a revolution in the transport world, see for example Haralambides and Veenstra (2000). The majority of the non-bulk cargo is nowadays trans-ported in containers that are standard sized. The containerization reduced the handling time in ports, and containers are also readily transferable to other modes of transportation, such as rail or road transport.

In the literature, for example in Lawrence (1972), the shipping industry is divided into three modes of operation: industrial, tramp and liner shipping. In industrial shipping, the cargo owner also controls the ship. Tramp ships follow the available cargo, like a taxi. Indus-trial and tramp shipping are usually involved in bulk cargo. Liner shipping mainly involves carrying containerized cargo according to time schedules, routes and tariffs advertised in advance. As this thesis focuses on liner shipping, the next section deepens this subject.

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2.1.1 Liner shipping

In this thesis, the terms ship and vessel are used interchangeably. Although vessel may refer to other means of transportation, we shall use it in the traditional sense, referring to a ship. A liner shipping company has a heterogeneous fleet of vessels, i.e. there are different ship types. Ships may differ by, amongst others, capacity, maximum sailing speed and opera-tional costs. With that fleet of vessels and under the assumption of an expected demand for container transportation, the liner shipping company tries to optimize its operations. It has to design a network of sailing lines, and to assign vessels to the sailing lines, such that the expected demand for container transportation is met. That is the ship routing and scheduling problem. Routing and scheduling in liner shipping is part of the wider subject of transporta-tion scheduling. Transportatransporta-tion scheduling has been widely discussed in the literature, but most of the attention has been devoted to scheduling vehicles. Ship routing and scheduling problems are different from those of other transportation modes, because ships operate under different conditions. Table 2.1 presents an overview of the differences in the operational envi-ronments among the four major freight transportation modes. Two important differences, also

Table 2.1: Comparison of operational characteristics of freight transportation modes. Source: Christiansen et al. (2004).

Mode

Operational characteristics Ship Aircraft Truck Train Fleet variety Large Small Small Small Power unit is an integral part of Yes Yes Often No

the transportation unit

Transportation unit size Fixed Fixed Usually fixed Variable Operating around the clock Usually Seldom Seldom Usually Trip (or voyage) length Days or weeks Hours or days Hours or days Days Operational uncertainty Larger Larger Smaller Smaller Right of way Shared Shared Shared Dedicated Pays port fees Yes Yes No No Route tolls Possible None Possible Possible Destination change while underway Possible No No No Port period spans multiple Yes No No Yes

operational time windows

Vessel-port compatibility depends Yes Seldom No No on load weight

Multiple products shipped together Yes No Yes Yes

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paid in the literature to ship routing and scheduling. Christiansen et al. (2004) give a review on ship routing and scheduling in the literature with focus on the recent years 1994-2003, and they also have included a section about this subject in liner shipping. Before we go into detail in the routing and scheduling problem in liner shipping, we first describe some important terminology and characteristics of liner shipping.

Terminology

Sailing lines of a liner shipper company are cyclic, i.e. the vessels sail in cycles on a fixed schedule, like in the sailing line depicted in Figure 1.2 on page 2. Such a cycle is called a trip in liner shipping. The trip time is the time (in days) it takes for one round-trip voyage. The sailing line in Figure 1.2 has a round-round-trip time of 42 days (6 weeks). Most liner shipping companies offer a weekly service on each sailing line, i.e. on a sailing line the vessels sail with an interval, called the “calling interval”, of one week. The number of ships that is deployed on a sailing line is then given by:

number of deployed ships = round-trip time (in days)

calling interval (in days). (2.1)

To avoid non-integer numbers of deployed ships, both the round-trip time and the calling interval are given in a number of days corresponding to whole weeks. Each sailing line has its own number of deployed ships, and it is assumed that more or less identical ships, identical in terms of length and capacity, sail on a sailing line.

When a ship operates on a sailing line, its two main operations are sailing and port operations. In the sailing phase, also called the sailing leg, a certain distance has to be sailed within a given time window. The fuel consumption is the most important variable cost parameter in the sailing phase. The fuel consumption is highly dependent on the sailing

speed. Sailing speed is measured in knots in the shipping world. One knot is equal to

one nautical mile per hour1. Figure 2.1 depicts the relation between service speed (sailing

speed) and fuel consumption for four types of container ships and nine different sailing speeds. These are typical figures that might slightly vary depending on factors such as the draft of the vessel and the engine condition, but the shape of the graphs holds in general for all container ships. Figure 2.1 shows the well known exponential relation between the sailing speed and the fuel consumption per day. The fuel consumption per day is on its own not very meaningful. Since a ship has to sail a certain distance, we are more interested in the fuel consumption per nautical mile. Doubling the service speed also doubles the sailed distance per day, so a doubling in the fuel consumption would then imply equal fuel consumption per nautical mile. As can be seen in Figure 2.1, doubling the service speed leads, roughly, to a multiplication by a factor six of the fuel consumption per day. Thus doubling the service speed leads to a tripled fuel consumption per nautical mile. Therefore, and because the fuel consumption is a considerable expense for a liner shipping company, the sailing speed is an important parameter for a liner shipping company to take into account when designing the sailing network. Moreover, the fuel price has fluctuated considerably in recent years, and has a long term upward tendency. Notteboom and Vernimmen (2009) discuss the effect of high fuel costs on sailing line configuration in container shipping, and explain the relation between the sailing speed and the fuel consumption in more detail.

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a nautical mile is 1852 meter

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Figure 2.1: Daily fuel consumption for four types of containers ships at different service speeds. Source: Notteboom and Vernimmen (2009).

Port operations are divided into piloting and container handling. In the piloting phase, a pilot comes on board and guides the vessel into the port to its berthing location. In case of large ships, tugboats are needed in the piloting phase to help large ships making sharp turns. Then the ship is berthed, possibly after waiting until its berthing location is idle. Next, the container handling starts. Containers are loaded and unloaded from the ship. Finally, the ship is unberthed and piloted out the port for the next sailing leg.

Characteristics

The size of container ships has increased steadily over the past few decades. In 1970 the maximum ship capacity was 2000 TEU, and nowadays ships with capacity up to 15000 TEU exists. Larger ships are expected to benefit from the economies of scale. In microeconomic theory, economies of scale are the cost advantages resulting from increased production. When considering the size of container ships, economies of scale are the cost reductions per trans-ported container that are expected to be achieved through the use of larger ships. However, there are some drawbacks for container mega ships (> 10000 TEU). For example, the low water depth of access channels to ports could be an impediment. Imai et al. (2006) analyze the economic viability of container mega ships and give more drawbacks of container mega ships.

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transferred to smaller container ships for further transportation. The choice between the two different network structures is important for a liner shipping company, and the possibility for transshipment should therefore be taken into account when designing the sailing network.

Figure 2.2: Network structures. Source: Imai et al. (2009).

Routing and scheduling problem in liner shipping

As explained before, a liner shipping company operates on a sailing network that consist of multiple sailing lines. The construction of the sailing network is known as the routing and scheduling problem for the liner shipping company. The routing and scheduling problem in liner shipping consists of multiple issues. Given a fleet of vessels and a certain demand for container transportation around the world, a liner shipper faces the following three decision problems:

1. Create the sailing lines; (2.2)

2. Assign vessels to sailing lines; (2.3)

3. Route the containers on the sailing network. (2.4)

Creating the sailing lines implies that sailing lines are proposed, and that for each sailing line the following is determined: the sequence of ports that are visited, the calling interval and the round-trip time. A sailing line has usually a one week calling interval. Assigning vessels to the sailing lines implies that the fleet of vessels is deployed to the sailing lines. Important characteristics of the vessels assigned to a sailing line are the number of vessels and the capacity of the vessels. The number of vessels assigned to a sailing line depends on the calling interval and on the round-trip time, as shown in Equation (2.1). Routing the containers on the sailing network deals with routing the containers on the available sailing

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lines and vessels. Containers move from their origin to their destination port, possibly after visiting some intermediate ports. Some of the intermediate ports that a container visits during its journey from the origin port to the destination port may act as transshipment ports.

2.2 Research objective and research questions

The three decision problems in (2.2)-(2.4) are interdependent. Decisions made in one problem affect decision making in the other problems. For example, the sailing lines proposed in

decision problem 1 affect the possible routes for the cargo in decision problem 3. Since

the three decision problems are highly interdependent, it is important to study them in an integrated framework. The three decision problems in (2.2)-(2.4) are, each on their own, complex problems. That makes the integrated problem a very complex problem, especially for large size problems that liner shipping companies face in practice.

Routing and scheduling in ocean shipping has traditionally often been done by pencil and paper, based on the planners’ knowledge and experience (Christiansen et al. (2004)). As container ships and fleets become larger, the planning becomes more complex. Therefore, optimization based decision support systems for ship routing and scheduling are needed. Since a couple of years, shipping companies have been employing planners with a more theoretical background. However, there is still a gap between researchers and routing and scheduling planners, and it is not well known how good the current sailing networks are. This leads to the objective of this research:

Analyze the performance of sailing networks in liner shipping. This objective leads to the following research questions:

• How can a mathematical model be formed for the planning problem behind sailing networks?

• How can that mathematical model be used for analyzing the performance of sailing networks?

• What is the performance of the current sailing network in the case study, and what are possible improvements for it?

2.2.1 Methodology

This section describes the methodology used in this research to answer the research questions stated in Section 2.2. First, the method to form a mathematical model is described. Second, the method for determining to what extent the mathematical model can be used for analyzing

the performance of sailing networks is discussed. Finally, the method for analyzing and

evaluating a current sailing network is described. Forming a mathematical model

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ship routing and scheduling problem is known, we are able to formulate a mathematical opti-mization model for the ship routing and scheduling problem. The ship routing and scheduling problem is formulated as a vehicle routing problem (VRP). Section 3.3.1 contains the math-ematical model that we formulate for the ship routing and scheduling problem. Since the computational complexity of the VRP is NP-hard, and the mathematical model we propose is a generalization of the VRP, the computational complexity of that mathematical model is also NP-hard. For more information about the theory of computational complexity, see Section 3.2.2.

Evaluating and solving the mathematical model

If the mathematical model for the ship routing and scheduling problem would be simple enough, the performance of a sailing network could be analyzed by comparing it to analytical solutions of the ship routing and scheduling problem and by applying sensitivity analysis to those solutions. However, because the ship routing and scheduling problem is NP-hard, there exist in general no efficient algorithms to solve it to optimality. An efficient algorithm is an algorithm that solves all problem instances of the ship routing and scheduling problem within reasonable time. Nevertheless, for an NP-hard problem, there may exist instances of the problem that are solvable within reasonable time. It follows from the literature that this is applicable to very small and simplified problem instances of the ship routing and scheduling problem. However, this is not the case for realistic size ship routing and scheduling

problems, like the case study that is considered in this research. Therefore we design a

heuristic to find solutions to the ship routing and scheduling problem. We propose a two-phase heuristic that first routes the containers over the network, and afterwards assigns vessels to the routed containers. This process is iterated to obtain a feasible solution, and afterwards again iterations are used when trying to find better solutions. Chapter 4 discusses the solvability of the ship routing and scheduling problem and the proposed two-phase heuristic.

Analyzing and evaluating a sailing network, and proposing improvements

The proposed two-phase heuristic is applied to the case study. We have the current network design of the liner shipping company, and we generate with the two-phase heuristic an alterna-tive sailing network for that liner shipping company. Both the current sailing network and the alternative sailing networks are analyzed and compared by means of simulation. Simulation modeling is an analysis method where computers are used to evaluate a model numerically in order to estimate the desired true characteristics of the model (Law (2007)).

Simulation is used for several reasons. First, it is straightforward that it impossible for the liner shipping company to analyze alternative sailing networks by applying them in practice. Simulation is a good tool to imitate the implementation of an alternative sailing network in

practice. Second, the system2 that represents a sailing network is highly complex. It contains

all kind of operations (e.g. sailing, container handling, transshipment activities) with a lot of stochasticity involved. The mathematical model that we formulate for the ship routing and scheduling problem is, as almost every mathematical model, a simplified representation of the reality. When formulating the mathematical model, many modeling assumptions have to be made. However, the mathematical model is still too complex to find an optimal solution

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In a simulation study, the system is the collection of entities that act and interact together. In this case, the system is the collection of all ports, vessels, and sailing legs in the sailing network.

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analytically. Simulation is often used when models are too difficult to analyze analytically, and when scenarios need to be evaluated in a more realistic setting than the mathematical model. Finally, simulation is used because TBA is specialized in simulation and already has a simulation model that is close to what we need in our research.

When evaluating and comparing the current sailing network and the newly generated sailing network, we get insights in the performance of the different sailing networks. Moreover, the simulation model enables us to trace bottlenecks in the sailing networks. Suggestions to resolve the bottlenecks and to improve the performance of the current sailing network are proposed, and the effects of those changes are determined by simulation.

2.2.2 Level and scope of the research

To answer the research questions, assumptions are made to limit the level and scope of the research, especially when forming a mathematical model for the ship routing and scheduling problem. We only take into account the tactical planning issues that a liner shipping com-pany faces when designing its sailing network. A liner shipping comcom-pany makes decisions at strategic, tactical and operational planning levels. Strategic planning issues are long term, and involve the determination of the fleet size and mix. Adjusting the fleet size and mix is a long term operation. When you decide either to build or to order a new ship, it will take a long time before that ship is actually ready for use. Tactical planning problems are medium term, and the ship routing and scheduling problem is the main tactical problem for a liner shipping company. Operational planning problems are short term, and deal, for example, with the demand for container transportation. The demand for container transportation may fluctuate from week to week. By only taking into account the tactical planning problem, it is assumed that the vessel fleet and the expected demand for container transportation are fixed.

2.3 Literature review

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(a) Level 1: ports

(b) Level 2: sailing lines (c) Level 3: container volumes

Figure 2.3: Three-level approach of a sailing network. Dotted lines indicate po-tential sailing legs. Source: Veenstra et al. (2005).

physical constraints, like roads and tracks in road and rail transport. The second level con-tains the sailing lines that are offered by the liner shipping company, and the deployment of the vessels among those sailing lines. The third level is the actual service level, and indicates, for each vessel and sailing leg in level two, which containers are on board. This three-level approach of a sailing network corresponds to the three decision problems involved in designing a sailing network as stated in (2.2)-(2.4). Decision problems (2.2) and (2.3) correspond to the step from level one to level two in the three-level approach, and decision problem (2.4) with the step from level two to level three in the three-level approach.

The ship routing and scheduling problem, consisting of the three decision problems in (2.2)-(2.4), is quite complex. Therefore, mostly tailor made models for specific, small problems with specialized constraints and objectives are available in the literature on ship routing and scheduling in liner shipping. We mention a few interesting articles about ship routing and scheduling in liner shipping. For a comprehensive review on recently published literature on this subject we refer to Christiansen et al. (2004).

In the now following discussion of the literature, several sizes of problem instances are mentioned. Recall the size of the case study: it involves about 60 ports and 125 vessels, see Section 1.3. Furthermore, the case study has about 2200 origin-destination (OD)-pairs. An OD-pair specifies the amount of containers (in TEU) that have to be transported from the

origin to the destination port. If a sailing network has n ports, there are at most n2 − n

OD-pairs, because for each port containers can be transported to all ports except to the port

itself. In the case study, about 2200 of the 602− 60 = 3540 OD-pairs exist. This implies that

there is no demand for container transportation between all pairs of ports.

Rana and Vickson (1991) formulate a large nonlinear integer programming model for the ship routing and scheduling problem for a liner shipping company. That model maximizes profit by finding an optimal sequence of ports to visit for each vessel and an optimal number of containers to be transported between each pair of ports by each vessel. However, their model does not allow for transshipment and requires a special network structure. That special network structure implies that the route of a vessel is fixed by two end ports, say A and B, and the vessel sails between that two end ports. In the end ports, the vessel reverses its direction and sails the opposite direction visiting the same ports. For example, when a vessel sails from A to B via C, D and E, the route of that vessel is A → C → D → E → B →

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E → D → C → A. That special network structure implies a large limitation on the possible routes of the vessels. Moreover, in reality the majority of the sailing routes does not have that special network structure. Rana and Vickson (1991) conclude that their model is too complex to find an optimal solution by an exact algorithm. Therefore, they decompose the problem in several subproblems and find solutions for problems with up to 3 vessels and 20 ports.

In Powell and Perakis (1997) an integer linear programming model is presented to deter-mine the optimal fleet deployment. The fleet deployment is the assignment of the available vessels to the different sailing lines. Powell and Perakis (1997) assume that the sailing lines are predetermined. That is a simplification of the problem that we consider, where the sailing lines are not predetermined but also form part of the decisions to be made. Several case studies are solved to optimality. The results show opportunities for cost reductions for the liner shipping company involved in the case studies.

Fagerholt (2004) proposes a multi-trip vehicle routing problem for the ship routing and scheduling problem of a liner shipping company. The model assumes that there is one depot from which all vessels start their sailing line, and no transshipment is allowed. The model is solved efficiently to optimality for small problem instances. The author states that the solution procedure is probably unable to solve larger, real life problems.

Agarwal and Ergun (2008) are one of the few that present a quite general mixed integer programming model that is not constrained to specific problems. Their model incorporates relevant constraints, such as transshipment, an heterogeneous fleet of vessels and multiple pick-up and delivery. Multiple pick-up and delivery implies that containers that are loaded at one port are allowed to have more than one port of destination. The authors discuss a greedy heuristic, a column generation based heuristic and a two-phase Benders decomposition based heuristic to solve the problem. They perform computational experiments with up to 20 ports and 100 ships, which are realistic size problems in terms of number of ports and vessels. However, their experiments are not realistic in terms of the number of OD-pairs. Only experiments with up to 114 OD-pairs are considered in Agarwal and Ergun (2008), and they conclude that increasing the number of OD-pairs considerably increases the complexity of the problem.

Also worthwhile to mention in this overview is McLean and Biles (2008). They discuss a simulation model to analyze the performance of sailing networks in liner shipping. They describe the characteristics of their simulation model, and discuss the performance measures that they use for analyzing the performance of the simulated sailing networks. A case study is simulated and the results of the simulation show the contribution of different parts of the sailing network to the performance of the sailing network as a whole. McLean and Biles (2008) use a quite simple simulation model. For example, they assume that the vessel speed is equal during a sailing leg, and they do not take into account congestion in ports.

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2.4 Performance measures

This section introduces the three key performance measures that are used to analyze the performance of sailing networks, and briefly discusses why they are important for the per-formance of a sailing network. The perper-formance measures are chosen after reading articles about sailing networks in liner shipping, and after discussions with subject-matter experts. The three key performance measures in this research are:

• Total costs of the sailing network; • Reliability of the sailing network;

• Utilization of the stowage capacity in the sailing network.

The total costs are the most important performance measure of a sailing network. It is as-sumed that the expected demand for container transportation is given and fixed, and therefore the total costs give direct information about the price level that has to be charged for the container transportation. A liner shipping company tries to keep the total costs as low as pos-sible, but not at the cost of everything else. Besides low prices for container transportation, customers of a liner shipping company are also eager for their containers to be picked up and delivered on time. Therefore, the reliability of a sailing network is an important performance measure. The reliability of a sailing network is defined, in this thesis, as the percentage of ship arrivals in ports that are not delayed, i.e. the percentage arrivals on time. The last key performance measure is the utilization of the stowage capacity in the sailing network. The utilization of the stowage capacity gives information about the suitability, in terms of capacity, of the vessel fleet. For example, when a sailing line has on all its sailing legs a very low utilization of the stowage capacity, that vessel type could be replaced by a smaller one, and costs are possibly reduced. The precise mathematical definitions of the key performance measures are given in Section 6.2.4.

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Chapter 3

Mathematical model

This chapter defines the mathematical model for the ship routing and scheduling problem. That model is used to get insights in the ship routing and scheduling problem and sailing networks. In Section 3.1, a verbal description of the model is given. Here we describe the objective, variables and constraints that are incorporated in the model. Section 3.2 treats the general concepts of the vehicle routing problem (VRP), as we formulate the ship routing and scheduling problem as a VRP. Finally, Section 3.3 gives the mathematical model for the ship routing and scheduling problem.

3.1 Verbal model description

As formulated before, a liner shipping company transports containers between ports. This has to be done against minimal costs, and given an expected demand for container transportation and a fleet of vessels. (2.2)-(2.4) gave an informal description of the decision problems that are then faced by the liner shipping company. This section describes the input and output, objective, and constraints that belong to mathematical model of the ship routing and schedul-ing problem for a liner shippschedul-ing company, and ends with an example of a ship routschedul-ing and scheduling problem and a corresponding sailing network.

Input and output

The input of the mathematical model consists of the following data:

• OD-matrix. The origin-destination (OD)-matrix contains for every port combination the amount of containers (in TEU) that has to be transported weekly between the two ports. The OD-matrix represents the expected demand for container transportation, as it is an estimation of the future container flows. It is assumed that the OD-matrix is equal in every week during the considered planning horizon.

• Distance matrix. The distance matrix contains for every port combination the sailing distance between the two ports. The sailing distance is given in nautical miles.

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minimal and maximal sailing speed (in knots), and the fixed costs for using a vessel of that vessel type (in USD per week ).

• Cost parameters for ports. Besides the costs for vessels, also costs for visiting ports and loading/unloading containers need to be incurred. When a ship visits a port is has to pay port fees. Costs for loading/unloading containers occur when a container is loaded at its origin port, unloaded at its destination port, and when it undergoes transshipment at an intermediate port.

With that input the sailing network has to be designed. The sailing network is the output of the ship routing and scheduling problem. A number of sailing lines are created. For each sailing line, the following is determined: the sequence of ports that are visited, the containers that are transported between that ports, which vessel type operates on the sailing line, and the speed at which the vessels sail between the ports. The sailing speed is assumed to be constant on a sailing leg, but may differ per sailing leg.

Objective

The objective of the mathematical model is to minimize the total costs of the sailing network. The costs of the sailing network are decomposed in components. We distinguish between sailing costs and operational costs, following the discussion in Section 2.1.1, and fixed costs. Sailing costs taken into account are the fuel costs. The fuel costs depend on the sailing speed. Operational costs are the port fees and the costs for loading/unloading containers. The fixed costs are the costs for using the ships.

Constraints

In forming a sailing network there is a variety of constraints that need to be taken into account. The mathematical model contains the following constraints that are essential in liner shipping:

• Cyclic sailing lines and one sailing line per ship. Cyclic sailing lines are a main charac-teristic of liner shipping. A ship is assigned to one sailing line which it sails continuously. • All containers in the OD-matrix are transported. The sailing network has to be designed in such a way that it is possible to transport all expected demand for container trans-portation. Partial satisfaction of demand is not allowed. This is due to the fact that if partial satisfaction of demand was allowed while the objective is to minimize costs, then no containers would be transported and the costs would be zero.

• Maximum round-trip time per vessel. Since the sailing lines are cyclic and it is assumed that ships of the same type sail on a sailing line, the maximal round-trip time for a ship is bounded by the number of ships of that type. This follows from Equation (2.1) and from the assumption that the sailing lines have a weekly frequency.

• Vessel capacity. The sum of all containers that are planned to be transported on a sailing leg on a vessel should never exceed the vessel’s capacity.

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to be very small when the individual demands for container transportation are small compared to the size of the ships, which is the case in a realistic sailing network. Example of a ship routing and scheduling problem

This section presents an example of the ship routing and scheduling problem to illustrate the above model description. Suppose that a liner shipping company transports containers between six ports. The OD-matrix and distance matrix are given in Tables 3.2(b) and 3.2(a). There a two vessel types, vessel I and II. Vessel I has a capacity of 1000 TEU and a maximum

Table 3.1: Instance of a ship routing and scheduling problem.

OD-matrix A B C D E F A 0 101 35 245 155 214 B 42 0 8 56 35 49 C 72 32 0 78 49 68 D 217 97 33 0 149 204 E 69 31 11 75 0 65 F 89 40 14 96 61 0

(a) OD-matrix in TEU.

Distance A B C D E F A 0 100 30 200 300 400 B 100 0 100 150 220 320 C 30 100 0 190 280 380 D 200 150 190 0 160 250 E 300 220 280 160 0 150 F 400 320 380 250 150 0 (b) Port to port distance in nautical miles.

trip time of 1 week, and vessel II has a capacity of 500 TEU and a maximum round-trip time of 2 weeks. All other input parameters are for the sake of simplicity not mentioned here. A possible sailing network for this ship routing and scheduling problem is then depicted in Figure 3.1. From this figure follows that the sailing lines have the following routes, with

Figure 3.1: A sailing network. Dashed lines: vessel I, solid lines: vessel II. between brackets the amount of TEU on board on that sailing leg:

Vessel I: A → D (689), D → F (779), F → C (318), C → A (516)

Vessel II: A → B (456), B → D (355), D → E (498), E → F (300), F → D(461),

D → A(378)

Notice that transshipment is involved in the sailing network in Figure 3.1. For example, the containers that have to be transported from C to B are transported from C to A on vessel I, transshipped in port A, and transported from A to B on vessel II.

We gave a verbal description of the ship routing and scheduling problem. The next step

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is the mathematical modeling of the ship routing and scheduling problem. It is chosen to model the ship routing and scheduling problem as a vehicle routing problem. The motivation for this choice is outlined in Section 3.3. The next section gives an introduction to vehicle routing problems.

3.2 Vehicle routing problem

In the literature, the ship routing and scheduling problem for a liner shipping company is usually modeled as a VRP. The VRP is a general description of a class of problems in the field of combinatorial optimization. Well know combinatorial optimization problems, like the traveling salesman problem and the Chinese postman problem, belong to the class of VRPs. VRPs consist of determining the routes to be used by a fleet of vehicles to deliver goods to a set of customers.

The general VRP is defined on a graph G = (V, A), where V = {0, 1, . . . , n} is a vertex set and A = {(i, j) : i, j ∈ V, i 6= j} is an arc set. Vertex 0 represents the depot, and the

remaining n vertices are customers. Each customer has a given demand qi of the good. A

nonnegative matrix C is defined on arc set A. For every arc in the directed graph G, cij is its

associated cost, time or distance to go from vertex i to vertex j. A fleet of m vehicles with identical capacity Q is located at the depot. Since the vehicles have a limited capacity, this is called the capacitated vehicle routing problem (CVRP). The CVRP is the most elementary version of a VRP. The CVRP consists of determining at most m routes satisfying the following conditions:

1. Every customer appears in exactly one route; (3.1)

2. Every route starts and ends at the depot; (3.2)

3. The total demand of the customers on any route does not exceed Q; (3.3)

4. The total routing costs, time or distance is minimized. (3.4)

The mathematical formulation of a VRP is an integer linear programming model. Introduce

n2 binary variables (xij; i, j ≤ n) to indicate if a vehicle traverses an arc in the optimal

solution.

xij =

1 if arc (i, j) ∈ A belongs to the optimal solution,

0 otherwise.

Given a set S ⊆ V \ {0}, we denote by r(S) the minimum number of vehicles needed to serve all customers in S, i.e. the optimal solution of the Bin Packing Problem (BPP) with item

set S. The BPP is the following. Given are an item set S, where each item has a weight qi,

and bins with identical capacity Q. Solving the BPP determines the minimal number of bins required to load all items.

The integer linear programming model for the CVRP is now as follows:

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X j∈V xij = 1, ∀i ∈ V \ {0}, (3.7) X j∈V x0j = X i∈V xi0≤ m, (3.8) X i /∈S X j∈S xij ≥ r(S), ∀S ⊆ V \ {0}, S 6= ∅, (3.9) xij ∈ {0, 1}, ∀i, j ∈ V. (3.10)

Equations (3.6) and (3.7) ensure that, respectively, the indegree and outdegree of every vertex are 1, i.e. every customer appears in one route. Equation (3.8) guarantees that at most m vehicles are used and that all vehicles start and end at the depot. Equation (3.9) ensures both the connectivity and the capacity constraints. For each cut (S, V \ S) the number of vehicles that goes from {V \ S} to S is not smaller than the minimum number of vehicles needed to transport all demands in S.

3.2.1 Extensions of the CVRP

There exist several extensions of the CVRP. This is due to the fact that the CVRP is a very basic model that only contains the three basic constraints (3.1)-(3.3). When modeling vehicle routing issues that occur in reality, often more constraints are required to form a vehicle routing problem that is a valid representation of reality. We describe two extensions of the CVRP that also play a role in the ship routing and scheduling problem. These extensions are generalizations of the CVRP.

VRP with time windows

In many practical applications customers require to be served within a predetermined time window. The VRP with time windows (VRPTW) is an extension of the CVRP in which for

each customer i, the service starts within the time window [ai, bi], and the vehicle stops for

si time instants. The VRPTW is a generalization of the CVRP (ai = 0, bi = ∞, si = 0 for

all i ∈ V ). In the ship routing and scheduling problem there may be restrictions on the the times that ships are allowed to enter ports.

VRP with pick-up and delivery

Customers may not only face demand for the good, but may also supply the good. Like in container transportation, where ports face demand and supply of containers. The VRP with

pick-up and delivery (VRPPD) is also an extension of the CVRP. di and pi represent the

demand to be delivered and picked up at customer i, respectively. Oi denotes the vertex that

is the origin of the delivery demand, and Di denotes the vertex that is the destination of the

pick-up demand. Additional constraints of the VRPPD compared to that of the CVRP are: 1. The current load of the vehicle along the route must be nonnegative and may never

exceed the vehicle capacity Q;

2. For each customer i, the customer Oi must be served in the same route and before

customer i;

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3. For each customer i, the customer Di must be served in the same route and after

customer i;

The VRPPD is also a generalization of the CVRP (pi = 0 and Oi = Di= 0 for all i ∈ V ).

3.2.2 Computational complexity

An important characteristic of a combinatorial optimization problem (COP) is its computa-tional complexity. The computacomputa-tional complexity of a problem gives information about the relationship between the problem size and the computation time needed to solve the problem. The problem size is determined by the size of the input data. For example, for the CVRP the problem size is determined by n (number of customers) and m (number of vehicles). The computation time is, given the problem size, the time needed by the fastest existing algorithm to solve that problem to optimality. Optimization problems can be classified into classes according to their computational complexity. Most COPs, including the VRP, belong to the class of NP-hard problems. The class of NP-hard problems is the class with the “most difficult” problems to solve to optimality. When an optimization problem belongs to the class of NP- hard problems, this implies that the time needed to solve the problem to optimality grows exponentially with the problem size. For instance the VRP with 10 customers and only 1 depot (so a traveling salesman problem) can be solved to optimality within a second, whereas solving a VRP with 100 customers and 1 depot to optimality may take months on an ordinary computer. For a more detailed description of computational complexity we refer to Garey and Johnson (1979).

3.3 Ship routing and scheduling problem as a VRP

We gave a verbal model description of the ship routing and scheduling problem, and the general concepts of VRPs were described. In this section, the ship routing and scheduling problem is formulated as a VRP. The choice for the VRP comes quite naturally, as the ship routing and scheduling problem deals with routing containers (goods in the VRP) to ports (customers in the VRP) with a fleet of vessels (vehicles in the VRP). In this section, some extra assumptions that are made in the VRP for the ship routing and scheduling model compared to the basic CVRP are explained, and afterwards, the mathematical model for the ship routing and scheduling problem is formulated.

3.3.1 VRP assumptions for the ship routing and scheduling problem

The CVRP is the basic formulation of a VRP. To formulate the ship routing and scheduling problem as a VRP, additional assumptions are made, which are listed below.

1. Containers are classified into cargo. A cargo is a collection of containers with the same origin and destination port. Each cargo is characterized by its origin port, destination port and number of containers. This has similarities with the VRP with pick up and deliveries.

2. Transshipment is allowed. Transshipment is an important aspect of liner shipping.

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3. There is no depot. The vessels do not start and end their routes at a depot, but sail continuously on cyclic routes.

4. Multiple vessels per port. Ports may be visited by more than one vessel. This is different from the CVRP, where each customer is visited by exactly one vehicle. Moreover, a vessel is allowed to visit a ports more than once in a round-trip.

3.3.2 Model specification

We are now able to formulate a VRP for the verbal description of the ship routing and schedul-ing problem given in Section 3.1. The parameters and decision variables are introduced, and subsequently a mixed integer nonlinear programming model for the VRP is presented. Parameters

All cost parameters are in USD and all speeds are in knots.

P Set of ports, indexed by i and j.

A Arc set, A = {(i, j) : i, j ∈ P, i 6= j}

C Set of cargo, indexed by c.

Oc, Dc, Bc Origin port, destination port and cargo volume (in TEU) of cargo c.

V Set of vessel types, indexed by v.

Nv Number of vessels that are of vessel type v.

Qv Capacity of vessel type v (in TEU).

Kci Parameter indicating whether there is supply (Kci= 1), demand (Kci= −1),

or neither (Kci= 0), of cargo c in port i.

dij Distance from port i to port j in nautical miles.

F (Sijv) Fuel consumption in tons per nautical mile when vessel type v sails at speed Sijv.

CF Costs of one ton fuel.

C_iT Costs for loading or unloading one TEU in port i.

C_ivP Fixed costs incurred when vessel type v visits port i.

C_vU Fixed costs (per week) incurred when vessel type v is used for transporting containers.

S−_v, S_v+ Minimum and maximum sailing speed of vessel type v.

Ei Time (in hours) needed to load or unload one TEU in port i.

M Maximum number of times a port may be visited in one round-trip.

Decision variables

xijv Binary variable indicating whether (xijv = 1) or not (xijv = 0) to sail directly

from port i to port j with vessel type v.

yiv Binary variable indicating whether (yiv= 1) or not (yiv= 0) port i is called

by vessel type v.

uijvc Binary variable indicating whether (uijvc = 1) or not (uijvc= 0) cargo c is

transported directly from port i to port j on vessel type v.

δivc Binary variable indicating whether (δivc = 1) or not (δivc = 0) cargo c is

unloaded in port i from vessel type v.

γivc Binary variable indicating whether (γivc = 1) or not (γivc = 0) cargo c is

loaded in port i on vessel type v.

Sijv Sailing speed of vessel type v when sailing directly from port i to port j.

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The ship routing and scheduling problem can be formulated as the following mixed integer nonlinear programming model, referred to as Problem (P):

(P) Minimize CF X v∈V X (i,j)∈A F (Sijv)dijxijv+ X i∈P X v∈V X c∈C C_iT(δivc+ γivc)Bc +X v∈V X (i,j)∈A C_ivPxijv+ X v∈V C_vUNv, (3.11) s.t. X j∈P ujivc− X j∈P uijvc≤ δivc, ∀v ∈ V, c ∈ C, i ∈ P, (3.12) X j∈P uijvc− X j∈P ujivc≤ γivc, ∀v ∈ V, c ∈ C, i ∈ P, (3.13) X v∈V X j∈P uijvc− X v∈V X j∈P ujivc= Kci, ∀c ∈ C, i ∈ P, (3.14) X c∈C

Bcuijvc ≤ Qvxijv, ∀v ∈ V, (i, j) ∈ A, (3.15)

xijvS−v ≤ Sijv ≤ Sv+, ∀v ∈ V, (i, j) ∈ A, (3.16)

X j∈P xijv− X j∈P xjiv = 0, ∀v ∈ V, i ∈ P, (3.17) X i∈S X j /∈S xijv ≥ yav+ ybv− 1, ∀S ⊆ P, S 6= ∅, a ∈ S, b ∈ {P \ S}, v ∈ V, (3.18) X j∈P xijv ≤ M yiv, ∀v ∈ V, i ∈ P, (3.19) X (i,j)∈A dijxijv Sijv +X c∈C X i∈P BcEi(δivc+ γivc) ≤ 7 × 24 × Nv, ∀v ∈ V, (3.20) δivc, γivc∈ {0, 1}, ∀v ∈ V, c ∈ C, i ∈ P, (3.21)

Sijv ≥ 0, xijv ∈ {0, 1} ∀v ∈ V, (i, j) ∈ A, (3.22)

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Equation (3.11) denotes the objective function and minimizes the total costs of the chosen sailing network. There are four terms in the objective function. The first term are the fuel costs of the chosen sailing lines. The second term are the container handling costs. The third term are the port fees, and the last term are the fixed costs incurred to the usage of the ships. Constraints (3.12) and (3.13) ensure that cargo can not be split, i.e. when a vessel arrives at a port, all containers of cargo c are loaded/unloaded or cargo c is not loaded/unloaded. Constraints (3.14) and (3.25) guarantee the flow of cargo. If cargo c is loaded or unloaded from vessel v in port i, and cargo c has neither origin or destination in port i, then the reverse action must take place for cargo c on another vessel in port i. Furthermore, the constraints ensure that cargo c is loaded at its origin and unloaded at its destination. Constraint (3.15) requires that the volume of all cargo on board vessel v from port i to port j is less or equal than the capacity of the vessel. Constraint (3.16) ensures that a vessel sails at a speed within its speed range. Constraint (3.17) ensures that a vessel arriving at a port also departs from that port. Constraint (3.18) guarantees that the routes are cyclic. Constraint (3.19) ensures the upper limit on the number of calls that a sailing line may have to the same port. Constraint (3.20) ensures that the round-trip time of each sailing line does not exceed the maximum possible round-trip time. The left hand side of this equation consist of the sailing time and the time for container handling in ports.

As Problem (P) is a VRP, it is an NP-hard problem. NP-hard problems are in general difficult to solve to optimality. The next chapter goes into further detail on solution techniques for Problem (P).

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Chapter 4

Solving the ship routing and

scheduling problem

This chapter treats solution techniques for the ship routing and scheduling problem. The mathematical model for the ship routing and scheduling problem (Problem (P)) was formu-lated in Section 3.3.2 as a mixed integer nonlinear programming (MINLP) model. First, we elaborate in more detail on mixed integer nonlinear programming, and thereafter we explain the consequences of being a MINLP for solving the ship routing and scheduling problem. Several possible solution techniques are discussed, including a two-phase heuristic.

4.1 MINLP

A MINLP model is a mathematical programming model with continuous and discrete decision variables, and with nonlinearities in the objective function and/or the constraints. The use of MINLP is a natural approach of formulating problems wherein it is necessary to simulta-neously optimize the discrete system structure and continuous decision variables. In case of the ship routing and scheduling problem, the system structure is the choice of the routes of the vessels and the cargoes, and the continuous decision variables are the sailing speeds on the sailing legs. The general form of a MINLP model is as follows:

Minimize Z = f (x, y)

subject to g(x, y) ≤ 0 (4.1)

x ∈ X ⊆ Rn, y ∈ Y ⊆ Zm,

where f (x, y) is a (non)linear objective function and g(x, y) is a (non)linear constraint

func-tion. The variables x and y are the decision variables, and Zm _{denotes the set of integer}

vectors in Rm. MINLP models are difficult to solve because they inherit the difficulties of

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problem.

4.2 Solution techniques

As already mentioned in the literature review in Section 2.3, general model formulations for the ship routing and scheduling problem are not yet available in the literature. Therefore there are also no general solution procedures available for the ship routing and scheduling problem, but only tailor made solution procedures for specific models. This section elaborates on three possible solution techniques for the ship routing and scheduling problem defined in Problem (P): complete enumeration, MINLP solvers and a two-phase heuristic.

4.2.1 Complete enumeration

Complete enumeration is an exact method to solve Problem (P). In complete enumeration all possible solutions are evaluated, and the one with the lowest objective value is chosen as the optimal solution. Complete enumeration for Problem (P) is an infinite procedure, because of

the continuous decision variables Sijv. However, suppose that all Sijv are fixed in advance.

Complete enumeration of Problem (P) is then a finite procedure, because all the remaining decision variables are binary, and thus bounded from below and above. However, in that case the number of possible solutions is tremendous, even for small problem instances op problem (P). For example, consider the problem with ten ports and three vessel types. When fixing

Sijv, we end up with around 25000 binary decision variables. That implies that the number

of solutions to be evaluated is at most 225000. It is obviously not possible to evaluate all those

candidate solutions, and therefore complete enumeration can not be used to solve Problem (P).

4.2.2 MINLP solver

There is also software available to solve MINLP models. That software uses smart algorithms and heuristics to find a global solution or an approximation to the global solution. The MINLP model of the ship routing and scheduling problem has been implemented in the software package AIMMS, which is one of the world’s leading optimization packages. AIMMS is capable to solve all kind of programming models, and contains several algorithms that can be used to solve the problems. AIMMS has a friendly user interface, in which problems are easily implemented and the progress of the optimization is shown. In this section, all optimization runs are performed on a Pentium 3 GHz PC with 1 GB of RAM, and all used problem instances have a feasible solution.

As already mentioned, solving MINLP models with more than 1000 decision variables is, in general, a big challenge. Therefore we start with a ship routing and scheduling problem with 5 ports and 2 vessels, which contains 1460 binary decision variables and 50 continuous decision variables. For this problem instance not even a feasible solution is found after 15 hours of run-time, and thus this problem instance is probably too complicated to solve with an optimization package. The main bottleneck is that the model is nonlinear, due to the fact that the first term in the objective (3.11) and the first term in constraint (3.20) are nonlinear. To make the model somewhat easier, the problem is transferred into an integer

linear programming (ILP) model by changing the continuous decision variables Sijv into fixed

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by the optimization procedure. This simplification implies that the model now has 1460 binary decision variables and no other decision variables. It was tried to solve several self generated problem instances, with different numbers of ports and vessels, to get insight in what problem sizes of Problem (P) are able to be solved with an optimization package. The

problem instances are divided into problem classes. The problem classes are denoted as

a − b − c − d, where a is the number of ports, b the number of vessel types, c on which indices the sailing speed may vary, and d how the range of the sailing speed is defined. For example, if c = (i, j, v), the sailing speed may differ per sailing leg and per vessel type, and if c = (v), the sailing speed may only vary between different vessel types and is equal for the same vessel type on different sailing legs. d = [x..y] implies that the sailing speed is a continuous decision variable on the interval [x, y], and d = {x..y} implies that the sailing speed is an integer valued decision variable on the interval [x, y]. Note that when the range of the sailing speed is a one point interval, i.e. d = {x..x} or d = [x..x], the sailing speed is actually a parameter, and the model is a ILP.

Table 4.1 provides an overview of the problem instances that have been tried to be solved to optimality in AIMMS. The second column in the table contains the number of decision variables in the models, with between brackets how many of them are binary valued. The

run-Table 4.1: Solving different problem instances of Problem (P) in AIMMS. The MINLP and ILP models were solved with the algorithms Baron 7.5.3 and CPLEX 12.1, respectively. See Appendix C for a short description of these algo-rithms.

Problem class No. of variables Type Run-time (s)

5 − 2 − Sijv− [10..25] 1510 (1460) MINLP > 15 hours

5 − 2 − Sv− [10..25] 1462 (1460) MINLP > 15 hours 5 − 2 − Sv− {10..25} 1462 (1460) MINLP 23340 (≈ 6.5 hours) 5 − 2 − Sv− {15..17} 1462 (1460) MINLP 5680 5 − 2 − Sv− {15..15} 1460 (1460) ILP 19 8 − 3 − Sv− {15..15} 13656 (13656) ILP > 15 hours 7 − 2 − Sv− {15..15} 5404 (5404) ILP > 15 hours 6 − 2 − Sv− {15..15} 2964 (2964) ILP 9156

time for solving a problem instance also depends on the specific input data of that problem instance. Therefore, for two different problem instances of the same problem class, the run-times for solving the problems are in general not equal. This implies that the run-run-times in Table 4.1 are only an indication for the order of the run-times needed to solve problem instances of the specified problem classes. From the table can be concluded that solving the ship routing and scheduling problem, as defined in Problem (P), can only be solved with an optimization package for very small problem instances. When it is a MINLP model, so with the speeds also as decision variables, the problem is extremely difficult to solve. When simplifying the problem to an ILP model, it is solvable within reasonable computational time (a few hours) for problem instances with maximal 6 ports and 2 vessels.

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4.2.3 Heuristics

As solving Problem (P) exactly is not possible, except for very small and thus unrealistic problem instances, the use of heuristics is needed to find solutions for Problem (P). Heuristics are approximating algorithms that try to generate good feasible solutions within reasonable time. Solutions provided by heuristics are usually not optimal, and they might be far from optimal. Heuristics for the ship routing and scheduling problem in Problem (P) are not yet available in the literature, except for some simplified problems. Therefore we developed a heuristic that is able to generate feasible solutions for Problem (P). We propose a two-phase heuristic that first routes the containers, and second routes the vessels and schedules the routed containers on the vessels. We first give a motivation for the heuristic, and afterwards give a verbal description of the heuristic.

Two-phase heuristic

A two-phase heuristic is proposed to manage the complexity of Problem (P). The two-phase heuristic splits Problem (P) in two parts: first routing the containers and second routing the vessels. Recall that the ship routing and scheduling problem consists of three decision problems, as given in (2.2)-(2.4) on page 9. Phase I of the heuristic is concerned with (2.4), and phase II with (2.2) and (2.3). The two-phase heuristic excludes some of the interdependencies between the three decision problem in (2.2)-(2.4). However, by repeatedly applying the two-phase heuristic and by using the output of an iteration as input for the next iteration, we aim to find good solutions for Problem (P).

The two-phase heuristic uses as input the OD-matrix, the fleet of available vessel types

V and the directed graph G0 = (P, A), where P is the set of ports and A is the arc set with

the distance matrix D = ||dij|| defined on it. Arc (i, j) exists if there is the possibility that a

vessel sails directly from port i to port j, i.e. it is a potential sailing leg. In its most general

form the graph G0 is a directed complete graph, because it is possible to sail directly between

every pair of ports, like in Figure 2.3(a). The weight of arc (i, j) ∈ A is given by the distance between port i and port j in nautical miles.

In phase I of the heuristic, routes are determined for each entry, called a cargo flow, in the OD-matrix. For each cargo flow the shortest path from the origin port to the destination

port is found. If the graph G0 is complete and the strict triangle inequality assumption1

holds for the distance matrix, this procedure will result in the arc from the origin port to the destination port for each cargo flow. Since there are only a limited number of vessels available and the vessel capacities are many times larger than the size of most cargo flows, there is no possibility that there are enough vessels such that all those arcs can be sailed. Furthermore, this will result in vessels sailing around with only a fraction of their capacity occupied.

To overcome this problem, first some notation is needed. For each cargo flow, the arcs in its shortest path are denoted as “cargo route segments”. A cargo route segment is denoted as

rijc, and is defined if the shortest path of cargo c traverses arc (i, j). rijcis characterized by

the size of cargo c. Furthermore, let an arc in G0 _{be an “used arc” if the arc occurs in at least}

one shortest path. The amount of cargo transported on an used arc is the sum of the cargo route segments on arc (i, j). To be able to find feasible solutions in phase II of the heuristic,

1_{The strict triangle inequality assumption is the following: ∀i, j, k ∈ P} _{=⇒ d}

ij < dik+ dkj. This is not

Analyzing sailing network performance in liner shipping

Analyzing sailing network performance in liner shipping

Analyzing sailing network performance in liner shipping

Master’s Thesis Econometrics, Operations Research and

Actuarial Studies

Preface

Contents

Chapter 1

Introduction

1.1

Sailing networks in liner shipping

1.2

Overview of thesis

1.3

Case study

1.4

TBA

Chapter 2

Research description

2.1

Context of research

2.2

Research objective and research questions

2.3

Literature review

2.4

Performance measures

Chapter 3

Mathematical model

3.1

Verbal model description

3.2

Vehicle routing problem

3.3

Ship routing and scheduling problem as a VRP

Chapter 4

Solving the ship routing and

scheduling problem

4.1

MINLP

4.2

Solution techniques