The liner shipping network design problem incorporating maximal transit times with a soft time window and slow-steaming

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The liner shipping network design problem

incorporating maximal transit times with a soft

time window and slow-steaming

Marc van der Wal

June 22, 2015

Student number: s2011190 Supervisor: Dr. E. Ursavas Second supervisor: Dr. X. Zhu Abstract

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1 Introduction

In transportation services, container shipping plays an important role. Due to rapid growth in the worldwide economy in the last decades, the need for trans-portation of products has increased enormously. Containerized shipping is a large scale operation, since 60% of the total value of goods transported by sea is done via containerized shipping (Stopford 2009). This comes down to the fact that the maritime liner shipping network has an additional value to the economy of more than 400 billion dollars (IHS Global Insight 2009). Furthermore, a vast amount of operators is attracted because of the fact that 90% of the world’s volume- and 70% of the worlds value of products is transported via shipping operations (Christiansen, Fagerholt 2011). The shipping operations market has a tight connection to the financial markets, where in the preceding years the financial crises have brought margins down tremendously (Christiansen, Fager-holt 2011). Combining the large number of operators and small margins, results in a competitive environment where mainly the skilful actors succeed in the long run (Christiansen, Fagerholt 2011).

For a long time the research described within the maritime routing and scheduling has lagged far behind air and land transport research (Christiansen, Fagerholt et al. 2004). Recently though, there has been an increase in research in this area. IHS Global Insight (2009) states that shipping the cargo of this network via air or land would increase the costs (in dollars and environmentally) tremendously. Therefore, shipping cargo via sea is a frequently used mode of transportation. Reducing the cost in liner shipping operations has therefore a great impact on the total cost of transportation worldwide.

The remaining part of the introduction consists of the following subjects. First, the liner shipping model of this research is placed within a classification of shipping operations and planning levels. Next the liner shipping network design problem and its components are discussed. Thereafter the concepts of slow steaming and soft delivery time windows are explained. In the second section, the large neighbourhood search algorithm is presented and finally the research question of this thesis will be stated.

1.1 The liner shipping network design problem

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shipping operations with, for instance, optimal assignment of cargo to ships and ship schedules. The lowest planning level is operational, where ship schedules, that for instance determine which ship visits which ports, are put together.

This thesis will focus on liner shipping at the operational planning level, specifically to the liner shipping network design problem (LSNDP). The purpose of the liner shipping network design problem is to find a set of non-simple cyclic sailing routes that will jointly transport multiple commodities via the use of a fleet of container vessels (Brouer, Alvarez et al. 2014). A non-simple cyclic sailing route is a route where a port may be visited twice, while the start and end of the route are the same. The objective of this problem is to maximize the revenue of the cargo transport and minimize the operational costs (Brouer, Alvarez et al. 2014). To illustrate the LSNDP, an example of a LSNDP is provided below.

The LSNDP can be seen in two parts (i.e. two dependant problems), the defi-nitions of rotations and the placement of containers on the generated routes. The container placement can be seen as a multi commodity flow problem (MCFP). These two problems are captured in this research by two mixed integer pro-gramming (MIP) models. The content of the two models is described in detail in chapter 4.

Figure 1: Example map

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port as it is visited twice in the rotation. In this version of the LSNDP, the starting- and ending point of a route must be the butterfly port if there is a butterfly port. The starting (and ending) port is hereafter referred to as the master port.

Next the MCFP receives the output of the auxiliary problem solution found, as an input. The MIP formulation of the MCFP minimizes the costs within the given constraints, which are defined separately for this problem. The MIP formulation places containers on the ships within the earlier found rotations. When there are multiple rotations, the MCFP may also use transshipment to switch the container of rotation at one of the ports in the current rotation. Transshipment is the unloading of container(s) off a ship, to later be reloaded on different ship(s) to finish the journey to its destination. The ability to use transshipment provides the opportunity for a container to reach its destination, even though its origin and destination are not in the same rotation. An optimal solution for the LSNDP may need for one- or even both of the problem parts to have local suboptimal objective values to enable the other problem part to gain more profit. Because the LSNDP exists out of these two separate problems, it is generally not possible to solve the LSNDP to optimality within polynomial time, which means it is NP-hard. This is proven in Brouer et al. (2014). Therefore, a search heuristic is needed to find good solutions. The large neighbourhood search (LNS) heuristic is such a heuristic and will be discussed in the next section. First though, the concepts of slow steaming and soft delivery time windows are presented.

The effect of introducing slow steaming as a method of (fuel) cost reduction for liner shipping carriers, is one of the parts of this research. Slow steaming is the reduction of sailing speed to reduce the fuel usage. Bunker fuel is revealed as the dominant cost in operating a liner shipping network and it may be more than 60% of the total cost for the carrier (Golias, Saharidis et al. 2009, Stopford 2009, Løfstedt, Alvarez et al. 2010). As fuel bunkering is a significant part of the total costs, incorporating it in LSNDP models would make them more realistic. Another implication of fuel usage reduction is the reduction of CO2 emission. Even if not measured directly, a reduction of CO2 emission via fuel consumption is considered important worldwide. The focus on environmental emissions has grown in the past decades, while maritime transportation accounts for 5% of total global CO2 emissions (Christiansen, Fagerholt 2011). Therefore it is easy to recognize the environmental benefits from reduced fuel consumption, which can be achieved by more efficient planning. Details as to the slow steaming are found in the literature review in chapter 2.

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set by the customer. The application of time windows is useful, as for instance perishable goods need to be transported or if there are contracts which state a latest possible arrival time of the goods transported.

In real life Mearsk has presented the LINER-LIB 12 dataset, which contains maximal transit times between ports based on historic data. That dataset is used in this research as a means to create results that can be used for analysis. The transit times provide a time in which the containers should be transported from the origin port to their destination port.

1.2 Large neighbourhood search heuristic

The LNS heuristic finds a good solution by performing the following steps. First a quick initialization is performed, where the auxiliary problem and thereafter MCFP are solved once, to find initial solutions. Next the LNS heuristic will iterate sessions of finding rotations and solving their MCFPs until all vessels are used, there is no more demand or no solution can be found. Every iteration, a cluster of ports is selected for solving and a removal strategy for a number of rotations in a solution is changed. By doing this, the heuristic searches in a wider number of solutions with the goal to find the best one. The exact functioning of the LNS heuristic will be explained in more detail in chapter 5.

In Pepin, Desaulniers et al. (2006) the effect of using a Large Neighbourhood Search (LNS) heuristic based on column generation on the search quality and computation time, is presented. They found that the LNS heuristic finds results faster than other heuristics, without deteriorating excessive solution quality, while column generation produces the best quality solutions when sufficient computational time is available and stability is required.

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1.3 Research question

This thesis extends from current literature by incorporating important factors of liner shipping in the LSNDP that have not been taken into account until now. To find novel results regarding the use of the LNS heuristic, slow steaming and soft delivery time windows, this thesis will answer the following research question:

What is the influence of incorporating slow steaming and maximal transit times with soft time windows in the LSNDP in a column based large neighbour-hood search heuristic on the performance of solving the LSNDP?

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2 Literature review

This chapter reviews the recent findings for the liner shipping network design problem (LSNDP) and literature relating to slow steaming and soft delivery time windows. That information will provide a solid basis for this research, to add value to the current knowledge base. Furthermore choices are made on the way the concepts of this research are applied.

The literature search is performed via a backward and forward literature search starting with the paper of Brouer et al. (2014). For this, the databases of Smartcat and Google Scholar are used. There, the keywords: liner, shipping, time windows, slow steaming, LSNDP, vehicle routing problem, LNS (large neighbourhood search) and column based heuristic are also used to search the databases for relevant literature. The most regularly cited papers by the found papers that were useful, are reviewed as well.

2.1 The liner shipping network design problem

As presented in the previous chapter, the focus of this thesis lays within the liner shipping classification, specifically the liner shipping network design problem (LSNDP). Rana and Vickson (1991) were the first to present a model for liner shipping with non-simple routes and created the first versions of what is now called the LSNDP. Their model does not allow for transshipments, which are essential in today’s liner-shipping, and contains non-linearity which results in a high complexity. Now, two decades later, the models have improved significantly by including important elements of the liner shipping and by decreasing non-linearity related complexity. An extensive literature review of this is presented in Brouer et al. (2014).

Recently, Argewal and Ergun (2008) have created a simultaneous ship schedul-ing and cargo routschedul-ing model, which uses a weekly frequency constraint by group-ing vessels into classes. Next to the practical relevance, this reduces the com-putation time when compared to continuous- or daily scheduling. Although the model does allow transshipment, the costs related to that are not taken into account. Information about different vessel classes is provided by Cariou (2011), where classes differ in the number of containers they can carry, or TEU (twenty feet equivalent unit) size. Per class is, amongst others, fuel consumption given and the effect that slow steaming has on that, which is useful later in this research if the dataset used does not include this.

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In area of LSNDP research, Brouer et al. (2014) presented a benchmark dataset for the LSNDP. That dataset is used in this study to test the heuristics and compare the results with other research that uses the benchmark dataset. Based on the framework of Alvarez (2009), Brouer et al. (2014) also provided a heuristic method to generate solution for the instances of the dataset. Their heuristic tries to maximize the profit for the liner shipping carrier, which rep-resents a simplified version of the real world problem. It takes into account butterfly ports (via non simple cyclic sailing routes), (bi)weekly frequencies, penalties for rejecting demand, canal- and port calling costs in their mixed inte-ger programming models. In this thesis the work of Brouer er al. (2014) is used as a means to research the effects of using a different solution search heuristic to find good solutions to the LSNDP.

In the more recent work Brouer, Desaulniers and Pisinger (2014), a math heuristic is used to generate solutions to the LSNDP on the LINER-LIB 12 benchmark dataset. Their results will also be used as a comparison for the results generated in this research.

2.2 Search heuristic

Search heuristics are used frequently for NP-hard problems. Therefore, review-ing more general problems that use search heuristics is a useful way to find a better search heuristic for the LSNDP. A more general problem found is the vehicle routing problem (VRP). In a VRP, vehicles are routed throughout a number of possible routes, where the goal is to minimize total cost. The VRP with hard constraints (all demand has to be fulfilled) can be viewed as the JRFDP. A VRP, where pick-ups are split, there are split deliveries, there is multiple cross docking, there are no depots and there is a heterogeneous vessel fleet, can generally be viewed as a LSNDP. Thus the LSNDP is a specific version of the JRFDP. Argewal and Ergun (2008) have proved the NP completeness of the simultaneous scheduling and cargo routing problem (which is a VRP). Hence the LSNDP is NP-hard, which is proved by Brouer et al. (2014) as well.

Shaw (1998) introduced the use of a LNS heuristic for the Vehicle Routing Problem (VRP), which has potential for the JRFDP and thus for the LSNDP. Therefore, the LNS heuristic based on column generation is recognized for its potential to find better solutions faster than other heuristics used in previous studies.

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to use a column based heuristic.

2.3 Slow steaming

In operating a liner shipping fleet, the dominant cost for the carrier company is bunker fuel cost(Stopford 2009). Since the fuel consumption is linked to the sailing speed, taking into account the sailing speed of vessels is essential to the LSNDP. This gives the carriers a strong incentive to reduce sailing speed, even though this might cause the use of more vessels and increase the buffer time in the schedule (Notteboom, Vernimmen 2009). Also delays in and before a port can increase the bunker fuel consumption, since fuel consumption does not stop when a vessel is in the port. It needs to be noted that time window violation and slow steaming are dependent on each other as a lower sailing speed can result in the violation of a time window. Slow steaming can only be made sustainable to reduce fuel costs for liner shipping if competitiveness can be maintained and the time windows are of sufficient size or may be violated inexpensively (Løfstedt, Alvarez et al. 2010). Therefore, later in this thesis, the effect of using soft delivery time windows is studied as well.

Furthermore Brouer et al. (2014) states that it is becoming increasingly important for companies to ensure transport with the lowest CO2 emission.

According to Christiansen et al. (2011) the influence of maritime shipping

emissions is 5% of the worldwide CO2 emission and hence slow steaming (in particular on back-haul connections) to reduce network bunker consumption and overall CO2 emission is essential to maritime transportation and the entire world.

Implementing the possibility to vary the sailing speed continuously would cause non-linearity in the model. When non-linearity is introduced in a MIP problem it becomes significantly harder to solve, that is, needing significantly more computational power. This is why recent models like Christiansen et al. (2004), Gelareh et al. (2010) and Meng et al. (2012) do not have continues sailing options in their models. There are methods to avoid the non-linearity caused by incorporating sailing speed.

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(Ronen 1993). This only holds for a certain speed range, which is why a more realistic model should include a wide range of speeds to include the different gradations of the effect of slow-steaming. Since the sailing speed is not assumed to be narrow in the model of this thesis and the aim is an optimal solution, the third approach is the one used in the models of this research.

When analysing the effect of slow steaming within the LSNDP, a method needs to be selected to implement slow steaming in the MIP formulation. The implementation is performed within the route generation model (auxiliary model). In the models of Brouer et. al (2014) and Brouer, Desaulniers and Pisinger (2014) the ships can sail at one speed that is part of a set of possible sailing speeds. That speed has to be held in the entire rotation (in between every port), the speeds needs to stay the same for every ship in that rotation. In that situ-ation, there might be need for a quick delivery in one of the first visited ports, then the entire rotation must be sailed at the speed needed for that part of the rotation. One can imagine that this results in unnecessary high costs.

The extension of slow steaming in this research is implemented in such a way that in between ports in a rotation different speeds may be used. The same range of speeds applies as in the earlier model, only fuel may be saved by the flexibility of the new speed options. Furthermore the entire model becomes more flexible as the number weekly or biweekly schedules may shift to a more optimal solution. The results found by the models of Brouer et. al (2014) and Brouer, Desaulniers and Pisinger (2014) are still in the pool of possible solutions. Even if the new possibilities for routes do not add a better solution, the original best solution is still in the solution pool. Therefore it is thought that a more flexible model has the potential to only perform better, with regard to the objective function. It needs to be noted that this extension will increase the computational power needed to solve the auxiliary problem. This increase will slow down the search for a best solution, which negatively effects the final objective function value. The exact influence of slow steaming on the search heuristic performance and objective value is analysed via the experiment sets in this thesis.

2.4 Soft delivery time windows

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in the possibility to check whether the maximal transit times are exceeded. On another note, it might be preferable for the transportation company to violate the time window if the profits are higher than the costs. Therefore soft delivery time windows are needed for the model to be able to exceed the maximal transit times.

The first implementation of time windows in scheduling and routing prob-lems was by Solomon (1987), who created an algorithm for the vehicle scheduling and routing problem. Time windows were classified by Ting and Tzeng (2003), who recognize hard time windows in busy ports and soft time windows in less busy ports. In most cases though, soft time windows are treated as a lack of time window constraints. Additionally to the first two classes of time windows, hard time windows where either the start or end is soft is considered as the third class. When entirely soft time windows are implemented there exists a possibility to penalize the time window violation. If the penalty is sufficiently high, soft time windows will become hard.

In the models of Brouer et. al (2014) and Brouer, Desaulniers and Pisinger (2014) the maximal transit times between ports are not taken into account. The maximal transit time data is delivered within the LINER-LIB 12 dataset for further research. That data will be used in this research.

In the experiment sets, the soft delivery time window concept is applied in the following way. The soft delivery time window computes the time window in which a ship can carry a load from one port to another. If this takes longer than the maximal transit time, a penalty cost in incurred. The height of the penalty of exceeding the maximal transit time influences the importance of being at a port in time. When the penalty is sufficiently large, the soft delivery time window will become hard as the cost of exceeding will be larger than the profit of transporting. By compensating for violating the maximal transit time with penalties, the model discourages violation of the maximal transit time, while allowing it for when a situation arises where the penalty costs are inferior to the advantages in the schedule. The exact formulation of these models are described in chapter 4.

This thesis extends from current knowledge by incorporating discretized slow steaming and maximal transit times with a soft time window in the LSNDP. The result is a new extensive model that is closer to the real life problem of the LSNDP, while a near optimal solution is searched using the LNS heuristic. First the LNS heuristic is created that finds solutions for the LSNDP. Next the slow steaming is implemented separately to be able to evaluate its performance before adding the more realistic, only less profitable and more computing time needing soft delivery time windows.

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3 Research methodology

The book of Karlsson (2009) provides an overview of the different research methodologies within operations management. Within the book its classifica-tion, this study is placed within the quantitative research classificaclassifica-tion, specifi-cally axiomatic prescriptive research, since this thesis research is driven by the (idealized) model itself. A further description of this research type is: pre-scriptive research is primarily interested in developing policies, strategies and actions to improve over the results available in the existing literature, to find an optimal solution for a newly defined problem or to compare various strategies for addressing a specific problem (Karlsson 2009). According to that book, the classification of research is directly linked to a research methodology. Namely the two step: modelling- model solving methodology, which is the methodology this thesis will follow. First though, a literature search is provided to guide the research of this thesis and to analyse what has been done in this area until now.

3.1 Sub questions

Solving the model will provide data about the performance of the model. These will be compared to other studies who used the LINER-LIB 12 dataset from the Maersk-line, which is the largest global liner shipping company. Solution quality and solving speed will be the most important factors to determine the performance of the model. The programming language used to create the mixed integer programming model and large neighbourhood search heuristic is C++, CPLEX is used to solve the MIP models.

To guide the research and give it more structure, the following sub questions are provided:

– In what time should the search heuristic be able to find a good solution and how is that accomplished?

– How well does the LNS heuristic perform in comparison to other heuristics in this area of research?

– What is the effect on performance, e.g. solving speed and solution quality of incorporating slow steaming?

– What is the effect on performance, e.g. solving speed and solution quality of incorporating soft delivery time windows?

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4 MIP formulation of LSNDP

4.1 Notation

The notation used in this chapter is based on the notation used in the auxiliary model of Brouer et al. (2014). Relevant changes have been made to the model of Brouer et al. (2014), which are discussed in this chapter. The model presented is based on the model including the corrections, with extensions to slow steaming and soft delivery time windows. Note that all variables needed for all experiment sets are in this section. The following sets are utilized in the formulation:

R All rotations in the model, indexed using r

P All ports in the model

E Set of all possible edges in the model. Edges are directed

and uncapacitated.

V Vessel classes in the model, indexed using ν

G Set of port pairs with demand for transport indexed using

(o, d)

S The set of sailing speeds available, indexed using s

The following parameters are utilized in the MIP formulation of the MCFP as well as in the auxiliary model:

aν

ij Canal costs for vessels of type ν when traversing arc (i, j)

cν _{Capacity (in FFE) of vessels of type ν}

dν_j Port call costs for vessels of type ν when entering port j.

e Fuel price per ton

fν Daily running cost for vessels of type ν over entire planning

horizon. ˜

fν _{Cost or revenue of excluding a vessel of type ν from operation.}

This can be the revenue obtained from chartering the vessel out over the entire planning horizon, or the cost of laying up the vessel. gν

s Fuel consumption (tons per mile) for vessels of type ν at speed s.

hv Fuel consumption (tons per day) for vessels of type ν when idle at

the port

kod Total demand (in FFE) to the liner company for transport from o

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lv_ij Length in nautical miles of direct sailing from port i to port j using vessel type ν.

pv

j Time at port j for vessels of type ν.

qod Revenue from transport of one FFE from o to d

˜

qod Penalty for failing to transport one FFE from o to d

uj Cost of lifting one FFE at port j

tj Cost of transhipping one FFE at port j

The following variables are exclusively used in the auxiliary model to generate new rotations.

ˆ

κod Residual demand (in FFE) to the liner company for transport form

o to d. At any iteration in the heuristic, the residual demand is computed by taking the original demand and subtracting the flow that is carried by existing rotations.

T Length of the planning horizon, in days.

δ Empirical parameter, estimates the amount of additional flow

flowing through a butterfly node, as compared to a regular node. ϕin

n , ϕoutn Empirical parameters that capture the importance of a port as an

exporter or importer.

mttod The maximal transit time between origin port o and destination

port d

eeod Penalty cost of exceeding the maximal transit time of transporting

from port o to port d.

γsν Speed vector containing the set of speeds s for vessel of type ν

κ Number of sister vessels on the new service.

ν Vessel type to be deployed.

In the auxiliary model the following decision variables are used:

Nj Binary variable, indicates whether port j is visited in the rotation.

Bj Binary variable, indicates whether port j is a butterfly port in the

rotation.

Ij Continuous variable, indicates the sequence of port j in a rotation

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Cj Binary variable, indicates if port j is the master port of the route.

Aijs Binary variable, indicates whether edge (i, j) forms part of the new

rotation and which speed s is used at that edge.

Qod Continuous variable, indicates the number of FFEs with origin at

port o and final destination at port d that will be carried per sailing of each vessel in the rotation.

Ari The arrival time at port i

Exod Binary variable, indicates if the maximal transit time between o

and d is exceeded.

W1, W2 Binary variables, respectively, indicate whether the new rotation

will have weekly or biweekly call frequency.

µ Inverse of the number of trips to be completed over the entire

planning horizon by each vessel on the new service.

ω Estimated cost per sailing per vessel of the new service.

4.2 Rotation generation model

For the experiments that do not contain slow steaming, the decision variables containing the set of sailing speed S are reduced. Only one sailing speed can be selected per auxiliary model being solved. The resulting model in that experiment set solves the rotation generating model (auxiliary model)

repeatedly for all speeds in the set of sailing speeds available and picks the one with the highest profit. In experiment sets that include slow steaming, the MIP model is applied as presented below. The model itself is stated next as AU X(ν, κ). M aximize ZAU X(ν,κ)= ω subject to ω = ( ˜fν− fν₎ ₍₁₎ − X (i,j)∈E X s∈S (ehνp ν j 24+ eg ν sl ν ij+ d ν j + a ν ij)Aijs (2) + X (o,d)∈G (qod− uo− ud+ ˜qod)Qod (3) − X (o,d)∈G Exodeeod (4)

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canal-and port call costs. Note that this expression is changed as it contained a inconsistency in the model of Brouer et al. (2014). The port staying time is defined in hours, as the remaining part of the equation has a unit of days. Therefore, a division by 24 is applied to resolve the inconsistency. Expression (3) takes the revenue of transport, cost of lifting and a penalty for failing to transport from origin to destination into account. Equation (4) exists of the penalty that occurs when a maximal transit time is exceeded between an origin and a destination. Note that equation (4) is only implemented in the experiment sets that include soft delivery time windows, the other experiment sets do not contain this part of the objective function.

The model is constrained by the following equations:

24T µ = X (i,j)∈E X s∈S (pν_j +l ν ij γν s )Aijs (5) X j∈P X s∈S Ajns= X j∈P X s∈S Anjs n ∈ P, (6) X s∈S Aijs≤ (Ni+ Nj) 2 ∀i, j ∈ E, (7) X j∈P Bj≤ 1 (8) Bi+ Ni = X j∈P X s∈S Aijs ∀i ∈ P, (9) Bj+ Nj= X i∈P X s∈S Aijs ∀j ∈ P, (10)

Equation (5) states that the decision horizon is equal to the time all the sailing trips take, including the port stay time (which was excluded of the dependency to Aijs in the model of Brouer et al. (2014)). For all ports in the model, the

number of entering and leaving edges in the model must be equal, this is what equation (6) enforces. Furthermore arc (i, j) can only be part of the rotation, if port i and j are in the rotation, which is stated in expression (7). Equation (8) enforces that every rotation can have maximally one butterfly port. If a port is in the rotation and is a butterfly node, this should be captured in the corresponding arc. Since it concerns binary variables, the number of exits (or entrances) plus the fact if the port is visited twice (butterfly node) can only be 0, 1 or 2, which is the same as the sum of the two binary variables, this is captured in equations (9) and (10).

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Qod≤ µkod κ ∀o, d ∈ G, (11) Qod≤ ˆkodNo ∀o, d ∈ G, (12) Qod≤ ˆkodNd ∀o, d ∈ G, (13) X d∈P Qod≤ ϕouto cν(No+ δBo) ∀o ∈ P, (14) X d∈P Qod≤ ϕoutd c ν (No+ δBd) ∀d ∈ P, (15) X (o,d)∈G lν_odQod≤ X (i,j)∈E X s∈S Aijslijνc ν ₍₁₆₎

Equation (11) limits the amount of cargo that can be transported between any o-d pair. The transportation of cargo is only possible if the origin and

destination are in the rotation, which is constrained by (12) and (13).

Equation (14) and (15) state that the amount of cargo loaded and unloaded at any port o or d is less than a certain fraction of vessel capacity that indicates the importance of the port. An approximation of the FFE miles to be

travelled should be lower than the total amount of FFE miles measured in the new rotation, this is stated in equation (16).

X j∈P Cj= 1 (17) Bj≤ Cj≤ Nj j ∈ P, (18) Nj ≤ Ij ≤ | P | Nj j ∈ P, (19) 1 + Ii− | P | Cj− | P | (1 − X s∈S Aijs) ≤ Ij ∀i, j ∈ E, (20)

In the model, there is always one master port, if there is a butterfly port, it must be the master port. This is what is captured in equations (17) and (18). Equations (19) and (20) identify the sequence in the rotation, variable Ij

indicates the sequence of port j in a rotation. Ij is needed for subtour

elimination, Ij is non-zero if, and only if, the port is present in the rotation.

W1+ W2= 1 (21) cν+ W1M ≥ 1200F F E (22) (W1− 1) + 0.91 ∗ 7κ T ≤ µ ≤ 7κ T + (1 − W1) (23) (W2− 1) + 0.91 ∗ 14κ T ≤ µ ≤ 14κ T + (1 − W2) (24)

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X s∈S Aijs≤ 1 ∀i, j ∈ E, (25) Ari≤ M (1 − Bi) ∀i ∈ E, (26) Ari≥ −M (1 − Bi) ∀i ∈ E, (27) Ari≤ M ( X j∈E X s∈S Aijs) ∀i ∈ E, (28) Ari≥ −M ( X j∈E X s∈S Aijs) ∀i ∈ E, (29)

Furthermore equation (25) limits the number of speeds between 2 ports, to one. This means that within a route, multiple speeds may be used, only between 2 ports, maximally one speed may be chosen. Expressions (26) until (29) constrain the arrival times to be zero if a port is in the model or is a butterfly port, otherwise they are free. Equations (26) until (33) and equation (35) are exclusively used in experiment sets containing soft delivery time windows, in the other experiment sets these equations are omitted.

Arj≥ Ari+ X s∈S ((pν_j +l ν ij γν s )Aijs) − M ∗ Bj − M (1 −X s∈S Aijs) ∀i, j ∈ E, (30) Arj≤ Ari+ X s∈S ((pν_j +l ν ij γν s )Aijs) + M ∗ Bj + M (1 −X s∈S Aijs) ∀i, j ∈ E, (31) Ard− Aro κ − ExodM ≤ mttod ∀o, d ∈ G, (32) P s∈S((p ν d+ lν od γν s)Aods) κ − ExodM ≤ mttod ∀o, d ∈ G, (33)

By equations (30) and (31) the difference between arrival times of port i and j has to be at least their transit times if the ports are in the model and are not the butterfly port. Since there, the difference is computed via the arrival times, equation (32) can determine whether the maximal transit time is exceeded via the decision variable Exod. If there is a shipping to the butterfly

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Aijs ∈ {0, 1} ∀i, j ∈ E, s ∈ S, (34) Exod∈ {0, 1} ∀o, d ∈ G, (35) Nj, Bj, Cj ∈ {0, 1} ∀j ∈ P, (36) W1, W2∈ {0, 1} (37) Qod≥ 0 ∀o, d ∈ G, (38) Ij≥ 0 ∀j ∈ P, (39) µ, ω ≥ 0 (40)

The remaining constraints (34) until (40) are non-negativity and integrality constraints.

4.3 MCFP model

The following Mixed Integer Programming (MIP) formulation to solve the Multi-Commodity Flow Problem (MCFP) is used, it is based on the work of Brouer et al. (2014).

4.3.1 Exclusive MCFP notation

The following notation is used:

R All rotations in the model, indexed using r.

P All ports in the model [P := P].

E Set of all possible edges in the model [E ⊂ P × P]. All

edges are directed and uncapacitated.

Er Set of edges used in rotation r.

Ωr Set of ordered port triples (h, i, j) in rotation r. The triples

are ordered in the same manner as they will be visited by a vessel in the rotation.

V Vessel classes in the model, indexed using ν [V := F ].

G Set of port pairs with demand for transport, indexed using

(o, d) [G ⊂ P × P].

The following variables are exclusively used in the MIP model to allocate containers to rotations.

mr Number of round trips performed by a vessel on rotation r during

the planning period. We do not impose an integral restriction on this parameter, because we assume that a fractional portion of the voyage can be completed at the beginning of the following

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νr Vessel type in rotation r.

zν _{Number of vessels of type ν available [q}F_].

In the MIP model the following decision variables are used:

X_(ij)dr Number of containers travelling to their final destination at port d along edge (i, j) on rotation r.

Urs

(hi)d Number of containers travelling to port d that arrive to port i via

edge (h, i) of rotation r for transshipment to rotation s.

Wr

(id) Number of containers travelling to port d reaching their final

destination via edge (i, d) using rotation r.

V_odr Demand from port o to port d that enters the network for the first

time because it is loaded to a vessel in rotation r.

Ood Demand from port o to port d that will not be serviced by the

liner company.

Yr Number of vessels assigned to rotation r.

4.3.2 The MCFP mathematical model

The MCFP mixed integer programming model minimizes ZM P as:

M inimize ZM P = θ subject to θ = X r∈R fνr_Yr₊X ν∈V ˜ fν(zν− X r∈R:νr=ν Yr) (41) +X r∈R mrYr X (i,j)∈E (ehνr_pνr j + eg νrsr_lνr ij + d νr j + a νr ij) (42) + X (o,d)∈G (˜qodOod− qod X r∈R V_odr) (43) +X r∈R X (h,i,j)∈Ωr (ui(W(hi)r + X d∈P d6=i V_idr) + ti X d∈P d6=i X s∈R s6=r U_(hi)drs ) (44)

The total cost function exists of expressions (41) until (44). Expression (41) consists of the daily running costs of the assigned vessels and the total cost- or revenue of excluding vessels from the operations. Next expression (42)

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destination and at transshipment points.

The model is constrained by the following equations: X_(hi)dr + Vidr+ X s∈R: (k,i,l)∈Ωs s6=r U_(ki)dsr = X_(ij)dr +X s∈R s6=r U_(hi)drs (45) Where: r ∈ R, (h, i, j) ∈ Ωr, d ∈ P : i 6= dh 6= d

X_(ij)jr = W_(ij)r (i, j) ∈ Er, r ∈ R, (46)

Ood+ X r∈R V_odr = kod o, d ∈ G, (47) X d∈P X_ijdr ≤ cνr_m rYr r ∈ R, (i, j) ∈ Er, (48) X r∈R:νr=ν Yr≤ zν _{ν ∈ V,} ₍₄₉₎

Equation (45) balances all containers that enter and leave any port. The number of containers reaching their destination at port j must be equal to the number of containers travelling to j of which the final destination is j, which is expressed in equation (46). Furthermore the fulfilled demand and unfulfilled demand must be equal to the total demand, this is stated in constraint (47). In equation (48) the number of containers transported to their final

destination, must be smaller or equal to the capacity of the assigned vessels, multiplied by the number of round trips the vessel takes. Furthermore equation (49) states that the total number of vessels of type v used, must be smaller or equal to the total number of vessels of type v available.

X_(ij)dr , U_(ij)drs , W_(id)r _{∈ R}+ (50)

Where: r, s ∈ R, r 6= s, (i, j) ∈ Er, D ∈ P,

Ood, Vodr ∈ R+ r ∈ R, (o, d) ∈ G, (51)

Yr∈ Z+ _{r ∈ R,} ₍₅₂₎

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5 Large neighbourhood search heuristic

description

This chapter illustrates the use of the large neighbourhood search (LNS) heuristic based on column generation. The LNS heuristic solves the LSNDP by looping through the auxiliary problems and MCFPs respectively, and searches through different set-ups to save the highest objective value, which represents the profit. To further explain this, the entire heuristic will be presented in blocks of pseudo code. The LNS heuristic, makes use of clusters, which will be explained first. Next, the rotation removal strategy of the heuristic is presented. In the end of this chapter, the blocks of pseudo code are presented to illustrate the steps taken in the LNS heuristic.

5.1 Data clustering

Since auxiliary problems of up to 15 ports can be solved to optimality quickly (Brouer, 2014), the ports are clustered together in clusters of up to 15 ports. The criteria used to cluster the ports, are geographic location and trade volumes. The clustering has no influence in the maps with less than 16 ports, only the larger instances are solved significantly quicker. The clustering of ports is performed in such a way that each cluster contains at least on overlapping port with another cluster. This assures that all ports are

reachable from every starting position. Before every iteration of the auxiliary loop, a cluster is selected to be solved. This is done based on an

approximation of the potential profit that is generated by using that cluster. The profit approximation is saved in the variable ηκ, which is calculated with

the following formula:

ηk = X (o,d)∈G:o,d∈κ ˜ ku_odqod lυ od

This formula defines the potential profit as the gains from shipping cargo from origin to destination per mile based on the (o,d) pairs, the residual demand in cluster k and the distances between the ports lυ

od. Summed over all (o,d) pairs

this results in the ηκ variable. Since the heuristic should not use the same

cluster every time, a cluster is picked randomly each time, where ηκrepresents

the weight that is put on the cluster. Thus clusters with a higher value of ηκ,

have a greater chance of being selected. This strategy is thought of to result in a good trade-off between solution quality and speed.

5.2 Rotation removal strategy

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depends on the iteration. The repeated pattern of removing 1, 3 and 2 rotations is used rather than a constant value, as the selection of the rotations is only based on the auxiliary model. Other effects, like transshipment, only appear in the MCFP, where the rotation cannot be changed any longer. Thus the risk of selecting rotations that are not the most profitable in total (MCFP and auxiliary), is high. To reduce this risk, the rotation removal strategy is used. This widens the search for an optimal profitability, as more options will be explored than if only local optima were used to find rotations. The used pattern is thought of to be a good compromise between solution quality and the broad search character of the LNS heuristic. To increase the solving speed in the experiments, the strategy is limited to maximally remove such a number of rotations that at least two rotations remain.

Next to the number of rotations to remove, the selection of rotations to remove has a strategy behind it as well. Pepin et al. (2006) defined a good strategy for this purpose, which selects rotations based on their properties. Three properties are used: random rotations, the oldest rotations and the closest rotations.

The random property simply selects a rotation completely random. The oldest property, selects rotations based on whichever rotation was added most recently to the set of current rotations. The closest property selects a rotations randomly and then computes the closeness of all other rotations with the picked rotation. The closeness is computed by dividing the number of ports that are in both rotations, by the number of ports that are only in one of the two rotations. The rotations with the highest closeness are selected for removal.

The strategies (based on which property a rotation is selected) are selected randomly at first. Each strategy has a value between 1 and 10, only they all start on 5. The strategy’s value divided by the sum of all strategy’s values represents the chance it is being picked. Every time a rotation is found that has a better total profit (MCFP and auxiliary total) on one of the strategies, the value is increased by one. Every time a rotation is found that is no improvement on the best profit found, the value is decreased by one. This results in the best working strategy from the past iterations, having the largest chance of being picked.

5.3 The AUX- and MCFP block

The LNS loops through the auxiliary problem and MCFP repeatedly. Within the LNS heuristic, there are more operations being performed though.

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Figure 2: AUX block Loop until one of these conditions are met:

Condition1: The number of vessels of type v available for new rotations in iteration u is 0 for all v.

Condition2: The residual demand between all ports in cluster ˜ku od

is 0.

Condition3: There is no solution found. Compute for each cluster ηκ and select a cluster;

for All vessel types selected do

for All number of vessels within the vessel type do Solve AUX(υ,κ);

Save the solution rotation; end

end

Compare the solution rotations and save the one with the highest profit;

if The highest profit solution > 0 then

Add the rotation to the set of rotations for the MCFP; Add all costs and revenues of the rotation to the total profit function;

Subtract all used vessels (type and number of vessels) from the available vessels in this loop;

Substract all demand transported from the remaining demand to be transported;

else

There is no solution found; end

End loop if one of the conditions is met;

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Figure 3: MCFP block Perform the following:

-Define a multi commodity flow graph based on the rotations created in this loop;

-Define a multi commodity flow problem from the graph; -Solve the MCFP and save its solution;

-Compute all related costs and profits;

-Update the total objective function for this iteration;

-Update the remaining demand for this iteration with the found MCFP solution;

Figure 4: LNS block Initialize:

Iteration= 0; Empty all rotations;

Set all remaining demand to total demand; AUX block;

MCFP block;

//Now there is a initial solution

//Stopping condition can either be a time limit or a number of iterations

if Stopping condition is not reached then Iteration= Iteration+1;

Set of new rotations = set of last found rotations;

Copy all costs and used vessels to new rotation’s variables; Select a rotation selection strategy according to the weights; According to the strategy picked, remove rotation(s) from current set of rotations;

Remove the related costs and add the newly available vessel to the rotation’s variables;

Compute the amount of cargo that can no longer be transported due to the removal of the rotation(s);

Add the newly untransported cargo to the remaining demand; AUX block; MCFP block; else STOP heuristic; end //End of heuristic

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5.4 Explanation of the LNS heuristic

The LNS heuristic starts by loading all data into the variables to be used in the heuristic. Next, new variables are created that are empty and to be filled when solutions are found, like rotations, cost, profit etcetera. Also the remaining (to be transported) demand, is set equal to the total demand. Initializing further, a first AUX- and MCFP block are ran respectively to create a starting point for the heuristic. The solutions are stored together with their respective costs, profits, fulfilled demand, used vessel etcetera.

After initialization the actual looping of the heuristic starts. If the stopping condition is reached, the loop will not start over again and the best found solution is returned. Two types of stopping conditions are used in this research; number of iterations and maximal running time. In the next chapter, the experiment sets will be defined. There, the choice for different stopping criteria is further explained.

In the LNS loop, first the iteration number is increased by one. One iteration consists of finding rotations until all demand is fulfilled. Therefore the iteration number represents the number of loops ran in which all demand is fulfilled. This can be seen in Figure 2, Figure 3 and Figure 4. The number of AUX- and MCFPs solved is captured in the number of columns evaluated, which is part of the output data when the heuristic finishes. The output data provides all information about the performed run. The objective value, cost components and key performance indicators are in the output report.

As seen in Figure 4, in the LNS loop all related variables to that iteration are updated first, before the AUX- and MCFP block are started. The removal strategy is picked next. After removing the rotations from the set of current rotations, the amount of cargo that can no longer be transported, is estimated and removed from the current value of demand, profit and cost variables. This not only includes direct transportation within a rotation, the effects of

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6 Experiment design

This chapter describes the performed experiment sets in this research. The goal is to get valid results for all settings that outperform the results of previous research. Outperforming means in the case of the LNS heuristic- and slow steaming experiments an increase in profit, while in the soft delivery time windows the goal is to create a more realistic version of reality. The results are compared with each other and with the results of model of Brouer et al. (2014). The results will also compared with the outcomes of the research of Brouer, Desaulniers and Pisinger (2014), only in less depth, as the first mentioned research shows more resemblance with the research of this thesis. Another reason why the comparison to Brouer, Desaulniers and Pisinger (2014) is performed less intensive, is that their paper did not specify any key performance indicators of their results. Comparison is therefore not possible. The outcomes of the experiment sets of this study, enable the answering of the sub research questions and finally the main research question.

6.1 Experiment settings

For the experiment sets, the computing time and -performance are crucial influences on the results. Therefore, the environment of the LNS heuristic is tested first, to be able to assess the difference in performance of this research and other researches. It is found that the solving of the auxiliary problems in this research is significantly slower than in the work of Brouer et al. (2014) and Brouer, Desaulniers and Pisinger (2014), while the same MIP formulation is used. To account for this difference, which is believed to be the result of differences between computing power, the experiment sets run a certain amount of iterations instead of time. The number of iterations ran is exactly the same as in the research of Brouer et al. (2014).

Preferably, the experiment sets would have been performed on all maps and scenarios in the dataset. However, for larger maps the auxiliary model would take over an hour in finding an optimal solution a small number of times in the heuristic. Since this problem only occurs in the largest maps, it is thought that the increase of the number of possible routes somehow causes this problem. Due to this problem it is decided that the experiment sets are only applied on the three smallest maps; Baltic, WAF and Mediterranean.

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6.2 Dataset

The dataset used in this research is the LINER-LIB 12 dataset. As it is also used in the work of Brouer et al. (2014) and Brouer, Desaulniers and Pisinger (2014), results can be compared and a good analysis of results is possible. The planning period T is set to 180 days, which is the equivalent to the

aforementioned research. The demand for this period is scaled accordingly. The dataset contains three scenarios for each map.

The Base scenario changes nothing to the data and thus solves the map as the data is defined. To be able to analyse the model’s sensitivity to the given data two different scenarios are ran per map.

The Low scenario represents a case where the vessel availability is lowered. Therefore the entire demand cannot be fulfilled, thus the vessels must be placed there where it is most profitable. In more detail; the number of vessels available is multiplied by 0.8 and rounded to the nearest integer value, thus representing a 20% decrease in available vessels. Furthermore the total cost rate is increased by 40% to reflect an increase in vessel costs.

In the third, High scenario, the number of vessels increases by 20%, while the total cost decrease by 20%. This scenario represents an overage of vessels where potential profits are high and the costs of using a vessel is low. To make sure that randomness does not significantly influences the results of this research, all maps and scenarios tested are repeated 10 times. The best and median results are presented for analysis and comparison with the results of Brouer et al. (2014) and Brouer, Desaulniers and Pisinger (2014).

6.3 Experiment set 1: LNS heuristic

The first experiment set that will be performed on the selected maps and scenarios, is to test the effect of the LNS heuristic on the performance. This test is executed without slow steaming or soft delivery time windows. The results gathered function as a base result to which the slow steaming included model can be compared. In that way an in- or decrease in performance can be appointed solely to the including of slow steaming. Furthermore the

performance of the LNS heuristic is compared to the results of Brouer er al. (2014), to view the effects of the LNS and the newly created C++ model. It needs to be noted that there are improvements made on the model of Brouer et al. (2014) (which contained some small errors, the changes are described in chapter 4). The changes are made in equations (2), (5), (9) and (10).

Especially in equation (2), the error causes higher fuel costs, which influences the results of Brouer et al. (2014) negatively.

6.4 Experiment set 2: Slow steaming

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the model of Brouer et al. (2014). In this experiment set number of iterations is used as the stopping criterion. In that way the results can compared in a good way.

6.5 Experiment set 3: Soft delivery time windows

Elaborating on the slow steaming model, soft delivery time windows are included in this experiment set. Since the goal of this experiment is to set a standard and starting point for further research, therefore maximal

computation time is used as the stopping criterion. In that way the model can be improved and compared based on computation time by further research. To give the model more time to calculate solutions with the current

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7 Computational results

This chapter presents the results that were the outcome of the experiment sets defined in the previous chapter. As already noted earlier, the time it takes to perform the number of iterations, is set longer than that of Brouer et al. (2014) and Brouer, Desaulniers and Pisinger (2014). The models of chapter 4 are solved using the heuristic described in chapter 5. In the tables the median run is given as the middle result of the 10 replicated runs. Since there are 10 runs, the median run represents average of the fifth- and sixth best run regarding the objective value.

Table 1 shows the performance in all cases in terms of total running time t, columns (rotations) evaluated, unique columns (rotations) evaluated and MCFPs evaluated. Of the 10 times the heuristic is ran, the median result is presented for all selected scenarios and maps of the LINER-LIB 12 dataset. In Table 2 the performance of the experiments regarding the number of

iterations, improving iterations and last improving iteration are presented for every model and instance.

Tables 3, 5 and 7 report the objective values in terms of profit and cost components for all experiment sets. The network key performance indicators of the experiment sets are presented in tables 4, 6 and 8. Figures 5 until 13 are located in the appendix and show the objective value for the best runs as a function of the computation time, for all experiments and scenarios.

Table 1: Performance for all versions and datasets tested in their median run

t sec Columns evaluated Unique columns MCFPs eval

Instance Low Base High Low Base High Low Base High Low Base High

Experiment set 1 Baltic 1236 1559 1722 3871.5 4378.5 4980 54 282.5 105 4418.5 4735.5 5438 WAF 11719 31237.5 36050 26742.5 83606.5 102135.5 12039.5 28048.5 31979.5 19623 50702.5 55645 Medirerranean 7573 12962.5 13427 2316 4356.5 4137.5 976.5 1633.5 1704.5 1490 2496 2520 Experiment set 2 Baltic 675 939 1815 752 947 1615.5 32.5 330 665.5 1299 1304 2073.5 WAF 5051 7278 9366 3483.5 6949 8976.5 1814 3040.5 3647.5 3064 5355 6383.5 Mediterranean 4856 5959.5 6218 392 722.5 653 220.5 336 330.5 320 467.5 460.5 Experiment set 3 Baltic 3000 3000 3000 1689.5 1342 1262 167.5 638.5 599.5 2781.5 1727 1610.5 WAF 9000 9000 9000 2707 3095 3679 1482.5 1609.5 1792.5 2257.5 2307.5 2496.5 Medirerranean 12000 12000 12000 401.5 533.5 712.5 227 302.5 342.5 321.5 380 476

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Table 2: Performance for all versions and datasets tested in their median run

Iterations Improving iter Last impr iter

Instance Low Base High Low Base High Low Base High

Experiment set 1 Baltic 548 358 459 3 5 6 5 10 17.5 WAF 267 337 271 15.5 13 10 155.5 146.5 178.5 Medirerranean 37 39 33 10 4 7.5 18 11 15.5 Experiment set 2 Baltic 548 358 459 7 9.5 9.5 15.5 30 235.5 WAF 267 337 271 17 12.5 12 151 80.5 124.5 Mediterranean 37 39 33 8.5 5.5 5 15.5 21 13 Experiment set 3 Baltic 1150 391 368 11 10.5 9.5 112.5 221.5 214 WAF 197 140.5 100 14 10.5 12.5 98.5 60.5 68.5 Medirerranean 36.5 38 26 7 6.5 2 20 13 6.5

Notes. Instance denotes the map and scenario combination, experiments are as defined in chapter 6, the iterations denotes the number of iterations performed by the LNS heuristic, Improving iter is the number of iterations in which the objective value found was higher than earlier in solutions, Last impr iter is the latest iteration in which an improvement on the objective function was made.

The differences between the scenarios become apparent directly. As more resources become available for less cost, more options become available and more rotations are needed to fulfil demand. This results in longer computation time needed for a set number of iterations, or less iterations where the time t is set as a stopping criterion. As the computation time increases, the number of columns evaluated increases as well. Where the computation time stays the same, the increase of options for the route generation allows less columns to be evaluated. The same trend is found for MCFPs evaluated and iterations.

For all experiments it is found that for the same number of iterations, the solution time increases as the map’s size increases. That is, the number of ports in the map and the number of transportation options between those ports. The experiments containing the WAF map show in Table 1, to need more time to perform the iterations set. This is concluded to be a consequence of the number of options for transport available. As the number of iterations to be performed do not decrease as significantly as in the Mediterranean map, the time needed for the WAF instances increased vastly. Due to the clustering applied, the number of ports being solved in an auxiliary problem decreases for the Mediterranean instances. The number of ports in the Baltic and WAF instances are not so large that they can benefit from clustering. The clustering in the Mediterranean map causes only a small increase in transportation options, and therefore computation time needed, per iteration in comparison to the WAF map. The number of iterations to be performed, along with the size of the auxiliary problems to be solved, causes the difference in

computation time needed for the different instances.

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limited number of total options for that scenario (there are no more unique columns). The decrease in the ratio is caused by the decreasing number of iterations available. The ratio increases when the scenarios have the same computational capacity allocated, this is seen in experiment 3, where the computation times are equal for the Low, Base and High scenarios. The same effect as in the map size increase is found, more solution space means the MIP can find more unique solutions.

It is observed that the solution space grows when the size of the instance increases, as more improving solutions are found and they appear in later iterations. An observation gained from the figures in the appendix, is that for most cases the convergence of the objective value towards a value close to the final solution, occurs within a short amount of time. When the maps and instances grow larger, the convergence happens less rapidly. Where in the largest map it is shown that the convergence of the objective value has not stabilized yet. Therefore it is possible that, given more time, the final objective value would increase significantly.

7.1 The effects of the LNS heuristic

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Table 3: The objective value seperated by profit and cost components

Instance Z Q cv cb cp cc cm ct Lv F

Baltic Low

Best -1.39E+07 7.47E+07 5.80E+06 2.63E+05 6.65E+06 0.00E+00 3.86E+07 0.00E+00 0.00E+00 3.73E+07 Median -3.86E+07 6.01E+07 5.80E+06 2.35E+05 5.61E+06 0.00E+00 2.87E+07 7.32E+04 0.00E+00 5.83E+07 Base

Best 2.36E+07 1.01E+08 6.48E+06 4.39E+05 1.22E+07 0.00E+00 5.76E+07 0.00E+00 0.00E+00 5.50E+05 Median 1.55E+07 9.57E+07 6.48E+06 4.22E+05 1.19E+07 0.00E+00 5.37E+07 0.00E+00 0.00E+00 8.15E+06 High

Best -1.20E+07 7.65E+07 5.18E+06 3.17E+05 7.06E+06 0.00E+00 4.07E+07 0.00E+00 0.00E+00 3.52E+07 Median -1.20E+07 7.65E+07 5.18E+06 3.17E+05 7.06E+06 0.00E+00 4.07E+07 0.00E+00 0.00E+00 3.52E+07 WAF

Low

Best 2.09E+08 3.66E+08 5.82E+07 7.71E+05 1.33E+07 0.00E+00 9.07E+07 7.24E+05 -1.13E+07 5.35E+06 Median 2.06E+08 3.67E+08 5.82E+07 8.17E+05 1.53E+07 0.00E+00 9.07E+07 6.10E+05 -1.06E+07 5.23E+06 Base

Best 2.22E+08 3.67E+08 5.29E+07 7.36E+05 1.41E+07 0.00E+00 9.07E+07 1.58E+05 -1.91E+07 5.24E+06 Median 2.19E+08 3.67E+08 5.29E+07 7.99E+05 1.49E+07 0.00E+00 9.07E+07 4.72E+05 -1.78E+07 5.23E+06 High

Best 2.30E+08 3.66E+08 4.95E+07 6.15E+05 1.39E+07 0.00E+00 9.06E+07 0.00E+00 -2.42E+07 5.55E+06 Median 2.23E+08 3.67E+08 4.95E+07 8.65E+05 1.63E+07 0.00E+00 9.07E+07 9.35E+05 -2.05E+07 5.23E+06 Mediterranean

Low

Best -7.58E+06 1.30E+08 2.66E+07 1.88E+06 2.41E+07 0.00E+00 6.69E+07 1.31E+07 0.00E+00 5.04E+06 Median -1.19E+07 1.25E+08 2.66E+07 1.65E+06 2.09E+07 0.00E+00 6.20E+07 1.32E+07 -1.35E+06 1.32E+07 High

Best 2.51E+06 1.32E+08 2.32E+07 1.79E+06 2.21E+07 0.00E+00 6.71E+07 1.20E+07 -1.87E+06 5.02E+06 Median -7.65E+06 1.25E+08 2.32E+07 1.77E+06 2.08E+07 0.00E+00 6.22E+07 1.28E+07 -2.09E+06 1.38E+07

Notes. Instance denotes the map and scenario combination with a separate row for the best and median result of the 10 runs performed, Z is the objective value in profit, note that in the heuristic a minimizing function is performed and a negative value is preferred, here the profit is displayed under Z, so a positive Z value is preferred in this table; Q is the total revenue, cvis the total vessel cost, cbis the total fuel cost, cpis the total port call cost, ccdisplays the total amount of canal fees, cmpresents the total move costs at the origin and destination port, ctis the cost related to transshipment, Lvis the income of chartered out vessels, F is the total penalty costs of not transporting cargo. All numbers are presented with a 2 digit precision. Note that due to rounding, Z might not always be exactly equal to the sum of all displayed variables.

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Table 4: Key performance indicators for all maps and instances

Instance Dep% |R| PCpW BPU% WPU% BAU% WAU% t/d% Rej%

Baltic Low Best 100.00 3.00 2.84 100.00 92.22 95.05 36.10 0.00 30.39 Median 100.00 2.00 3.80 100.00 99.81 62.12 33.45 0.56 47.56 Base Best 100.00 4.00 3.56 100.00 91.41 76.00 17.46 0.00 0.45 Median 100.00 3.00 4.56 100.00 96.33 59.82 24.47 0.00 6.65 High Best 100.00 2.00 5.14 100.00 100.00 41.13 38.93 0.00 28.69 Median 100.00 2.00 5.14 100.00 100.00 41.13 38.93 0.00 28.69 WAF Low Best 81.82 9.00 2.78 100.00 64.90 85.15 20.39 3.64 2.51 Median 81.82 8.50 3.10 100.00 62.32 77.33 21.35 2.80 2.45 Base Best 59.52 9.00 2.65 100.00 62.90 100.00 24.19 0.87 2.45 Median 61.90 7.00 3.70 100.00 43.64 62.96 21.38 3.70 2.45 High Best 44.90 6.00 3.32 100.00 69.51 55.94 22.31 3.53 2.60 Median 54.08 7.50 3.62 100.00 49.29 55.73 22.48 3.57 2.45 Mediterranean Low Best 100.00 8.00 4.88 53.15 20.42 6.59 1.65 56.61 11.85 Median 100.00 8.00 4.95 63.52 15.39 9.41 1.64 57.34 16.76 Base Best 100.00 8.00 6.76 45.37 21.55 8.11 1.73 62.49 2.67 Median 95.00 8.00 5.91 59.07 17.86 8.05 1.68 62.03 7.00 High Best 90.91 8.00 6.38 45.37 16.49 8.62 1.71 61.44 2.66 Median 90.91 8.00 6.28 55.74 14.10 6.89 1.59 61.36 7.29

Notes. Instance denotes the map and scenario combination with a separate row for the best and median result of the 10 runs performed, Dep% is the percentage of the total available fleet deployed, |R| is the number of rotations in the final solution, PCpW is the average number of port calls per service per week, BPU% is the best peak utilization, WPU% is the worst peak utilization, BAU% is the best average utilization percentage, WAU% is the worst average utilization percentage, t/d% is the percentage of transshipments performed of all delivered units of demand, Rej% is the percentage of rejected cargo.

The results presented in Table 4, show that in the Baltic map, the

percentage of the fleet deployed is 100% in all instances. As the percentage of rejection is high, the lack of capacity to transport all cargo becomes apparent. When comparing to the results of Brouer et al. (2014) it appears that, where the objective function is lower, the rejection rate is significantly higher. A higher rejection rate simply means that less cargo is transported. The origin of this problem lays in the fact that less port calls are serviced per week

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available to service the demand.

The solutions from Table 3 contain a higher objective value than in Brouer et al. (2014) in the maps of WAF and Mediterranean, the reason for the higher objective value differs. Where in WAF the departure rate, rejection rate and the port calls per service per week are lower, more cargo is

transported using less ships. In the Mediterranean map the departure rate is higher, the rejection rate is higher and the difference in port calls per service per week is small. Thus where in WAF generally more efficient rotations are found, in Mediterranean this is not the reason why the improved objective value is found. In Mediterranean the reason for the higher profit can be found in Table 3, where as a result of the correct use of the fuel cost component (described in chapter 4 in equation (2)), the fuel costs are significantly lower. This is not the only reason for the higher profit of the rotations in the Mediterranean map. The amount of transshipment (t/d%) is found to be higher than in Brouer et al. (2014). Transshipping cargo is a cost effective way to transport cargo between ports that are not within one rotation of the solution. It excludes the need for extra, costly, rotations with a low utilization.

The best average utilization (BAU%) and worst average utilization (WAU%) provide useful information about how the ships are utilized. When these percentages are low, the ships are carrying a low number of containers on average. This means that there is room for improvement, i.e. taking on more cargo. The cost of the ship sailing the route is already made, therefore creating a higher utilization would only increase the profit. In the smaller maps, all the utilizations of the ships are generally higher, which leads to lower costs or more revenue from fulfilling demand. In the largest map the average utilizations are significantly lower than in the work of Brouer et al. (2014). As the number of ports visited per service per week is also lower, this leads to a lower trade volume between all ports, which can been seen in the total revenue. The differences in revenue are about 10%, the increase of

transshipment compensates for the lower trade volume and low utilizations (as the utilizations are sometimes 80% lower). The rotations created have a higher cost efficiency as the profits are higher, while the revenue is lower.

The LNS heuristic has, for most cases, improved the total objective

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7.2 The slow steaming extension results

The results of the model including slow steaming are presented two tables. In Table 5 the objective value and its components are displayed. The KPI’s are presented in Table 6. The results of this experiment are compared with the results of experiment 1. The only change between the two experiments is the implementation of slow steaming as described in chapter 4. By comparing both results, the differences can all be appointed to slow steaming.

Table 5: The objective value seperated by profit and cost components

Instance Z Q cv cb cp cc cm ct Lv F

Baltic Low

Best 2.08E+07 9.77E+07 6.48E+06 4.20E+05 1.04E+07 0.00E+00 5.53E+07 0.00E+00 0.00E+00 4.34E+06 Median 1.76E+07 9.72E+07 6.48E+06 4.19E+05 1.11E+07 0.00E+00 5.48E+07 0.00E+00 0.00E+00 6.49E+06 High

Best 2.40E+07 1.00E+08 5.18E+06 4.34E+05 1.15E+07 0.00E+00 5.70E+07 0.00E+00 0.00E+00 1.91E+06 Median 2.17E+07 9.85E+07 5.18E+06 4.19E+05 1.12E+07 0.00E+00 5.58E+07 0.00E+00 0.00E+00 4.37E+06 WAF

Low

Best 2.09E+08 3.67E+08 5.82E+07 7.62E+05 1.52E+07 0.00E+00 9.09E+07 4.23E+05 -1.23E+07 4.98E+06 Median 1.98E+08 3.67E+08 5.82E+07 8.33E+05 1.64E+07 0.00E+00 9.07E+07 1.88E+05 -3.78E+06 5.25E+06 Base

Best 2.21E+08 3.67E+08 5.29E+07 8.10E+05 1.43E+07 0.00E+00 9.09E+07 1.60E+06 -1.89E+07 4.98E+06 Median 2.12E+08 3.67E+08 5.29E+07 8.73E+05 1.62E+07 0.00E+00 9.07E+07 6.71E+05 -1.25E+07 5.23E+06 High

Best 2.24E+08 3.65E+08 4.95E+07 7.97E+05 1.44E+07 0.00E+00 9.02E+07 1.07E+05 -1.97E+07 6.05E+06 Median 2.19E+08 3.67E+08 4.95E+07 8.45E+05 1.59E+07 0.00E+00 9.07E+07 9.09E+04 -1.48E+07 5.23E+06 Mediterranean

Low

Best -2.68E+07 1.15E+08 2.80E+07 1.30E+06 1.72E+07 0.00E+00 5.55E+07 9.25E+06 -3.78E+06 3.41E+07 Median -3.45E+07 1.10E+08 2.80E+07 1.44E+06 1.72E+07 0.00E+00 5.32E+07 1.08E+07 -1.26E+06 3.49E+07 Base

Best -8.21E+06 1.24E+08 2.66E+07 1.69E+06 1.99E+07 0.00E+00 6.19E+07 1.05E+07 -1.80E+06 1.34E+07 Median -1.40E+07 1.22E+08 2.66E+07 1.63E+06 1.93E+07 0.00E+00 5.96E+07 1.18E+07 -1.98E+06 1.86E+07 High

Best -5.50E+06 1.26E+08 2.32E+07 1.63E+06 1.97E+07 0.00E+00 6.20E+07 1.40E+07 -1.58E+06 1.26E+07 Median -1.23E+07 1.23E+08 2.32E+07 1.63E+06 2.01E+07 0.00E+00 6.05E+07 1.24E+07 -1.87E+06 1.85E+07

Notes. Instance denotes the map and scenario combination with a separate row for the best and median result of the 10 runs performed, Z is the objective value in profit, note that in the heuristic a minimizing function is performed and a negative value is preferred, here the profit is displayed under Z, so a positive Z value is preferred in this table; Q is the total revenue, cvis the total vessel cost, cbis the total fuel cost, cpis the total port call cost, ccdisplays the total amount of canal fees, cmpresents the total move costs at the origin and destination port, ctis the cost related to transshipment, Lvis the income of chartered out vessels, F is the total penalty costs of not transporting cargo. All numbers are presented with a 2 digit precision. Note that due to rounding, Z might not always be exactly equal to the sum of all displayed variables.

The liner shipping network design problem incorporating maximal transit times with a soft time window and slow-steaming