CONTAINERSHIP ROUTING PROBLEM IN LINER SHIPPING WITH EMPHASIS ON SUSTAINABILITY
Master thesis, Msc, Supply Chain Management University of Groningen, Faculty of Economics and Business
June 23, 2014
Ting Xu
Student number: S2437023 e-mail: t.xu.2@student.rug.nl
Supervisor/ university:
E. Ursavas Co-assessor/ university:
X. Zhu
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Abstract
In this paper the concept of sustainability is applied to the containership routing problem to attract the attention of researchers in this field to the environmental issues. A mathematical model is developed to find out the optimal routes for a homogeneous containership fleet traveling between the depot and the surrounding small ports with the objective of minimizing the total CO2
emission. The concept of slow steaming is integrated into the ship routing problem by arranging the strategic speed that captures the economic advantage of slow steaming for the containership fleet. Based on the result of the real-life test instances, the proposed model formulation can help to generate optimal routes for liner companies in reasonable computational time when the number of ports involved does not exceed 70. In addition, it is suggested that increasing the size of the assigned ships can reduce more CO2 emission than raising the number of ships.
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Table of Contents
Table of Contents ... 2
1. Introduction ... 3
2. Literature Review ... 4
2.1 Sustainability in shipping industry ... 4
2.2 Ship Routing Model ... 5
3. Problem Description ... 7
4. Model ... 12
4.1 Objective function ... 12
4.2 Constraints ... 13
5. Numerical experiments ... 15
5.1 Test instance overview ... 15
5.2 Parameter description ... 15
5.3 Results and analysis ... 16
6. Conclusions ... 20
References ... 21
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1. Introduction
With the increased pace of global warming and climate change during the past decade, a greater emphasis has been placed on the environmental management issue (Lirn, Lin & Shang, 2014). The international shipping industry is no exception to this green trend, with a similar emphasis being placed on sustainability (Lirn et al., 2014), which is mainly about the reduction of CO2 emission in this context. In response to this increasing concern over environmental issues in shipping industry, the International Maritime Organization (IMO) set a goal to gradually reduce the amount of CO2 emission from existing ships by 20% to 50% by 2050 (IMO, 2009).
Accordingly, how to effectively reduce CO2 emission during shipping operation has become an urgent task for shipping companies, especially the companies involved in the liner shipping. This is because most of the carriage of goods in this transport mode is accomplished by containerships, the largest source of CO2 emission (Cheng, Lai, Lun, & Wong, 2013). As suggested by many researchers, the application of slow steaming can significantly reduce the amount of CO2 emission (Lu, Liu & Wooldridge, 2014). Also, IMO is considering reducing CO2 emission during ship operation through the managerial measures in the operational level such as speed control, namely, slow steaming (Du, Chen, Quan, Long & Fung, 2011). Therefore, addressing sustainability in the ship routing field by integrating the concept of slow steaming will make a large contribution to alleviate the problem faced by shipping lines since the route and speed of the ship together exert a significant influence on the amount of CO2 emission during ship voyage.
During recent decades, many researchers have been focusing on mathematical models for containership routing problems (Christiansen, Fagerholt & Ronen, 2004; Kjeldsen, 2011; Meng, Wang, Andersson & Thun, 2013). In their studies, different factors such as stowage planning and ship capacity as well as different objectives such as cost minimization and profit maximization were addressed.
Despite the large number of researches done in this field, very few studies took into consideration green factors such as slow steaming. Slow steaming is quite useful in reducing the amount of CO2 emission. According to Woo and Moon (2013), when the voyage speed is decelerated by 20%, the reduction in CO2 emission can exceed 20%. In fact, slow steaming has reduced CO2 emissions by around 11% between 2009 and 2011 (Cariou, 2011), close to the target of a 15% reduction by 2018 that is proposed by IMO. Based on these studies, it is obvious that slow steaming can largely reduce the emission of CO2 during ship operation. However, few studies in containership routing problems take this factor into account. As mentioned before, the route and speed of the ship together has a huge impact on the emission of CO2, therefore it is extremely necessary to further study containership routing models involving slow steaming so as to help the shipping lines determine the routing of containerships considering sustainability.
In this paper, we investigate the ship routing problem in liner shipping for a fleet of homogeneous containerships performing short-distance pick-up and deliveries in a feeder network in an attempt to develop the routes for containerships with the objective of minimizing the total CO2 emission during the voyage under strategic voyage speed. The problem is first analyzed and then a quantitative model is developed, which can be directly used in shipping companies. On the one hand, the study aims to contribute to the existing literature by developing a model for
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containership routing problems including the concept of slow steaming; on the other hand, a decision support system is thus developed to help shipping lines optimize the routing for containerships. In addition, the research attempts to attract the attention of researchers in the field of ship routing problems to the environmental issues and create awareness that the emission should also be considered in the routing problems.
The remainder of the paper is structured as follows: in section 2 literature dealing with ship routing problems and green shipping management is introduced and explained; in section 3 problem description is given; section 4 proposes the related mathematical model and corresponding constraints; in section 5 a computational study is displayed; in section 6 conclusions are made for the paper and suggestions are given for future research.
2. Literature Review
2.1 Sustainability in shipping industry
During recent decades, there is an increasing concern over green shipping management. It has been proven that green shipping management is positively related to the financial and service performance in shipping companies (Cheng, et al., 2013). Lirn, Lin and Shang (2014) proposed in their research that there are three key factors of green shipping management capability, namely, greener policy, greener ships, and greener suppliers. Greener policy is concerned with implementing an environmental policy to create a vision or culture of environmental protection (Lai et al. 2011). The greener ship addresses the prevention of air pollution (BSR 2010; The A.P.
Moller-Maersk Group, 2011), which is mainly caused by the emission of CO2 in the shipping industry. The third factor, greener suppliers, includes many aspects. The focal company may assist suppliers in building up their own environmental systems, require suppliers to provide certification of testing for green product conformance, pressure suppliers to take environmental action, provide suppliers with design specifications that include environmental requirements for purchased items, as well as choosing suppliers according to environmental criteria (Lin et al., 2014).
The major concern of this paper is to integrate the factor of greener ships, namely, CO2
emission reduction, into the containership routing problems. Reduction in the amount of CO2
emission during ship operation calls for a large number of measures. Increasing the operational efficiency of the fleet in terms of CO2 emission is one of them. Among all the measures that improve the operational efficiency of the fleet, slowing down the sailing speed is the one that brings significant improvement and does not need extra modification to the ships (Faber, Freund, Koepke & Nelissen, 2010). In shipping industry, the CO2 emission is proportional to the fuel consumption, which is a cubic function of the sailing speed of the ship (Ronen, 1982). Therefore, the sailing speed of vessels is a crucial decision with regard to the CO2 emission.
The concept of slow steaming was introduced and adopted by many shipping companies since 2008 to slow down the voyage speed so that the CO2 emission can be lowered (Qi & Song, 2012). Slow steaming means that the average speed of a voyage is set at the level less than the commercial speed (Kjeldsen, 2011) in an attempt to reduce the fuel consumption and then the CO2
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emission. Slow steaming can also reduce the operating cost to a large extent under conditions of high fuel prices and low freight rates (Woo & Moon, 2013), the very situation in the current liner shipping industry. Accordingly, integrating the concept of slow streaming into the daily operation of ships is a wise option at the moment for the shipping lines.
Psaraftis and Kontovas (2009) conducted a study in which a fleet of homogeneous vessels load from port A, traverse to and discharge at port B and then travel back to port A, with a specific speed. The result of the study shows that the total emission from ships would be always reduced by lowering the sailing speed.
Faber et al. (2010) expected that the emissions of containerships, tankers and bulkers could be reduced by 30% at maximum in the following years by slowing down the sailing speed, compared to the situation in 2007. They also mentioned that though currently vessels are slowing down the speed there is additional potential to sail even slower.
Lindstad, Asbjørnslett and Strømman (2011) presented a study that examine the impact of speed reduction on the cost and emission of the world fleet and suggested that there is a large room for the CO2 emission reduction by slow steaming. Moreover, they recommended that speed limits can by employed as the best means of slowing down the sailing speeds to the desired levels for all classes of vessels.
Carious (2011) investigated the effect of slow steaming on theemission of containerships and concluded that slow steaming has reduced emission by around 11% during 2009 and 2010.
Furthermore, the paper proposed that slow steaming can be cost beneficial when bunker price is no lower than $350-$400 per ton.
According to the study carried out by Woo and Moon (2013), the strategic range of sailing speed that makes full use of the advantages of slow steaming is from 14 to 22 knots while the optimal voyage speed that both maximizes the reduction of CO2 emission during ship operation and minimizes the operating cost is at 17.4 knots.
Du et al. (2011) proposed an approach by regression analysis that easily handles the general nonlinear function between fuel consumption rate and vessel sailing speed. In this paper, the approach proposed by them is applied to calculate the fuel consumption.
Since slow steaming is related to the sailing speed of the ship, which in turn affects the fuel consumption, in this paper the concept of slow steaming is operationalized into the voyage speed of ships. Each ship is arranged to operate under the strategic sailing speed that captures the political and economic advantages of slow steaming according to literature. Under this circumstance, the CO2 emission is to be lowered.
2.2 Ship Routing Model
Much research has been done in the area of ship routing. When it comes to containership routing and scheduling, however, less attention has been assigned to containership routing problems despite the increase in container transportation (Sigurt, Ulstein, Nygreen & Ryan, 2005).
Rana and Vickson (1991) formulated a mixed integer – non-linear model for optimally
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routing a fleet of containerships along a trade route. In this model, the shipping lines have the option to deny part of the demand at the port if it is not profitable to transport such cargo or more cost-effective cargo is available in another port. Each vessel can only be assigned to one route that has to be travelled an integer number of times in the planning period. They then used a decomposition method for the model as well as for the network in order to solve the problem, which lays a foundation for the solutions of such models.
Cho and Perakis (1996) proposed a linear programming (LP) model with regard to fleet size and design of optimal liner routes for a container shipping company. They generated a set of candidate routes and applied linear programming for selecting the optimal set of routes for containerships.
Sambracos, Paravantis, Tarantilis and Kiranoudis (2004) developed a model to design ship routes in a feeder network in the Aegean Sea, from one depot (Piraeus) to 12 other ports (islands).
The model assumed an identical ship fleet with a known cost structure, ship capacity and sailing speed for each vessel. They applied a list-based threshold acceptance meta-heuristic method.
Karlaftis, Kepaptsoglou and Sambracos (2009) later extended this model to establish the optimal routes for containerships performing simultaneous deliveries and pick-ups between a hub and several spoke ports, which is the basic model for the paper. The model was formulated according to the vehicle routing problem with simultaneous pick-ups and deliveries. In this research, the problem was solved using a hybrid genetic algorithm.
Reinhardt, Kallehauge and Nørrelund (2007) suggested that in order to better model the real world of liner shipping, loops should be allowed in the generated routes for each ship. However, when the model is formulated linear, only simple routes can be generated. Therefore, the model only permits one route for each ship and then this route is traversed several times by a specific ship within the planning period. The model proposed in their study allows multiple commodities since various types of containers are treated as distinct commodities and a vessel has different capacities for different types of containers. Since the model made in this paper is also linear, this research gives some hints on the model formulation.
Norstad, Fagerholt and Laporte (2011) developed a model including speed optimization, based on the traditional tramp ship routing and scheduling problem, and proposed a heuristic solution method for the model. Since the major goal of this research is to include the concept of slow steaming in the containership routing problems, this research provides some insight into this topic.
Christiansen et.al (2004) provided a review of the status of ship routing and scheduling problems at that time. They focused on routing problem at different levels, namely, strategic level, tactical level and operational level. They separately discussed literatures handling different transport modes: industrial, tramp and liner shipping. Kjeldsen (2011) classified ship routing and scheduling problems in liner shipping according to the major characteristics of the problems dealt with in the studies and briefly introduced studies under each category. Meng et.al (2013) reviewed the studies in the containership routing problems over the past 30 years. They classified and summarized these studies based on model formulations, assumptions and solution algorithms.
These review papers provide an overview of the current status of the containership routing problems and the gaps in the literature.
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In this paper, the model is elaborated upon containership routing problems in feeder network with the aim of minimizing CO2 emission during ship voyage.
3. Problem Description
Ship routing is defined as the assignment of the sequences of ports to be visited by a ship (Ronen, 1993). The problem is to generate a feasible set of routes while achieving the objective. A solution defines the route for every ship in the fleet where the route is a sequence of ports at which the ship has to call to provide the specific services. On the one hand, the routing problem is part of the network design; on the other hand it is also a separate problem with a shorter time horizon of usually six to eighteen months. Generally, in shipping industry there are three modes of transportation: industrial, tramp, liner shipping. In industrial shipping, the cargo owners or shippers who own the ship try to ship all their cargoes at minimal cost (Christiansen et al., 2004).
Tramp shipping companies transport a set of contracted bulk cargoes between specified ports within a specific time period while also attempting to generate additional revenue from carrying optional cargoes (Norstad, Fagerholt & Laporte, 2011). Liners operate according to a published itinerary and schedule similar to a bus line (Christiansen et al., 2004) and most of the cargoes in liner shipping are carried in containers. This paper focuses on the ship routing problem for containerships in liner shipping, which is quite different from that in other two types of shipping operation in the following aspects:
1) Tramps usually wait until they are full before departure while liners depart based on planned schedule regardless of whether the ship is full (Agarwal & Ergun, 2008);
2) Tramp shipping mainly deals with bulk cargo while the cargo in liner shipping is containerized.
In addition, the ship routing problem for liner shipping includes decisions at different planning levels (Agarwal & Ergun, 2008) as shown below. The problem scope of the paper is within tactical planning- designing the service network.
1) Strategic planning, i.e., acquiring resources, determining fleet size and mix
2) Tactical planning, i.e., designing the service network (frequency of routes, port selection, port rotation), assigning ships to routes
3) Operational planning, i.e., choosing which cargo to accept or reject for routing, routing the selected cargo.
The aim of liner container shipping network design is to determine which ports the ships should visit and in what order (Meng et al., 2013). Researches done on this topic can be classified into four categories as shown in figure 1. The first category is a feeder network, which consists of a hub port and many feeder ports as shown in figure 1(a). Containers either originate from or are destined for the hub port, and transshipment is excluded within the feeder network (Meng et al., 2013). The second category attempts to design one or a few shipping routes without container transshipment operations (Meng et al., 2013), as shown in figure 1(b). The third group of studies aims to design a hub and spoke liner shipping network similar to those used by airlines and in telecommunication systems (Meng et al., 2013), as shown in figure 1(c). The fourth line of research seeks to develop the general liner shipping network which usually includes more ports in
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the network and allows for container transshipment operations (Meng et al., 2013), as shown in figure 1(d). This paper focuses on the first category of the liner container shipping network design.
Figure 1 - Liner Shipping Network Design Categories (Meng, et al., 2013)
In liner shipping, feeder network is an important segment. In order to achieve economy of scale, cargo in small ports will be transported to major ports so that the large containerships departing from major ports can carry together cargo from different origins with the same destination. Under this circumstance, these major ports serve as transit ports, delivering cargo from other major ports to small ports while also carrying cargo from these small ports to other major ports. Since these pick-ups and deliveries between transit port and small ports occur quite often, it deserves further investigation and a quicker and better ship routing plan can be significantly helpful. In this paper, we concentrate on the transport between one major port and the surrounding small ports. More specifically, the problem in this paper considers deliveries and pick-ups between transit port, namely the depot, and its surrounding ports. This means that transshipment between surrounding ports is not allowed. In other words, all containers have to either start from, or end up, at the transit port. Therefore, the following assumptions are made so as to formulate the model (Zachariadis, Tarantilis & Kiranoudis, 2009; Liu, Xie, Augusto &
9 Rodriguez, 2013):
1) every vessel begins and terminates at the transit port/depot;
2) each port is visited exactly once, by exactly one vessel, which means all the demands at the port should be delivered by exactly one vessel;
3) every route transports the delivery quantities from the transit port to the visited ports;
4) every route transports the pick-up quantities from the visited ports back to the transit port.
In fact, the containership routing problem in feeder service addressed in this paper can be transformed into the vehicle routing problem with simultaneous pick-ups and deliveries (VRPSPD) (Karlaftis et al., 2009). In order to make clearer the problem dealt with in this article, the major characteristics of the problem are described in more details in the following paragraphs.
1) The number of starting points
The starting point refers to the actual starting position at sea or in port for vessels at the beginning of the route (Fagerholt, Johnsen & Lindstad, 2009). There are two choices for this characteristic, that is, one or multiple. In this study, the number of starting points imposed on each route is one. Each route starts from the transit port/depot;
2) The number of ending points
The ending point refers to the actual ending position at sea or in port for vessels at the end of the route. Also, there are two choices for this characteristic, namely, one or multiple. In this study, the number of ending points imposed on each route is one. Each route ends at the transit port/depot;
3) Type of operation
There are various types of operation such as delivery, pick-up and the combination of these two. For the combination of the two, there are two alternatives- simultaneous deliveries and pick-ups and separated deliveries and pick-up. In this paper, the former one is applicable with regard to the characteristic of type of operation, which means that loading and discharging take place in each port of call, closer to the reality;
4) Nature of demand
The choices of nature of demand include deterministic demand, stochastic demand and demand dependent on service. In this paper, the demand is deterministic, which means the demand is an input and known in advance during the assignment of the ports of call for each vessel;
5) Number of vessels
This characteristic is related to the fleet size in each case. In order to simplify the problem and the model, many articles only deal with one vessel, which is not always the case in real life. Therefore, to better model the reality, in this study the number of vessels can be multiple and depends on the real data from the benchmark;
10 6) Fleet composition
There are two choices regarding to this characteristic, namely, homogeneous or heterogeneous fleet. Homogeneous fleet means that all the vessels in the fleet are identical in all the important aspects such as ship type, ship capacity and sailing speed while heterogeneous fleet implies that all the vessels in the fleet differ from each other in some or all the important aspects. In this study, a homogeneous fleet composition is applicable, which implies that all the vessels belong to the same ship type and have the same speed and capacity;
7) Sailing speed
This characteristic is concerned with whether the sailing speed of the vessels is a decision variable. In this study, the sailing speed of the vessels is an input and determined in advance.
The strategic sailing speed that best secures the economic and political advantages of slow steaming is applied to the homogeneous fleet so as to address sustainability in the routing problem;
8) Demand splitting
Demand splitting means that the demand of the port is satisfied by more than one vessel. In this study demand splitting is not allowed. This is because the demands at the surrounding ports are relatively small and thus there is no need to split demand;
9) Partial satisfaction of demand
This characteristic is concerned with whether it is allowed to only satisfy part of the demand at the port and deny other part of the demand. The reason why shipping lines deny some parts of the demand is that sometimes the cost to transport certain type of cargo may exceed the income to do so. There are two choices for this characteristic, namely, allowed or not allowed.
In this study, it is not allowed to deny any demand at the port. This is because shipping lines cannot afford to deny any demand at the currently sluggish shipping market. They have to accept as many demands as possible to maintain market share;
10) Capacity types
The capacity for containership is often measured in TEU (Twenty-foot Equivalent Unit) or FEU (forty-foot equivalent unit) while the capacity for bulker is generally measured in tonnage. Since the model in this paper deals with containerships, the capacity type is TEU or FEU;
11) Cargo transshipment
This characteristic is concerned with whether it is allowed to transship cargo between ports.
As mentioned above, the transshipment between small surrounding ports is not allowed in this model;
12) Number of routes per ship
The number of routes assigned to each ship can be one or multiple. If the ship can only be allocated to one route, then it will remain on the route performing multiple voyages during
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the planning period (Ronen, 1983). If the ship can be allocated to more than one route, then it will become hard for the shipping lines to make a fixed schedule. Therefore, in the containerized liner industry the former one is always the case in order to make a clear schedule for the cargo owners. As in this paper the model is applicable to the liner industry, the number of routes per ship should be exactly one;
13) Planning horizon
This characteristic is concerned with whether the vessel needs to complete the route within a predetermined planning horizon. In this study, an allowable travelling time is assigned to all the vessels in the fleet, which means the vessel has to finish the voyage within a certain time period;
14) Ships required to be empty
According to the definition of liner shipping by Ronen (1983), ships may not be empty.
However, the majority of the available articles addressing liner shipping include the requirement that at certain points the ships should be empty in order to simplify the problem.
Without this condition, the calculation of ship capacity becomes quite complicated and lengthens the computational time when searching for solutions since if the ships are not allowed to be empty in one or more ports the capacity calculation needs to take into consideration the amount of cargo delivered to and picked up from previous voyage. In this study, the vessels need to be empty at the starting point/depot since all the cargo should be discharged at the depot when the ships finally return to the depot;
15) Port precedence requirement
The port precedence requirement implies that due to some specific reasons one port should be visited before the other. In this study, there is no explicit port precedence requirement between the ports except that the first and last ports visited in each route should be the transit port/depot;
16) Requirement for compatibility between ships and ports
This characteristic is concerned with whether there are constraints with regard to the compatibility between ships and ports. For instance, a certain ship may not be allowed to call at a specific port due to its large size. In this study, there is no requirement related to the compatibility between ships and ports since it is assumed that they are compatible;
17) Objective
There are multiple objectives such as profit maximization, cost minimization, distance minimization in the ship routing problem. Though in the liner shipping the objective mainly focuses on monetary issues, the purpose of this paper is to integrate the green factor into the ship routing problem. Therefore, the objective is to minimize the environmental impact, namely, the CO2 emission;
The major characteristics of the problem dealt with in this paper are described clearly and thoroughly in the above paragraphs. According to these characteristics, the following model meeting all the conditions is presented.
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4. Model
Hereafter a model assigning multiple containerships to a set of ports is presented, with an objective to minimize the total CO2 emission during the voyage. The constraints are related to voyage time duration, speed limitation, ship capacity, sub-tour elimination.
Decision variables:
𝑥𝑖𝑗𝑘 = 1 𝑖𝑓 𝑝𝑜𝑟𝑡𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡𝑒 𝑟𝑜𝑢𝑡𝑒 𝑜𝑓 𝑡𝑒 𝑠𝑖𝑝 𝑘; 0 𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒
Parameters:
P − the set of ports (1,2 … 𝑛)
i, j, t − nodes belonging to P (𝑖, 𝑗 = 0 𝑓𝑜𝑟 𝑡𝑒 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 𝑝𝑜𝑟𝑡/𝑑𝑒𝑝𝑜𝑡) V − the set of ships indexed by k (1,2 … 𝑚)
EF − emission factor for 𝐶𝑂2 (𝑔/𝑘𝑔 − 𝑓𝑢𝑒𝑙)
F𝑖𝑗𝑘− the amount of fuel consumption for ship k traveling from port i to port j a𝑖𝑗− the sailing distance from port i to port j
(the distance from port i to port j and that from port j to port i are the same) T𝑘− maximum route travel time for ship k
S𝑖𝑘− the necessary time to serve node i by ship k
𝑣𝑘− strategic sailing speed for ship k secureing the economic advantages of slow steaming 𝑄𝑘− 𝑡𝑒 𝑐𝑎𝑟𝑔𝑜 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑓𝑜𝑟 𝑠𝑖𝑝 𝑘
4.1 Objective function
In literature, there is a general method that can approximately calculate the CO2 emission, which is shown in the following equation:
Vessel 𝐶𝑂2 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛(𝑔) = 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟(𝑔/𝑘𝑔 − 𝑓𝑢𝑒𝑙) ∗ 𝑓𝑢𝑒𝑙 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛(𝑘𝑔) Based on this formula, the objective function of the model in this paper is as follows:
Minimize total 𝐶𝑂2 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = 𝐸𝐹 ∙ ∑ ∑ ∑ 𝐹𝑖𝑗𝑘∙ 𝑥𝑖𝑗𝑘
𝑗∈𝑃 𝑖∈𝑃 𝑘∈𝑉
𝐹𝑖𝑗𝑘=𝛽0∙ 𝑎𝑖𝑗
𝑣𝑘 + 𝛽1∙ 𝑎𝑖𝑗∙ 𝑣𝑘𝜇−1 ∀𝑖, 𝑗 ∈ 𝑃, 𝑘 ∈ 𝑉
β0 and β1 in the above formula are the regression coefficients. According to the suggestions of MAN Diesel & Turbo (2004), the value of μ should be determined based on the following rules:
μ= 3.5 for feeder containerships and μ = 4 for medium-sized containerships.
𝐸𝐹 = 3110 (𝑔/𝑘𝑔 − 𝑓𝑢𝑒𝑙)
The value of the emission factor EF is based on the standard adopted by COSCO (2009). The fuel consumption function is based on the study from Du et.al (2011).
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4.2 Constraints
In the following section, the constraints defining the problem are presented and briefly explained.
4.2.1 Flow constraints
∑ ∑ 𝑥𝑖𝑗𝑘
𝑘∈𝑉
= 1 ∀𝑗 ∈ 𝑃/*0+ (1)
𝑖∈𝑃
∑ ∑ 𝑥𝑖𝑗𝑘
𝑘∈𝑉
= 1 ∀𝑖 ∈ 𝑃
𝑗∈𝑃
/*0+ (2)
∑ 𝑥𝑖𝑡𝑘
𝑖∈𝑃
= ∑ 𝑥𝑡𝑗𝑘
𝑗∈𝑃
∀𝑡 ∈ 𝑃, 𝑘 ∈ 𝑉 (3)
∑ 𝑥0𝑗𝑘
𝑗∈𝑃/*0+
= 1 ∀𝑘 ∈ 𝑉 (4)
∑ 𝑥𝑖0𝑘
𝑖∈𝑃/*0+
= 1 ∀𝑘 ∈ 𝑉 (5)
Constraints (1) and (2) ensure that every demand node is served by exactly one ship.
Constraint (3) guarantees that a ship exits the demand node it enters.
Constraints (4) and (5) ensure that each ship starts from and ends at the depot.
4.2.2 Time constraints
∑ ∑ (𝑠𝑖𝑘+𝑎𝑖𝑗 𝑣𝑘)
𝑗∈𝑃
∙ 𝑥𝑖𝑗𝑘 ≤ 𝑇𝑘 ∀𝑘 ∈ 𝑉
𝑖∈𝑃
(6)
Constraint (6) handles maximum allowable voyage time for each ship.
4.2.3 Capacity constraints
∑ ∑(𝑑𝑗)
𝑗∈𝑃
∙ 𝑥𝑖𝑗𝑘≤ 𝑄𝑘 ∀𝑘 ∈ 𝑉
𝑖∈𝑃
(7)
Constraint (7) ensures that the ship capacity constraint is not violated.
Since the model allows simultaneous deliveries and pick-ups between the transit port and its surrounding ports, the following constraints are included to replace the original capacity constraint (7) formulated above. These constraints are elaborated on those suggested by Dethloff (2001).
𝑙𝑘= ∑ ∑ 𝑑𝑗∙ 𝑥𝑖𝑗𝑘
𝑗∈𝑃 𝑖∈𝑃
∀𝑘 ∈ 𝑉 (8)
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𝑙𝑗≥ 𝑙𝑘− 𝑑𝑗+ 𝑝𝑗− 𝑀 ∙ (1 − 𝑥0𝑗𝑘) ∀𝑘 ∈ 𝑉 , 𝑗 ∈ 𝑃/*0+ (9)
𝑙𝑗≥ 𝑙𝑖− 𝑑𝑗+ 𝑝𝑗− 𝑀 ∙ (1 − ∑ 𝑥𝑖𝑗𝑘
𝑘∈𝑉
) ∀𝑖, 𝑗 ∈ 𝑃/*0+ , 𝑗 ≠ 𝑖 (10)
𝑙𝑘≤ 𝑄𝑘 ∀𝑘 ∈ 𝑉 (11) 𝑙𝑗≤ 𝑄𝑘 ∀𝑘 ∈ 𝑉 , 𝑗 ∈ 𝑃/*0+ (12)
𝑙𝑘− the load for ship k when leaving the transit port 𝑙𝑗− the load of ship after having served port j 𝑑𝑗− the demand for node j
𝑝𝑗− the pick − up load in port j M − the arbitrarily large number
Constraint (8) determines the initial ship load.
Constraints (9) and (10) correspond to ship loads for the first and successive ports.
Constraints (11) and (12) guarantee that the ship capacity is not exceeded.
4.2.4 Sub-tour elimination constraints
Although the flow constrains are used to determine the starting and ending point of the vessel, they do not necessarily eliminate sub-tours. Figure 2 presents an example of sub-tour. In order to prohibit such circumstances, sub-tour elimination constraints (SECs) are introduced, which are based on the formulation by Kara and Bektas (2006). In this formulation, bounding constraints are first introduced that further shortens the computational time. In this way, Kara and Bektas improved the sequential formulation of SECs developed by Miller, Tucker and Zemlim (1960), which is known as the MTZ formulation of traveling salesman problem (TSP). The major advantage of this formulation is that it introduces small size of SECs, which in turn largely reduces the computational time.
Figure 2 - An Example of Sub-tour
𝑢𝑖− (𝑛 − 𝑚 − 2) ∙ ∑ 𝑥0𝑗𝑘
𝑘𝜖𝑉
− ∑ 𝑥𝑖0𝑘
𝑘𝜖𝑉
≤ 𝑛 − 𝑚 − 1 ∀𝑖, 𝑗 ⊆ 𝑃/*0+ (13)
15 𝑢𝑖− ∑ 𝑥0𝑗𝑘
𝑘𝜖𝑉
≥ 2 ∀𝑖 ⊆ 𝑃/*0+ (14)
𝑢𝑖− 𝑢𝑗+ (𝑛 − 𝑚) ∙ ∑ 𝑥𝑖𝑗𝑘 𝑘∈𝑉
+ (𝑛 − 𝑚 − 2) ∙ ∑ 𝑥𝑗𝑖𝑘 𝑘∈𝑉
≤ 𝑛 − 𝑚 − 1 ∀𝑖, 𝑗 ⊆ 𝑃/*0+, 𝑖 ≠ 𝑗 (15)
𝑢𝑖− variable used to eliminate subtours, representing the position of node i ∈ P in the route n − total number of the ports including the transit port/depot
m − total number of the vessels
Constraints (13) and (14) are the bounding constraints that determine the upper and lower bound of the variable 𝑢𝑖 separately.
Constraint (15) serves as the SEC ensuring that there are no sub-tours in the route for every vessel.
5. Numerical experiments
5.1 Test instance overview
In order to test the proposed model formulation, several experiments have been performed to measure the effect and computational time using the real-life instances from Maersk introduced by Brouer, Alvarez, Plum, Pisinger and Sigurd in 2013. Since this paper concentrates on the feeder service between one major port and its small surrounding ports, the multi-hub instance summarized in the benchmark is separated into 4 independent instances to fit the problem in this paper and the two single-hub instances of Baltic Sea and West Africa are directly used as input.
Table 1 summarizes the test instances applied in this paper. The optimization models concerned are formulated and solved in GAMS environment. All the numerical experiments are performed on a personal computer with Intel Core i5 1.70 GHz CPU and 8 GB RAM.
Table 1 - Test instance description
5.2 Parameter description
Some basic parameters such as EF (emission factor) and the regression coefficients 𝛽0 and 𝛽1 for the fuel consumption function and the value of 𝜇 are chosen according to the regression for a 5700 TEU (2850 FEU) containership arriving at Tianjin Port (Du et.al, 2011). The strategic sailing speed that captures the economic advantages of slow steaming is 17.4 knots as proven by
Instance 1 2 3 4 5 6
Area Baltic sea West Africa Mediterranean Mediterranean Mediterranean Mediterranean Transit port/Depot Bremenrhaven,
Germany Algeciras, Spain Gioia Tauro, Italy Malaga,Spain Tangier, Morocco Said, Egypt
# of ports 12 20 11 19 21 29
# of demands 22 38 20 36 40 56
# of ships 3 3 3 3 3 5
ship capacity (FEU) 2850 2850 2850 2850 2850 2850
16
Woo and Moon (2013) and is applied in the numerical experiments. The below table summarizes the values for these parameters. Other parameters such as the demand and pick-up demand at each port and the distance between two ports are determined according to the Maersk benchmark mentioned above.
Table 2 - Basic parameters
5.3 Results and analysis
5.3.1 Computational time and size limit
The model formulation with the improved MTZ sub-tour elimination formulation can normally solve the instances within 2 minutes. Due to the large number of variables in the MTZ formulation, the largest size of the problem that can be solved with this formulation is up to 70 ports. Based on the results of these six instances, we find out that with the number of ports increases the length of computational time increases. The results show that the proposed model formulation can be simply used to solve instances of moderate size to optimality in reasonable computational time. When the number of ports exceeds 70 cities, it takes GAMS considerable amount of time to find the optimal solution. Table 3 summarizes the response time of all the six instances. The average computational time is 22 seconds.
Karlaftis et.al (2009) proposed a model to deal with a similar problem handled in this paper, but with different objective function and sub-tour elimination constraints. However, only small-sized problems can be solved directly by this model in reasonable time and additional solution algorithm is needed to solve even medium-sized instances (more than 15 ports). Due to this fact, only test instance 1 and 3 which are of small size are tested to search for the differences in results between the two models. These two small instances can be solved in about 40 seconds by their model while with the model in this paper they can be solved in only 10 seconds. The differences in computational time between the two models are shown in table 4.
Table 3 - Computational time
Strategic sailing speed EF(emission factor) for CO2 β0 β1 μ
Value 17.4 knots 3110 g/kg-fuel 699 0.004238 4
Instance 1 2 3 4 5 6
Area Baltic sea West Africa Mediterranean Mediterranean Mediterranean Mediterranean
# of ports 12 20 11 19 21 29
Computational time
(seconds) 3 17 3 6 15 87
17
Table 4 - Differences in computational time between two models
5.3.2 Routes and CO2 emission
To test the effect of slow steaming on the total CO2 emission, the six instances are also tested with the normal sailing speed of 22 knots. The total emission reduction for each instance is shown in table 5. The average reduction rate is 99.46%. According to the results, it is obvious that the total CO2 emission during ship voyage can be largely reduced with the concept of slow steaming and that the proposed model can be useful for the shipping lines who attempt to contribute to the environmental issues.
Table 5 - Total emission reduction
In order to better compare the results of the instances, table 6 illustrates the routes for each ship in details for instance 1, 2 and 5 and figure 3, 4 and 5 present these optimal routes on maps.
In each of the instances, three ships of same capacity are assigned to complete the transport between the transit port and the small surrounding ports and thus there are three routes for each instance. Based on the demand information from the benchmark, there are more demands between the depot and the small surrounding ports for instance 1 and 2 and the demand is quite small for instance 5. Judging from table 4, it seems obvious that among the three routes in each instance there is one route that includes relatively more ports, compared with the other two routes. Since the demand between ports in instance 5 is relatively small, the route with more ports in this instance includes more demand points compared with other two instances. Therefore, we suggest that it emits less CO2 to incorporate more ports in one route. Since there is capacity limit for ships, we propose that larger ships can be assigned to area with more demands so as to reduce the number of ships used, which in turn lowers the CO2 emission during ship voyage.
Instance 1 Instance 3 The model of Karlaftis,
Kepaptsoglou and Sambracos 39 36
Model in this paper 3 3
Computational time (seconds)
Instance 1 2 3 4 5 6
Area Baltic sea West Africa Mediterranean Mediterranean Mediterranean Mediterranean Transit port/Depot Bremenrhaven,
Germany
Algeciras,
Spain Gioia Tauro, Italy Malaga,Spain Tangier, Morocco Said, Egypt CO2 emission under speed of
17.4 knots (ton) 9178259 48900945 20804051 10806951 11238868 9160262
CO2 emission under speed of
22 knots (ton) 14371351578 7656932442 3257506357 1692156201 1759786090 1434317245
Reduction rate 99.94% 99.36% 99.36% 99.36% 99.36% 99.36%
18
Table 6 - Results for instance 1, 2 and 5
Table 6 explains the route for each ship clearly and the following figures depict these routes.
In the figures, the routes for ship 1 are depicted in black lines, routes for ship 2 in red lines and routes for ship 3 in orange lines. The arrow represents the proceeding direction of the ship and the blue points stand for the ports to visit.
Figure 3 - Optimal ship routes for instance 1
Instance area # of ships Transit port/Depot Route for ship 1 Route for ship 2 Route for ship 3
1 Baltic sea 3 Bremenrhaven,
Germany
Kristiansand-Bergen-Rauma- Kotka-Aarhus
Gothenburg-St Petersburg- Kaliningrad-Gdynia-
Alesund
Stavanger
2 West Africa 3 Algeciras, Spain Freetown-Apapa-Monrovia- Takoradi
Conakry- Port Gentil -Douala-Pointe Noire-
Matadi-Luanda
Dakar-Bissau- Libreville-Abidjan-
Cotonou-Lome- Djibouti-Boma-
Lobito
5 Mediterranean 3 Tangier, Morocco
Agadir-Trieste-Thessaloniki- Poti-Varna-Odessa-Mersin- Izmir-Ambarli-Tunis-Leixoes-
Haifa-Vigo
Valencia-Barcelona
Casablanca- Beirut-Genoa- Ashdod-Piraeus
19
Figure 4 - Optimal ship routes for instance 2
Figure 5 - Optimal ship routes for instance 5
20
6. Conclusions
In this paper, with focus on sustainability a mathematical model is formulated and solved, generating routes for containerships in a feeder network. The proposed model formulation provides optimal route for each vessel, subject to several real-life constraints, at reasonable computational time.
The contribution of this paper is two-fold: firstly, from theoretical aspect, it extends the mathematical model in literatures to include the concept of slow steaming in the containership routing problem; secondly, the practical contribution is that the proposed model helps solve the routing problem in reasonable time, which can serve as a quick decision support system for shipping lines operating in the feeder service.
In addition, since the model formulated in this paper takes into consideration the concept of slow steaming, it can reduce the CO2 emission from containerships during operation to some extent, meeting the urgent needs of shipping lines to contribute to the environmental causes.
Furthermore, the objective of the model is to minimize the total CO2 emission during ship voyage, which is quite novel in the routing problems. Generally, existing researches in routing problems always focus on monetary issues when it comes to the objective of the model. This research aims to attract the attention of researchers to the environmental issues in the routing problems. Finally, based on the results of the instances, it is recommended to assign large ships to area with more demands to reduce the number of ships needed so that the CO2 emission can be further reduced.
This is because increasing the size of the ships can better reduce CO2 emission than increasing the number of vessels, judging from the results of this study.
Also there are some limitations of the study presented in this paper. First of all, the model assumes that there is a homogeneous fleet and the fleet is always fixed, which exclude the possibility of fleet deployment. Secondly, the model assumes that the ports and the vessels are always compatible, which is sometimes not the case in reality. Thirdly, the model does not take into consideration the situation that there are different types of containers at port. Finally, due to the non-linear relationship between the sailing speed and the fuel consumption the sailing speed is treated as an input in this model. The future research can extend the model to include the sailing speed as a decision variable so as to make better use of the concept of slow steaming.
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