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Container routing in liner shipping
Transportation Research Part E 49 (2013) 1–7
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Container routing in liner shipping Shuaian Wang a,b, Qiang Meng b,⇑, Zhuo Sun c a b c
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore Centre for Maritime Studies, National University of Singapore, Singapore 118411, Singapore
a r t i c l e
i n f o
Article history: Received 23 November 2011 Received in revised form 7 March 2012 Accepted 5 April 2012
Keywords: Liner shipping Container routing Maritime cabotage Transit time
a b s t r a c t Container paths play an important role in liner shipping services with container transshipment operations. In the literature, link-based multi-commodity flow formulations are widely used for container routing. However, they have two deficiencies: the level of service in terms of the origin-to-destination transit time is not incorporated and maritime cabotage may be violated. To overcome these deficiencies, we first present an operational network representation of a liner shipping network. Based on the network, an integer linear programming model is formulated to obtain container paths with minimum cost. Finally, we add constraints to the integer linear programming model, excluding those paths already obtained, so as to find all the container paths. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Container transshipment is a common practice in liner shipping: approximately 27% of the world’s container throughput consists of transshipment containers (UNCTAD, 2008). Transshipment not only enables cargo consolidation for the deployment of large container ships but also expands the service scope of liner shipping companies as any port-to-port delivery service is feasible, even if there is no liner service route connecting the two ports. Nevertheless, container transshipment operations are difficult to manage in practice. From the origin port to the destination port, containers may be transported on different liner service routes and transshipped at many intermediate ports. To control the delivery of containers, liner shipping companies define a set of paths for each origin–destination (O–D) port pair. A container path contains all the information on how containers are transported (see, e.g., Wang and Meng, 2012a, 2012b). To keep this paper relatively self-contained, we first use the liner service network shown in Fig. 1 to illustrate the concept of the container path. The liner service network in Fig. 1 has three liner service routes (ship routes or SR for short), the set of which is denoted by R ¼ f1; 2; 3g, linking a set of ports P. The itinerary of each ship route r 2 R forms a loop. Let Nr represent the number of portcalls on a round trip. Define I r :¼ f1; 2; ; N r g and let pri 2 P represent the port of the ith portcall on ship route r 2 R, i 2 I r . We can arbitrarily define one portcall as the first portcall, e.g., in Fig. 1 Hong Kong is the first portcall on Ship Route 1 (SR1), Jakarta and Singapore are the second and third portcalls, respectively, and the number of portcalls on SR1 N1 = 3. It should be mentioned that, although Singapore is visited twice during the round trip of SR2, these two calls can easily be differentiated by employing the port calling sequence to refer to a portcall. With the above definitions and notation, a possible container path in Fig. 1 is defined as:
þXM SR2ð2; 4Þ þ CB-SR3ð1; 2Þ þ CN
ð1Þ
Containers on this path are loaded at Xiamen onto ships deployed on SR2, and delivered to Colombo. At Colombo, these containers are discharged and reloaded onto ships deployed on SR3, and subsequently delivered to Chennai where they ⇑ Corresponding author. Tel.: +65 6516 5494; fax: +65 6779 1635. E-mail address: [email protected] (Q. Meng). 1366-5545/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tre.2012.06.009
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S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
Fig. 1. An illustrative liner shipping service network. Source: Wang and Meng, 2012a.
are discharged. Therefore, these containers originate from Xiamen and end up at Chennai, having been transshipped at Colombo. The numbers 2 and 4 in the notation SR2(2, 4) stand for the port calling sequences of the load port and discharge port on this ship route, respectively. That is, containers are loaded at the 2nd portcall (XM) and discharged at the 4th portcall (CB) on SR2. Liner shipping companies adopt the concept of container paths to manage the transportation of containers in a service network with transshipment. In practice, a liner shipping company frequently alters its shipping network because of changes in container shipment demand. Such alterations may include the launch or cancellation of a feeder service, the addition or removal of a portcall on a ship route, or swapping two portcalls on a ship route. In addition, the company will reshuffle its services when new ships are delivered. Therefore, the set of container paths should be updated frequently. According to our discussion with a global liner shipping company, the container paths are actually designed manually by experienced managers. Evidently, manual planning is inefficient and may not find all possible container paths. Consequently, it is of practical significance to build a model that automates the generation of container paths for liner shipping companies. The literature on liner shipping operations can be categorized into four streams (see Ronen, 1983, 1993; Christiansen et al., 2004, 2007, for extensive reviews). The first stream focuses on a single liner ship route (e.g., Shintani et al., 2007; Ronen, 2011; Meng and Wang, 2011b; Qi and Song, 2012) or a few liner ship routes (e.g., Meng and Wang, 2011c) without container transshipment operations, or a butterfly ship route with transshipment at a single port that is visited twice in a round trip (e.g., Reinhardt and Pisinger, 2012). The second stream is devoted to feeder service networks, in which all the feeder ship routes connect to a hub (e.g., Fagerholt, 1999, 2004; Fagerholt and Lindstad, 2000; Sambracos et al., 2004; Karlaftis et al., 2009; Halvorsen-Weare and Fagerholt, 2011) and all containers are transported between feeder ports and the hub. The third stream concentrates on the liner hub-and-spoke (H&S) shipping network services (e.g., Imai et al., 2009; Gelareh et al., 2010; Gelareh and Pisinger, 2011), where containers to be moved between two feeders assigned to different hubs are usually transshipped at hub ports. The fourth stream looks at more general liner shipping networks where containers can be transshipped at any port. Container routing in such general networks is the most challenging. Most of the studies in this stream have adopted link-based multi-commodity flow formulations (e.g., Agarwal and Ergun, 2008; Alvarez, 2009; Brouer et al., 2011; Jepsen et al., 2011; Wang and Meng, 2012c) or segment-based formulations where a segment is a sequence of consecutive links (e.g., Bell et al., 2011; Meng and Wang, 2011a). A few works have applied path-based formulations without mentioning how to generate the container paths (e.g., Wang and Meng, 2012a, 2012b). The link-based formulation is widely used because it is much more compact than the path-based formulation. However, it is difficult (if not impossible) for link-based formulations to incorporate service-level considerations in terms of the O–D transit time, which is an important competition factor (Notteboom, 2006). Moreover, link-based formulations can hardly incorporate the constraint of maritime cabotage, which forbids a liner shipping company to load and discharge the same container at two ports within a country. For example, Maersk Line and MSC are not allowed to transport containers from Shanghai to Yantian because both ports are located in China. Moreover, they are not allowed to transship containers at Yantian that are heading to or destined for Shanghai, either. Only local shipping companies, such as COSCO and CSCL, are allowed to provide shipping services from Shanghai to Yantian. The objective of this study is to develop a mathematical model for generating container paths while considering operational constraints. We will first propose an operational network representation of a liner shipping network in the next section. Based on the operational network, in Section 3 an integer programming model is formulated that obtains all the container paths for each O–D pair. Section 4 reports the findings of a case study. We conclude with a summary in Section 5. 2. Liner shipping operational network To facilitate the generation of container paths, we reformulate a liner shipping network as an operational network. For example, Fig. 2 shows the corresponding operational network for the O–D pair Xiamen-Singapore in the network from
S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
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Fig. 2. A liner shipping operational network for the O–D Xiamen-Singapore.
Fig. 1. The operational network (N, A) for the O–D pair ðo; dÞ 2 P P, where N represents the node set and A is the arc set, is constructed as follows. First, add a virtual source node and a virtual sink node to N. Each node in N except for the virtual source and sink nodes corresponds to a portcall on a liner ship route. Hence, every node in N except for the source and sink nodes can be represented by (r, i), i.e., the ith portcall on ship route r 2 R. Define the arc set A :¼ Av [ At [ Asource [ Asink , comprising the voyage arcs, transshipment arcs, source arcs, and sink arcs, respectively. A voyage arc a e Av can be represented by its tail node (r, i). In other words, voyage arc (r, i) is the voyage from node (r, i) to node (r, i + 1). Note that voyage arc (r, Nr) is the voyage from node (r, Nr) to node (r, 1) because each ship route forms a loop. To simplify the notation, in the following we define node (r, i + 1), i = Nr, to be node (r, 1). A transshipment arc a e At can be represented by ((r, i), (s, j)) where pri = psj. In other words, containers are transshipped at port pri = psj from a ship on ship route r to a ship on ship route s. A source arc is an arc from the source node to a node (r, i) referring to the origin port o, i.e., pri = o. A sink arc is defined similarly. To summarize, we can use a e A to refer to an arc of any type. We can also use (r, i) e Av to refer to the voyage arc from node (r, i) to node (r, i + 1), and use ((r, i), (s, j)) e At to refer to the transshipment arc from the ith portcall of ship route r to the jth portcall of ship route s. Each arc a e A is associated with a duration, denoted by ta, which is determined by the liner service schedules. Suppose that each ship route provides a weekly service frequency and consider a particular ship on each ship route. Suppose that the ship on ship route r 2 R visits the first portcall at time tr1 (the time 0 can be arbitrarily defined, e.g., 00:00:00 01/01/2011), and then visits the second portcall at time tr2, etc. In this context, the duration of a voyage arc a = (r, i) e Av is the time between the arrival at the ith portcall and the arrival at the (i + 1)th portcall, i.e., ta = tr,i+1 tri. The duration of a transshipment arc a = ((r, i), (s, j)) e At can be formulated as follows. Suppose that, at the port pri = psj, the minimum dwell time is Tp, that is, if the incoming ship arrives at least Tp earlier than the outgoing ship, then the containers can be transshipped from the incoming ship to the outgoing ship. Considering the weekly service frequency, we have (see, e.g., Alvarez, 2012; Wang and Meng, 2012a):
ta ¼ minftsj tri þ 7kg; where k 2 Z and t sj t ri þ 7k P T p ; a ¼ ððr; iÞ; ðs; jÞÞ 2 At k
ð2Þ
In Eq. (2), k is an integer reflecting the weekly service frequency and the value 7 means that a portcall is visited every 7 days. The duration of a source/sink arc is 0. Similar to ta, a e A, each arc a e A is also associated with a cost, denoted by ca. The cost of a source arc is the cost of loading containers at their origin port, and the cost of a sink arc is the cost of discharging containers at their destination port. The cost of a voyage arc can be set to 0 because the marginal cost of transporting one more container is insignificant compared with the handling cost. The cost of a transshipment arc is mainly comprised of the container transshipment cost at the port associated with the transshipment arc. In the operational network, a container path is a path from the source node to the sink node. Theoretically, there may be an infinite number of paths for each O–D pair. However, practically, the number of paths is quite limited because of operational constraints and business considerations, which will be elaborated in the next section. 3. Mathematical model We first formulate an integer programming model to obtain the container path with the minimum cost for an O–D pair ðo; dÞ 2 P P. Practical considerations are explained together with the constraints. After that, we describe how to generate all container paths for the O–D pair.
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3.1. Notation asource (r, i) asink(r, i) Atp
The source arc from the virtual source node to node (r, i) The sink arc from node (r, i) to the virtual sink node Set of transshipment arcs at port p 2 P, Atp :¼ fððr; iÞ; ðs; jÞÞ 2 At : pri ¼ psj ¼ pg
Avp þ
Set of voyage arcs entering port p 2 P, Avp þ :¼ fðr; iÞ 2 Av : pr;iþ1 ¼ pg
v
Set of voyage arcs leaving port p 2 P, Avp :¼ fðr; iÞ 2 Av : pri ¼ pg Maximum cost associated with transporting a container between the O–D pair (o, d) Maximum allowable transit time for the O–D pair ðo; dÞ 2 P P The set of countries to which the set of ports P belong The set of countries in which maritime cabotage is imposed on the liner shipping company
Ap Cod Tod
H b H
The set of countries in which the liner shipping company can freely provide maritime transport services The country to which port p 2 P belongs The set of ports located in country h, h e H
b HnH hp Ph
3.2. Decision Variables
xa 2 f0; 1g; a 2 A
ð3Þ
xa = 1 if and only if arc a is contained in the container path. Hence, the values of all xa fully represent a container path. For example, in Fig. 2 we use numbers 1, 2 . . .25 to represent the arcs and a solution can be represented by vector x: =(xa)aeA = (x1, x2 x25). If we obtain a solution in which x1 = x3 = x17 = 1 and all other entries are 0, then we know that the container path is composed of the three arcs 1, 3, and 17. In other words, this container path is direct shipment from Xiamen to Singapore on ship route 2, which can be represented similarly to Eq. (1):
þXM SR2ð2; 3Þ þ SG
ð4Þ
3.3. Objective function Minimize the total cost of transporting one container:
X min ca xa xa
ð5Þ
a2A
3.4. Constraints Flow conservation:
X
xa ¼ 1
ð6Þ
a2Asource
X
xa ¼ 1
ð7Þ
a2Asink
xðr;i1Þ þ
X
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
X
xasource ðr;iÞ þ xðr;i1Þ þ
X
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
xðr;i1Þ þ
X
xððr;iÞ;ðs;jÞÞ ;
8r 2 R; 8i 2 I r ; pri 2 P n fo; dg
ð8Þ
ððr;iÞ;ðs;jÞÞ2At
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
X
xððr;iÞ;ðs;jÞÞ ;
8r 2 R; 8i 2 I r ; pri ¼ o
ð9Þ
ððr;iÞ;ðs;jÞÞ2At
X
xððr;iÞ;ðs;jÞÞ þ xasink ðr;iÞ ;
8r 2 R; 8i 2 I r ; pri ¼ d
ð10Þ
ððr;iÞ;ðs;jÞÞ2At
Containers never visit their origin port from other ports and containers never go from their destination port to other ports (we define arc (r, i 1), i = 1, to be arc (r, Nr)):
xðr;i1Þ ¼ 0; 8r 2 R; 8i 2 I r ; pri ¼ o
ð11Þ
xðr;iÞ ¼ 0; 8r 2 R; 8i 2 I r ; pri ¼ d
ð12Þ
S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
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Containers are never transshipped at their origin or destination port and containers can be transshipped at a port other than their origin or destination at most once:
xððr;iÞ;ðs;jÞÞ ¼ 0; 8ððr; iÞ; ðs; jÞÞ 2 Ato [ Atd X
xððr;iÞ;ðs;jÞÞ 6 1; 8p 2 P n fo; dg
ð13Þ ð14Þ
ððr;iÞ;ðs;jÞÞ2Atp
Maritime cabotage has to be respected. If the liner shipping company is subject to the maritime cabotage restriction in the ^ , then it cannot transship the containers at ports in the same country as the origin country of the origin port, namely, ho 2 H port. That is,
xððr;iÞ;ðs;jÞÞ ¼ 0; 8p 2 Pho ; 8ððr; iÞ; ðs; jÞÞ 2 Atp
ð15Þ
Similarly, if the liner shipping company is subject to the maritime cabotage restriction in the country of the destination port, ^ , we have, namely, hd 2 H
xððr;iÞ;ðs;jÞÞ ¼ 0; 8p 2 Phd ; 8ððr; iÞ; ðs; jÞÞ 2 Atp
ð16Þ
For other countries that impose the maritime cabotage restriction, the liner shipping company can transship the containers at no more than one port within the country. Otherwise, the company would have to transport containers from one transshipment port to another transshipment port within the country, and this would violate maritime cabotage. Therefore, we have
X
X
^ n fho ; hd g xððr;iÞ;ðs;jÞÞ 6 1; 8h 2 H
ð17Þ
p2P h ððr;iÞ;ðs;jÞÞ2At
p
If a transshipment arc ((r, i), (s, j)) e At is visited, then the voyage arcs (r, i 1) e Av, (s, j) e Av must also be visited, and none of the other voyage arcs entering or leaving the port associated with the transshipment arc is visited:
xðr;i1Þ P xððr;iÞ;ðs;jÞÞ ; 8ððr; iÞ; ðs; jÞÞ 2 At
ð18Þ
xðs;jÞ P xððr;iÞ;ðs;jÞÞ ; 8ððr; iÞ; ðs; jÞÞ 2 At
ð19Þ
xa 6 1 xððr;iÞ;ðs;jÞÞ ; 8p 2 P; 8ððr; iÞ; ðs; jÞÞ 2 Atp ; 8a 2 ðApv þ [ Apv Þ n fðr; i 1Þ; ðs; jÞg
ð20Þ
The transit time cannot exceed the maximum allowable transit time Tod:
X
t a xa 6 T od
ð21Þ
a2Av [At
The total cost cannot be larger than the maximum allowable cost:
X ca xa 6 C od
ð22Þ
a2A
A few comments should be made regarding the above model. Constraint (21) is an important business consideration because it defines the level of service offered to customers. In practice, Tod is determined according to the competition in the liner shipping market: if other liner shipping companies provide shorter transit times, the focal company must also set a smaller value of Tod. In the constraint (22), Cod could be, for example, the freight rate for the O–D since otherwise the company would not profit from transporting containers. Of course, we can also drop constraint (22) and compare the optimal value of the model with Cod to check whether constraint (22) is violated. After generating the container path with the lowest cost that satisfies the transit time constraint and maritime cabotage, we can generate the container path with the next lowest cost as follows. Let the binary vector ðxa Þa2A 2 f0; 1gjAj represent the container path with the lowest cost. Add to the above model the following constraint to exclude this container path from being generated again:
X a2A1
ð1 xa Þ þ
X
xa P 1
ð23Þ
a2A0
where
A1 :¼ fa 2 A : xa ¼ 1g
ð24Þ
A0 :¼ fa 2 A : xa ¼ 0g
ð25Þ
The other container paths can be generated one by one in a similar manner by excluding all previously generated container paths until the model is infeasible.
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S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
The above model is an integer (binary) linear programming model and can be solved efficiently by state-of-the-art integer linear programming solvers. Moreover, the container paths for different O–D pairs can be generated simultaneously on different computers. 4. Case study We apply the integer programming model to a liner shipping network provided by a global liner shipping company. The network has a total of 166 ports all over the world, as shown in Fig. 3. A total of 75 ship routes are operated in the network, with 538 voyage legs. We examine the number of container paths from Shanghai to San Pedro. We assume that the minimum dwell time Tp = 1 day for all ports. Since the loading and discharge costs are constant, only transshipment costs are considered. We further assume that the transshipment costs at all ports are the same. Therefore, the maximum allowable cost in Eq. (22) can be represented by the maximum allowable number of transshipments. The model which is integrated into MicroCity (2012) is programmed with C++ and solved by CPLEX. Table 1 shows the number of container paths subject to no transshipments or at most one transshipment, and different maximum allowable transit time constraints. The rows ‘‘Impose cabotage’’ denote that all countries have the maritime cabotage restrictions and the rows ‘‘No cabotage’’ mean that the liner shipping company can provide maritime transportation services freely. Evidently, the number of container paths reduces dramatically when the maritime cabotage restriction is imposed because many of the possible transshipment ports are no longer available. The number of container paths that do not involve transshipment does not change because such container paths are not affected by maritime cabotage. As expected, the number of container paths increases when we allow more transshipments or longer transit time. Finally, we observe that the transit time of containers is usually longer when they are transshipped. The two container paths without transshipment in Table 1 have very short transit times (12 days and 17 days, respectively). 5. Summary We have presented a mathematical model for generating container paths while considering the O–D transit time and maritime cabotage constraints. The container paths generated could be used by liner shipping companies to manage the container flow in liner shipping networks with transshipment. At the tactical planning level, the resulting container flow could help liner shipping companies to analyze their ship capacity utilization, deploy suitable types of ships and alter shipping services. At the operational level, the container paths could be employed to determine which cargo to accept/reject and how to transport the accepted containers at the minimum cost. A container path must satisfy practical constraints, among which the most important are the level of service, represented by the O–D transit time, and maritime cabotage. The proposed integer linear programming model can easily capture these
Fig. 3. A global liner shipping network of 166 ports.
Table 1 Number of container paths in different settings for the O–D Shanghai-San Pedro. Number of container paths
Impose cabotage No cabotage
Maximum transit time (days)
No transshipment allowed Maximum 1 transshipment No transshipment allowed Maximum 1 transshipment
20
25
30
35
40
2 2 2 5
2 7 2 16
2 15 2 30
2 20 2 37
2 28 2 46
S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
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constraints and obtain all the container paths satisfying them. Hopefully, the model will assist liner shipping companies in managing container transportation more economically. Acknowledgements This study is supported by the research Grants WBS No. R-264-000-244-720 and WBS No. R-302-000-014-720 from the NOL Fellowship Programme of Singapore. References Agarwal, R., Ergun, O., 2008. Ship scheduling and network design for cargo routing in liner shipping. Transportation Science 42 (2), 175–196. Alvarez, J.F., 2009. Joint routing and deployment of a fleet of container vessels. Maritime Economics & Logistics 11, 186–208. Alvarez, J.F., 2012. Mathematical expressions for the transit time of merchandise through a liner shipping network. Journal of the Operational Research Society 63 (6), 709–714. Bell, M.G.H., Liu, X., Angeloudis, P., Fonzone, A., Hosseinloo, S.H., 2011. A frequency-based maritime container assignment model. Transportation Research 45B, 1152–1161. Brouer, B.D., Pisinger, D., Spoorendonk, S., 2011. Liner shipping cargo allocation with repositioning of empty containers. INFOR 49 (2), 109–124. Christiansen, M., Fagerholt, K., Ronen, D., 2004. Ship routing and scheduling: status and perspectives. Transportation Science 38 (1), 1–18. Christiansen, M., Fagerholt, K., Nygreen, B., Ronen, D., 2007. Maritime transportation. In: Barnhart, C., Laporte, G. (Eds.), Handbook in OR & MS, vol. 14. Elsevier, pp. 189–284. Fagerholt, K., 1999. Optimal fleet design in a ship routing problem. International Transactions in Operational Research 6, 453–464. Fagerholt, K., 2004. Designing optimal routes in a liner shipping problem. Maritime Policy and Management 31 (4), 259–268. Fagerholt, K., Lindstad, H., 2000. Optimal policies for maintaining a supply service in the Norwegian Sea. Omega 28, 269–275. Gelareh, S., Nickel, S., Pisinger, D., 2010. Liner shipping hub network design in a competitive environment. Transportation Research 46E, 991–1004. Gelareh, S., Pisinger, D., 2011. Fleet deployment, network design and hub location of liner shipping companies. Transportation Research 47E, 947–964. Halvorsen-Weare, E.E., Fagerholt, K., 2011. Robust supply vessel planning. Lecture Notes in Computer Science 6701, 559–573. Imai, A., Shintani, K., Papadimitriou, S., 2009. Multi-port vs. Hub-and-Spoke port calls by containerships. Transportation Research 45E, 740–757. Jepsen, M.K., Løfstedt, B., Plum, C.E.M., Pisinger, D., Sigurd, M.M., 2011. A path based model for a green liner shipping network design problem. In: Proceedings of the International Multi Conference of Engineers and Computer Scientists, 2011. Karlaftis, M.G., Kepaptsoglou, K., Sambracos, E., 2009. Containership routing with time deadlines and simultaneous deliveries and pick-ups. Transportation Research 45E, 210–221. Meng, Q., Wang, S., 2011a. Liner shipping service network design with empty container repositioning. Transportation Research 47E, 695–708. Meng, Q., Wang, S., 2011b. Optimal operating strategy for a long-haul liner service route. European Journal of Operational Research 215, 105–114. Meng, Q., Wang, T., 2011c. A scenario-based dynamic programming model for multi-period liner ship fleet planning. Transportation Research 47E, 401–413. MicroCity. (accessed 04.07.11). Wang, S., Meng, Q., 2012a. Liner ship route schedule design with sea contingency time and port time uncertainty. Transportation Research 46B, 615–633. Wang, S., Meng, Q., 2012b. Sailing speed optimization for container ships in a liner shipping network. Transportation Research Part 48E, 701–714. Wang, S., Meng, Q., 2012c. Liner ship fleet deployment with container transshipment operations. Transportation Research 48E, 470–484.
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Container routing in liner shipping Shuaian Wang a,b, Qiang Meng b,⇑, Zhuo Sun c a b c
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore Centre for Maritime Studies, National University of Singapore, Singapore 118411, Singapore
a r t i c l e
i n f o
Article history: Received 23 November 2011 Received in revised form 7 March 2012 Accepted 5 April 2012
Keywords: Liner shipping Container routing Maritime cabotage Transit time
a b s t r a c t Container paths play an important role in liner shipping services with container transshipment operations. In the literature, link-based multi-commodity flow formulations are widely used for container routing. However, they have two deficiencies: the level of service in terms of the origin-to-destination transit time is not incorporated and maritime cabotage may be violated. To overcome these deficiencies, we first present an operational network representation of a liner shipping network. Based on the network, an integer linear programming model is formulated to obtain container paths with minimum cost. Finally, we add constraints to the integer linear programming model, excluding those paths already obtained, so as to find all the container paths. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Container transshipment is a common practice in liner shipping: approximately 27% of the world’s container throughput consists of transshipment containers (UNCTAD, 2008). Transshipment not only enables cargo consolidation for the deployment of large container ships but also expands the service scope of liner shipping companies as any port-to-port delivery service is feasible, even if there is no liner service route connecting the two ports. Nevertheless, container transshipment operations are difficult to manage in practice. From the origin port to the destination port, containers may be transported on different liner service routes and transshipped at many intermediate ports. To control the delivery of containers, liner shipping companies define a set of paths for each origin–destination (O–D) port pair. A container path contains all the information on how containers are transported (see, e.g., Wang and Meng, 2012a, 2012b). To keep this paper relatively self-contained, we first use the liner service network shown in Fig. 1 to illustrate the concept of the container path. The liner service network in Fig. 1 has three liner service routes (ship routes or SR for short), the set of which is denoted by R ¼ f1; 2; 3g, linking a set of ports P. The itinerary of each ship route r 2 R forms a loop. Let Nr represent the number of portcalls on a round trip. Define I r :¼ f1; 2; ; N r g and let pri 2 P represent the port of the ith portcall on ship route r 2 R, i 2 I r . We can arbitrarily define one portcall as the first portcall, e.g., in Fig. 1 Hong Kong is the first portcall on Ship Route 1 (SR1), Jakarta and Singapore are the second and third portcalls, respectively, and the number of portcalls on SR1 N1 = 3. It should be mentioned that, although Singapore is visited twice during the round trip of SR2, these two calls can easily be differentiated by employing the port calling sequence to refer to a portcall. With the above definitions and notation, a possible container path in Fig. 1 is defined as:
þXM SR2ð2; 4Þ þ CB-SR3ð1; 2Þ þ CN
ð1Þ
Containers on this path are loaded at Xiamen onto ships deployed on SR2, and delivered to Colombo. At Colombo, these containers are discharged and reloaded onto ships deployed on SR3, and subsequently delivered to Chennai where they ⇑ Corresponding author. Tel.: +65 6516 5494; fax: +65 6779 1635. E-mail address: [email protected] (Q. Meng). 1366-5545/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tre.2012.06.009
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Fig. 1. An illustrative liner shipping service network. Source: Wang and Meng, 2012a.
are discharged. Therefore, these containers originate from Xiamen and end up at Chennai, having been transshipped at Colombo. The numbers 2 and 4 in the notation SR2(2, 4) stand for the port calling sequences of the load port and discharge port on this ship route, respectively. That is, containers are loaded at the 2nd portcall (XM) and discharged at the 4th portcall (CB) on SR2. Liner shipping companies adopt the concept of container paths to manage the transportation of containers in a service network with transshipment. In practice, a liner shipping company frequently alters its shipping network because of changes in container shipment demand. Such alterations may include the launch or cancellation of a feeder service, the addition or removal of a portcall on a ship route, or swapping two portcalls on a ship route. In addition, the company will reshuffle its services when new ships are delivered. Therefore, the set of container paths should be updated frequently. According to our discussion with a global liner shipping company, the container paths are actually designed manually by experienced managers. Evidently, manual planning is inefficient and may not find all possible container paths. Consequently, it is of practical significance to build a model that automates the generation of container paths for liner shipping companies. The literature on liner shipping operations can be categorized into four streams (see Ronen, 1983, 1993; Christiansen et al., 2004, 2007, for extensive reviews). The first stream focuses on a single liner ship route (e.g., Shintani et al., 2007; Ronen, 2011; Meng and Wang, 2011b; Qi and Song, 2012) or a few liner ship routes (e.g., Meng and Wang, 2011c) without container transshipment operations, or a butterfly ship route with transshipment at a single port that is visited twice in a round trip (e.g., Reinhardt and Pisinger, 2012). The second stream is devoted to feeder service networks, in which all the feeder ship routes connect to a hub (e.g., Fagerholt, 1999, 2004; Fagerholt and Lindstad, 2000; Sambracos et al., 2004; Karlaftis et al., 2009; Halvorsen-Weare and Fagerholt, 2011) and all containers are transported between feeder ports and the hub. The third stream concentrates on the liner hub-and-spoke (H&S) shipping network services (e.g., Imai et al., 2009; Gelareh et al., 2010; Gelareh and Pisinger, 2011), where containers to be moved between two feeders assigned to different hubs are usually transshipped at hub ports. The fourth stream looks at more general liner shipping networks where containers can be transshipped at any port. Container routing in such general networks is the most challenging. Most of the studies in this stream have adopted link-based multi-commodity flow formulations (e.g., Agarwal and Ergun, 2008; Alvarez, 2009; Brouer et al., 2011; Jepsen et al., 2011; Wang and Meng, 2012c) or segment-based formulations where a segment is a sequence of consecutive links (e.g., Bell et al., 2011; Meng and Wang, 2011a). A few works have applied path-based formulations without mentioning how to generate the container paths (e.g., Wang and Meng, 2012a, 2012b). The link-based formulation is widely used because it is much more compact than the path-based formulation. However, it is difficult (if not impossible) for link-based formulations to incorporate service-level considerations in terms of the O–D transit time, which is an important competition factor (Notteboom, 2006). Moreover, link-based formulations can hardly incorporate the constraint of maritime cabotage, which forbids a liner shipping company to load and discharge the same container at two ports within a country. For example, Maersk Line and MSC are not allowed to transport containers from Shanghai to Yantian because both ports are located in China. Moreover, they are not allowed to transship containers at Yantian that are heading to or destined for Shanghai, either. Only local shipping companies, such as COSCO and CSCL, are allowed to provide shipping services from Shanghai to Yantian. The objective of this study is to develop a mathematical model for generating container paths while considering operational constraints. We will first propose an operational network representation of a liner shipping network in the next section. Based on the operational network, in Section 3 an integer programming model is formulated that obtains all the container paths for each O–D pair. Section 4 reports the findings of a case study. We conclude with a summary in Section 5. 2. Liner shipping operational network To facilitate the generation of container paths, we reformulate a liner shipping network as an operational network. For example, Fig. 2 shows the corresponding operational network for the O–D pair Xiamen-Singapore in the network from
S. Wang et al. / Transportation Research Part E 49 (2013) 1–7
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Fig. 2. A liner shipping operational network for the O–D Xiamen-Singapore.
Fig. 1. The operational network (N, A) for the O–D pair ðo; dÞ 2 P P, where N represents the node set and A is the arc set, is constructed as follows. First, add a virtual source node and a virtual sink node to N. Each node in N except for the virtual source and sink nodes corresponds to a portcall on a liner ship route. Hence, every node in N except for the source and sink nodes can be represented by (r, i), i.e., the ith portcall on ship route r 2 R. Define the arc set A :¼ Av [ At [ Asource [ Asink , comprising the voyage arcs, transshipment arcs, source arcs, and sink arcs, respectively. A voyage arc a e Av can be represented by its tail node (r, i). In other words, voyage arc (r, i) is the voyage from node (r, i) to node (r, i + 1). Note that voyage arc (r, Nr) is the voyage from node (r, Nr) to node (r, 1) because each ship route forms a loop. To simplify the notation, in the following we define node (r, i + 1), i = Nr, to be node (r, 1). A transshipment arc a e At can be represented by ((r, i), (s, j)) where pri = psj. In other words, containers are transshipped at port pri = psj from a ship on ship route r to a ship on ship route s. A source arc is an arc from the source node to a node (r, i) referring to the origin port o, i.e., pri = o. A sink arc is defined similarly. To summarize, we can use a e A to refer to an arc of any type. We can also use (r, i) e Av to refer to the voyage arc from node (r, i) to node (r, i + 1), and use ((r, i), (s, j)) e At to refer to the transshipment arc from the ith portcall of ship route r to the jth portcall of ship route s. Each arc a e A is associated with a duration, denoted by ta, which is determined by the liner service schedules. Suppose that each ship route provides a weekly service frequency and consider a particular ship on each ship route. Suppose that the ship on ship route r 2 R visits the first portcall at time tr1 (the time 0 can be arbitrarily defined, e.g., 00:00:00 01/01/2011), and then visits the second portcall at time tr2, etc. In this context, the duration of a voyage arc a = (r, i) e Av is the time between the arrival at the ith portcall and the arrival at the (i + 1)th portcall, i.e., ta = tr,i+1 tri. The duration of a transshipment arc a = ((r, i), (s, j)) e At can be formulated as follows. Suppose that, at the port pri = psj, the minimum dwell time is Tp, that is, if the incoming ship arrives at least Tp earlier than the outgoing ship, then the containers can be transshipped from the incoming ship to the outgoing ship. Considering the weekly service frequency, we have (see, e.g., Alvarez, 2012; Wang and Meng, 2012a):
ta ¼ minftsj tri þ 7kg; where k 2 Z and t sj t ri þ 7k P T p ; a ¼ ððr; iÞ; ðs; jÞÞ 2 At k
ð2Þ
In Eq. (2), k is an integer reflecting the weekly service frequency and the value 7 means that a portcall is visited every 7 days. The duration of a source/sink arc is 0. Similar to ta, a e A, each arc a e A is also associated with a cost, denoted by ca. The cost of a source arc is the cost of loading containers at their origin port, and the cost of a sink arc is the cost of discharging containers at their destination port. The cost of a voyage arc can be set to 0 because the marginal cost of transporting one more container is insignificant compared with the handling cost. The cost of a transshipment arc is mainly comprised of the container transshipment cost at the port associated with the transshipment arc. In the operational network, a container path is a path from the source node to the sink node. Theoretically, there may be an infinite number of paths for each O–D pair. However, practically, the number of paths is quite limited because of operational constraints and business considerations, which will be elaborated in the next section. 3. Mathematical model We first formulate an integer programming model to obtain the container path with the minimum cost for an O–D pair ðo; dÞ 2 P P. Practical considerations are explained together with the constraints. After that, we describe how to generate all container paths for the O–D pair.
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3.1. Notation asource (r, i) asink(r, i) Atp
The source arc from the virtual source node to node (r, i) The sink arc from node (r, i) to the virtual sink node Set of transshipment arcs at port p 2 P, Atp :¼ fððr; iÞ; ðs; jÞÞ 2 At : pri ¼ psj ¼ pg
Avp þ
Set of voyage arcs entering port p 2 P, Avp þ :¼ fðr; iÞ 2 Av : pr;iþ1 ¼ pg
v
Set of voyage arcs leaving port p 2 P, Avp :¼ fðr; iÞ 2 Av : pri ¼ pg Maximum cost associated with transporting a container between the O–D pair (o, d) Maximum allowable transit time for the O–D pair ðo; dÞ 2 P P The set of countries to which the set of ports P belong The set of countries in which maritime cabotage is imposed on the liner shipping company
Ap Cod Tod
H b H
The set of countries in which the liner shipping company can freely provide maritime transport services The country to which port p 2 P belongs The set of ports located in country h, h e H
b HnH hp Ph
3.2. Decision Variables
xa 2 f0; 1g; a 2 A
ð3Þ
xa = 1 if and only if arc a is contained in the container path. Hence, the values of all xa fully represent a container path. For example, in Fig. 2 we use numbers 1, 2 . . .25 to represent the arcs and a solution can be represented by vector x: =(xa)aeA = (x1, x2 x25). If we obtain a solution in which x1 = x3 = x17 = 1 and all other entries are 0, then we know that the container path is composed of the three arcs 1, 3, and 17. In other words, this container path is direct shipment from Xiamen to Singapore on ship route 2, which can be represented similarly to Eq. (1):
þXM SR2ð2; 3Þ þ SG
ð4Þ
3.3. Objective function Minimize the total cost of transporting one container:
X min ca xa xa
ð5Þ
a2A
3.4. Constraints Flow conservation:
X
xa ¼ 1
ð6Þ
a2Asource
X
xa ¼ 1
ð7Þ
a2Asink
xðr;i1Þ þ
X
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
X
xasource ðr;iÞ þ xðr;i1Þ þ
X
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
xðr;i1Þ þ
X
xððr;iÞ;ðs;jÞÞ ;
8r 2 R; 8i 2 I r ; pri 2 P n fo; dg
ð8Þ
ððr;iÞ;ðs;jÞÞ2At
xððs;jÞ;ðr;iÞÞ ¼ xðr;iÞ þ
ððs;jÞ;ðr;iÞÞ2At
X
xððr;iÞ;ðs;jÞÞ ;
8r 2 R; 8i 2 I r ; pri ¼ o
ð9Þ
ððr;iÞ;ðs;jÞÞ2At
X
xððr;iÞ;ðs;jÞÞ þ xasink ðr;iÞ ;
8r 2 R; 8i 2 I r ; pri ¼ d
ð10Þ
ððr;iÞ;ðs;jÞÞ2At
Containers never visit their origin port from other ports and containers never go from their destination port to other ports (we define arc (r, i 1), i = 1, to be arc (r, Nr)):
xðr;i1Þ ¼ 0; 8r 2 R; 8i 2 I r ; pri ¼ o
ð11Þ
xðr;iÞ ¼ 0; 8r 2 R; 8i 2 I r ; pri ¼ d
ð12Þ
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Containers are never transshipped at their origin or destination port and containers can be transshipped at a port other than their origin or destination at most once:
xððr;iÞ;ðs;jÞÞ ¼ 0; 8ððr; iÞ; ðs; jÞÞ 2 Ato [ Atd X
xððr;iÞ;ðs;jÞÞ 6 1; 8p 2 P n fo; dg
ð13Þ ð14Þ
ððr;iÞ;ðs;jÞÞ2Atp
Maritime cabotage has to be respected. If the liner shipping company is subject to the maritime cabotage restriction in the ^ , then it cannot transship the containers at ports in the same country as the origin country of the origin port, namely, ho 2 H port. That is,
xððr;iÞ;ðs;jÞÞ ¼ 0; 8p 2 Pho ; 8ððr; iÞ; ðs; jÞÞ 2 Atp
ð15Þ
Similarly, if the liner shipping company is subject to the maritime cabotage restriction in the country of the destination port, ^ , we have, namely, hd 2 H
xððr;iÞ;ðs;jÞÞ ¼ 0; 8p 2 Phd ; 8ððr; iÞ; ðs; jÞÞ 2 Atp
ð16Þ
For other countries that impose the maritime cabotage restriction, the liner shipping company can transship the containers at no more than one port within the country. Otherwise, the company would have to transport containers from one transshipment port to another transshipment port within the country, and this would violate maritime cabotage. Therefore, we have
X
X
^ n fho ; hd g xððr;iÞ;ðs;jÞÞ 6 1; 8h 2 H
ð17Þ
p2P h ððr;iÞ;ðs;jÞÞ2At
p
If a transshipment arc ((r, i), (s, j)) e At is visited, then the voyage arcs (r, i 1) e Av, (s, j) e Av must also be visited, and none of the other voyage arcs entering or leaving the port associated with the transshipment arc is visited:
xðr;i1Þ P xððr;iÞ;ðs;jÞÞ ; 8ððr; iÞ; ðs; jÞÞ 2 At
ð18Þ
xðs;jÞ P xððr;iÞ;ðs;jÞÞ ; 8ððr; iÞ; ðs; jÞÞ 2 At
ð19Þ
xa 6 1 xððr;iÞ;ðs;jÞÞ ; 8p 2 P; 8ððr; iÞ; ðs; jÞÞ 2 Atp ; 8a 2 ðApv þ [ Apv Þ n fðr; i 1Þ; ðs; jÞg
ð20Þ
The transit time cannot exceed the maximum allowable transit time Tod:
X
t a xa 6 T od
ð21Þ
a2Av [At
The total cost cannot be larger than the maximum allowable cost:
X ca xa 6 C od
ð22Þ
a2A
A few comments should be made regarding the above model. Constraint (21) is an important business consideration because it defines the level of service offered to customers. In practice, Tod is determined according to the competition in the liner shipping market: if other liner shipping companies provide shorter transit times, the focal company must also set a smaller value of Tod. In the constraint (22), Cod could be, for example, the freight rate for the O–D since otherwise the company would not profit from transporting containers. Of course, we can also drop constraint (22) and compare the optimal value of the model with Cod to check whether constraint (22) is violated. After generating the container path with the lowest cost that satisfies the transit time constraint and maritime cabotage, we can generate the container path with the next lowest cost as follows. Let the binary vector ðxa Þa2A 2 f0; 1gjAj represent the container path with the lowest cost. Add to the above model the following constraint to exclude this container path from being generated again:
X a2A1
ð1 xa Þ þ
X
xa P 1
ð23Þ
a2A0
where
A1 :¼ fa 2 A : xa ¼ 1g
ð24Þ
A0 :¼ fa 2 A : xa ¼ 0g
ð25Þ
The other container paths can be generated one by one in a similar manner by excluding all previously generated container paths until the model is infeasible.
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The above model is an integer (binary) linear programming model and can be solved efficiently by state-of-the-art integer linear programming solvers. Moreover, the container paths for different O–D pairs can be generated simultaneously on different computers. 4. Case study We apply the integer programming model to a liner shipping network provided by a global liner shipping company. The network has a total of 166 ports all over the world, as shown in Fig. 3. A total of 75 ship routes are operated in the network, with 538 voyage legs. We examine the number of container paths from Shanghai to San Pedro. We assume that the minimum dwell time Tp = 1 day for all ports. Since the loading and discharge costs are constant, only transshipment costs are considered. We further assume that the transshipment costs at all ports are the same. Therefore, the maximum allowable cost in Eq. (22) can be represented by the maximum allowable number of transshipments. The model which is integrated into MicroCity (2012) is programmed with C++ and solved by CPLEX. Table 1 shows the number of container paths subject to no transshipments or at most one transshipment, and different maximum allowable transit time constraints. The rows ‘‘Impose cabotage’’ denote that all countries have the maritime cabotage restrictions and the rows ‘‘No cabotage’’ mean that the liner shipping company can provide maritime transportation services freely. Evidently, the number of container paths reduces dramatically when the maritime cabotage restriction is imposed because many of the possible transshipment ports are no longer available. The number of container paths that do not involve transshipment does not change because such container paths are not affected by maritime cabotage. As expected, the number of container paths increases when we allow more transshipments or longer transit time. Finally, we observe that the transit time of containers is usually longer when they are transshipped. The two container paths without transshipment in Table 1 have very short transit times (12 days and 17 days, respectively). 5. Summary We have presented a mathematical model for generating container paths while considering the O–D transit time and maritime cabotage constraints. The container paths generated could be used by liner shipping companies to manage the container flow in liner shipping networks with transshipment. At the tactical planning level, the resulting container flow could help liner shipping companies to analyze their ship capacity utilization, deploy suitable types of ships and alter shipping services. At the operational level, the container paths could be employed to determine which cargo to accept/reject and how to transport the accepted containers at the minimum cost. A container path must satisfy practical constraints, among which the most important are the level of service, represented by the O–D transit time, and maritime cabotage. The proposed integer linear programming model can easily capture these
Fig. 3. A global liner shipping network of 166 ports.
Table 1 Number of container paths in different settings for the O–D Shanghai-San Pedro. Number of container paths
Impose cabotage No cabotage
Maximum transit time (days)
No transshipment allowed Maximum 1 transshipment No transshipment allowed Maximum 1 transshipment
20
25
30
35
40
2 2 2 5
2 7 2 16
2 15 2 30
2 20 2 37
2 28 2 46
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constraints and obtain all the container paths satisfying them. Hopefully, the model will assist liner shipping companies in managing container transportation more economically. Acknowledgements This study is supported by the research Grants WBS No. R-264-000-244-720 and WBS No. R-302-000-014-720 from the NOL Fellowship Programme of Singapore. References Agarwal, R., Ergun, O., 2008. Ship scheduling and network design for cargo routing in liner shipping. Transportation Science 42 (2), 175–196. Alvarez, J.F., 2009. Joint routing and deployment of a fleet of container vessels. Maritime Economics & Logistics 11, 186–208. Alvarez, J.F., 2012. Mathematical expressions for the transit time of merchandise through a liner shipping network. Journal of the Operational Research Society 63 (6), 709–714. Bell, M.G.H., Liu, X., Angeloudis, P., Fonzone, A., Hosseinloo, S.H., 2011. A frequency-based maritime container assignment model. Transportation Research 45B, 1152–1161. Brouer, B.D., Pisinger, D., Spoorendonk, S., 2011. Liner shipping cargo allocation with repositioning of empty containers. INFOR 49 (2), 109–124. Christiansen, M., Fagerholt, K., Ronen, D., 2004. Ship routing and scheduling: status and perspectives. Transportation Science 38 (1), 1–18. Christiansen, M., Fagerholt, K., Nygreen, B., Ronen, D., 2007. Maritime transportation. In: Barnhart, C., Laporte, G. (Eds.), Handbook in OR & MS, vol. 14. Elsevier, pp. 189–284. Fagerholt, K., 1999. Optimal fleet design in a ship routing problem. International Transactions in Operational Research 6, 453–464. Fagerholt, K., 2004. Designing optimal routes in a liner shipping problem. Maritime Policy and Management 31 (4), 259–268. Fagerholt, K., Lindstad, H., 2000. Optimal policies for maintaining a supply service in the Norwegian Sea. Omega 28, 269–275. Gelareh, S., Nickel, S., Pisinger, D., 2010. Liner shipping hub network design in a competitive environment. Transportation Research 46E, 991–1004. Gelareh, S., Pisinger, D., 2011. Fleet deployment, network design and hub location of liner shipping companies. Transportation Research 47E, 947–964. Halvorsen-Weare, E.E., Fagerholt, K., 2011. Robust supply vessel planning. Lecture Notes in Computer Science 6701, 559–573. Imai, A., Shintani, K., Papadimitriou, S., 2009. Multi-port vs. Hub-and-Spoke port calls by containerships. Transportation Research 45E, 740–757. Jepsen, M.K., Løfstedt, B., Plum, C.E.M., Pisinger, D., Sigurd, M.M., 2011. A path based model for a green liner shipping network design problem. In: Proceedings of the International Multi Conference of Engineers and Computer Scientists, 2011. Karlaftis, M.G., Kepaptsoglou, K., Sambracos, E., 2009. Containership routing with time deadlines and simultaneous deliveries and pick-ups. Transportation Research 45E, 210–221. Meng, Q., Wang, S., 2011a. Liner shipping service network design with empty container repositioning. Transportation Research 47E, 695–708. Meng, Q., Wang, S., 2011b. Optimal operating strategy for a long-haul liner service route. European Journal of Operational Research 215, 105–114. Meng, Q., Wang, T., 2011c. A scenario-based dynamic programming model for multi-period liner ship fleet planning. Transportation Research 47E, 401–413. MicroCity.