Analysis of three container routing strategies
Introduction
Global liner shipping companies transport containerized cargoes between ports to make a profit. Since the financial crisis in 2008, the container shipping market has been at the trough of the market cycle as indicated by overcapacity, low freight rates, and low profit margins. Although the container shipment demand has shown signs of recovery, for instance, the containerized trade volume in 2013 was 160 million TEUs (twenty-foot equivalent units), 17% higher than 2008 (UNCTAD, 2009), the freight rates remain very low. In 2013, the freight rate from Shanghai to Northern Europe was only $1084/TEU and from Shanghai to US West Coast was $2033/FEU (forty-foot equivalent unit) (UNCTAD, 2014); by contrast, in the fourth quarter of 2007, the freight rate from Asia to Europe was averaged at $2054/TEU and from Asia to the US was averaged at $3414/FEU ($1707/TEU) (UNCTAD, 2008).
In an effort to deal with low freight rate levels and to leverage some earnings, liner shipping companies have taken measures to improve efficiency and optimize operations to increase revenue and curb costs. For instance, some companies settle agreements to reduce capacity on certain trade lanes and thereby increase freight rate (Bloomberg, 2012). This is possible because the container shipment demand is inelastic to the freight rate which constitutes only a small proportion of the value of the cargoes. Some companies form alliances to expand their service scope and increase service frequency. For instance, the G6 Alliance, formed at the end of 2011 to bring members of the New World Alliance and the Grand Alliance together, expanded cooperation to the Asia–North America East Coast trade lane in May 2013 (UNCTAD, 2014). Efficient repositioning of empty containers can also reduce the operating costs for liner shipping companies (Akyüz and Lee, 2014). In addition, slow-steaming and trim optimization of containerships are adopted by liner shipping companies due to the high bunker cost (Ng, 2014, Ng, 2015, Du et al., 2015b).
This study considers an operational-level container routing problem. Here the container routing problem is mainly concerned with the decisions on which container shipment demand to fulfill in case of limited capacity and how to transport the accepted containers in the network. Broadly speaking, the most profitable containers should be accepted considering capacity limitations. Operational and tactical-level container routing problems are different from each other. The tactical-level container routing models are usually part of liner service design models (Zheng et al., 2015a, Zheng et al., 2015b, Du et al., 2015a). For instance, a liner company first predicts the container shipment demand for the next three to six months. Then, a shipping network is designed to fulfill the demand at minimum cost (Tran and Haasis, 2015b). In tactical-level container routing models the demand is usually expressed as the volume of containers for each port pair; in other words, the demand is assumed to be the same in different weeks because capturing the volatility of container shipment demand in service design is too difficult. It should be mentioned that the container routing decisions obtained from tactical-level models are not implemented but discarded after the liner services are designed. After the liner services are designed, the operational-level container routing decisions are made based on the revealed demand, and the container routing decisions are implemented. This is the focus of our study.
The operation-level container routing problem is challenging as the operational-level transaction between the customer and the liner shipping company is very complex, including booking, picking up an empty container, and delivery of the loaded container to the export port (Du et al., 2011, Chen et al., 2013, Li et al., 2015). Space reservation in shipping does not require prior payment and is not binding. As a result, cancellations are very common (30% according to Leach, 2011) and no-show is not penalized. Moreover, the number of containers received by a liner shipping company from the spot market is unpredictable. Consequently, the container shipment demand in each week is highly random, as shown in Fig. 1. When a liner shipping company makes decisions on which container shipment demand to fulfill and how to transport the accepted containers, the exact future demand for each week is unknown. It takes up to several weeks for a container to be transported from its origin port to its destination port. Then the liner company may accept a less profitable container while having to reject a more profitable one in the future. In this study, the term “non-myopic heuristic container routing” or “non-myopic heuristic routing” is used to represent the operational-level container routing problem considering uncertain demand. Next, a toy example is presented to illustrate the challenges of non-myopic heuristic container routing.
Example 1 Fig. 2 shows a route that includes four ports and . Four ships, each with a capacity of one container, are deployed on the route to provide a weekly service frequency, and each port is visited on Sunday. The voyage time plus the time spent at ports between two adjacent ports is one week. Suppose that the container shipment demands in different weeks are independently and identically distributed: in each week there are two demand scenarios with the same probability: “low” and “high”; when the demand is low, the numbers of containers for port pairs and are ; when it is high, the numbers are . That is, in each week the probability that there is one container from port to port is 1, and the probability that there is one container from to is 0.5. Suppose customers require that containers must be delivered to their destinations in two weeks, otherwise the containers are rejected by the shipping company. The profit for delivering one container from to is , and from to is . That is, containers from to are much more profitable than those from to . At the beginning of each week, the liner shipping company observes the realized demand and then makes the container routing decisions. Note that the company only knows the probability distribution of the demand based on historical data and does not know the exact realizations of the demand in the future weeks. An intuitive approach for making container routing decisions in each week is the myopic strategy: the decision is made in the manner to maximize the profit of only that particular week without considering the demand in the future weeks. In Example 1, if the myopic strategy is adopted, then if the demand in week 1 is low, we will accept the cargo from to . In week 2, if the demand is low again, we will still accept the cargo from to . If the demand is high in week 2 (one container from to and one from to ), since the ship that visits port in week 2 already carries the container from to in week 1, it cannot carry the container from to any more. As a result, to maximize the profit in week 2, we will still accept the container from to and reject the container from to . Hence, using the myopic strategy, the container from to is always accepted and the one from to is always rejected in all of the following weeks. Table 1 shows other examples of the realizations to elaborate the myopic approach. When the myopic strategy is implemented, the expected profit will be only per week. From this toy example, the shortcoming of the myopic strategy can be identified: the container from to is much more profitable than that from to . However, even if the demand is high in a particular week, we still cannot transport the container from to because the container from to loaded in the previous week already occupies the ship slot. The shortcoming, i.e., the inability to take future decisions into account, is rooted in the conservative nature of the myopic strategy. In essence, the myopic strategy makes container routing decisions in a particular week based on the most conservative assumption that there is no demand in the future weeks. Based on this example, we can easily prove:
Using the myopic routing strategy, (i) when the demand between a port pair increases, it is possible that the total profit decreases; (ii) when the capacity of a leg increases, it is possible that the total profit decreases; (iii) when the freight rate for a port pair increases, it is possible that the total profit decreases.
To overcome the shortcoming of the myopic strategy for this toy example, an improved strategy is given: always rejecting the container from to and always accepting the container from to when the demand is high. The profit in each week using the improved strategy (called a non-myopic heuristic routing strategy as it accounts for the future random demand) for the four scenarios in Table 1 is shown in Table 2. Since the probability of obtaining a unit demand at a high demand week is 0.5, the expected profit is per week, much higher than that of the myopic strategy.
It should be noted that when we make container routing decisions in a week, the future demand realizations are unknown, although in this example we know it is either low or high. If we had known the exact demand realizations for all of the future weeks, we could make the best container routing decisions using the full information strategy. Evidently, the full information strategy for this example is: (i) if the demand is high, (i.1) deliver the container from a to c as well as the one from b to d if the subsequent week has low demand, (i.2) and deliver the container from b to d if the demand in the subsequent week is high. (ii) if the demand is low, (ii.1) accept the container from to if the demand in the subsequent week is also low, and (ii.2) reject the container if the subsequent week has high demand. The expected profit using the full information strategy is . The profit in each week for the four scenarios using the full information strategy is also shown in Table 2. The full information strategy outperforms the non-myopic heuristic routing strategy: when the demand is low, the non-myopic heuristic routing strategy has to reserve the capacity for potential demand from to in the following week; whereas the full information strategy can take advantage of the exact demand information in the subsequent weeks. By continuing the example in Table 1, the comparison of the profits between the three strategies under different demand patterns is summarized in Table 2.
The objective of our study is to develop a non-myopic heuristic routing strategy for liner shipping companies to increase their profit. The non-myopic heuristic routing strategy outperforms the myopic routing strategy and could bring more profits to the liner shipping companies. The contributions of our study are fourfold:
- (1)
The worst case ratio of the profit using the myopic routing strategy and the profit using the full information strategy can be proved to approach zero; however, when all containers must be delivered to their destinations in weeks, there exists a non-myopic heuristic routing strategy whose profit is at least of the profit using the full information strategy.
- (2)
It is proved that, even if all containers must be delivered to their destinations in weeks, to make container routing decisions in week , the possible demands in all of the future weeks should be taken into account, including the demands in weeks after .
- (3)
It is affirmed that no practical routing strategy guarantees an expected total profit of more than 2/3 of the profit using the full information strategy for all problems.
- (4)
A non-myopic heuristic routing strategy that combines the myopic strategy and the full information strategy is proposed. Extensive numerical experiments show that the non-myopic heuristic routing strategy considerably outperforms the myopic strategy.
The remainder of the paper is organized as follows. Section 2 reviews relevant literature. Section 3 describes the container routing problem with stochastic demand. Section 4 presents models for the myopic and full information strategy, and then proposes a non-myopic heuristic routing strategy. Section 5 reports the results of numerical experiments. Conclusions are presented in Section 6.
Section snippets
Related works
There are three categories of relevant research: operational-level container routing, container slot allocation, and pricing. Readers interested in broader works could refer to Christiansen et al., 2013, Lee and Fransoo, 2013, Meng et al., 2014, and Tran and Haasis (2015a).
In the category of research on operational-level container routing, Brouer et al. (2011) considered a planning horizon of many periods and assumed that the container shipment demand in each period is known. In other words,
Backgrounds
This study considers a liner shipping company that operates a set of routes for to transport containers over a group of ports denoted by the set . Each route has a fixed port rotation, and fixed arrival and departure times at each port of call. Fig. 3 shows an illustrative example for a liner shipping service network, which includes three routes denoted by and eight ports denoted by . The itinerary of each route forms a loop. Let denote the number
Model formulation
This section presents mathematical models. Only one type of container (TEU) is considered, and there is no long-term contractual demand. In Section 4.1 we list the notations used in the models. Sections 4.2 Myopic strategy, 4.3 Full information strategy present the myopic model and the full information model, respectively. Section 4.4 investigates properties of practical routing models and proposes a non-myopic heuristic routing strategy that combines the myopic and full information strategies.
Numerical experiments
We conduct extensive numerical experiments that are randomly generated to test the performance of the proposed non-myopic heuristic routing strategy. All of the linear programming models are solved by CPLEX12.5.1 with the technology of C# (VS2008) on a PC (Intel Core i5, 1.70 GHz; Memory, 8G).
The test instances are generated as follows. The ports are randomly selected from major Asian ports. The port rotations on the services are randomly chosen. Two types of ships—5000-TEU and 6000-TEU—are
Conclusions and future works
As containers need several weeks to be delivered to their destinations and the future container shipment demand is uncertain, a myopic strategy for container routing that maximizes the immediate profit may lead to insufficient remaining capacity for transporting more profitable containers in the future. Therefore, it is of high interest to develop a non-myopic routing strategy that incorporates information on the future demand distribution for liner shipping companies to increase their profit.
Acknowledgements
The authors thank the editors and reviewers for their valuable comments and constructive suggestions. This research is supported by the National Natural Science Foundation of China (71671107, 71422007) and Shanghai Social Science Research Program (2014BGL006).
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