3.1 Slot co-allocation with a joint fleet

Slot co-allocation is defined in this paper as the cooperation of the shipping alliance members to assess the utilization of shared ship capacities in a round trip.

The cooperative model of a joint fleet, generally two or more ship operators, has an operating agreement for one or several shipping routes. In the agreement of liner service, frequency of port calls, shipping schedule, the number of ships offered by each alliance member and other issues are clearly defined. Cooperative entities are jointly responsible for operating the liner route, but the companies are independently responsible for marketing their own slot share in the voyage.

The main concept of slot co-allocation with joint a fleet agreement between the different parties of the alliance is illustrated in Figure 1 (Numbers 1 to 10 are calling ports). In this cooperative mode, in a round trip the ship operator is responsible for the seaworthiness and operation of the vessel. However, each partner is responsible for its own slot allocation. Each party independently and individually conducts marketing and documentation and is expected to maintain and continue all its commercial, operational and administrative activities in order to sustain and run its own business. For the cooperative route, each party offers a different number of ships. Each party receives the allotted slot space from the total slot space available in the voyage based on its contribution to the route slot space sale. Even though vessels are operated by different ship operators, for each round trip each party has the same schedule, calling ports, and share of space on board the ship.

Figure 1. Slot co-allocation in a round voyage and joint fleet.

3.2 Factors affecting slot co-allocation

As indicated before, slot utilization in the liner containership is vitally important; liner shipping companies try to avoid unused slots in a voyage to generate the highest possible revenue from containership slots. The main aspect of joint service in a joint fleet operating environment of slot co-allocation is subject to each party's slot cost, pricing, market demand, and type of container.

In this study, slot cost is for a single container ship transport. This is a function of the variable cost of using the ship's space to transport a container. Shipping alliance members' slot costs are the driving factor for the optimization of slot co-allocation.

Pricing is one of the core elements of the shipping slot allocation system. In a fiercely competitive shipping market, freight rates cannot increase easily and must cover the costly repositioning of empty containers. Therefore, liner companies can have difficulty generating sufficient revenue and must optimize slot allocation and ensure their pricing and space utilization is as efficient as possible.

In the various market segments, different types of containers have their own freight rate. Obviously, carriers prefer to select containers with the highest return in order to increase their income. However, pricing is also influenced by market demand. To determine freight rates, carriers often engage in historical data analysis to forecast cargo demand, and adjust the prices with the evolving booking data from the agents or customers.

Carriers usually declare available space for slot allocation according to agents' space requests and demand for cargo. A Basic Slot Allocation table (BSA table) by country is often prepared for the local agents' freight services. However, the actual volume of container cargo a ship carries is limited by container and weight capacities. Therefore, space should be effectively allocated to maximize the freight, considering the market demand and the running ship's utilization.

Many different containers are used in liner shipping: 20-ft, 40-ft, refrigerated, and special containers. This variety makes slot co-allocation for alliance members very complex. For example, a 20-ft container occupies a standard TEU (Twenty-foot Equivalent Unit) slot, and a 40-ft container takes two standard TEU slots. A refrigerated container is restricted to space that includes electric outlets. Furthermore, each vessel has a specific number of slots for each container category of the 1 TEU, 2 TEU, etc. Even though there are no revenues generated from each member's own empty containers (there are revenues from the empty if the container belongs to another firm), empty container repositioning is an important and inevitable task because of the imbalance of import and export trade. Shipping companies move empty containers to support future voyages and to sustain the operation of the shipping service.

4.1 Model Assumptions

The model is subject to the following assumptions:

• • The cooperative route in the model is an ocean liner trunk route. Thus, the slot co-allocation model ignores the impact on the regional transportation network.

• • Under the joint service agreement the cooperative liner shipping route is known; therefore, the calling ports and port sequence of the route are determined in advance.

• • In a round trip, there is only one ship running for all the alliance members. Total number of slots of each party for the voyage is determined in the agreement.

• • The freight rates and variable slot costs of each origin-destination port pair for the single voyage for a weekly shipping service have been estimated in advance for the round trip.

• • The minimum/maximum container cargo demand of each origin-destination port pair for the local shipping agents or customers has been estimated based on the method identified above.

• • The common types of containers and empty containers transported have been considered in the model. There are 12 major types of containers including laden and empty (i.e. 20' laden and empty dry container, 20' laden and empty reefer container, 20' laden and empty open top container, 40' laden and empty dry container, 40' laden and empty reefer container, 40' laden and empty open top container). A 20' laden/empty container occupies one slot, and a 40' laden/empty container occupies two slots.

4.2 Slot Co-Allocation Optimization Model

The purpose of slot resource sharing for the shipping alliance members is to maximize revenue generated. The shipping alliance business model indicates that each member of the alliance arranges for their own freight by container category. The model includes all served port pairs and origin-destination port pairs are chosen randomly from the pairs of ports available with the objective to maximize revenues.

The model's aim is to generate maximum revenue for the alliance members in the cooperative round trip while satisfying capacity, weight, demand and other constraints. It is formulated as follows (full notation is contained in the Appendix):

(1)$$Max\;Z = \sum\limits_{\forall i \in SA} \, {\sum\limits_{(o,d) \in \Omega} \, {\sum\limits_{k \in CT} {w_i(F^k_{iod}}} - C^k_{iod})X^{hk}_{iod}} $$

such that,

(2)$$\sum\limits_{(o,d) \in \Omega} \, {\sum\limits_{k \in CO} {a^s_{od}t^k X^{hk}_{iod} \leqslant U^h_{ai}\quad \forall s \in S,\forall i \in SA}} $$

(3)$$\sum\limits_{(o,d) \in \Omega} \, {\sum\limits_{k \in CO} {a^s_{od}w_{od}^k X^{hk}_{iod} \leqslant DW^h \quad \forall s \in S,\forall i \in SA}} $$

(4)$$\sum\limits_{(o,d) \in \Omega} \, {\sum\limits_{k \in RF} {a^s_{od}t^k X^{hk}_{iod} \leqslant RP^h_{ai}\quad \forall s \in S,\forall i \in SA}} $$

(5)$$DL^{kh}_{iod} \leqslant X^{hk}_{iod} \leqslant DU^{kh}_{iod}, \quad \forall k \in CT\,,\forall (o,d) \in \Omega, \forall i \in SA$$

(6)$$X^{hk}_{iod} \geqslant EC^{kh}_{iod} \quad \forall k \in EC,\forall (o,d) \in \Omega, \forall i \in SA$$

(7)$$U^h_{ai} = \displaystyle{{\sum\limits_{\forall h \in VS} {N^h_iU_i^h}} \over {\sum\limits_{\forall h \in VS} {N^h_i U_i^h} + \sum\limits_{\forall h \in VS} {N_j U_j^h}}} \times U_i^h \quad \forall i,j \in SA,\;\;i \ne j$$

(8)$${RP}^h_{ai} = \displaystyle{{\sum\limits_{\forall h \in VS} {N^h_i RP_i^h}} \over {\sum\limits_{\forall h \in VS} {N^h_i RP_i^h} + \sum\limits_{\forall h \in VS} {N_j RP_j^h}}} \times RP_i^{h\;} \;\;\forall i,j \in SA,\;i \ne j$$

(9)$$X^{hk}_{iod} \in N \cup \left\{ 0 \right\}\quad \forall k \in CO\,,\forall (o,d) \in \Omega, \forall i \in SA$$

The objective of equation (1) is to maximize the total estimated revenues from various container categories for the alliance. In equation (1), the weight restriction is proposed as ∑_∀i∈SAw _i=1. The shipping alliance business and agreement indicate that each member of the alliance can use slots on the ship; therefore, $w_i = \textstyle{1 \over G}$ (where G is the number of shipping alliance members). Equations (2) and (3) are vessel capacity constraints. There are two major restrictions on the vessel capacity. Equation (2) implies that all the allocated slots for all kinds of laden and empty containers of all the alliance members cannot exceed the contractual capacities available (TEUs) for each member. Equation (3) ensures that the total weight of containers cannot exceed the vessel deadweight tonnage. Equation (4) indicates that all the slots for laden reefer containers cannot exceed the number of the vessel's reefer plugs. Equation (5) is cargo demand constraint for each alliance member; it provides the lower and upper bounds of allocated slots for laden containers of various categories and port pairs. Equation (6) is the empty container reposition demand constraint, which ensures that the total number of slots of empty containers must be greater than the reposition demand of k-type containers for each alliance member between different port pairs. Equations (7) and (8) are the constraints of the shipping alliance agreements for slot sharing under the joint fleet operation, which means that each member gets the slot capacity to use in a round trip according to his contribution to the route from the total capacity available to him. The integrity restrictions on the decision variables are represented in equation (9).

For a round trip, there is just one ship for the cooperative route; therefore, the number of variables in the model is the product of the number of shipping alliance members, the number of container categories and the number of port pairs. The model is a large scale integer programming (IP) model. This IP model can be solved by the traditional algorithm, such as the branch-and-bound method. For model solving, the optimization software LINGO11.0 can be utilized.

LINGO is a comprehensive optimization tool designed to make building and solving Linear, Nonlinear and Integer optimization models faster, easier and more efficient and can help cut development time. It formulates integer optimization problems quickly in a highly readable form. LINGO11.0 provides a completely integrated package that includes a powerful language for expressing optimization models, a full featured environment for building and editing problems, and a set of fast built-in solvers.

5.1 Background for the Case Study

The tested cooperative route serves 8 ports in China, Hong Kong, Saudi Arabia, Germany, Britain and Belgium with 8 sailing legs. The port rotation is: XMN (Xiamen) – NSH (Nansha) – YTN (Yantian) – HKG (Hong Kong) – JED (Jeddah) – HAM (Hamburg) – FXT (Felixstowe) – ANR (Antwerp) – XMN (Xiamen). This loop is divided between East and West bounds (Figure 2). For the joint fleet in this cooperative route, the two companies deployed seven full-container vessels to provide weekly service for every calling port.

For a round trip, just one ship is used for slot co-allocation. According to alliance agreements and constraints equations (7) and (8) above, in a voyage, the two companies get their own slot space and reefer plugs for their container transport service. For this voyage, the available reserved space for the companies is 2,500 TEUs for COSCO, and 1,500 TEUs for HANJIN, respectively, without considering weight limitations. The maximum available deadweight tonnage for this voyage is 65,500 tons. The number of reefer plugs for COSCO is 300 and for HANJIN 100.

Besides the general types of containers, the two companies also accepted some special containers, such as 20-ft and 40-ft open top, and reefers. To simplify the problem, these out-of-gauge containers are a part of the special containers. Though the weights of loaded cargo in each type of container maybe different, for a realistic trip, shippers often load their cargoes into the container to a similar weight. Therefore, the similar value for each category as provided by the companies is used. Table 1 shows the container categories and their weights and volumes. For the case study and a cooperative route, a market-leg relation parameter, a_od^s, with values of 1 or 0 is used. The values (of 1 and 0) are defined as whether the delivery passage was of port pair (o, d); passing the sailing leg s or not, 1 stands for yes and 0 stands for otherwise (Table 2). The analysis uses the two companies' weekly data of September 2009. Cost, revenue, number of ships in the joint fleet, dead weight tonnage, container demand and other databases variables are used to calculate the needed related model parameter data, such as: freight revenue, variable costs, container flow, and others.

Table 1. Weight and volume for different kinds of containers (COSCO and HANJIN).

Table 2. Relationship between port pairs and legs (a_od^s) for the cooperative route.

Note: The number for the port pairs indicates: 1stands for XMN, 2 for NSN, 3 for YTN, 4 for HKG, 5 for JED, 6 for HAM, 7 for FXT, and 8 for ANR; for example, port pair (1,2) means XMN- NSN.

5.2 Results Analysis and Discussion

LINGO11.0 was used to solve the mathematical optimization model. The results are reported in the corresponding tables. The optimization problem has 1,536 variables and 801 constraints. The optimal slot allocation results in TEU for all sailing legs of a round trip are shown in Table 3, and the number of box results is displayed in Table 4.

Table 3. Slot occupation for all sailing legs for each member (TEU units).

Table 4. Slot occupied results for all sailing legs for each member (number of boxes).

Tables 3 and 4 indicate that some voyage legs are very busy, such as YTN-HKG, HKG-JED, JED-HAM, ANR-XMN and HAM-FXT, whose slots occupied by laden and empty containers reach almost 100% of the allotted space for each alliance member. The above legs play important roles in container cargo transport for the cooperative route. The leg YTN-HKG is the only way to transit goods from China's mainland area at Hong Kong port, and empty containers from Europe to China. Saudi Arabia and other regions transit through HKG-JED and JED-HAM; they are important westbound legs in the route for laden containers from China, Hong Kong to the Middle East and Europe. ANR-XMN is mainly used for laden container transport from the Middle East to China and Hong Kong, and for empty container repositioning from Europe to the Middle East, Hong Kong, and China. XMN-NSH and NSN-YTN are in China's mainland; their loading rates do not reach 100% and their space is mainly shared by empty containers. These two legs transit laden containers at Hong Kong from Xiamen and Nansha Ports and empty containers from Middle East and Europe to China's mainland.

In the cooperative route environment, slot allocation at the ports is critical for each alliance member in the joint fleet. The slot distribution between port pairs, the allocated results for different ports and the alliance members are summarized in Tables 5 and 6. The allocated number of slots and TEUs of the main 12 types of containers are reported in the two tables. The optimum slot allocation in Tables 5 and 6 is different between ports. Due to the different marketing strategies of the two alliance members, the slot allocation of laden and empty containers for COSCO and HANJIN are different at the ports. COSCO's and HANJIN's slot numbers and TEUs for four ports (HKG, JED, HAM, and FXT) are larger than those for the other ports in the cooperative route, because these ports are important international transhipment ports or regional hubs.

Table 5. Number of boxes allocated for each member at the ports in the cooperative route.

Table 6. Slot allocation results in TEU for each member at each port (TEU units).

The total slot allocation in TEU for each member at each port (Table 6) is the sum of the number of boxes allocated for each of the two members at each port (Table 5).

The slot allocation for the container categories provides revenue maximization, during a period of trade imbalance in the route. As shown in Tables 4 and 5, the results indicate:

• • More slots are allocated for GP and OT containers, which contribute marginally more to the revenues per unit for the two members.

• • Empty containers do not provide benefits to the members. However, the demand for empty containers by the two members should be guaranteed for the next voyage.

• • Both COSCO and HANJIN, due to trade imbalance (China's exports to Europe are more than China's imports from Europe), allocate laden slots for the westbound voyage, and empty slots for the eastbound voyage.

• • This study of cooperative routes focused on multinational trade of goods.

In short, a few slots are arranged for laden containers within the region. The laden container cargo flow is mainly inter-regional, i.e., space for laden containers is mainly allocated for China, the Middle East and Europe inter-regional ports.