Optimization Model for Direct Transshipment Operations

1.1         Problem Description

The proposed mathematical model extends the research done by Zeng et al. (2017) with the objective of minimizing the total cost. The model handles a transshipment hub terminal which is visited by both feeder and mother vessels with estimated arrival times and specific containers capacity.

1.1.1        Transshipment and Non-Transshipment Containers

The incoming vessels might have containers to be transshipped to another vessel arriving at the terminal within the predetermined planning horizon. These transshipment containers might be unloaded and transferred to the yard storage area for some time before being loaded onto another vessel. Alternatively, the calling schedule for both vessels can be adjusted so that the containers can be directly transshipped. One of the most important features of the proposed mathematical model is to counteract this situation and decide the transshipment method that should be used for each pair of vessels individually by comparing the associated total cost of both scenarios.

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In addition to transshipment containers, vessels might have other non-transshipment operations exchanged with the terminal yard area and considered as import/export containers as suggested by Liang et al. (2012). Unloaded containers might leave the terminal either by being loaded to another vessel arriving after the specified planning horizon or by leaving through the terminal gate. Loaded containers could come to the terminal from its gate or could be dropped by a previously arrived vessel. However, handling sources and sinks of those non-transshipment containers are beyond the scope of this study.

1.1.2        Representation of Vessels Service Order

For instance, consider feeder vessel

jwhich has containers to be transshipped to mother vessels

iand

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gin addition to the non-transshipment containers. Thus, there are three possible service orders for feeder vessel

j while each operation can be handled at any order. As an example, the non-transshipment containers can be handled at the first, second, or even the third service order. The only requirement to be considered is to finish handling all containers included at a specific operation before working on the following one.

One of the possible scenarios for feeder vessel

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jservice order is as follows with a graphic representation shown at Figure 5:

• First, unload containers for mother vessel

  g(1^st service order)

• Then, handle non-transshipment containers (2^nd service order)
• Finally, unload containers for mother vessel

  i(3^rd service order)

  Figure 5. Feeder vessel j service order possible sequence (a)

  As illustrated in Figure 5, the vessel is represented by a rectangle with a horizontal dimension equal to its length and a vertical dimension expressing berthing time (Meisel and Bierwirth, 2006). Each bordered box within the vessel rectangle indicates the start and end times of each handling operation. Figure 6 shows another feasible scenario for the same feeder vessel:

  • First, handle non-transshipment containers (1^st service order)
  • Then, unload containers for mother vessel

    g(2^nd service order)

  • Finally, unload containers for vessel

    i(3^rd service order)

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    Figure 6. Feeder vessel j service order possible sequence (b)

    The concept of “service order” has been commonly addressed in literature (see Imai et al., 2001). Container terminals studied in most of these research papers have a specified number of discrete berths. Since each berth can only accommodate one vessel at a time, vessels arriving at the terminal need to be allocated to those berths with a service order that guarantees no overlapping between vessels at the same berth. The same concept is applied on vessels that have transshipment relationship with other vessels. In this research, an optimization model is proposed to handle this critical decision. The proposed model employs a mixed integer linear program (MILP) and focuses on the dynamic nature of how berth allocation planning and sequencing of service orders may be co-handled by adjusting transshipment operation schedule. As soon as a transshipment method is chosen, the model generates service orders for vessels that minimizes total turnaround time of vessels and the associated waiting cost.

    1.1.3        Berth Allocation Planning and Sequencing of Service Orders

    Berth allocation assigns berthing position and berthing time for incoming feeders and mother vessels along the terminal quay. The next sections explain each problem individually in detail.

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    • Berthing Position Allocation:

    The model assigns a berthing location while maintaining minimum total travel distance for both non-transshipment and transshipment containers. Non-transshipment containers move from quayside area to the assigned yard blocks. As shown in Figure 7, import containers are unloaded from mother vessels and transported to the designated yard blocks. Similarly, feeder vessels load export containers from yard blocks. Meanwhile, travelled distance for transshipment containers is determined according to the transshipment method chosen by the model, which will be either direct or traditional. By applying the traditional method, containers arrive and depart on vessels while being temporarily stored in the yard (Giallombardo, et al., 2010), such as the non-transshipment containers. The vessel must be allocated to a berthing position close enough to the designated yard blocks in order to speed up the vessel handling time. An example of the service flow for the indirect-transshipment containers case is shown in Figure 7, where the transshipment containers for mother vessel A will be discharged from feeder vessel C and stored at the designated yard block. Once mother vessel A is ready for the loading process, containers will be loaded to it. However, for the direct-transshipment case, containers are transferred from one vessel to another directly. Transferring containers from feeder D to mother vessels A and B is an example of a direct-transshipment case (see Figure 7).

    The proposed model allocates a berthing position for each incoming vessel in such a way that minimizes the overall travelled distance for all containers. It is worthwhile to mention that each group of containers is assigned randomly to a yard block within the terminal. In other words, yard block assignment is an input parameter not a decision variable. For the example shown in Figure 7, if containers from feeder C to mother vessel A are transshipped through the terminal, they will be stored at yard block (3,1). Note that two decision variables are generated randomly by the proposed model; berthing positions along the quay and the transshipment method. Accordingly, the travelled distances as well as the incorporated costs are calculated based on model constraints and assumptions. The following input parameters determine the terminal layout dimensions and contribute to the distance calculations (Readers are referred to Chapter 4 for more detailed information):

    • yard block length and width
    • passing lane width
    • horizontal distance between yard blocks,
    • distance between quayside area to yard-side area