Model development
This study uses a twostage model comprising shipping lines and shippers. As decisionmakers in the upper stage, shipping lines design two different networks comprising one service for direct shipment and multiple services for transshipment. In the lower stage, the shippers’ generalized cost for direct shipment and transshipment is calculated. The notations of the model are as follows:
i,j

Port name (i ≠ j, i,j ∋ o: origin, h: hub, and d: destination)

k

Name of the transport system (DR: direct shipment or TS: transshipment)

\( {GC}_{ij}^k \)

Generalized cost of the shippers using transport system k from port i to j [USD/TEU (20ft equivalent unit)]

Cs_{ij}

Cost of the shipping lines from port i to j [USD/month]

vc_{ij}

Vessel costs (capital and operating costs) from port i to j [USD/day]

fc_{ij}

Fuel cost for transport from port i to j [USD/day]

pc_{i}

Port charges for transport from port i to j [USD/time]

hc_{i}

Handling charges in port i [USD/TEU]

f_{ij}

Frequency from port i to j [time/month]

s_{ij}

Vessel size deployed from port i to j [TEU]

v_{ij}

Navigation speed from port i to j [knot]

u_{ij}

Navigation distance between port i and j [nm]

Q_{ij}

Total container cargo demand from port i to j [TEU/month]

q_{ij}

Cargo flow at one call from port i to j [TEU/time]

σ_{ij}

Freight rate for transport from port i to j [USD/TEU]

L_{i}

Loading/unloading volume per hour in port i [TEU/hour]

T_{i}

Time taken to enter a port, including pilotage and waiting in anchorage, in port i [hour]

Wt_{i}

Wait time of shippers in port i [day]

α

Value of time of shippers [USD/hourTEU]

γ

Unknown parameter for slot utilization

μ

Unknown parameter for distance of the hub port

δ_{hd}

Unknown parameter of cargo demand from the hub to destination port, to reveal the competitiveness of the hub port

Shipping lines
The behavior of shipping lines as carriers is characterized by a model with the objective of maximizing their own profit. The freight rate to shippers, comprising the main revenue component for shipping lines, is influenced by several market factors, such as competition between services operated by different carriers. This study assumes strong competition between shipping lines, leading to lower freight rates, and diminishing the differences in freight rates among shipping lines. For maximizing their profit, shipping lines need to minimize the cost of each service. Therefore, in this study, the shipping lines’ objective is to minimize their own cost for the given inputs. The shipping lines determine the vessel size (s_{ij}), vessel speed (v_{ij}), and frequency (f_{ij}) for each route (from port i to j). Equation (1) expresses the shipping lines’ cost, calculated as the product of the frequency of calls and the operation cost per call.
According to Hsu and Hsieh (2007), operation cost comprises three elements: capital and operating costs, fuel cost, and port charges. Capital and operating costs include the total expenses paid for vessel operation each day. We renamed them vessel costs (vc_{ij}) in this study, comprising the cost of chartering the vessel and operating costs, including maintenance, repair costs, and so on. The shipping lines incur the vessel costs for the total voyage time that includes navigation time, expressed as the relationship between navigation distance and navigation speed (u_{ij}/v_{ij}), the time taken for loading or unloading in the port (q_{ij}/L_{i}), and the time taken to enter the port, including pilotage and waiting in anchorage (T_{i}). Fuel cost (fc_{ij}) comprises the expense related to fuel consumption. Wang and Meng (2012) conclude that fuel consumption is the third power of navigation speed. We multiply the fuel consumption by navigation time to take the sailing time into consideration. Port charges can be categorized into the charges for the vessel and for handling cargo. In this study, the former is termed port charges (pc_{i}) and comprises the charges for servicing the vessel—including, for example, pilotage and towage—paid twice, that is, at the origin and destination port. The latter is termed handling charges (hc_{i}), paid for handling cargo in the container yard, and charged per container. Thus, we calculate the handling charges per call (q_{ij}).
There are two constraints. First, per eq. (2), the vessel capacity of the shipping lines exceeds total cargo demand (Q_{ij}). Parameter (γ) representing slot utilizations is set based on Wang and Meng (2012), as described in Input Values and Network Conditions section. Equation (3) indicates that the cargo volume at one port of call is calculated based on the total cargo volume and frequency. Parameter (δ_{hd}) expresses the aggregated cargoes with the same destination but different origins, in the hub port. For example, if parameter δ_{hd} equals one, there are no services from the hub to destination port without a simulation target origin–destination (OD) pair. If parameter δ_{hd} equals five, then four times the cargo volume between the simulation target OD pair is aggregated in the hub port, to transport five times the cargo volume. Additionally, this parameter indicates the port’s competitiveness as a hub, because a larger hub port would aggregate more cargo, thus expressing a larger value. We can simulate the effects of the differences in port competitiveness by modifying this parameter.
$$ \underset{f_{ij},{s}_{ij},{v}_{ij}}{\min }{Cs}_{ij}={f}_{ij}\left(\ {vc}_{ij}\left({T}_i+{q}_{ij}/{L}_i+{u}_{ij}/{v}_{ij}+{T}_j+{q}_{ij}/{L}_j\right)+{fc}_{ij}{v}_{ij}^3{u}_{ij}/{v}_{ij}+{pc}_i+{pc}_j+\left({hc}_i+{hc}_j\right){q}_{ij}\right) $$
(1)
subject to:
$$ \gamma \bullet {f}_{ij}\bullet {s}_{ij}\ge {Q}_{ij} $$
(2)
$$ {q}_{hd}={\delta}_{hd}\bullet {Q}_{od}/{f}_{hd} $$
(3)
Second, vessel costs, fuel cost, and port charges depend on the vessel size, and are calculated using eqs. (4), (5), and (6), respectively. The coefficients of these equations are estimated in previous studies (Tran 2011; Kim et al. 2019), using several methods, including the least squares method based on actual data of container vessels. The term BP in eq. (4) indicates the bunker price, set at 387.5 USD/ton, indicating the average in 20 major global ports from July 31, 2018 to August 29, 2019, sourced from Ship and Bunker (2019).
$$ {fc}_{ij}= BP\bullet \left(0.0392\bullet {s}_{ij}+5.582\right)/{\left(5.4178\bullet {s}_{ij}^{0.1746}\right)}^3 $$
(4)
$$ {vc}_{ij}=108.05\bullet {s}_{ij}^{0.6257}+1.4095\bullet {s}_{ij}+6125.9 $$
(5)
$$ {pc}_{ij}=0.3936\bullet \left(12.556\bullet {s}_{ij}+1087.2\right)+5356 $$
(6)
Shippers
In this study, we calculate the generalized cost per TEU of shippers, to enable them to choose between direct shipment and transshipment. Equation (7) expresses the generalized cost for direct shipment. Wt_{i} indicates the waiting time at port i, such as for the customs procedure. The term 30/2f_{od} represents the average waiting time for calling the vessel. As the unit of frequency is times per month, the number 30 is multiplied. As for denominator, the number 2 is used to calculate the average value (Wang et al. 2014). Other timerelated factors considered include the navigation time and loading or unloading time. The monetary cost of shippers includes the freight rate (σ_{ij}) calculated in eq. (8), imposed by the shipping lines. As expressed in eq. (8), the shipping lines determine the freight rate based on avoiding a deficit. The smallest integer value is calculated as the freight rate, exceeding the total cost by cargo demand. Equation (9) expresses the generalized cost for transshipment. The generalized cost for direct shipment includes origin to destination port, while the generalized cost for transshipment includes origin to hub port, and hub to destination port. In principle, the generalized costs for direct shipment and transshipment have the same components.
$$ {GC}_{od}^{DR}=\alpha \left({Wt}_o+\frac{30}{2{f}_{od}}+\frac{q_{od}}{L_o}+\frac{u_{od}}{v_{od}}+\frac{q_{od}}{L_d}+{Wt}_d\right)+{\sigma}_{od} $$
(7)
$$ {\sigma}_{ij}=\left\{\kern.4em \min\ {\sigma}_{ij}{Q}_{ij}\bullet {\sigma}_{ij}\ge {Cs}_{ij},{\sigma}_{ij}\in \mathbb{N}\operatorname{}\right\} $$
(8)
$$ {GC}_{od}^{TS}=\alpha \left({Wt}_o+\frac{30}{2{f}_{oh}}+\frac{q_{oh}}{L_o}+\frac{u_{oh}}{v_{oh}}+\frac{30}{2{f}_{hd}}+\frac{u_{hd}}{v_{hd}}+\frac{q_{hd}}{L_d}+{Wt}_d\right)+{\sigma}_{oh}+{\sigma}_{hd} $$
(9)
Calculation flow
In this study, we consider a virtual maritime network instead of a real one, so that we can easily modify the two factors—geographical distance and cargo demand—as illustrated in the examples in Fig. 1. The virtual network has one origin port, two destination ports, and one hub port. These are the minimum required values to investigate the influence of geographical changes in one port on another. For instance, focusing on the shipment to Destination 2 in Fig. 1, we can simulate the influence of geographical changes in Destination 1 on Destination 2, by comparing examples 1 and 2.
Figure 2 illustrates the calculation flow of this study. First, we determine the number of OD pairs and the four factors, that is, cargo demand (Q_{ij}), distance (u_{ij}), value of time (α), and bunker price (BP), as inputs for the virtual maritime network. Second, we implement the twostage model in direct shipment and transshipment separately. At the upper stage, the shipping lines decide the variables including vessel size, navigation speed, and frequency to minimize their cost in each network, as expressed in eq. (1). In transshipment, the calculations are further categorized as from origin to hub port and from hub to destination port. Theoretically, we can calculate one of the decision variables, that is, the optimum navigation speed, because the cost of the shipping lines is a concave function of the navigation speed, as expressed in eq. (1). The partial differentiation of eq. (1) can be calculated to obtain the optimum navigation speed in eq. (10). As vessel and fuel costs are positive, we can calculate the optimum navigation speed based on these costs, as expressed in eq. (11). The optimum navigation speed depends on vessel and fuel costs; especially, as fuel cost is in proportion to bunker price (BP), the optimum navigation speed decrease when bunker price increase. This matches the property of the relationship between operating speed of vessels and fuel price (Notteboom and Vernimmen 2009). Furthermore, as vessel and fuel costs depend on the vessel size, the optimum navigation speed also depends on it, thus enabling a direct calculation of the optimum navigation speed using the optimum vessel size. Owing to the relatively low computational complexity required to obtain optimum values of the other decision variables, including vessel size and frequency, we implement a bruteforce search as the solution algorithm.
In the lower stage, the shippers’ generalized cost is calculated based on the shipping lines’ decisions on vessel size, navigation speed, and frequency. Subsequently, we compare the generalized cost of each network. We compare all patterns of the inputs, such as distance and cargo demand of OD pairs, to reveal costeffectiveness of direct shipment or transshipment. These calculations are coded in MATLAB, which is a multiparadigm programming language and numerical computing environment, and processed on an Intel® Core™ i58265U processor with 8 gigabytes of random access memory.
$$ \frac{\partial {Cs}_{ij}}{\partial {v}_{ij}}=\frac{f_{ij}{u}_{ij}}{v_{ij}^2}\left(2\bullet {fc}_{ij}\bullet {v}_{ij}^3{vc}_{ij}\right)\ \mathrm{where}\ vc>0\ and\ fc>0 $$
(10)
$$ {v}_{ij}={\left(\frac{v{c}_{ij}}{2\bullet {fc}_{ij}}\right)}^{\frac{1}{3}} $$
(11)