Methodology

Model development

This study uses a two$-$stage 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$)$

GCk

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

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

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

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

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

hci Handling charges in port i $[$USD$/$TEU$]$

fij Frequency from port i to j $[$time$/$month$]$

Tagawa et al$.$ Journal of Shipping and Trade $(2021)6$:$4$ Page $3$ of $15$

Model development $($Continued$)$

sij Vessel size deployed from port i to j $[$TEU$]$

vij Navigation speed from port i to j $[$knot$]$

uij Navigation distance between port i and j $[$nm$]$

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

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

$\backslash$sigma ij Freight rate for transport from port i to j $[$USD$/$TEU$]$

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

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

Wti Wait time of shippers in port i $[$day$]$

$\backslash$alpha Value of time of shippers $[$USD$/$hour$-$TEU$]$

$\backslash$gamma Unknown parameter for slot utilization

$\backslash$mu Unknown parameter for distance of the hub port

$\backslash$delta hd Unknown parameter of cargo demand from the hub to destination port, to reveal the competitiveness of

the hub port

summarize