Table 1 Description of parameters and variables

I

Collection of all sites (i, j ∈ I), I₁ represents a collection of Chinese sites, I₂ represents a collection of European sites.

I₁

Collection of train companies (k, g ∈ I₁).As the train company is based on the Chinese city, it is represented by the collection of Chinese stations.

T

Decision period length, weeks.

\( {\mathrm{b}}_{\mathrm{ik}}^{\mathrm{t}} \)

The initial volume of k company’s containers at station i at the end of t period, FEU.

\( {\mathrm{d}}_{\mathrm{ijk}}^{\mathrm{t}} \)

The demand of k company’s container transportation from station i to station j in t period, FEU.

c_ij

Single case transportation cost from station i to station j, USD / FEU.

cj

Rental of single container and single time cost of CRCT, USD / FEU.

cy

Single case and single time cost of COSCO Shipping Group, USD / FEU.

cj^outlast

Rental of overdue single container of CRCT, USD / FEU * week.

cy^outlast

Rental of COSCO Shipping Group’s single case overdue cost, USD / FEU * week.

\( c{j}_i^{trans} \)

Transportation cost of single container from station i to the nearest rental and return point of CRCT, USD / FEU.

\( \mathrm{c}{\mathrm{y}}_{\mathrm{i}}^{\mathrm{trans}} \)

Transportation cost of single container from station i to the nearest rental and return point of COSCO Shipping Group, USD / FEU.

\( {\mathrm{c}}_{\mathrm{i}}^{\mathrm{store}} \)

Single container and single cycle storage cost of site i, USD / FEU * week.

hj

Leasing the container of CRCT for a single time, week.

hy

Lease the container of COSCO Shipping Group for a single time, week.

\( {\mathrm{x}}_{\mathrm{ijkg}}^{\mathrm{t}} \)

In t period, K company calls the number of containers from G company at site i to site j, FEU.

\( \mathrm{l}{\mathrm{j}}_{\mathrm{ik}}^{\mathrm{t}} \)

In t period, K company rented the container volume of CRCT at site i, FEU.

\( \mathrm{l}{\mathrm{y}}_{\mathrm{ik}}^{\mathrm{t}} \)

In t period, K company rented COSCO Shipping Company’s container volume at site i, FEU.

\( \mathrm{h}{\mathrm{j}}_{\mathrm{ik}}^{\mathrm{t}} \)

In t period, K company returns the container quantity of CRCT at site i, FEU.

\( \mathrm{h}{\mathrm{y}}_{\mathrm{ik}}^{\mathrm{t}} \)

In t period, K company returns the container quantity of COSCO Shipping Company at site i, FEU.

hjo^t

The container quantity of China Railway Container Co., Ltd. not returned due to overdue period in t period, FEU.

\( \mathrm{hy}{\mathrm{o}}_{\mathrm{dom}}^{\mathrm{t}} \)

Container volume of COSCO Shipping Company in t period due to overdue return in China, FEU.