Table 1 Notations for the model

Parameters of nodes

  S

set of harbors, s∈S

  D

set of inland depots

  R

set of harbors and inland depots, \(R=S\bigcup D\), m,n∈R

  C

set of customers, k∈C

  β_kn

1, if customer k is coverd/served by depot n; 0, otherwise.

Parameters of costs

  CF_mn

unit transportation cost of full container between depot m and n

  CE_mn

unit transportation cost of empty container between depot m and n

  CYF_kn

unit transportation cost of full container between customer k’s location and depot n

  CYE_kn

unit transportation cost of empty container between customer k’s location and depot n

  CZ_n

unit leasing cost of empty container at depot n

  CG_k

penalty cost of transportation enterprise if customer k’s demand for empty container is delayed

Other parameters

  a_k

quantity of import-full containers of customer k

  b_k

quantity of export-full containers of customer k

  YIF_nk

quantity of full containers for import from depot n to customer k’s location

  YEF_kn

quantity of full containers for export from customer k’s location to depot n

  YE_nk

quantity of empty containers from depot n to customer k’s location

  YE_kn

quantity of empty containers from customer k’s location to depot n

  W_kn

unit revenue for transporting customer k’s full container at depot n

decision variables

  IF_mn

quantity of import-full containers that transported from depot m to depot n

  EF_mn

quantity of export-full containers that transported from depot m to depot n

  XE_mn

quantity of empty containers that transported from depot m to depot n

  GEF_k

quantity of customer k’s abandoned demand for empty container by transportation enterprise

  Z_n

quantity of empty containers leasing at depot n

  SE_s

quantity of empty containers leasing at harbor s