4.2. Symbol Description
(1) Parameters
L: The set of CRE lines.
: The origin and destination set of the CRE.
: The city set of CRE in China.
: The city set of CRE in foreign regions.
: The port set of the international liner company.
: The port set of the international liner companies in China.
: The port set of liner companies in foreign regions.
T: The set of decision period.
: The origin of the CRE; .
: The destination of the CRE; .
: Empty container demand of the CRE company in city i in t period, , .
: Empty container demand of the international liner company in port p in t period, , .
: Empty container demand of the international liner company in city i in t periods; , .
: Empty container supply of the CRE company in city i in t period, , .
: Empty container suppky of the international liner company in port p in t period, , .
: Empty container supply of the international liner company in city i in t periods; , .
: The maximum volume of empty containers allowed by the CRE company at the city i, .
: The maximum volume of empty containers allowed by the international liner company at the port p, .
: The maximum volume of empty containers allowed by the international liner company at the city i, .
: Empty container repositioning unit cost from city i to city j in China; .
: Empty container repositioning unit cost from city i to city j in foreign area; .
: Empty container repositioning unit cost from city i to city j by sea-land intermodal transport; , .
: Empty container repositioning unit cost of the CRE l, .
: Empty container repositioning unit cost of the from city
i to city
j the CRE
l;
,
. Due to the existence of one-way trains,
has different values.
: Empty containers unit rental cost at the city i; .
: The mutual rental cost between CRE company and liner company.
: Booking price.
: Unit empty container repositioning cost between city i and port p in China; , .
: Unit empty container repositioning cost between city i and port p in foreign area; , .
: Unit empty container repositioning between port p and port q in China; .
: The maximum booking capacity that the international company can provide for the CRE company in t period; .
: the empty container transportation capacity between port P and port Q in t period. ∀.
:the empty container transportation capacity of CRE line l in t period, ∀
(2) Decision variables
: The volume of empty containers transported from node i to node j through line l in the t period; , .
: The volume of empty containers transported from node i to node j by sea land intermodal transportation in the t period; , .
: Empty container inventory of the CRE company at the end of period t at city i; , .
: Empty container inventory of port p at the end of period t; , .
: Empty container inventory of the international liner company at city node i at the end of period t; , .
: The volume of containers rented by the CRE company in the city node i at the t period; , .
: The volume of containers rented by the international liner company in the city i at the t period; , .
: The rental volume of the international liner company at port p in t period; , .
: The empty container volume of the international liner company rented by the CRE company at the city node i in the t period; , .
: The empty container volume of the CRE company rented by the international liner company at the city i in the t period; , .
: The empty container volume of the international liner company at port p rented by the CRE company at city i in the t period; , , .
: The empty container volume of the international liner company renting the CRE company in the city i at port p in the t period; , , .
: Empty container repositioning volume of the international liner company between port p and port q in the t period; , .
: Empty container repositioning volume of the international liner company from port p to city i in the t period; , , .
: Empty container repositioning volume of the international liner company from city i to port p in the t period; , , .
4.3. Mathematical Model
The mathematical model of this paper mainly considers three constraints: empty container inventory balance, empty container transportation capacity limit and empty container storage capacity limit. Among them, the balance of empty container inventory mainly depends on the empty container volume of inflow and outflow nodes in the two periods.
Figure 3 shows the diagram of empty container in and out of each node in the adjacent cycle, which is before and after the cooperation between the international liner company and the CRE company. The left side of the node is the inflow of empty containers, and the right side is the outflow of empty containers. Under both of cooperation, this paper constructs the empty container flow balance constraint according to
Figure 3, as shown in
Section 4.3.1,
Section 4.3.2 and
Section 4.3.3.
4.3.1. The Empty Container Repositioning Optimization Model of CRE
The objective function is established to minimize the total empty container repositioning cost of the CRE company in the planning period, including the cost of empty container repositioning of inter regional city (cost of CRE transportation + cost of sea land combined transport), the cost of empty container repositioning of China regional cities, the cost of empty container repositioning of European regional cities, the rental cost, the storage cost.
Subject to: (1) The empty container flow balance in the Europe cities
Storage volume at the end of the previous period + the volume of empty containers repositioned from other cities in the same region in this period + the volume of containers rented by the CRE company from container rental company in this period + empty container supply in the period = Storage volume in the period + the demand of empty containers in this period + the volume of empty containers repositioned to other city nodes in this region in this period + the volume of empty containers repositioned by the CRE in this period + the volume of empty containers repositioned by sea land combined transport in this period.
(2) The empty container flow balance in the China cities
Storage volume at the end of the previous period + the volume of empty containers rented by the CRE company from the international liner company at all ports in this period + the volume of empty containers repositioned from other cities in this region in this period + the volume of empty containers repositioned by sea and land combined transportation in the period + the volume of empty containers rented by CRE company from the rental company in the period + the empty container supply in this period = Storage volume in this period + the empty container demand in this period + the volume of empty containers repositioned to other cities in this region in this period.
(3) Storage capacity constraints
Constraint (4) is the limit of the storage capacity of the CRE company in the city.
(4) Transport capacity constraints
Constraint (5) is the capacity restriction of empty containers transported by CRE line.
4.3.2. The Empty Container Repositioning Optimization Model of the International Liner Company
The objective function is established to minimize the total empty container repositioning cost of the international liner company in the planning period, including empty container repositioning between ports, the rental cost, the storage cost of empty containers at ports and cities, the cost of empty container repositioning and rental by the international liner companies in China, the cost of empty container repositioning between ports and cities in China, and the cost of empty container repositioning between ports and cities in Europe.
(1) The empty container flow balance in the cities
Storage volume at the end of the previous period+ the volume of empty container repositioned by the international liner company from all ports in this period+ the volume of empty container from rental company in this period + empty container supply in this period= Storage volume in this period + the demand of empty container in this period+ the volume of empty container repositioned to the port by the international liner company in this period.
(2) The empty container flow balance in the china ports
Storage volume at the end of the previous period + the rental volume of the international liner company from the CRE company at the city in this period+ the repositioning volume of the international liner company from other ports during this period + the rental volume of the rental company in this period+ empty container supply in this period+ the repositioning volume of the international liner company from the city to all ports in this period = Storage volume in this period+ the volume of containers rented by the CRE company from the international liner company at the city in this period+ the demand of empty container in this period+ the repositioning volume of the international liner company from this port to the others in this period.
(3) The empty container flow balance in the Europe ports
Storage volume at the end of the previous period + the repositioning volume of the international liner company from other ports during this period + the rental volume of the rental company in this period+ empty container supply in this period+ the repositioning volume of the international liner company from the city to all ports in this period = Storage volume in this period+ the volume of containers rented by the CRE company from the international liner company at the city in this period+ the demand of empty container in this period+ the repositioning volume of the international liner company from this port to the others in this period.
(4) Storage capacity constraints
Constraint (10) is the limit of the storage capacity of the international liner company at the city node. Constraint (11) is the limitation of the storage capacity of the international liner company at the port.
(5) Transport capacity constraints
Constraint (12) is the capacity restriction of empty containers transported by the international liner companies between ports.
4.3.3. The Empty Container Repositioning Optimization Model of the Cooperation between CRE and the International Liner Company
The objective function is established to minimize the total empty container repositioning cost of the cooperation in the planning period. The first part is the total cost of empty container repositioning of the CRE company, including the cost of empty container repositioning of inter regional city nodes (cost of CRE transportation + cost of sea land combined transport), the cost of empty container repositioning at China cities, the cost of empty container repositioning of at European cities, the external rental cost o, the storage cost, the transportation cost and the cost of container rental of the CRE company at all ports from the international liner company, transportation cost and container rental cost of the CRE company from the international liner company at the cities. The second part is the total cost of empty container repositioning of the international liner company, including empty container repositioning between ports, the external rental cost, the storage cost of at ports and cities, the cost of empty container repositioning and rental by the international liner companies from CRE company in China, the cost of empty container repositioning and rental by the international liner companies from CRE company in Europe, the cost of empty container repositioning between ports and cities in China, and the cost of empty container repositioning between ports and cities in Europe.
(1) The empty container flow balance in the Europe cities
Storage volume at the end of the previous period + the volume of empty containers repositioned from other cities in the same region in this period + the volume of containers rented by the CRE company from the international liner company at all ports in this period + the volume of containers rented by the CRE company from container rental company in this period + empty container supply in the period + the volume of containers rented by the CRE company from the international liner company at the city in this period = Storage volume in the period + the volume of containers rented by the international liner company from the CRE company at the city in this period + the demand of empty containers in this period + the volume of empty containers repositioned to other city nodes in this region in this period + the volume of empty containers repositioned by the CRE in this period + the volume of empty containers repositioned by sea land combined transport in this period + the volume of empty containers rented by the international liner company from the CRE company at all ports in this period.
(2) The empty container flow balance in the China cities of CRE
Storage volume at the end of the previous period + the volume of empty containers repositioned by the CRE in the period + the volume of empty containers rented by the CRE company from the international liner company at all ports in this period + the volume of empty containers repositioned from other cities in this region in this period + the volume of empty containers repositioned by sea and land combined transportation in the period + the volume of empty containers rented by CRE company from the rental company in the period + the volume of empty containers rented by the CRE company from the international liner company at the city in this period + the empty container supply in this period = Storage volume in this period + the empty container demand in this period + the volume of empty containers repositioned to other cities in this region in this period + the empty container rented by international liner company from the CRE company at all ports in this period +the volume of empty containers rented by the international liner company from the CRE company at the city in this period.
(3) The empty container flow balance in the cities of the international liner company
Storage volume at the end of the previous period+ the volume of empty container repositioned by the international liner company from all ports in this period+ the volume of empty container from rental company in this period+ the volume of empty containers rented by the international liner company from the CRE company at the city in this period + empty container supply in this period= Storage volume in this period+ the volume of containers rented by the CRE company from the international liner company at the city in this period+ the demand of empty container in this period+ the volume of empty container repositioned to the port by the international liner company in this period.
(4) The empty container flow balance in the china ports of the international liner company
Storage volume at the end of the previous period + the rental volume of the international liner company from the CRE company at the city in this period+ the repositioning volume of the international liner company from other ports during this period + the rental volume of the rental company in this period+ empty container supply in this period+ the repositioning volume of the international liner company from the city to all ports in this period = Storage volume in this period+ the volume of containers rented by the CRE company from the international liner company at the city in this period+ the demand of empty container in this period+ the repositioning volume of the international liner company from this port to the others in this period+ the volume of containers rented by the CRE company from the international liner company at all port.
(5) The empty container flow balance in the Europe ports of the international liner company
(6) Storage capacity constraints
Constraints (4), (10)–(11).
(7) Transport capacity constraints
Constraints (5), (12).
(8) Booking capacity constraint
Constraint (19) is the maximum booking capacity that the international liner company can provide for the CRE company in each cycle.