Spatial Optimization Model for US Imports

The model developed for US imports includes major shipping destinations and container ports in the United States and ports in exporting countries. Major container shipping destinations will be identified as one of each states of the US mainland, based on population. Major container exporting countries are identified on the basis of US import volume. The model includes several inland transportation modes in the United States to evaluate the effects of intermodal competition

transit of containers from ports in exporting countries to shipping destinations in the United States.

Figure 3.6. Regions associated with container trade and transportation modes for US imports.

Figure 3.6 represents the supply chain of container shipments from export ports in other countries to shipping destinations at US imports. Ocean vessels are the transportation mode used for shipping containers from ports in exporting countries to ports in the United States. The model for import includes barges, as one of the inland transportation modes, which are available to ship containers from the US Gulf ports to water access points along major US river systems. Since barges are the cheapest mode of inland transportation for long distances, barges are used for long-distance hauling. Containers imported in the Gulf ports in the United States can be moved to water access points by barge and then moved to shipping destinations by truck, called a truck-barge combination. For shipment from other US import ports, either rails or trucks are used to

Inland Transit Ocean Transit

Export Port f

Panama Canal Import Port u

Shipping Destination

d

𝑚_𝑢𝑑 – Rail/Truck/Truck-Barge Combination 𝑣_𝑓𝑢 – Container Ship

ship containers since trucks are the cheapest transportation mode for shipping containers short distances, followed by rail in general.

The international transportation system from exporting countries to the United States is optimized by using a linear programing model. The objective function of the spatial optimization model is to minimize transportation and handling costs in shipping containers from export ports to US ports. The objective function is specified as follows:

𝑊_𝑖𝑜 = ∑ ∑ 𝑄_𝑓𝑢^𝑜 (𝑇_𝑓𝑢^𝑜

where f is the index for ports in exporting countries, u is an index for container ports in the United States, 𝑄_𝑓𝑢^𝑜 is containers (TEUs) shipped from export port f to US port u without the Panama Canal, 𝑄_𝑓𝑢^𝑜𝑐 is containers (TEUs) shipped from export f to US port u through the Panama Canal, 𝑇_𝑓𝑢^𝑜 is ocean transportation costs by ship size from export port f to US port u without using the canal, 𝑇_𝑓𝑢^𝑜𝑐 is ocean transportation costs by ship size from export port f to US port u through the canal, ℎ_𝑓𝑢 is handling charges at ports f and u , 𝜌 is the toll rate per TEU at the Panama Canal, and 𝜎 is the delay cost per TEU at the Panama Canal.

Ocean transportation costs between exporting countries and the United States consist of costs from export ports to US ports. Since two ocean routes for container shipments are available, the least expensive route is chosen for container shipments. The first term of equation 3.17

represents the sum of ocean transportation costs from export ports to US ports without the Panama Canal and handling costs at the ports in exporting and importing countries. The next term indicates the sum of transportation costs from export ports to US ports through the Panama Canal, handling costs at ports, the PNC toll, and delay costs at the Panama Canal.

The domestic transportation system in the United States is optimized by using a linear programing model. The objective function of the spatial optimization model is to minimize transportation and handling costs in shipping containers from US ports to shipping destinations in the United States. The objective function is specified as follows:

𝑊_𝑖𝑙 = ∑ ∑ 𝑄_𝑢𝑑^𝑡 𝑇_𝑢𝑑^𝑡

where u is an index for the import ports in the United States, d is an index for shipping

destinations in the United States, t is an index for truck transportation mode, r is an index for rail transportation mode, b is an index for barge transportation mode, e is an index for barge access points, and T represents transportation costs. 𝑄_𝑢𝑑^𝑡 is the containers (TEUs) shipped from import port u to destination d by trucks, 𝑄_𝑢𝑑^𝑟 is the containers (TEUs) shipped from import port u to destination d by rails,𝑄_𝑢𝑒^𝑏 is containers (TEUs) shipped from import port u to barge point e by barges, 𝑄_𝑒𝑑^𝑡 is containers (TEUs) shipped from barge access point e to destination d by trucks, 𝑇_𝑢𝑑^𝑡 is transportation costs from import port u to destination d by trucks, 𝑇_𝑢𝑑^𝑟 is transportation costs from import port u to destination d by rails, 𝑇_𝑢𝑒^𝑏 is transportation cost from import port u to barge access point e by barges. 𝑇_𝑒𝑑^𝑡 is transportation cost from barge access point e to destination d by trucks.

Inland transportation costs in the United States are accumulated from import ports to shipping destinations by multi-inland transportation modes. When more than one transportation mode is available, the least expensive mode is chosen for container shipments. The first and second terms of equation 3.18 represent the sum of transportation costs for shipping containers from import ports to shipping destinations by trucks and rails, respectively; the third term represents the sum of transportation costs from import ports to barge access points by barges.

The last term indicates the sum of the transportation costs from barge access points to shipping destinations by trucks.

Since one of the objectives of this study is to optimize container flows from exporting countries to the United States under the spatial optimization and the objective function of the spatial optimization model for US import the sum of equations 3.17 and 3.18 is as follows:

𝑊_𝑖 = ∑ ∑ 𝑄_𝑓𝑢^𝑜 (𝑇_𝑓𝑢^𝑜

Transportation costs are divided into ocean transportation and domestic transportation costs.

Ocean shipments are divided into shipments from exporting countries to the United States, whether or not the shipments pass through the Panama Canal. Handling charges, such as loading and unloading charges, at the ports in exporting and importing countries, the PNC toll rate, and delay costs at the canal are included in the objective function.

Major constraints of the model are import demand in the United States, supply in exporting countries, cargo handling capacities of the U.S ports and the Panama Canal and the inventory-clearing condition at export and import ports. The constraints are presented as follows:

∑ 𝑄_𝑢𝑑^𝑡

∑ ∑ 𝑄_𝑓𝑢^𝑜𝑐 (TEUs) at destination d, 𝑄_{𝐺𝑢𝑙𝑓,𝑒}^𝑏 is the number of containers from the Gulf ports to barge access points e. Other variables are defined previously.

Equations 3.20 and 3.21 represent import demand and export supply constraints respectively. Equation 3.20 represents the import demand constraints in the United States.

Equation 3.21 indicates that the total containers shipped from ports in exporting countries should be equal to or smaller than the quantities of containers shipped to the United States. Equations 3.22 and 3.23 represent container handling capacities at ports in the United States and the Panama Canal, respectively. The total quantities of containers handled by each port and the Panama Canal should be equal to or smaller than their annual handling capacities. Equation 3.24 indicates that the quantity of containers received at import port u is equal to the sum of

containers shipped from export ports. Equation 3.25 describes that the quantity of containers received at destination d is equal to sum of the containers from import ports and barge access

points. Equation 3.26 describes an inventory-clearing condition at ports, indicating that the container quantities received by each import port from exporting countries must equal to sum of containers shipped to destinations and barge access points. Equation 3.27 is an inventory clearing condition at barge access points indicating that the quantity of containers received in each barge access point by barge from the Gulf ports should be equal to the quantity shipped out through trucks to destinations.