schematically in Fig. 4.
4.3. Shipper-carrier problem
1) Bi-level model
The carrier-shipper interactions are captured by a bi-level modelling approach when the carrier model
is the upper level model and the shipper model is the lower level model. Fig. 5 shows the shipper-carrier
interactions model and their related decisions. Fig. 6 summarizes the bi-level modelling approach and the
equilibrium concepts.
At the upper level, carriers make pricing and routing decisions with the knowledge of shipper flow.
Carriers investigate or forecast the shipper demand and determine service charges and delivery routes
through competitions and interactions among them.
At the lower level, shippers determine the production, shipment and consumption pattern based on the
transportation service information presented by the carriers. Shippers consider the ocean/land
transportation and port service charges and carrier link flows.
For the carrier competitive game, the Nash equilibrium principle is used. Shipper behaviour is
modelled using the SPE principle. The concept of multi-leader-follower game is applied to a multi-level
game due to multiple and competitive leaders.
2) Heuristic Algorithm
Both carrier and shipper problems are formulated using Variational Inequalities (VI). The individual
carrier and shipper models are solved employing the projection algorithm which resolves VI into a
sequence of sub-problems by the general iterative scheme and the three-level model is solved by a
heuristic algorithm. The heuristic to solve the bi-level model is as follows: Carriers make pricing and
routing decisions. Shippers determine production, shipment and consumption pattern, considering
carriers' pricing and routing. Carriers examine the reactions of shippers for every feasible scenario. From
all these reactions, carriers choose the one that gives them the maximum profit.