Adopting a Bi-level Optimization Method for the Freight Transportation Problem: A Multi-objective Programming Approach

Nowadays, various companies find it a crucial task to elevate their distribution system efficiency to survive in what is an intensely competitive environment. Freight transportation via road systems may not always be a practical option for the shipper because of financial, social, and environmental aspects. So, in addition to the road system, the shipper can benefit greatly from the rail system. Thus, taking advantage of different transportation modes can help shippers to tackle, to some extent, arising issues. In this paper, we aim at investigating a freight transportation network including rail and road transportation modes in which the shipper's choice affects the decisions deeply. Given that this problem as a bi-level programming model, the rail operator acts as a leader and the shipper is treated as a follower. Once the rail operator reveals the rail transportation price, an appropriate response is received from the shipper and this pattern is repeated until an equilibrium is achieved. From the sustainability view, a multi-objective mixed-integer programming model is developed to simulate their interactions considering a sustainable triple bottom line. Employing the Karush-Kuhn-Tucker approach, we converted the formulated bi-level programming model into a single one. Several numerical examples are investigated to evaluate the validity of the developed model and approaches. Different in-depth sensitivity analyses are carried out and the impacts of several factors, such as line and train capacity, on the rail operator and shipper's decisions are investigated. The results reveal that the rail operator could increase their market share by a wise long-term investment in the rail infrastructure.