A simulated annealing approach for the multi-periodic rail-car fleet sizing problem

In this paper, we proposed a new formulation and a solution procedure for optimizing the fleet size and freight car allocation wherein car demands and travel times are assumed to be deterministic and unmet demands are backordered. We assume that unmet demands become zero at the end of the planning horizon, i.e., the car demands are totally satisfied through the horizon. There are important interactions between decisions on sizing a rail car fleet and utilizing that fleet. Consequently, the optimum use of rail-cars for demands response in the length of the time periods is one of the main advantages of the proposed model. The model also provides rail network information such as yard capacity, unmet demands, and number of loaded and empty rail-car at any given time and location. Computational tests showed that small-size instances can be solved by the exact approach in a fair amount of CPU time, but it is not feasible for medium and large-size instances. To tackle this problem, a Simulated Annealing (SA) algorithm is proposed to solve the model. The algorithm works efficiently on a neighborhood search within solution space, acceptance probability, and inferior solutions to escape from trap (i.e., local optimal solution). Numerical examples are solved to check for the efficiency and validity of the SA algorithm.