Sparrow search algorithm with adaptive t distribution for multi-objective low-carbon multimodal transportation planning problem with fuzzy demand and fuzzy time

In this paper, we present a multi-objective low-carbon multimodal transportation planning problem with fuzzy demand and fuzzy time (MOLCMTPP-FDFT) that minimizes both cost and time while incorporating mandatory carbon emission, carbon tax, carbon trading, and carbon offset policies. Chance constrained programming and interval programming are introduced to formulate the fuzzy demand chance constrained programming model and the fuzzy time interval programming model, respectively. Then, the mathematical model of uncertain MOLCMTPP-FDFT is transformed into a deterministic model. Based on the sparrow search algorithm, t distribution and the concept of Pareto optimality for multi-objective optimization, we also propose a solution strategy for the proposed model. In this algorithm, the number of iterations is used as the degree of freedom of -to update the sparrow location, which strikes a balance between the capabilities of global search and local search. Finally, the proposed algorithm and MOLCMTPP-FDFT are applied to a real case, resulting in a minimum cost of 260730.48 and a time duration of 13.044, which outperform the minimum cost of 268874.88 and minimum time of 18.32 obtained using single-mode transportation. The carbon emissions resulting from the lowest-cost solution obtained using single-mode transportation are 3,198.48, which are significantly more than the allowed emissions. Therefore, the proposed algorithm and mathematical model of MOLCMTPP-FDFT are valuable tools for optimizing multimodal transportation route. Additionally, the experimental results not only validate the superior efficiency and energy-saving benefits of the proposed multimodal transportation routes in comparison to the actual single-modal transportation, but also demonstrate the applicability of different low-carbon policies.