## 1 Introduction  
Intermodal transportation (IMT), in which goods are transported in containers, can improve the quality of transportation services in terms of economics, timeliness, and environmental sustainability compared with unimodal transportation. In China, IMT has been in a period of rapid development since 2015; it exhibited an annual growth rate of 30% and reached a freight volume of 263.9 million Twenty feet equivalent unit (TEU) containers in 2021 [1]. IMT has already become the backbone of the comprehensive transportation system in China. It requires the coordination of various transportation modes (e.g., rail, road, and water) that have different capacities, operational times, in-transit speeds, and cost structures in in-transit and transfer processes. As a result, its organization is more difficult than that of unimodal transportation. Consequently, quantitatively supporting the intermodal operator (IMO) in planning the best intermodal route to satisfy the receiver's requirements and improve the service level and environmental sustainability of transportation in long-distance and large-volume IMT scenarios remains a key issue [2]; hence, studies have focused on the intermodal routing problem (IMRP). The transportation industry accounts for more than 30% of total carbon emissions (CEs) and is acknowledged to be the key contributor to CEs [3]. Furthermore, the rapid development of the automobile industry has led to a significant increase in carbon emissions [4]. Although IMT is a low-carbon means of transportation compared with unimodal transportation, its application over long distances and with large volumes still results in a large amount of carbon emissions. Solving the green intermodal routing problem (GIMRP) is a promising approach to achieve low-carbon IMT. Many relevant studies have explored the GIMRP, in which carbon tax regulation (CTR) is the most widely adopted method to reduce the CEs of routing [5, 6, 7, 8, 9]. [PERSON] et al. [10] pointed out that CTR can reduce CEs more effectively than carbon cap-and-trade regulation in the GIMRP for food grain transportation. However, existing studies preferred to model the carbon tax rate in the CTR as a deterministic parameter, and did not consider its uncertainty caused by various factors, including government policy modifications, regional differences, and market changes. Simultaneously, a carbon tax pilot project has not been implemented by the Chinese government and thus it is difficult to determine the value of the carbon tax rate exactly, and this uncertainty could influence the performance of CTR in reducing CEs as well as the optimization result of the GIMRP. The uncertainty of the carbon trading price in carbon cap-and-trade regulations has been discussed in a few studies [11, 12], and has been demonstrated to significantly influence the optimization results. Similarly, formulating the uncertainty of the carbon tax rate can enhance the feasibility of a GIMRP approach that considers the reduction in CEs as an important issue. Furthermore, the uncertainty of the carbon tax rate might influence the feasibility of CTR in reducing the CEs of the planned intermodal route (PIMR), and it is meaningful to investigate such a feasibility in different decision-making scenarios. To date, a few studies, e.g., [PERSON] et al. [7, 10], have explored the feasibility of CTR in different IMRP decision-making scenarios where the receiver holds different attitudes on the reliability of transportation (RT). However, [PERSON] et al. [7, 10] referred to a deterministic carbon tax rate, and there are no existing studies that verify the feasibility of this in the condition that the carbon tax rate is considered uncertain. Another concern in the IMRP is the requirement for on-time transportation that could better coordinate with post-transportation processing.