## 1 Introduction
The least-time transportation considered by the IMRP [13] is not suitable in this situation, and time windows are widely employed in existing studies to reduce or avoid both early and late container delivery to the receivers [1]. An appropriate time window helps to achieve on-time transportation and strengthens the use of advanced manufacturing strategies, e.g., just-in-time production, by receivers in the post-transportation processing of the delivered goods. Currently, theconcept of a soft time window (STW) that allows but penalizes early and late deliveries has been commonly applied in IMRP modeling [1; 5; 6; 14; 15]. The timeliness of the PIMR can be optimized through reducing the penalty costs included in the economic objective of the routing. However, the STW allows the delivery of containers to be accomplished at any time within the planning horizon on the condition that the economic advantage can be achieved [10]. In this case, the STW might lead to the delivery of containers being too early or too late, thereby violating the receiver's tolerance. Meanwhile, the hard time window (HTW), which restricts the delivery of containers to within the time window, is not flexible and could lead to a significant increase in costs [16]. Under this situation, in the modeling of a vehicle routing problem, [PERSON] et al. [17] proposed the mixed time window (MTW, also known as the flexible time window), combining a STW and HTW to maintain the flexibility of delivery and simultaneously preventing overly early or late deliveries. In the MTW, the time interval of the HTW covers that of the STW. The delivery of containers must be accomplished within the predetermined HTW; however, delivery can occur outside of the STW interval, but would incur a penalty. At present, a few studies, e.g., that by [PERSON] et al. [16], have formulated such a time window in the IMRP. However, [PERSON] et al. [16] did not consider the reduction in CEs and modeled the IMRP in an uncapacitated intermodal network. Still, this work provided a solid foundation for our research. Last but not least, capacity is the key factor influencing the reliability of the IMRP. The IMRP should be conducted before the actual transportation where the PIMR is implemented. Effective communication is usually lacking between the IMO, carriers, and terminal operators. Moreover, various factors summarized by [PERSON] et al. [1] influence the stability of the capacities of transportation modes and transfer nodes in an intermodal network. Therefore, it is difficult for the IMO to exactly determine or predict the intermodal network capacity in the advanced IMRP, which results in capacity uncertainty. The IMO can only obtain the interval of the possible values of the capacity of a transportation mode or a transfer node in the intermodal network. To date, relevant studies have preferred to employ triangular or trapezoidal fuzzy numbers [1; 10; 18; 19; 20] to formulate capacity uncertainty when modeling the capacitated IMRP. However, in practical planning, the IMO cannot obtain enough data to exactly determine the most likely value or interval to establish a triangular or trapezoidal fuzzy capacity [7]. Therefore, it is more feasible for the IMO to model the uncertainty using interval fuzzy numbers (IFNs) that only use the minimum and maximum possible values to formulate uncertainty. Currently, only a few studies on intermodal transportation planning, e.g., [PERSON] et al. [7], have adopted IFNs in the IMRP under capacity uncertainty. Additionally, to the best of our knowledge, there is no research that has comprehensively considered the twofold uncertainty of both the carbon tax rate and intermodal network capacity and modeled them by IFNs. Above all, the existing relevant research has the following limitations, which motivate the present study:  
1. Although CTR has been widely adopted in the GIMRP, the uncertainty of its carbon tax rate has not been considered.