## 1 Introduction
Additionally, the feasibility of CTR under uncertain carbon tax rate has not been verified in the different decision-making scenarios of the GIMRP;
2. The uncertainty of carbon tax rate and capacity has not been fully formulated in the IMRP literature, which limits the improvements to the RT through solving the IMRP. The widely used triangular and trapezoidal fuzzy numbers are not applicable for modeling uncertainty in cases where limited data are available for the IMRP;
3. Although the MTW has been formulated in IMRP research, its feasibility in the GIMRP under capacity and carbon tax rate uncertainty remains unclear. Therefore, it is necessary to demonstrate its feasibility in this problem scenario. In order to address the limitations of the existing literature, this study was established based on research by [PERSON] et al. [16] and [PERSON] et al. [7], and focuses on a green intermodal routing problem with MTW under interval fuzzy capacity and carbon tax rate (GIMRPMTWIFCCTR). The main contribution of this study is that it combines the MTW and the twofold uncertainty of capacity and carbon tax rate in the GIMRP to achieve improved timeliness and reliability. The improvement will be demonstrated in a numerical experiment. Another contribution is that we conduct a systematic numerical experiment to summarize many managerial insights to help the IMO to provide the receiver with a high-quality IMT service that yields reduced costs, improved timeliness and reliability, and less CEs. The following research is detailed in this article:  
1. An MTW that is a mixture of HTW and STW is integrated into the GIMRP to improve the timeliness of transportation planning. Furthermore, CTR is employed to reduce the CEs of transportation, in which the carbon tax rate uncertainty is modeled by an IFN. The interval fuzzy capacity of the intermodal network is also included in the GIMRP discussed in this study;
2. An interval fuzzy linear optimization model (IFLOM) is established to address the proposed problem, and its equivalent crisp reformulation is then given, in which the optimization level is introduced to reflect the receiver's attitude towards the RT. Instead of developing heuristic algorithms, as done in works by [PERSON] et al. [13] and [PERSON] et al. [14], which cannot ensure a global optimum solution (GOS) is obtained, this study uses the branch-and-bound algorithm, an exact solution method that is suitable for solving mixed-integer linear optimization models [21], to obtain the GOS to the GIMRPMTWIFCCTR;
3. A numerical experiment is presented to verify the feasibility of the proposed methods that includes uncertainty modeling, MTW, and CTR under carbon tax rate uncertainty. Some managerial conclusions are further detailed to provide decision support for the IMO to organize the IMT in such a way that helps the receiver to obtain an improved service level in terms of economics, timeliness, and reliability, while also achieving environmental benefits. The remaining sections of this study are organized as follows. In Section 2, the studied routing problem is briefly described, and an IFLOM is then proposed to address the GIMRPMTWIFCCTR. In Section 3, defuzzification of the proposed IFLOM is carried out to make the GIMRPMTWIFCCTR solvable, in which the optimization level is introduced into the processing to reflect the receiver's attitude towards the RT. In Section 4, a numerical experiment is presented to demonstrate the feasibility of the proposed method and reveal the relationships between the economic, carbon emission, and reliability objectives in the GIMRPMTWIFCCTR. The advantages of the uncertainty modeling and the use of the MTW are verified, and the feasibility of CTR under an uncertain carbon tax rate is analyzed. The influence of the shipper's release time and the interval fuzzy capacity's stability on the routing optimization is investigated at the end of this section. Finally, the conclusions are summarized in Section 5.