## 1 Introduction
In order to optimize the gearbox efficiency and reduce the gearbox volume for a two-stage bevel helical gearbox, the Taguchi and GRA techniques were also utilized in [42]. Furthermore, in order to improve the efficiency and lower gearbox bulk, these techniques were applied to the optimization of a TSHG with in [43]. [PERSON], [PERSON], and [PERSON], [44] used Taguchi and GRA methods to solve the MOOP. The aim of this work was to reduce the gearbox height and enhance the gearbox efficiency. The Taguchi and GRA techniques were also applied in [45] to minimize the gearbox cross-section area and maximize the gearbox efficiency. The TOPSIS method was utilized to address the MOOP of a TSHG with SDGSs by [PERSON] et al. [46]. The objective of this work was to decrease the cross-sectional area of the gearbox and improve its efficiency. In decision-making procedures, two distinct approaches are employed: MCDM and multi-objective optimization (MOO). MCDM is a highly utilized technique for ranking and selecting options across several domains. It assists in determining the optimal solution that fulfills various requirements from an extensive list of options. On the other hand, the primary aim of MOO is to employ optimization techniques in order to concurrently identify the optimal solutions for numerous separate objectives. MCDM focuses on the process of prioritizing choices, while MOO focuses on identifying the most optimal solutions that maintain a balance between several objectives. In this work, the MAIRCA technique is utilized as a multi-criteria decision-making (MCDM) method to address the multi-objective optimization problem (MOOP). The MAIRCA method is a novel MCDM approach that was first proposed in 2014 by [PERSON], [PERSON], and colleagues [47]. This method relies on the evaluation of theoretical and empirical alternative ratings through comparison. The MAIRCA technique provides reliable results and utilizes a robust and well-organized analytical framework for ranking the possibilities. This method exhibits superior stability in comparison to other methods, such as MOORA, TOPSIS, ELECTRE, and COPRAS [48]. The tool is a straightforward mathematical instrument that exhibits a remarkable level of stability when it comes to variations in the criteria's nature and characteristics [49]. Although helical gearbox multi-objective optimization has been thoroughly investigated, the best main design parameters for these gearboxes have not been determined using the MCDM method. Furthermore, the power loss during idle motion, or the power loss re sulting from idle rotating gears submerged in lubricant in the event of bath lubrication, was not included in the aforementioned research. The results of a multi-objective optimization study on a TSHG with double gear sets in the second stage are presented in this work. This optimization effort's two primary goals are to improve the gearbox efficiency and decrease the gearbox volume. Three additional optimal fundamental design factors for the gearbox are examined: the CWFW for both stages and the gear ratio of the first stage. In addition, the MAIRCA method is applied to the optimization task, and the entropy method is used to establish the weights of the criteria. The novelty of this study is that, for the first time, a MOOP involving a TSHG with SDGSs is successfully solved using the MCDM approach (MAIRCA technique). Second, in the process of multi-objective gearbox optimization, the power losses during idle motion have been considered for the first time in determining the gearbox efficiency. Furthermore, the solutions to the challenge outperform those from previous research.