Solving a Multi-Objective Optimization Problem of a Two-Stage Helical Gearbox with Second-Stage Double Gear Sets Using the MAIRCA Method

Abstract

This paper provides a novel application of the multi-criteria decision-making (MCDM) method to the multi-objective optimization problem (MOOP) of creating a two-stage helical gearbox (TSHG) with second-stage double gear sets (SDGSs). The aim of the study is to determine the optimum major design components for enhancing the gearbox efficiency while reducing the gearbox volume. In this work, three primary design parameters are chosen to accomplish this: the gear ratio of the first stage and the coefficients of the wheel face width (CWFW) of the first and second stages. Additionally, the study is conducted with two distinct objectives in mind: the lowest gearbox volume and the maximum gearbox efficiency. Moreover, phase 1 and phase 2, respectively, are the two stages of the MOOP. Phase 2 handles the MOOP to identify the ideal primary design factors as well as the single-objective optimization problem to minimize the difference between the variable levels. Additionally, the Multi-Attributive Ideal–Real Comparative Analysis (MAIRCA) approach is selected to deal with the MOOP. The results of the study are utilized to determine the ideal values for three crucial design parameters in order to create a TSHG with SDGSs.

1. Introduction

A gearbox is a crucial part of a mechanical drive system. It is employed to improve the torque and decrease the speed from the input to the output of the gearbox. In reality, there are a wide variety of forms of gearboxes: worm, planetary, bevel, and cylindrical gearboxes (with spur or helical gears). Because they operate more smoothly and silently than spur-gear gearboxes, helical gearboxes are highly common and exceed other types of gearboxes in terms of efficiency [1]. This is the rationale behind the efforts of numerous academics to optimize the helical gearbox.

Up to now, there has been quite a lot of research on optimizing a helical gearbox. Research on helical gearbox optimization has been fairly extensive up to this point on both single-objective and multi-objective optimization problems. The single-objective optimization problems have been conducted with different objectives, including increasing gearbox efficiency [2,3], reducing radiated noise [4,5,6], gear box length [7,8,9,10], gearbox height [11], gearbox volume [8,9,10,11,12], gear mass [11,13,14], gearbox cross-section area [15,16,17,18], and gearbox cost [19,20,21]. The research was conducted for two-stage [2,3,7,8,9,10], three-stage [14,18,22], and four-stage gearboxes [9,21]. Solving the MOOP for the helical gearbox, in particular, is highly studied as, in practice, a gearbox can only be satisfied by a large number of objective functions.

Various techniques have been employed to address the multi-objective optimization issue with a gearbox. Wang, Y., et al. [23] provided an optimal multi-objective analysis of a cycloid pin gear planetary reducer. The study focused on analyzing the reducer volume, turning arm bearing force, and pin maximum bending stress using Pareto optimal solutions in order to minimize all three of these objectives. The study’s findings demonstrate that the modified algorithm can generate Pareto optimal solutions that surpass those generated by the regular design. Chong, T.H., et al. [24] conducted a study on multi-objective optimization problem (MOOP) to reduce the size of the gears and minimize the vibration caused by their meshing in cylindrical gear pairs. The design method minimized the size and vibration of cylindrical gear pairs by taking into account the strength and geometric limitations. The method’s validity will be verified by the design findings presently employed in an elevator reduction drive. Park C.I. [25] reported the outcomes of their study on addressing the MOOP for reducing helical gear noise. The study employed the design of an experiment and the response surface method as the approach to address the MOOP. The MCDM method was utilized by Chrystopher, V.T., et al. [26] to determine the optimal gear material for a gearbox with the goal of improving the wear resistance and surface fatigue. The objective of this study is to improve the efficiency of the surface fatigue resistance when it is applied to gearboxes. Jawaz Alam and Sumanta Panda [27] developed an approach for spur gear set design optimization that aims to reduce the gear weight, contact stress throughout the contact path, and suitable film thickness at the contact point. This work combined particle swarm optimization, particle swarm optimization-based teaching learning optimization, and Jaya methods to ensure a significant decrease in the weight and contact stress of a profile-modified spur gear set with sufficient film thickness at the site of contact. The results of the study show that the gear design is significantly better with optimum addendum coefficient values inside the design space than with traditional designs. M. Patil et al. presented a novel multi-objective optimization of a two-stage spur gearbox using the NSGA-II approach [28]. This attempt involved two aim functions: minimum gearbox power losses and minimum gearbox volume. The results of the study indicate that there is a high probability of wear failure in solutions that are obtained using single-objective minimization. Furthermore, the total power loss is reduced by half when multi-goal optimization is used instead of single-objective optimization.

Using the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) approach, A. Parmar et al. [29] carried out an optimization analysis of a planetary gearbox while taking into account considerable regular mechanical and tribological constraints. When comparing the results of single-objective optimization with and without tribological limits, using the study’s recommendations led to a significant reduction in the weight and power loss. With the aid of the NSGA-II method, D. Miler et al. [30] performed a multi-objective optimization of the gear pair characteristics with the aim of lowering the transmission volume and power losses. This method was also used in [31] to solve a MOOP, including the reduction of both the gearing mass and the flank adhesive wear speed, and in [32] to solve an optimization problem involving the design of a multi-speed gearbox that has four competing objectives. The NSGA-II was combined with the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach in [33] to solve a multi-objective optimization problem of a small gearbox with three objectives: the gearbox volume, the power output, and the center distance. Three distinct techniques were used in the multi-objective optimization of a two-stage helical gear train by R.C. Sanghvi et al. [34]: genetic algorithms (GAs), NSGA-II, and the MATLAB optimization toolbox. The study had two aims, which were to determine the minimal gearbox volume and the load-carrying capacity. The comparison of the findings shows that the NSGA-II approach produced superior outcomes than the other techniques for both objectives. Sabarinath, P., et al. [35] looked into the multi-objective design of the transmission using helical gear pairs. The opposing number of overlap ratios and gear volumes served as indicators of the goal functions. The study employed the Parameter Adaptive Harmony Search Algorithm (PAHS) to resolve the optimization problem. Using a mono-objective self-adaptive algorithm, the optimization of tooth modifications for spur and helical gears was resolved in [36]. The particle swarm optimization (PSO) technology served as the foundation for this tactic. The multi-objective optimization enhanced the maximal contact pressures and the transmission error signal’s root mean square values. A multi-objective micro-geometry optimization of the gear tooth with response surface methods was conducted by J.A. Korta et al. [37]. H. Wang et al. [38] also employed the response surface methodology approach to multi-objectively optimize the helical gear in a centrifugal compressor. Liu, S., et al. [39] presented a MOOP work using two-stage helical gear sets in a helical hydraulic rotary actuator. The optimization was performed via the NSGA-II method, incorporating four permutations of the three objectives: volume, transmission efficiency, and maximum contact stress. The findings suggest that there is a trade-off between the maximal contact stress and the volume in the ideal region. Also, taking into account various combinations enhances the durability of the optimization outcomes. Liu, X., and Pan, Q., [40] utilized a Discrete Non-Dominated Sorting Genetic Algorithm III (NSGA-III) to optimize a two-stage gearbox. The study investigated three objectives: volume, center distance, and power loss. It also analyzed several design characteristics, such as the number of gear teeth, module, and helix angle.

Gray relation analysis (GRA) and Taguchi techniques were recently applied by X.H. Le and N.P. Vu [41] to the MOOP of designing a TSHG. The lowest gearbox mass and the highest gearbox efficiency were the two objectives selected for this investigation. The optimal values for the five crucial design components that contribute to making a TSHG were established using the study’s findings. 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]. Hung, T.Q., and Huong, T.T.T., [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 V.T. Dinh 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 Dragan, P., 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 resulting 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.

2. Optimization Problem

This section builds on the optimization problem by first calculating the gearbox efficiency and volume. The given objective functions and constraints are then provided.

2.1. Determination of Gearbox Volume

Assume that the gearbox is in a rectangular shape. Therefore, the gearbox volume $V_{gb}$, can be determined by (Figure 1):

$$V_{gb}=L·B·H$$

(1)

where L, B, and H are the length, the width, and the height of the gearbox, which are calculated by [50]:

$$L={(d}_{11}+d_{21}/2+d_{12}/2+d_{22}/2+22.5)/0.975$$

(2)

$$H=max\left(d_{21};d_{22}\right)+8.5·S_G$$

(3)

$$B=b_1+2·b_2+7·S_G$$

(4)

$$S_G=0.005·L+4.5$$

(5)

In the above equations, b₁, and b₂ are the gear width (mm) of stage i; and $d_{1i}$_, $d_{2i}$ are the pitch diameter of the pinion and the gear (mm), which can be calculated by [51]:

$$b_1={\psi }_{ba1}{·a}_1$$

(6)

$$b_2={\psi }_{ba2}{·a}_2$$

(7)

$$d_{1i}=2·a_i/(u_i+1)$$

(8)

$$d_{2i}=2·a_i·u_i·/(u_i+1)$$

(9)

In the above equations, ${\psi }_{ba1}$ and ${\psi }_{ba2}$ are the CWFW of stage 1 and 2; u_i, and $a_i$ are the gear ratio, and the center distance (mm) of stage i; i = 1÷2. It is assumed that the gear trains are well lubricated. In this instance, the primary failure of each gear stage is tooth surface peeling failure; hence, the helical gear sets will be constructed based on the contact strength [16]. Therefore, $a_i$ is determined by [51]:

$$a_i=k_a·(u_i+1)·\sqrt[3]{T_{1i}·k_{H\beta }/({\left[{\sigma }_i\right]}^2·u_i·{\psi }_{bai})}$$

(10)

where $T_{1i}$ is the torque on the pinion of stage i (Nmm) (i = 1 ÷ 2), which is calculated by the following equations:

$$T_{11}=T_{out}/\left(u_g·{\eta }_{hg}^2·{\eta }_b^3\right)$$

(11)

$$T_{12}=T_{out}/\left({2·u}_2·{\eta }_{hg}·{\eta }_{be}^2\right)$$

(12)

where $T_{out}$ is the output torque (Nmm); $u_g$ is the gearbox ratio; ${\eta }_{hg}$, is the efficiency of a helical gear unit; and ${\eta }_b$ is the efficiency of a pair of rolling bearings.

2.2. Determination of Gearbox Efficiency

The efficiency of the gearbox (%), ${\eta }_{gb}$ is calculated by:

$${\eta }_{gb}=100-\frac{100·P_l}{P_{in}}$$

(13)

Here, P_l is the total power loss, which can be determined by [52],

$$P_l=P_{lg}+P_{lb}+P_{ls}+P_{Z0}$$

(14)

where P_lg, P_lb, P_ls, and P_zo are the power loss in the gears, in the bearings, in the seals, and in the idle motion (kW), which can be found as follows:

(+) Determination of P_lg:

$${\mathrm{P}}_{lg}={\sum }_{i=1}^2{\mathrm{P}}_{\mathrm{l}\mathrm{g}\mathrm{i}}$$

(15)

where

$$P_{lgi}=P_{gi}·\left(1-{\eta }_{gi}\right)$$

(16)

Here, ${\eta }_{gi}$ is the efficiency of the i stage, which is found by [53]:

$${\eta }_{gi}=1-\left(\frac{1+1/{\mathrm{u}}_{\mathrm{i}}}{{\mathsf{\beta }}_{\mathrm{a}\mathrm{i}}+{\mathsf{\beta }}_{\mathrm{r}\mathrm{i}}}\right)·\frac{{\mathrm{f}}_{\mathrm{i}}}{2}·\left({\mathsf{\beta }}_{\mathrm{a}\mathrm{i}}^2+{\mathsf{\beta }}_{\mathrm{r}\mathrm{i}}^2\right)$$

(17)

In the above equation, ${\beta }_{\mathrm{a}\mathrm{i}}$ and ${\beta }_{\mathrm{r}\mathrm{i}}$ are the arc of approach and recess on i stage, which can be calculated by [53]:

$${\mathsf{\beta }}_{\mathrm{a}\mathrm{i}}=\frac{{\left({\mathrm{R}}_{\mathrm{e}2\mathrm{i}}^2-{\mathrm{R}}_{02\mathrm{i}}^2\right)}^{1/2}-{\mathrm{R}}_{2\mathrm{i}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\alpha }}{{\mathrm{R}}_{01\mathrm{i}}}$$

(18)

$${\mathsf{\beta }}_{\mathrm{r}\mathrm{i}}=\frac{{\left({\mathrm{R}}_{\mathrm{e}1\mathrm{i}}^2-{\mathrm{R}}_{01\mathrm{i}}^2\right)}^{1/2}-{\mathrm{R}}_{1\mathrm{i}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\alpha }}{{\mathrm{R}}_{01\mathrm{i}}}$$

(19)

where $\mathsf{\alpha }$ is the pressure angle (rad). It is assumed that the gear has an involute gear profile with a pressure angle of 20 degrees; f is the friction coefficient, which can be determined by [11]:

-

if v ≤ 0.424 (m/s):

$$\mathrm{f}=-0.0877·\mathrm{v}+0.0525$$

(20)

-

if v > 0.424 (m/s):

$$\mathrm{f}=0.0028·\mathrm{v}+0.0104$$

(21)

where v is the sliding velocity of the gear (m/s).

(+) Determination of P_lb [52]:

$$P_{lb}={\sum }_{i=1}^6{f}_b·F_i·v_{bi}$$

(22)

where i = 1 ÷ 6, and $f_b=0.0011$ is the friction coefficient of the bearing (for the radical ball bearings with angular contact) [52]; Fi is the load of the bearing (N); and $v_{\mathrm{b}i}$ is the peripheral velocity of the bearing (m/s).

(+) Determination of P_ls [52]:

$$P_{\mathrm{l}\mathrm{s}}={\sum }_{i=1}^2{\mathrm{P}}_{\mathrm{s}\mathrm{i}}$$

(23)

Here, i = 1 ÷ 2, is the ordinal number of the seal; and $P_{si}$ is the power loss in seal i (kW), which can be determined by:

$$P_{si}=\left[145-1.6·t_{oil}+350·\mathit{log}\mathit{log}\left({VG}_{40}+0.8\right)\right]·d_s^2·n·{10}^{-7}$$

(24)

where d_s and n are the diameter (mm) and the revolution of the shaft; and VG₄₀ is the ISO Viscosity Grades number.

(+) Determination of P_zo [52]:

$$P_{Z0}={\sum }_{i=1}^k{T}_{Hi}·\frac{\pi ·n_i}{30}$$

(25)

In (25), k = 2, is the total gear pair number; n is the driven gear revolution; and $T_{Hi}$ is the hydraulic torque of power loss (Nm), which is found by [52]:

$$T_{Hi}=C_{Sp}·C_1·e^{\frac{C_2·v}{v_{t0}}}$$

(26)

Here, $C_{Sp}=1$, for stage 1 in the case of the involved oil having to pass until the mesh, and for stage 2, $C_{Sp}$ can be found by (Figure 2):

$$C_{Sp}={\left(\frac{4·e_{max}}{3·h_C}\right)}^{1.5}·\frac{2·h_C}{l_{hi}}$$

(27)

With l_hi, which is calculated by [52]:

$${\mathrm{l}}_{\mathrm{h}\mathrm{i}}=\left(1.2÷2.0\right)·{\mathrm{d}}_{\mathrm{a}2\mathrm{i}}$$

(28)

In (26), C₁ and C₂ are calculated by [52]:

$${\mathrm{C}}_1=0.063·\left(\frac{{\mathrm{e}}_1+{\mathrm{e}}_2}{{\mathrm{e}}_0}\right)+0.0128·\left(\frac{\mathrm{b}}{{\mathrm{b}}_0}\right)$$

(29)

$${\mathrm{C}}_2=\frac{{\mathrm{e}}_1+{\mathrm{e}}_2}{80·{\mathrm{e}}_0}+0.2$$

(30)

With e₀ = b₀ = 10 (mm).

2.3. Objective Functions and Constrains

2.3.1. Objectives Functions

Two single objectives compose the MOOP in this paper:

-

Minimizing the gearbox volume:

$$\mathrm{m}\mathrm{i}\mathrm{n}{f}_1\left(X\right)={\mathrm{m}}_{gb}$$

(31)

-

Maximizing the gearbox efficiency:

$$\mathrm{m}\mathrm{a}\mathrm{x}{f}_2\left(X\right)={\mathsf{\eta }}_{\mathrm{g}\mathrm{b}}$$

(32)

where X is the vector that denotes the design variables. A TSHG with SDGSs has five main design factors: u₁, ${\psi }_{ba1}$, ${\psi }_{ba2}$, and AS₁, AS₂. Moreover, it is found that the maximum values of AS₁ and AS₂ match their ideal values [43]. Consequently, the optimization problem is solved with u₁, ${\psi }_{ba1}$, and ${\psi }_{ba2}$, the three main design elements in this work, as variables. The outcome is:

$$X=\left\{u_1,{\psi }_{ba1},{\psi }_{ba2}\right\}$$

(33)

2.3.2. Constrains

The multi-objective function needs to adhere to the following restrictions:

$$1\le u_1\le 9, \mathrm{and} 1\le u_2\le 9$$

(34)

$$0.25\le {\psi }_{ba1}\le 0.4, \mathrm{and} 0.25\le {\psi }_{ba2}\le 0.4$$

(35)

3. Methodology

3.1. Method to Solve the Multi-Objective Optimization

As stated in Section 2.3, three main design factors are selected as variables for the MOOP in this work. These variables are listed in

Table 1. Input parameters.

Parameter	Symbol	Lower Limit	Upper Limit
Gearbox ratio of stage 1	u1	1	9
CWFW of stage 1	${\psi }_{ba1}$	0.25	0.4
CWFW of stage 2	${\psi }_{ba2}$	0.25	0.4

, along with their lowest and maximum values. In actuality, it is challenging to address the multi-objective optimization (MOO) problem using an MCDM method. The reason for this is that there are a lot of options or potential solutions when it comes to dealing with a MOO problem. To ensure the accuracy of the parameters and avoid missing the optimization problem’s solution, the three parameters in this work have limits, as shown in Table 1, and the step between the variables is 0.02. In this case, the number of options (or experimental runs) that must be determined and compared is $(9-1)/0.02·(0.4-0.25)/0.02·(0.4-0.25)/0.02=22.500$ (runs). Therefore, it is not viable to address the MOO problem directly using the MCDM approach due to the sheer amount of possibilities. The MAIRCA method is used to solve the MOOP in this work and find the optimal values for the three main design variables. Minimizing the gearbox volume and optimizing the gearbox efficiency are the two goals. A simulation experiment is constructed in order to address the MOOP for a TSHG with SDGSs. Furthermore, by adopting the full factorial design, the number of experiments can be increased without taking into account the budget for each experiment because this is a simulation experiment. There will be 5³ = 125 experiments as a result of the three experimental variables and five levels for each variable. But out of the three variables that are listed, Table 1 shows that u₁ has the widest distribution (ranging from 1 to 9). Consequently, there is a considerable difference between the levels of this variable (in this case, (9 − 1)/4 = 2), even with five levels). The values of u₁ are typically rounded to the hundredths in practice. As a result, in order to make the computation easier and simpler, a way to lessen the vast distance between the variables of u₁ must be found. A solution to multi-objective problems is suggested in this work (Figure 3). The following are the two components of this procedure: phase 1 factors determine the best primary design by solving the multi-objective optimization issue; and phase 2 factors minimize the gap between the levels by solving the single-objective optimization problem. Furthermore, if the levels of the variable are not sufficiently close to one another or if the best solution is not appropriate for the requirement, the MAIRCA issue will be rerun using the smaller distance between two levels of the u₁ in order to solve the multi-objective problem (see Figure 3).

3.2. Method to Solve MCDM Problem

In this paper, the MCDM problem is handled via the MAIRCA technique. The following actions must be taken in order to apply this approach [47]:

-

Step 1: Building the initial matrix:

$$X=\left[\begin{array}{c}\begin{array}{ccc}{x}_{11}& \cdots & x_{1n}\\ x_{21}& \cdots & x_{2n}\\ ⋮& \cdots & ⋮\end{array}\\ \begin{array}{ccc}{x}_{m1}& \cdots & x_{mn}\end{array}\end{array}\right]$$

(36)

Here, the result of criterion n in variant m is represented by x_mn.

-

Step 2: Determining options based on various selections $P_{A_j}$ through:

$$P_{A_j}=\frac{1}{m}$$

(37)

-

Step 3: Calculating the $t_{p_{ij}}$ elements using:

$$t_{p_{ij}}=P_{A_j}· w_j$$

(38)

where i = 1, 2,…, m; j = 1, 2, …, n.

-

Step 4: Finding $t_{r_{ij}}$ by:

(+)

For the gearbox efficiency objective:

$$t_{r_{ij}}=t_{p_{ij}}·\left(\frac{x_{ij}-x_i^{-}}{x_i^{+}-x_i^{-}}\right)$$

(39)

(+)

For the gearbox volume objective:

$$t_{r_{ij}}=t_{p_{ij}}·\left(\frac{x_{ij}-x_i^{+}}{x_i^{-}-x_i^{+}}\right)$$

(40)

-

Step 5: Calculating the complete gap matrix g_ij by:

$$g_{ij}=t_{p_{ij}}-t_{r_{ij}}$$

(41)

-

Step 6: Determining the final values of the criteria functions, Qi:

$$Q_i={\sum }_{i=1}^m{g}_{ij}$$

(42)

3.3. Method to Find the Weight of Criteria

The weights of the criteria in this research are determined using the entropy technique. This technique can be implemented by following the steps outlined below [54].

-

Finding the indicator normalized values:

$$p_{\mathrm{ij}}=\frac{x_{\mathrm{ij}}}{m+{\sum }_{i=1}^m{x}_{\mathrm{ij}}^2}$$

(43)

-

Calculating the entropy of each indicator:

$$m{e}_j=-{\sum }_{i=1}^m\left[p_{ij}\times ln\left(p_{ij}\right)\right]-\left(1-{\sum }_{i=1}^m{p}_{ij}\right)\times ln\left(1-{\sum }_{i=1}^m{p}_{ij}\right)$$

(44)

-

Determining each indicator’s weight:

$${\mathrm{w}}_{\mathrm{j}}=\frac{1-{\mathrm{m}\mathrm{e}}_{\mathrm{j}}}{{\sum }_{\mathrm{j}=1}^{\mathrm{m}}\left(1-{\mathrm{m}\mathrm{e}}_{\mathrm{j}}\right)}$$

(45)

4. Single-Objective Optimization

In this work, the single-objective optimization problem is solved using the direct search approach. Additionally, two single-objective tasks are solved with a MATLAB R2024a (License No. 41224477) computer program: maximizing the gearbox efficiency and minimizing the gearbox volume. The figures and observations from the program include the following. The link between ηgb and u₁ is depicted in Figure 4. It is clear that for an ideal value of u₁, η_gb reaches its maximum. Figure 5 shows the relationship between u₁ and V_gb. It was found from the figure that V_gb reaches its minimum value when u₁ is at its ideal value. The relationship between η_gb and V_gb, and ${\psi }_{ba1}$ and ${\psi }_{ba2}$, is depicted in Figure 6 and Figure 7, respectively. Figure 6 illustrates that an increase in ${\psi }_{ba1}$ results in an increase in η_gb and a decrease in V_gb. As ${\psi }_{ba2}$ increases, both η_gb and V_gb also drop (Figure 7). The relationship between the overall gearbox ratio, u_t, and the optimal gear ratio of the first stage u₁ is shown in Figure 8. Additionally,

Table 2. New constraints of u₁.

ut	u1
	Lower Limit	Upper Limit
10	2.2	2.36
15	2.79	4.76
20	3.32	5.75
25	3.76	6.66
30	4.19	7.51
35	4.58	8.31
40	4.93	8.88

presents the newly derived constraints for the variable u₁ that are calculated from the solutions of single-objective problems.

5. Multi-Objective Optimization

To conduct simulation experiments, a computer program has been developed. For the purpose of the investigation, the gearbox ratios of 10, 15, 20, 25, 30, 35, and 40 are all considered. The solutions to this u_t = 35 problem are shown below. For the first 125 testing cycles, this total gearbox ratio is employed (as stated in Section 3.1). The gearbox mass and efficiency, which are the experiment’s output values, will be entered into the MAIRCA as input parameters in order to solve the MOOP. With each MAIRCA step, the gap between the two levels of each variable will decrease. For example, when u_g = 35 in step 1, u₁ rises from 4.58 to 8.31 (Table 2). Consequently, the distance between the two levels of u1 is (8.31 − 4.58)/4 = 0.933. Until there is less than 0.02 between the two levels of the variable, this process will be repeated.

Table 3. Main design factors and output results for u_t = 35 in the 4th run of the MAIRCA. The background color describes the best option.

Trial.	u1	${\mathsf{\psi }}_{\mathbf{b}\mathbf{a}1}$	${\mathsf{\psi }}_{\mathbf{b}\mathbf{a}2}$	Vgb (dm3)	ηgb (%)
1	6.99	0.25	0.25	21.23	93.44
2	6.99	0.25	0.29	20.82	93.38
3	6.99	0.25	0.33	20.52	93.32
4	6.99	0.25	0.36	20.30	93.26
5	6.99	0.25	0.40	20.12	93.21
6	6.99	0.29	0.25	21.37	92.26
7	6.99	0.29	0.29	20.93	92.21
26	7.01	0.25	0.25	21.21	93.42
27	7.01	0.25	0.29	20.81	93.36
28	7.01	0.25	0.33	20.51	93.30
54	7.02	0.25	0.36	20.27	93.23
55	7.02	0.25	0.40	20.10	93.18
56	7.02	0.29	0.25	21.35	92.22
71	7.02	0.40	0.25	21.87	87.72
72	7.02	0.40	0.29	21.34	87.66
73	7.02	0.40	0.33	20.94	87.60
101	7.05	0.25	0.25	21.17	93.33
102	7.05	0.25	0.29	20.77	93.27
103	7.05	0.25	0.33	20.47	93.21
123	7.05	0.40	0.33	20.92	87.53
124	7.05	0.40	0.36	20.60	87.48
125	7.05	0.40	0.40	20.35	87.42

displays the main design parameters and output responses for u_t = 35 in the MAIRCA experiment’s fourth and final iteration. The weights of the criteria have been determined through the application of the entropy approach (see Section 3.3). To begin with, obtain the normalized values of p_ij using Equation (43). The entropy value of each indicator (me_j) is found using Equation (44). Lastly, determine the weight of the criteria w_j using Equation (45). For the latest MAIRCA experiment, the weights of V_gb and η_gb are found to be 0.4508 and 0.5492, respectively. Section 3.2 provides guidelines for using the MAIRCA approach for multi-objective decision-making. Adopting the measures recommended in Section 2.1, in particular, after the starting matrix is put up, the priority, or criterion PAj, is calculated by applying Equation (2). As a result, V_gb and η_gb have the same priority, which is 1/125 = 0.008. In addition, the value of parameter t_pij is determined using Equation (38) and the weight of the criterion is determined in Section 2.2. The tp_ij values for η_gb and V_gb are 0.0036 and 0.0044, respectively. The values of tr_ij are then obtained using Equations (39) and (40), and the values of g_ij are then determined using Equation (41). In the end, the values of the criteria functions Qi are determined using Equation (42). The results of the option ranking and the different parameters that the MAIRCA technique computed (for the last run of MAIRCA work) are displayed in

Table 4. Calculated results and ranking of the options via the MAIRCA method for u_t = 35. The background color describes the best option.

Trial.	trij		gij		Qi	Rank
	Vgb	ηgb	Vgb	ηgb
1	0.0016	0.0036	0.0028	0.0000	0.0028	54
2	0.0026	0.0036	0.0018	0.0000	0.0018	30
3	0.0034	0.0035	0.0010	0.0001	0.0011	20
4	0.0039	0.0035	0.0005	0.0001	0.0006	10
5	0.0043	0.0035	0.0000	0.0001	0.0002	3
6	0.0013	0.0029	0.0031	0.0007	0.0038	75
7	0.0024	0.0029	0.0020	0.0007	0.0028	55
…
26	0.0017	0.0036	0.0027	0.0000	0.0027	51
27	0.0027	0.0036	0.0017	0.0000	0.0018	29
28	0.0034	0.0035	0.0010	0.0001	0.0011	19
…
54	0.0040	0.0035	0.0004	0.0001	0.0005	7
55	0.0044	0.0035	0.0000	0.0002	0.0002	1
56	0.0013	0.0029	0.0031	0.0007	0.0038	73
…
71	0.0001	0.0002	0.0043	0.0034	0.0077	123
72	0.0014	0.0001	0.0030	0.0035	0.0065	118
73	0.0023	0.0001	0.0021	0.0035	0.0055	108
…
101	0.0018	0.0035	0.0026	0.0001	0.0027	46
102	0.0028	0.0035	0.0016	0.0001	0.0017	26
103	0.0035	0.0035	0.0009	0.0001	0.0010	16
…
123	0.0024	0.0001	0.0020	0.0035	0.0055	106
124	0.0032	0.0000	0.0012	0.0036	0.0048	91
125	0.0038	0	0.0006	0.0036	0.00422	83

. The table shows that option 55 is the best option out of all the options provided. Accordingly, u₁ = 7.02, ${\psi }_{ba1}$ = 0.25, and ${\psi }_{ba2}$ = 0.4 are the ideal values for the primary design features (see Table 3).

The MOO problem is handled using two additional MCDM techniques, Evaluation by an Area-based Method of Ranking (EAMR) and Measurement of Alternatives and Ranking According to COmpromise Solution (MARCOS), in order to evaluate the reliability of the outcomes that are found (option 55 is the best). To determine the weights, the entropy approach is also used. When both of these methods are used to address the MOO problem, option 55 finally performs the best (see

Table 5. Rankings of the options when using the MAIRCA, EAMR, and MARCOS methods. The background color describes the best option.

	Ranking
Trial.	MAIRCA	EAMR	MARCOS
1	54	55	65
2	30	35	38
3	20	20	20
4	10	10	10
5	3	3	3
6	75	77	85
7	55	52	55
…
26	51	54	64
27	29	33	36
28	19	19	19
…
54	7	7	7
55	1	1	1
56	73	74	83
…
71	123	123	123
72	118	118	115
73	108	108	108
…
101	46	48	59
102	26	27	31
103	16	16	16
…
123	106	106	106
124	91	91	91
125	83	81	71

). Using all three MCDM methods, the best option, 55, is produced, demonstrating that the best option is independent of the decision-making process selected.

Continuing with the preceding discussion,

Table 6. Optimum values of the main design factors.

No.	ut
	10	15	20	25	30	35	40
u1	2.4	4.12	5.02	5.79	6.63	7.02	7.66
${\psi }_{ba1}$	0.25	0.25	0.25	0.25	0.25	0.25	0.25
${\psi }_{ba2}$	0.4	0.4	0.4	0.4	0.4	0.4	0.4

displays the ideal values for the primary design parameters that match the remaining u_t values of 10, 20, 25, 30, 35, and 40. With the data in this table, the following conclusions can be drawn. Whereas ${\psi }_{ba2}$ chooses the greatest value (${\psi }_{ba2}$ = 0.4), ${\psi }_{ba1}$ chooses the lowest value (${\psi }_{ba1}$ = 0.25). This is because the smallest gearbox volume can only be produced by a small box with a cross-sectional area (LxH). This requires d_w21 and d_w22 to be roughly equal in order to be accomplished [55]. In addition, a larger ${\psi }_{ba2}$ is needed to reduce the diameter of d_w22 due to the higher torque of the second stage. On the other hand, because the first stage has a lower torque, a smaller ${\psi }_{ba1}$ needs to be selected in order to raise d_w21.

In order to evaluate the effectiveness of applying the formula for the power loss in the idle motion when calculating power loss in gears, the calculated values of the gearbox efficiency in this work are compared with the results in [33]. From Table 4 [33], when the total gearbox ratio is 7.5, the gearbox efficiency will be 99.33–99.71% (determined from input power of 10 kW, maximum output power of 9.971 kW and minimum output power of 9.933 kW). In addition, in this work, when u_t changes from 10 to 40, the gearbox efficiency will have the values of 87.42–93.44% (Table 3). In fact, the efficiency of a helical gear train is 0.93–0.98 (93–98%) and the efficiency of a pair of bearings is 0.99–0.995 (99–99.5%) [51]. From these data, the efficiency of a TSHG can be determined as follows:

$${\mathsf{\eta }}_{\mathrm{g}\mathrm{b}}={\mathsf{\eta }}_g^2·{\mathsf{\eta }}_b^3={0.93}^2·{0.99}^3÷{0.98}^2·{0.995}^3=0.83÷0.95 \mathrm{or} {\mathsf{\eta }}_{\mathrm{g}\mathrm{b}}=83÷95$$

(46)

By comparing the above computed values with (1), it is evident that the gearbox efficiency in this work (87.42–93.44%), is fairly close to reality (83–95%). Moreover, the gearbox efficiency in [33] (99.33–99.71%) is extremely high and not consistent with reality. This proves that using the formula to calculate the power loss in the idle motion in this work is a valuable new point and should be applied.

To evaluate the accuracy of finding the optimal values of the proposed method with other methods, like the Taguchi and Gray Relational Analysis (GRA) methods, the result of finding the optimal gear ratio of the first stage u₁ in this work is compared with that in [41]. In this work, the solutions to the optimization problem, for example, u1, are found by directly comparing (over multiple steps) 22,500 values (with u₁ = 1–9) to choose from (as mentioned in Section 3.1). In this case, the step between the variables of u₁ is only 0.02. Meanwhile, when using the Taguchi and GRA methods, the total number of options to choose the optimal value of u₁ is only 25 (Table 3 [41]). In this case, the step between the variables is 0.775 (much larger than when using the proposed method). As a result, the value u₁ found using the proposed method will be more accurate than using the Taguchi and GRA methods. In other words, the proposed method is better than the Taguchi and GRA methods.

The association between the ideal values of u₁ and u_t is depicted in Figure 9. Additionally, it is discovered that the ideal values of u₁ may be determined using the regression equation that follows (with R² = 0.9974):

$$u_1=3.7098·ln\left(u_t\right)-6.0709$$

(47)

After finding u₁, the optimal value of u₂ is calculated by the following formula:

$$u_2=u_t/u_1$$

(48)

6. Conclusions

In this study, the multi-objective optimization issue associated with the design of a TSHG with a SDGSs is solved using the MAIRCA approach. The objective of the study is to determine the optimal essential design parameters that minimize the gearbox volume while maximizing the gearbox efficiency. Three crucial design elements are selected to accomplish this: the first-stage gear ratio and the CWFW for the first and second stages. Furthermore, the method of solving the multi-objective optimization issue involves two parts. Phase 2 is concerned with identifying the ideal fundamental design factors, while phase 1 is devoted to addressing the single-objective optimization issue of minimizing the difference between the variable values. This work produced the following conclusions:

By bridging the gap between the variable levels, the single-objective optimization problem speeds up and simplifies the MOOP’s solution.

-

Based on the study’s findings, the best values for the three primary design parameters of a two-stage helical gear gearbox with an SDGS ${\psi }_{ba1}$ = 0.2, ${\psi }_{ba1}$ = 0.4 and u₁ determined according to (47).

-

Two individual targets are evaluated in relation to the main design parameters.

A precise solution to the MOOP can be obtained by repeatedly applying the MAIRCA technique until the desired outcomes are obtained (variables have an accuracy of less than 0.02).

-

The remarkable degree of agreement between the experimental data and the u₁ model suggests that the data are reliable.