Optimization Method for Assembly Sequence Evaluation Based on Assembly Cost and Ontology of Aviation Reducers

Featured Application

The research in this paper can provide a feasible technical means for the assembly sequence optimization of aviation reducers with complex structures, effectively promote the progress of the complex assembly sequence optimization technology of aviation reducers, and realize the comprehensive evaluation of the overall assembly quality and performance index of aviation reducer products. It can be applied to the assembly control of tanks, armored vehicles, ships, and other complex equipment to improve their assembly quality.

Abstract

An assembly sequence evaluation is one of the most important research directions of assembly sequence planning (ASP) for complex mechanical transmission products. Currently, aviation reducers lack a multi-perspective and multi-level evaluation of their assembly sequence. The existing evaluation indicators vary. The evaluation methods have low effectiveness and poor practicability. Therefore, a comprehensive multidimensional evaluation method for complex assembly sequences is proposed in this paper. A multidimensional comprehensive evaluation of the overall assembly quality and performance indices of aviation reducer products is realized. Firstly, the main factors affecting assembly sequence planning are considered: the attributes of the basic unit parts and the cost control of the assembly process. An evaluation index system of assembly sequence planning based on the two dimensions of assembly cost and ontology is constructed. Then, according to the multidimensional evaluation index, fuzzy evaluation theory is used to establish a fuzzy set and a matrix for each dimensional evaluation index. The index weight is divided. A comprehensive evaluation model and the function of each dimension are established. After a comprehensive evaluation, the multidimensional assembly sequence evaluation method for aviation reducers is formed. Finally, the method is applied to the assembly process of the primary reducer of a helicopter’s main reducer, and a comprehensive evaluation of its assembly sequence scheme is completed to verify the feasibility of the proposed method. This article constructs a complex assembly sequence evaluation method that includes 12 evaluation indicators, improves the assembly sequence planning evaluation index system of aviation reducers, and can effectively promote the progress of optimization technology for complex assembly sequences of aviation reducers.

1. Introduction

As a complex mechanical transmission product, aviation reducers have a large number of components and a complicated assembly process. This paper’s assembly process mainly guides the assembly. In research on the assembly sequence planning of mechanical products with complex structures, samples are tested according to the requirements of design drawings before trial production. After repeated trial loading, the design parameters are adjusted based on the newly processed samples, and trial loading is conducted again, with this being repeated until the design requirements are met. This repeated trial assembly method not only lengthens the product’s R&D cycle but also increases costs. Therefore, researchers have carried out research on the product assembly model. By establishing a three-dimensional model of product parts, the trial assembly of products is completed in a virtual assembly environment. This method greatly speeds up the product’s R&D speed and saves costs. However, it cannot solve the problem concerning the order in which product parts are installed. Aiming to solve this problem, researchers have gradually carried out research on product assembly sequence planning and design, the generation and representation of assembly sequences, and the evaluation and optimization of assembly sequences based on assembly models [1].

Assembly sequence planning consists of the development of a high-quality, short-cycle, low-cost, and simple assembly sequence based on an assembly model and the requirements of product design drawings; this is achieved by analyzing the structural and cooperative relationships between various parts and by determining whether the product assembly meets the geometric direction and existing technical conditions according to the constraints of the product assembly process. In research on assembly sequence planning, researchers have proposed a combination of the cut-set method, matrix operation method, knowledge solution method, particle swarm optimization algorithm, bacterial chemotaxis algorithm, and genetic algorithm, taking into account the convergence speed of the algorithm, the reversibility of the assembly process, the matrix of connection and interference moments, and the rules of knowledge bases. Research on product assembly sequence planning has been conducted from the perspective of the correlation function in extension set theory [2,3,4,5,6,7].

Assembly sequence planning has been studied since as early as the 1980s. Bourjault et al. constructed a computerized diagram search method to comprehensively analyze the connection and contact relationships among the major components, determine the feasibility of the assembly sequence with the assembly contact diagram, and develop the priority relationships among the parts in a question-and-answer manner to finally derive the assembly sequence [8,9]. Ying, K.C. proposed a cyber–physical assembly system (CPAS)-based metaheuristic for fully automated assembly sequence planning, taking into account the physical characteristics of robotic arms [10]. A novel assembly subset prediction method based on a precedence graph was proposed to solve the ASP problem; this proposed method adopts the local-to-whole idea and integrates a simplified firework algorithm [11]. To improve the efficiency of complex assemblies in large-scale assembly sequence planning, an intelligent sequence planning method for constructing an assembly hybrid G-diagram model was proposed to realize the hierarchy of assembly structures [12]. In order to realize the automatic planning of complex assembly product sequences and improve assembly efficiency, Zhao, M.H. proposed an assembly sequence planning system for workpieces (ASPW) based on deep reinforcement learning; the proposed ASPW-DQN unites curriculum learning and parameter transfer, which can avoid the explosive growth of assembly relations and improve system efficiency [13,14]. Aiming to solve the problem of insufficient individual intelligence in the evolutionary algorithm of assembly sequence planning, according to the disadvantage of the particle swarm optimization algorithm easily falling into local optimization, a various population strategy was adopted to shorten the evolution stagnation time, improve the evolution efficiency of the particle swarm optimization algorithm, and enhance the optimization ability of the algorithm [15]. In automotive body assembly systems, Shahi, V.J. proposed a methodology based on the modeling of dimensional error propagation in MSA with a batch of compliant non-ideal parts to improve product dimensional quality by optimizing ASP and the assembly line configuration [16]. To reduce human intervention in the process of manually teaching assembly sequences to industrial robots, Tariki, K. developed an automated sequence planning method using only a 3D CAD model [17]. The assembly sequence planning problem of large space structures (LSSs) was investigated from the perspective of structural vibration. Adopting a plate-type LSS and a truss-type LSS as specific objects, a dynamic multi-constrained assembly sequence planning algorithm was proposed based on structural dynamics modeling and a modified genetic algorithm (GA) [18].

Product assembly path planning and optimization refers to the optimal route of producing product parts from the starting point to the end point of assembly; the optimal route is determined by analyzing the actual product assembly process based on product assembly models, combined with the planned assembly sequence and the actual assembly conditions on site. In research on product assembly path planning and optimization, researchers have proposed the artificial potential field method, visual graph method, positional space method, virtual assembly method, moving boundary line method, and directional polyhedral cone algebra method, taking into account the motion relationship based on the gravitational field, spatial obstacles based on the spherical graph method, point obstacles based on free space, virtual reality-based technology, boundary division based on face–edge graphs, the direction of motion based on polyhedral cones, and other aspects, thereby expanding the planning research on product assembly paths [19,20,21,22,23,24,25,26,27,28,29].

In research on the planning of assembly sequences for mechanical transmission products with complex structures, regarding the evaluation of the assembly sequences, Wang Xiaoyi et al. used the sequence comparison method with directed graphs to evaluate the assembly sequences from the part- and sequence-level perspectives [30]. Zhang et al. introduced the concepts of the overall and unit levels for a comprehensive analysis of the assembly process, and they chose the optimal solution reference set method to evaluate the results of assembly sequence acquisition [31]. Li Lei et al. introduced a fuzzy evaluation in assembly sequence planning and integrated part quality and manufacturing quality to analyze the assembly sequence results [32]. Yuan Baoxun et al. proposed part- and system-level evaluation indices for the assembly process to analyze the assembly sequence comprehensively [33]. Zhou Kaijun et al. proposed the entropy-weighted fuzzy comprehensive evaluation method by integrating the indicators of assembly resources, cost, time, and part-level design to facilitate the reasonable planning of the assembly sequence [34].

In summary, evaluation indices for the assembly sequence of complex structural assemblies are grouped or isolated, and a complete evaluation system has not been developed for the evaluation of the assembly sequence of aviation reducers. Additionally, there is a lack of research on the assembly sequence planning of the complex assembly process of aviation reducers from both multiple perspectives and multi-level perspectives. This paper fills the gap in the evaluation method for the assembly sequence of aviation reducers, including the improvement of the evaluation index system and the optimization of evaluation methods.

Therefore, this study proposes a comprehensive evaluation method for the complex assembly sequences of aviation reducers based on cost and ontology dimensions. This method is a complex assembly sequence evaluation method with twelve evaluation indices based on the two perspectives of time cost and economic cost in the assembly cost dimension and the two perspectives of individual assembly base unit parts and the finished assembly as a whole in the assembly ontology dimension. It is possible to determine the goodness of the assembly solution through the comprehensive evaluation results of multiple factors in the cost dimension, as well as optimize the evaluation indices through the evaluation results of individual influencing factors. The evaluation results of the eight evaluation indices in the ontology dimension are combined to divide the weights for the calculation of the comprehensive evaluation method for the assembly sequence of aerospace gearboxes, which makes the evaluation results more objective and closer to the optimal results of the complex assembly sequence. This method provides a feasible technical means for the optimization of the assembly sequence of aviation reducers with complex structures, and it can effectively promote the progress of the optimization technology for the complex assembly sequence of aviation reducers. The following is the structure of the rest of the paper:

Firstly, an evaluation index system of assembly sequence planning based on the two dimensions of assembly cost and ontology is constructed. Then, fuzzy evaluation theory is used to establish a fuzzy set and a matrix for each dimensional evaluation index. A comprehensive evaluation model and the function of each dimension are established. After a comprehensive evaluation, the multidimensional assembly sequence evaluation method for aviation reducers is formed. Finally, the method is applied to the assembly process of the primary reducer of a helicopter’s main reducer, and a comprehensive evaluation of its assembly sequence scheme is completed to verify the feasibility of the proposed method.

2. Construction of Assembly Multidimensional Evaluation Index

In the assembly process of an aviation reducer transmission system, according to the actual situation of an assembly workshop in a reducer factory, the factors affecting the assembly of the reducer include the following: the manufacturing accuracy of the assembly unit, the business ability of the assembly personnel, the assembly tools, the assembly environment, the assembly path, the assembly logistics, the implementation of the assembly process instructions, the transit of the assembly unit, feedback on the assembly problems, the length of the assembly time, the assembly cost, the digitalization of the workshop, and the process level. Effective assembly sequence planning requires a sound evaluation index system. The assembly of an aerospace gearbox is based on a component unit, which consists of several components with different functions, and the final assembly is completed according to the overall design requirements of the gearbox and the assembly process instructions. Therefore, according to the assembly process of an aerospace gearbox, the bottom-up perspective from the basic unit to the complete assembly of the gearbox assembly, and the control of the assembly time and economic cost, an evaluation index system of the assembly sequence planning of an aerospace gearbox is built.

The composition of the evaluation index for the assembly sequence planning of an aviation reducer with a complex structure mainly consists of two parts: the reducer assembly cost factor and the reducer assembly ontology factor. In the design and manufacture of high-end weapon and equipment systems, the manufacturing reserve cycle of aviation reducer products directly affects the combat units’ response time to the enemy’s combat motives, and the rapid emergency response directly determines the ability to meet the enemy in combat, as well as the survival capability. The main indicators related to assembly cost are the assembly operation method, auxiliary time cost, tooling cost, and key parts positioning cost. The assembly body of a reducer is mainly considered from two perspectives, namely, the individual assembly base unit and the finished assembly as a whole; the evaluation index of an individual assembly base unit mainly considers its quality, size, symmetry, and fit, while the evaluation index of the finished assembly as a whole mainly considers the operability of the assembly space, the stability of the assembly, changes in the fixtures, and the adjustment times of the assembly direction. Figure 1 shows the structure of the multidimensional evaluation index for the aviation reducer assembly sequence.

2.1. Evaluation Index of Assembly Cost Dimension

In terms of the characteristics of complex structure assembly and aviation reducer manufacturing, currently, in assembly workshops, gearboxes are mainly assembled using a combination of manual and semi-automatic methods. Combined with an actual assembly cost control scheme of a factory, this section summarizes the main factors related to assembly cost that affect assembly sequence planning.

2.1.1. Assembly Operation Method

In an assembly plant, the time required for different assembly operation modes varies. Because aerospace gearbox products are manufactured via single-piece and small-batch production, an assembly plant can plan the assembly of gearboxes using two models, serial and parallel, according to the assembly process. Usually, the serial assembly planning operation mode requires a longer assembly time. In the actual assembly task arrangement of aviation reducers, a reducer assembly workshop adopts both serial and parallel models, and multiple models can be assembled at the same time in serial planning schemes. Figure 2 shows the assembly operation mode of a particular type of aviation reducer assembly workshop.

In the process of aviation reducer assembly, the larger the number of assembly units that can participate in assembly at the same time, the lower the assembly time cost of the reducer product and the more effective the specified assembly operation mode. Thus, the efficiency L of the aviation reducer assembly operation mode can be defined, that is, the number of assembly units that participate in the assembly operation of reducers at the same time per unit time, as shown in Equation (1):

$$Li={\sum }_{i=1}^m\frac{Ni}{Ti}$$

(1)

• m indicates that there are m parallel processes in the total assembly process.
• i indicates the i-th parallel process in the total parallel processes.
• Ni denotes the total number of unit parts involved in assembly at the same time in the i-th parallel process, and there may be several assembly process tasks in a parallel process.
• Ti denotes the time required to complete the assembly task in the i-th parallel process.

2.1.2. Auxiliary Time Cost

To complete the aerospace reducer assembly work, in addition to the necessary assembly operations for the installation of each component unit, there are essential auxiliary operations, such as the selection and installation of jigs and fixtures, the necessary recording and inspection of the assembly process, weighing, the provision of problem feedback, and communication. The more auxiliary operations there are, the longer the required assembly cycle time, indicating that there is room for further improvement in assembly sequence planning to enhance assembly efficiency. The auxiliary time cost T is defined as shown in Equation (2):

$$T={\sum }_{i=1}^{m}ti$$

(2)

• m denotes the total number of auxiliary operations.
• ti denotes the time taken for the i-th auxiliary operation.

2.1.3. Tooling Cost

In addition to the time cost of the fixtures mentioned above, there is also the economic cost of fixtures. Each assembly process uses different types of jigs and fixtures, and the quantity is not necessarily the same. In different assembly processes, the more types and quantities of fixtures used, the more complex the process, which, in turn, makes it more difficult to change the process; this also causes a longer assembly cycle time, which causes the assembly itself to have a higher economic cost. The economic cost of fixture C is defined as shown in Equation (3):

$$C={\sum }_{i=1}^{m}ci$$

(3)

• m denotes the total number of tooling fixtures used in the assembly process.
• ci denotes the economic cost of the i-th tooling fixture.

2.1.4. Positioning Cost of Key Parts

Due to the complex assembly structure and components of an aerospace reducer, the assembly accuracy and performance of each component must be ensured before entering the small space of the box to complete the final assembly. The higher the frequency of adjusting the positioning of the key parts, the longer the assembly cycle, which will lead to a lower assembly efficiency. If the sequence planning of the key parts’ assembly is not optimal, it may lead to more positioning adjustments of the key parts, which will lead to more fixtures and jigs, thereby increasing the economic cost of aero-gear assembly. The positioning cost of the key parts is defined as shown in Equation (4):

$$D_i=N_i\times T_i$$

(4)

• Di indicates the positioning cost of the i-th key parts; the equation can calculate the positioning cost of all key parts, and the larger the value, the higher the positioning cost, that is, the higher the assembly cost.
• Ni indicates the number of positioning adjustments of the i-th key parts.
• Ti indicates the time required for the positioning adjustment of the i-th key parts.

2.2. Evaluation Index of Assembly Ontology Dimension

The assembly ontology dimension of an aviation reducer is mainly considered from two perspectives, namely, the individual assembly base unit and the finished assembly as a whole; the evaluation index of an individual assembly base unit mainly considers its quality, size, symmetry, and fit relationship, while the evaluation index of the finished assembly as a whole mainly considers the operability of the assembly space, assembly stability, fixture changeability, and assembly direction.

2.2.1. Individual Indicators of Assembly Base Unit Parts

The factors affecting the evaluation of the sequence of the individual base unit of an aviation reducer are mostly related to its ontological properties, mainly including the quality, size, symmetry, and fit relationship of the individual base unit. Because of the complex structure of aerospace gearbox assembly, not all the factors influencing the evaluation of the sequence of individual base units are necessarily considered in the actual assembly process. Additionally, the evaluation system of the sequence of individual base units can be selected according to the actual needs of the assembly sequence planning of an aerospace gearbox. The factors influencing the sequence evaluation of the individual base units of aeronautical reducer gear are described below.

1

Quality of individual base units

In the assembly process of an aviation reducer, the larger the mass of an assembly unit individual, the more difficult it is for field craft personnel to carry out the assembly operation, resulting in the low operability of the assembly unit. For example, the mass of the main gearbox of a helicopter can only be moved to a tooling assembly station for the next assembly step by lifting it with a sling. This type of base unit, which requires additional auxiliary tools for assembly, will lead to an increase in the assembly cycle of the reducer and the assembly costs.

In the assembly process of an aviation reducer, the smaller the mass of the assembly unit individual, the easier it can be freed from auxiliary tools and the easier it is for the craftsman to carry out direct assembly. Thus, according to fuzzy evaluation theory, the partial trapezoidal distribution fuzzy set function is selected to calculate the fuzzy evaluation value of the base unit quality [35]. The calculation method of the quality index of individual base units is shown in Equation (5), where the closer the value of q(x) to 1, the easier the base unit is to assemble. That is, the higher the evaluation value of the quality index of an individual base unit, the easier it is to assemble, and a base unit with a low evaluation value can be considered to be installed first.

$$q\left(x\right)=\left\{\begin{array}{cc}0& x>q_{\max }\\ 1& x<q_{\min }\\ \frac{q_{\max }-x}{q_{\max }-q_{\min }}& q_{\min }\le x\le q_{\max }\end{array}\right.$$

(5)

• x indicates the current quality of an individual base unit.
• q_max indicates the maximum mass of the base unit that can be lifted by a technician for smooth installation, and this is determined according to experience.
• q_min indicates the minimum mass of the foundation unit that can be lifted by a technician for smooth installation.

2

Individual size of the base unit

The overall size of an aeronautical reducer is relatively large. Generally, the reducer box is fixed on the workstation first, and then components of different sizes are assembled sequentially. The site is effectively organized according to the actual number of craftsmen participating in the parallel assembly of multiple components.

In the assembly process of an aviation reducer, the size of an individual assembly unit can be too large or too small to be suitable for assembly. According to fuzzy evaluation theory, the trapezoidal fuzzy set function is used to calculate the index value of the size of an individual base unit, as shown in Equation (6):

$$w\left(x\right)=\left\{\begin{array}{cc}0& x<d_{\min },x\ge d_{\max }\\ 1& d_1<x<d_2\\ \frac{x-d_{\min }}{d_1-d_{\min }}& d_{\min }<x<d_1\\ \left|\frac{x-d_{\max }}{d_{\max }-d_2}\right|& d_2<x<d_{\max }\end{array}\right.$$

(6)

• x denotes the current shape size of an individual base unit.
• d₁ and d₂ denote the intervals of the size of an individual base unit suitable for tight space assembly, and the fuzzy evaluation value w(x) in this interval is 1, i.e., suitable for the assembly process to be carried out.
• d_min and d_max denote the minimum and maximum sizes of the base unit parts that can be assembled in the small assembly space of reducers, and they are determined empirically. The fuzzy evaluation value w(x) is 0 if the maximum size is exceeded or the minimum size is lower, which means that the assembly process is not suitable.

When the size of the assembly base unit is in the intervals [d_min, d₁] and [d₂, d_max], the fuzzy evaluation value w(x) is in the interval [0, 1], and the assembly base unit needs to be assembled using auxiliary tools. Therefore, the smaller the fuzzy evaluation value of the assembly base unit, the poorer the operability, and the assembly should be prioritized.

3

Symmetry of individual base units

Typically, the individual base units of aerospace reducers have a complex structure, and many components are symmetrical. Base units are assembled according to whether the symmetry of their structure is good or bad. Assembly base units with poor symmetry in their structure are given priority.

In the assembly process of an aviation reducer, the better the symmetry of the base unit body, the fewer adjustments to the assembly direction are required and the greater the operability of the assembly. Generally, the symmetry of an assembly base unit is determined according to the distance between the center of mass of the unit and the ideal symmetry center of mass. Thus, the basic idea of an enclosed box is introduced to calculate the symmetry index of individual base units, and the center of the enclosed box is taken as the ideal symmetry center of mass; the calculation method is shown in Equation (7):

$$d\left(x\right)=\left\{\begin{array}{c}1-\frac{\left|x-u\right|}{l_u}\\ 1-\frac{\left|y-v\right|}{l_v}\\ 1-\frac{\left|z-w\right|}{l_w}\end{array}\right.$$

(7)

• x, y, and z denote the coordinates of the center of mass of the assembly base unit.
• u, v, and w denote the coordinates of the center of the enclosed box of the assembly base unit.
• l_u, l_v, and l_w denote the maximum values of the side lengths of the enclosed box of the assembly base unit in the u, v, and w directions.

In Equation (7), the closer the center of mass of the assembly unit is to the ideal symmetric center of mass, the better the symmetry. x, y, and z represent the symmetry of any three dimensions of the assembly unit, assuming that the distance between the two centers in a certain direction is zero, which means that the unit is symmetrical in that direction. As the distance between the two centers increases, the individual symmetry value d(x) of the base unit becomes smaller, which means that the symmetry of the base unit in that direction is worse, and, thus, it should be given priority.

4

The fit relationship between individual base units

The fit relationship between individual base units mainly includes a gap fit, a transition fit, and an interference fit, resulting in three types of fit couplings. In an aviation reducer base unit, some individual units have more than one fit; that is, there are a variety of fit relationships. The higher the fit relationship value of the individual base unit, the more it should be prioritized for assembly.

The evaluation value of the fit relationship between base unit individuals is determined according to the experience of the craftsman, as shown in

Table 1. Evaluation value of fit relationship index of assembly basic unit.

Fitting Relationship	Gap Fit	Transition Fit	Overfill Fit
Empirical evaluation value	0.5	0.3	0.2

. If the individual base unit has both a gap fit and an interference fit or a transition fit in the assembly process, then the lowest evaluation value is selected. From the assembly base unit fit relationship index value table, it can be seen that base units with a lower value should be assembled first.

2.2.2. Assembly Integrity Evaluation Index

The evaluation index of the overall assembly sequence planning of an aviation reducer is mainly used to determine the operability of process personnel in a small assembly space and the influence of each assembly process on the completion of the final product. There are many factors involved in the overall evaluation index of aviation reducer assembly completion, and this paper mainly analyzes four of them, namely, assembly space operability, assembly stability, fixture changeability, and assembly direction adjustment times, so as to realize the evaluation of the overall assembly sequence of reducer assembly completion, as shown below.

1

Assembly space operability

The internal space of an aerospace reducer is limited, so the first consideration is determining how to ensure the operability of the assembly in a small space. The assembly space operability depends on the geometric configuration of each assembly unit of the reducer. A user-friendly geometric configuration is easier to assemble without interference, and the greater the user-friendliness of the geometric configuration, the easier it is to assemble the assembly unit in a small space. Additionally, delayed assembly can be considered in the sequence planning.

Aviation reducer assembly space operability means that, when a part is assembled in a certain direction, it does not interfere with other basic assembly units in its assembly path, and, if interference occurs, then the path cannot be selected. The assembly unit interference matrix mainly includes an interference matrix based on the standard orthogonal axis and an extended interference matrix [5]. According to the definition of the interference matrix, it can be determined whether the assembly base unit has assembly space operability in more than one direction, and, if so, then it will be easier to complete the process of aviation reducer assembly. The calculation of the assembly space operability index is shown in Equation (8):

$$g=1-\frac{k}{l+k}$$

(8)

• k indicates the number of times interference occurs during the assembly of the assembly base unit.
• l indicates the number of assembly base units assembled with the direction of freedom.

In Equation (8), it can be seen that the more interference occurs in the assembly process of the assembly base unit, the smaller the value of the assembly space operability index, and, thus, it is not easy to complete the assembly; the larger the number of degrees of freedom, the larger the value of the assembly space operability index, and, thus, it is easier to complete the assembly.

2

Assembly stability

The stability of aerospace reducer assembly refers to the degree of stability that each assembly unit can maintain under the constraint of the relevant fit relationship by its own gravity alone. In the assembly process of a reducer, the higher the assembly stability of an assembly relying on its own gravity, the more secure the fit relationship between the assembly units and the more stable the sequence planning of the assembly. In general, the more parts of the assembly unit that can be assembled by their own gravity, the fewer parts of the assembly unit that need to be reoriented, the less auxiliary tools are needed, the lower the assembly time and economic costs, and the better the overall assembly sequence planning of the reducer. In the assembly process of an aerospace reducer, the higher the assembly stability of an assembly body relying on its own gravity, the more secure the fit relationship between its assembly units and the more stable the sequence planning of the assembly body. Thus, from the two perspectives of self-gravity and the fit relationship, the assembly stability evaluation index is established, as shown in Equation (9):

$$Wd=\frac{1}{1+W{d}_g+W{d}_p}$$

(9)

• Wd_g indicates the stability of the assembly in the direction of its own gravity, and $W{d}_g={\sum }_{i=1}^n{g}_{ij}$, where g_ij indicates the number of any two unit parts in the assembly with n unit parts that cannot support each other in the direction of gravity; the greater the mutual support, the more unstable the direction of gravity.
• Wd_p indicates that the assembly depends on the stability of the fit relationship, and $W{d}_p={\sum }_{i=1}^n{p}_{ij}$, where p_ij indicates that the assembly with n unit parts in any two unit parts with instability, the worse the stability of the assembly.

3

Tooling fixture changeability

Fixture changeability refers to the frequency of replacing auxiliary tools in the assembly process of an aerospace reducer. The lower the frequency of fixture changes, the higher the evaluation of the assembly sequence of an aero-gear reducer, the more compact the assembly sequence, and the lower the assembly cost. Therefore, based on the principle that the higher the frequency of tooling fixture changes, the lower the assembly efficiency of an aero-gear reducer, a calculation method for the tooling fixture changeability index is established, which is shown in Equation (10):

$$T=\frac{1}{1+t}$$

(10)

• t indicates the number of fixture changes in the assembly process of an aviation reducer.

4

The number of assembly direction adjustments

In the process of aviation reducer assembly, the number of assembly direction adjustments should be reduced as much as possible to ensure that most parts are assembled in one assembly direction. In this way, the higher the assembly stability, the fewer fixtures are used, and, thus, the reducer assembly sequence planning becomes easier to optimize. Assuming that the assembly sequence of the reducer is fixed, then the assembly direction is also fixed in the established assembly sequence, so it is possible to derive all the adjustment times in the assembly direction of the established assembly sequence, i.e., the calculation of the index of the number of assembly direction adjustments, as shown in Equation (11):

$$D=\frac{1}{1+d}$$

(11)

• d indicates the number of all adjustments in the assembly direction of the established assembly sequence during the assembly of an aviation reducer.

3. Evaluation Method

In this section, to begin the study of the evaluation method for aero-reducer assembly sequence planning, we mainly focus on the cost control of the assembly of an aero-reducer and the control of the assembly of the body from the four perspectives of assembly time, economy, the assembly of individual basic units, and the completion of the whole assembly. From the definition of each evaluation index of the assembly sequence planning evaluation index system for aero-gear reducers, it can be seen that the values are basically non-numerical, and, thus, the results cannot be objectively evaluated but rather only judged subjectively by the relevant process personnel. Additionally, different process personnel give different evaluation results. The assembly sequence evaluation of all aero-gear reducer assembly cost and ontology dimensions is a fuzzy evaluation problem. In general, the solution to the fuzzy evaluation problem is to establish a fuzzy set function, but the evaluation index for the assembly sequence of the cost dimension of aero-gear reducers cannot be measured using numerical values. Therefore, fuzzy evaluation theory is applied to the evaluation index of the cost and body dimensions in order to establish a multidimensional assembly sequence evaluation method for aero-gear reducers. In this section, the value of each evaluation index of the cost dimension assembly sequence of an aero-gear reducer is defined in the form of a data table.

3.1. Evaluation Index Fuzzy Set

3.1.1. Cost Dimension

In terms of the cost dimension, the evaluation indices of the aviation reducer assembly sequence mainly include assembly operation methods, auxiliary time cost, tooling cost, and the positioning cost of key parts. In this paper, through the fuzzy evaluation method and according to the actual assembly process in a reducer assembly workshop, these four indices are divided into five fuzzy levels, of which level 1 is the lowest and has the worst evaluation effect, and level 5 is the highest and has the best evaluation effect. The fuzzy set is represented by the letter L, which is shown in Equation (12):

$$L=(l_{i1},l_{i2},\dots ,l_{ij})$$

(12)

• i denotes the cost dimension evaluation index, i = {1, 2, 3, 4}.
• j denotes the cost dimension fuzzy rank, j = {1, 2, 3, 4, 5}.
• l_ij denotes the result of the j-th rank of the i-th evaluation index.

As shown in

Table 2. Comparison of fuzzy evaluation indices of aviation reducer cost dimension.

Evaluation Indices	Level 1 Worst	Level 2 Poor	Level 3 Fair	Level 4 Good	Level 5 Best
Assembly operation methods	Slowest	Slow	Average	Fast	Fastest
Auxiliary time cost	Most	More	Average	Less	Least
Positioning cost of key parts	Longest	Long	Average	Short	Shortest
Tooling cost	Highest	High	Average	Low	Lowest

, each index is divided into five fuzzy levels, and the four evaluation indices in the table are scored by experienced assembly process personnel according to the current assembly cost execution scheme, giving the evaluation results of the corresponding levels to provide favorable support for the optimization of the assembly sequence.

3.1.2. Ontology Dimension

The evaluation index factors of the aviation reducer assembly sequence based on the ontology dimension mainly include the quality of individual base units, dimension, symmetry, fit relationship, the operability of the assembly space, assembly stability, the changeability of tooling and fixtures, and the assembly direction. The fuzzy set of evaluation indices for the ontology dimension is established and is shown in Equation (13):

$$B=(b{p}_{i1},b{p}_{i2},\dots ,b{p}_{ij})\cup (b{a}_{i1},b{a}_{i2},\dots ,b{a}_{ij})$$

(13)

• bp_ij denotes the index factors of the ontology dimension of the individual assembly base units.
• ba_ij denotes the overall index factor of the ontology dimension of assembly completion.
• i denotes the number of ontology dimension evaluation indicators, i = {1, 2, 3, 4}.
• j denotes the number of ontology dimension fuzzy levels, j = {1, 2, 3, 4, 5}.
• l_ij denotes the j-th rank result of the i-th evaluation index.

As shown in

Table 3. Comparison of fuzzy evaluation indices of aviation reducer body dimension.

Evaluation Indices	Level 1 Worst	Level 2 Poor	Level 3 Fair	Level 4 Good	Level 5 Best
Quality of individual unit	Maximum	Large	Average	Small	Minimum
Form dimensions	Maximum	Large	Average	Small	Minimum
Symmetry	Lowest	Low	Average	High	Highest
Fitting relationship	Worst	Poor	Average	Good	Best
Assembly Operability	Worst	Poor	Average	Good	Best
Assembly stability	Worst	Poor	Average	Good	Best
Tooling fixture Variability	Most	More	Average	Less	Least
Assembly direction	Most	More	Average	Less	Least

, each index is divided into five fuzzy levels. Experienced assembly process personnel score the eight evaluation indices in the table, namely, the quality of individual base units, form dimension, symmetry, fit relationship, assembly space operability, assembly stability, tooling fixture changeability, and assembly direction, according to the current assembly cost implementation scheme, to give the corresponding level evaluation results for the optimization of the assembly sequence.

3.2. Indicator Evaluation Matrix

Because an aviation reducer is an important part of the weapon and equipment transmission system, at the beginning of the reducer design, the guidance of rich manual experience is relied upon, but its high quality and high performance during use must be guaranteed, which requires evaluation by very experienced design, processing, and assembly process personnel. In the fuzzy set of cost and ontology dimension evaluation indices, the five fuzzy levels correspond to five scores: 1 point corresponds to level 1, 2 points correspond to level 2, 3 points correspond to level 3, 4 points correspond to level 4, and 5 points correspond to level 5. The corresponding scores are obtained after the experts’ comments, and, finally, all the experts’ scores are calculated to obtain the evaluation results of the fuzzy set of each evaluation index of the cost dimension, i.e., matrix C_M (4 rows × 5 columns), which is shown in Equation (14):

$$C_M=\left[\begin{array}{cccc}{c}_{i1}& c_{i2}& \dots & c_{ij}\end{array}\right]=\left[\begin{array}{cccc}{c}_{11}& c_{12}& \dots & c_{15}\\ .& .& \dots & .\\ .& .& \dots & .\\ c_{41}& c_{42}& \dots & c_{45}\end{array}\right]$$

(14)

• c_ij denotes the expert evaluation result of the i-th evaluation index; c_ij = n × g_ij, where n denotes the number of people who scored the corresponding level of the index, and g_ij denotes the score.

Similarly, the fuzzy set evaluation results of each evaluation index of the ontology dimension, i.e., matrix B_M (8 rows × 5 columns), are obtained using Equation (15):

$$B_M=\left[\begin{array}{cccc}{b}_{i1}& b_{i2}& \dots & b_{ij}\end{array}\right]=\left[\begin{array}{cccc}{b}_{11}& b_{12}& \dots & c_{15}\\ .& .& \dots & .\\ .& .& \dots & .\\ b_{81}& b_{82}& \dots & c_{85}\end{array}\right]$$

(15)

• b_ij denotes the expert evaluation result of the i-th evaluation index; b_ij = n × g_ij, where n denotes the number of people who scored the corresponding level of the index, and g_ij denotes the score.

3.3. Evaluation Index Weights

3.3.1. Cost Dimension Weights

The evaluation of the assembly cost of aviation reducers mainly includes two dimensions: assembly time cost and economic cost. The assembly time cost dimension mainly includes three indicators: assembly operation method, auxiliary time cost, and key parts positioning cost. The economic cost dimension includes one indicator: tooling cost.

1

Weight of assembly time cost dimension

It is assumed that the weights of the three indicators of the assembly cost dimension are Z₁, Z₂, and Z₃, and the weight values are given according to the expert evaluation, as shown in

Table 4. Weights of assembly time cost of aviation reducer.

Evaluation Indices	Assembly Operation Method Z1	Auxiliary Time Cost Z2	Key Parts Positioning Cost Z3
Weight value	0.5	0.3	0.2

.

2

Assembly economic cost dimension weights

It is supposed that the weight of one index of the assembly economic cost dimension is Q₄, which is mainly determined by the ratio of the expert-predicted cost B₁ to the actual cost B₂ of the reducer. Different cost ratios correspond to different weights, as shown in

Table 5. Weight of assembly economic cost of aviation reducer.

B1/B2	0–0.03	0.03–0.08	0.08–0.15	0.15–0.2	0.2–0.3	>0.3
Q4	0	0.1	0.2	0.4	0.5	1

.

3

Overall indicator weights for the cost dimension

The sum of the weights of the three indicators in the assembly time cost dimension is 1; i.e., Z₁ + Z₂ + Z₃ = 1. In order to simultaneously evaluate the weights of the assembly economic cost and the three indicators in the time cost dimension for the gearbox assembly sequence, the weight coefficient k = 1 − Q₄ is introduced to obtain the weights of the three indicators in the time cost dimension in the aviation reducer assembly sequence as Q₁ = k × Z₁, Q₂ = k × Z₂, and Q₃ = k × Z₃. That is, the weight Q_A of each indicator of the cost dimension can be determined as shown in Equation (16):

Q_A = (Q₁, Q₂, Q₃, Q₄)

(16)

3.3.2. Ontology Dimension Weights

The evaluation index of the aviation reducer assembly sequence based on the ontology dimension mainly includes eight factors: the quality of individual base units, overall dimension, symmetry, fit relationship, the operability of the assembly space, assembly stability, the changeability of tooling fixtures, and the assembly direction. According to the actual assembly situation and experience value of craft personnel, different weights are obtained for each index. The set of weight values for the ontological dimension evaluation indices is shown in Equation (17), and the weights of each index satisfy Equation (18).

$$B_Q=\left[\begin{array}{cccc}{b}_{q1}& b_{q2}& \dots & b_{qi}\end{array}\right]$$

(17)

Here, i = 8 indicates the eighth evaluation index, and b_qi indicates the i-th evaluation index weight value.

$${\sum }_{i=1}^n{b}_{qi}=1$$

(18)

3.4. Comprehensive Evaluation Method

3.4.1. Comprehensive Evaluation Method of Cost Dimension Evaluation Indices

The comprehensive evaluation of the cost dimension of aviation reducer assembly allows one to examine the feasibility of reducer assembly sequence planning from the perspective of assembly cost. The results of each evaluation index of the cost dimension and the weight of each evaluation index are comprehensively considered, and, after fuzzy transformation, the overall impact of each evaluation index can be determined from the assembly cost dimension. Thus, the comprehensive evaluation model of the cost dimension is D_C, as shown in Equation (19):

$$D_C=Q_A\times C_M=\left(Q_1,Q_2,Q_3,Q_4\right)\times \left(c_{i1},c_{i2},\dots ,c_{ij}\right)=\left(D_{c1},D_{c2},\dots ,D_{cs}\right)$$

(19)

$$D_{cs}=\left(\begin{array}{c}\begin{array}{c}{Q}_1\times c_{15}\\ Q_2\times c_{25}\end{array}\\ \begin{array}{c}{Q}_3\times c_{35}\\ Q_4\times c_{45}\end{array}\end{array}\right)$$

(20)

• i denotes the i-th evaluation index.
• D_cs denotes the score of the sth fuzzy level, s = (1, 2, 3, 4, 5), and the level with the highest evaluation score among the five fuzzy levels of the comprehensive evaluation model D_C is the result of this comprehensive evaluation of the cost dimension of the reducer, i.e., the comprehensive evaluation function of the cost dimension.

3.4.2. Comprehensive Evaluation Method of Ontology Dimension Evaluation Index

Considering the results of the ontology dimension evaluation indices and the weights of each evaluation index, after fuzzy transformation, the ontology dimension comprehensive evaluation model is obtained as D_b, as shown in Equation (21):

$$D_b=B_Q\times B_M=\left[\begin{array}{cccc}{b}_{q1}& b_{q2}& \dots & b_{qi}\end{array}\right]\times \left[\begin{array}{cccc}{b}_{11}& b_{12}& .& b_{15}\\ .& .& .& .\\ .& .& .& .\\ b_{81}& b_{82}& .& b_{85}\end{array}\right]=\left[\begin{array}{cccc}{D}_{b1}& D_{b2}& \dots & D_{b5}\end{array}\right]$$

(21)

3.4.3. Multidimensional Evaluation Index Comprehensive Evaluation Method

For the evaluation of the assembly sequence planning of aviation reducers with complex structures, this paper studies the evaluation method of a complex assembly sequence of aviation reducers from the two dimensions of assembly cost and assembly ontology. The assembly cost dimension studies four evaluation indices and methods from the perspectives of time and economy, and the assembly ontology dimension studies eight evaluation indices and methods from the perspectives of the assembly basic unit and assembly integrity. However, the results obtained from the evaluation of the assembly sequence planning of aviation reducers through a one-dimensional single perspective or one-dimensional multiple perspectives cannot fully meet the expectation of the optimal results of a complex assembly sequence, and the evaluation results are too one-sided. Therefore, in order to evaluate and analyze the complex assembly sequence of aviation reducers more objectively, it is necessary to comprehensively evaluate the evaluation indices of the assembly cost and assembly body, as shown in Equation (22):

$$D_z=m\ast D_c+n\ast D_b=\left[\begin{array}{cccc}m\ast D_{c1}+n\ast D_{b1}& m\ast D_{c2}+n\ast D_{b2}& \dots & m\ast D_{c5}+n\ast D_{b5}\end{array}\right]$$

(22)

• m denotes the cost dimension weight parameter.
• n denotes the ontology dimension weight parameter.

4. Evaluation Example and Results

Based on the assembly process of the first-stage reducer box of a helicopter’s main reducer, the assembly sequence scheme is comprehensively evaluated from the cost dimension. Six technicians with rich industry experience, mainly from the design process room, assembly workshop, and process planning room, participated in the evaluation. An assembly model of the first-stage reducer box is shown in Figure 3. According to the above evaluation methods, the specific evaluation process is described below.

4.1. Personnel Involved in the Evaluation of Each Evaluation Index of the Cost Dimension

The process personnel involved in the evaluation combined their actual knowledge of the current first-stage reducer box assembly program and evaluated the aviation reducer cost dimension according to the fuzzy evaluation data sheet. The number of participants in each index is shown in

Table 6. The number of participants in each index.

Evaluation Indices	Level 1 Worst	Level 2 Poor	Level 3 Fair	Level 4 Good	Level 5 Best
Assembly operation methods	1	1	1	3	0
Auxiliary time cost	0	1	3	1	0
Positioning cost of key parts	0	1	3	2	0
Tooling cost	0	1	4	1	0

below.

Table 6 shows the following: Regarding assembly operation methods, one expert gives the worst evaluation, one expert gives a poor evaluation, one expert gives a fair evaluation, and three experts give a good evaluation. Regarding auxiliary time cost, one expert gives a poor evaluation, three experts give a fair evaluation, and one expert gives a good evaluation. Regarding the positioning cost of key parts, one expert gives a poor evaluation, three experts give a fair evaluation, and two experts give a good evaluation. Regarding tooling cost, one expert gives a poor evaluation, four experts give a fair evaluation, and one expert gives a good evaluation.

4.2. Evaluation Results of Each Cost Dimension Index

$$C_M=\left[\begin{array}{ccccc}1& 2& 3& 12& 0\\ 0& 2& 9& 4& 0\\ 0& 2& 9& 8& 0\\ 0& 2& 12& 4& 0\end{array}\right]$$

4.3. Weights of Each Indicator of Cost Dimension

According to the table of the economic cost weights of aviation reducer assembly in Section 3.3, the economic cost weight of the first-stage reducer box is 0.2, so the weights of each index are as follows:

Q_A = (0.4, 0.24, 0.16, 0.2)

4.4. Comprehensive Evaluation Results of Cost Dimensions

D_C = (0.4, 2, 7.2, 7.84, 0)

According to the comprehensive evaluation results, f_DC = 7.84; i.e., the comprehensive evaluation level corresponds to level 4—“good”—which means that the current assembly scheme of first-stage reducer boxes is feasible. To further optimize the assembly sequence, it can be analyzed from the results of the individual evaluation indices; i.e., the situation of each evaluation index can be analyzed using the experts’ evaluation results in the table of the aviation reducer cost dimension. For example, the evaluation results of tooling cost are as follows: one expert rated it as poor, four experts rated it as fair, and one expert rated it as good. This means that five out of six experts rated it as “average”. Therefore, the tooling cost has the possibility of further optimization. If the cost optimization of tooling cost is carried out according to the expert evaluation results, then the cost dimension needs to be evaluated again afterward.

5. Conclusions

In view of the lack of effective methods evaluating the assembly sequence of aviation reducers from multiple perspectives and levels, the authors of this paper study evaluation methods for the assembly sequence of aviation reducers from the two dimensions of assembly cost and assembly ontology. The following conclusions are drawn:

(1) The main factors affecting the assembly sequence planning of aviation reducers with complex structures are considered to be the ontological attributes of the assembly base unit parts. Thus, combined with the assembly process cost control problem, an aviation reducer assembly sequence planning evaluation index system based on assembly cost and ontological elements is constructed, which provides a guidance direction for the development of a sound aviation reducer evaluation index system.

(2) Based on the newly constructed evaluation index system for aviation reducer assembly sequence planning, an assembly sequence evaluation method based on the cost dimension is proposed, in combination with fuzzy set evaluation theory. This method uses a data table to define the values of the four evaluation indicators of the cost dimension of the assembly sequence of aviation reducers. Process personnel first evaluate each indicator and then carry out a comprehensive evaluation based on the different fuzzy level scores; i.e., a two-level fuzzy evaluation method is used. This method allows for both sets of comprehensive results of the cost dimension multi-factor evaluation to be used to determine the goodness of the assembly solution and the optimization of the evaluation indicators using the evaluation results of the individual influence factor evaluation indicators. By combining the evaluation results of eight evaluation indicators of the ontology dimension and dividing the weights, a multidimensional and multi-perspective (12 dimensional members) comprehensive evaluation method for the aviation gearbox assembly sequence is formed.

(3) Based on the assembly process of the primary gearbox of a helicopter’s main gearbox, a comprehensive evaluation of its assembly sequence scheme is carried out from the cost dimension. From the comprehensive evaluation results, it can be seen that the current assembly scheme of the primary reducer gearbox is feasible. However, an analysis of the results of the individual evaluation indicators shows that there are possibilities to further optimize the tooling costs. If the cost optimization of the tooling fixture is carried out according to the results of an expert evaluation, then the cost dimension needs to be evaluated again afterward.

This paper proposes a comprehensive evaluation method for the assembly sequence of aviation reducers based on cost and ontology dimensions, which provides a feasible technical means to optimize the assembly sequence of aviation reducers with complex structures. The method is firstly applicable to the assembly process of a certain series of helicopter reducers, and also applicable to other aviation reducers with similar structures and similar assembly processes. The evaluation results can be more objective and closer to the optimal results of complex assembly sequences. The research in this paper can effectively promote the progress of complex assembly sequence optimization technology for aviation reducers. Furthermore, it further improves the evaluation index system for the assembly sequence planning of aviation reducers, and it realizes a comprehensive evaluation of the overall assembly quality and performance indices of aero-gearbox products.