**1. Introduction**

Symmetry is an issue that largely affects artificial neural networks. Symmetry can be found in the structure of the neural network itself; symmetric weight and many other related issues are also in the presented article.

The process of assembling parts into assembly units until the final product is achieved is one of the most important stages of the production process. During the process, specific features characterizing the product are created. This process also makes a major contribution to the product development itself. Planning the assembly process requires a series of analyses, i.e., the separation of assembly units and the determination of the relationships between them. The result of these analyses is the selection of the base parts for the separate assembly units and the specific assembly sequences, which are the basis for estimating assembly difficulties. Product quality and manufacturing costs depend mainly on the product structure; this structure describes the functionally imposed layout and the geometrically possible assembly sequences. It is natural to choose which one allows us to obtain a finished product in the shortest time.

**Citation:** Suszy ´nski, M.; Peta, K.; Cernohlávek, V.; Svoboda, M. ˇ Mechanical Assembly Sequence Determination Using Artificial Neural Networks Based on Selected DFA Rating Factors. *Symmetry* **2022**, *14*, 1013. https://doi.org/10.3390/ sym14051013

Academic Editors: Peng-Yeng Yin, Ray-I Chang, Youcef Gheraibia, Ming-Chin Chuang, Hua-Yi Lin and Jen-Chun Lee

Received: 11 April 2022 Accepted: 12 May 2022 Published: 16 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Obtaining the most advantageous technology of assembling parts in given production conditions is an extremely important, multi-criteria and difficult to model task, which is mainly based on determining the assembly sequence plan (ASP) and selecting all the components of the production process, but also applies to balancing the assembly line (ALB). These issues are fundamentally related to the degree of automation of the considered process, but also to the production conditions in a given enterprise. Due to the complexity of the assembly sequence planning problem, its optimization is required in order to obtain an efficient computational approach. The aim is therefore to include the evaluation process and selecting the assembly sequence as early as possible in the product development phase. The high complexity of this process (great number of components) makes it difficult to determine the set of permissible assembly sequences and leads to a phenomenon that is difficult to solve, "combinatorial explosion"; therefore, one of the solutions to this problem is the heuristic approach presented in this publication, where neural networks based on selected DFA rating factors describing assembly sequences allow us to forecast a mechanical assembly time.

The paper, one by one, presents the optimization approach to the problem of assembly sequence generation with the use of heuristic methods, the issues of assembly sequence evaluation criteria and artificial neural networks in an assembly sequence planning problem. The most important part of the work is the concept of the system supporting the determination of the assembly sequence based on artificial neural networks and DFA rating factors for manual mechanical assembly.

## **2. Optimization Approach**

Bio-inspired algorithms are quite modern, increasingly used, and efficient tools for the industry. However, from a mathematical point of view, these problems belong to the most general of non-linear optimization problems, where these tools cannot guarantee that the best solution will be found. The numerical cost and the accuracy of these algorithms depend, among others, "on the initialization of their internal parameters which may themselves be the subject of parameter tuning according to the application" [1].

In the literature on the subject, various optimization algorithms are proposed to optimize the ASP problem. The most common classes of algorithms used to optimize ASP are meta-heuristic approaches. The meta-heuristic algorithm provides a solution framework to optimize different optimization problems with relatively few modifications to adapt them to a specific problem and limits the general computation time. This approach turns out to be, in many cases, sufficient due to the quality of the obtained results at appropriate costs in such a multi-criteria task as assembly sequence planning, where the set of feasible solutions to be analyzed is large [2–4].

Studies on ASP have implemented different heuristic optimization algorithms, such as genetic algorithm (GA) [5–7]. The authors of [6] presented the application of genetic algorithms (GA) and the ant colony algorithm (ACO), using the example of reflector antenna assembly. The accurate simulation of the assembly was required to evaluate and optimize the ASP. The initial population was created by ACO and optimized by GA operators to generate an optimal solution. By releasing the information on the optimal solution to the ant search paths of ACO, convergence towards a globally optimal solution was accelerated. A model of the finite element simulation was used to simulate the dynamic assembly process of the reflectors according to the algorithm results of the proposed approach, which can improve the quality of the optimal solution and reduce the probability of finding a local optimal solution.

Another method based on a genetic algorithm [7] was used in the process of planning the assembly sequence for satellites in the space industry. This method takes into account a process with a large number of connections in a multi-stage and parallel assembly. Priority relationships are established between the assembly units, and the assembly sequence is represented by a directed acyclic graph. Original, a two-part crossover and mutation operators for assembling sequence were proposed.

Another approach to generate acceptable assembly sequences is an algorithm based on a simulated annealing (SA) process [8,9]. The method is derived from an analogy with thermodynamics, and more specifically, with how a liquid solidifies to form a crystal structure. In [8], the authors proposed a novel method under the name of genetic simulated annealing algorithm (GSAA) and ant colony optimization (ACO) algorithm for assembly sequence planning (ASP), which assist the planner in generating an effective assembly sequence with respect to a large constraint assembly perplexity.

The ant colony optimization (ACO) algorithm was also presented in [6,8,10–12]. In [12], an improved ASP-oriented ant colony algorithm was proposed in order to achieve an optimal or close to optimal assembly sequence. In this algorithm, the assembly operation constraint is introduced into the state transfer function as heuristic information, while the feasible transition area is set up by obtaining assembly relationships of the components. In addition, a dynamic change in pheromone trail persistence was also used.

Evolutionary algorithms (EA) for connector-based assembly sequence planning were also analyzed [13], but they generated many erroneous searches and it was necessary to build a multi-agent connector-based evolution algorithm. Competition, crossover-mutation and learning were designed as the behaviors of agents that locate a lattice-like structure environment. The presented metaheuristic algorithms are highly efficient and seem to be interesting tools in solving ASP problems.

In [14], the authors proposed a three-stage integrated approach with some heuristic working rules to assist the planner in generating an effective assembly sequence. In this work, the back-propagation neural network (NN) was employed to optimize the available assembly sequence.

The results show that the proposed model can facilitate assembly sequence optimization and allows the designer to recognize the contact relationship and assembly constraints of three-dimensional components.

Other approaches to this problem were presented of the basis of immune algorithm (IA) [15,16], scatter search algorithm (SSA) [17], particle swarm optimization (PSO) [15] and other heuristic methods (HM) [4,18–20]. Figure 1 shows the published research on optimization ASP from analyzed articles and conference papers from the last twenty years based on the Google Scholar database.

**Figure 1.** Frequency of the algorithms of the optimization method usage in ASP.

#### *2.1. Assembly Sequence Assessment Criteria*

The planning of assembly sequences (ASP) consists of determining the feasible and, at the same time, the most advantageous order of joining assembly units due to certain criteria. The grea<sup>t</sup> complexity of the task of selecting the appropriate one from among all the acceptable and, at the same time, feasible ones, due to the constraints of the structural nature of the assembly sequence, is a consequence of the large number of possible combinations. It is, therefore, necessary, in the first stage, to limit the set of possible solutions, and then to evaluate them and select the most advantageous assembly sequence. This is inextricably linked with the use of certain criteria for this purpose, due to which the discussed process can be optimized. It is very often necessary to assign appropriate priorities/weights to the criteria used for the assessment that are tailored to the specific assembly process. In the literature on the subject, the assessment and selection of the most favorable sequence can be made according to various criteria, depending on the specifics of a process in the plant where the assembly takes place. Such criteria may be: the assembly time, the number of changes in the assembly directions, the number of tool changes, the stability of the assembly states that arise, the degree of difficulty in reaching the next process state, the complexity of the assembly unit movements, the correctness of the assembly itself or even the cost-effectiveness of the process. Selected data on the criteria for evaluating assembly sequences can be obtained automatically from the electronic construction documentation or supplemented on the basis of a case-by-case analysis.

#### *2.2. Assembly Sequence Assessment Criteria Based on DFA Rating Factors*

Design For Assembly (DFA) is one of the methodologies supporting the design of the assembly process. By introducing design changes, in line with the guidelines of the DFA methodology, we can achieve, above all, shorter assembly times, by reducing unnecessary components and the assembly tasks necessary to assemble the product. DFA analysis also highlights the possible weak points of the structure and helps to create a product that does not require major changes in further phases of the product lifecycle. Thanks to the introduction of DFA to the design process, the product development team proposes improved design solutions that are characterized by better indicators, such as a simple structure, which directly simplifies the assembly operations. The benefits of using this methodology are mainly due to the systematic review of functional requirements and the replacement of groups of elements by single integrated units–assembly modules. Generally, the designer carries out the presented analyses in a series of assessment charts. Next, the designer assesses the component functionality, manufacturing processes, form and assembly characteristics using values extracted from the charts according to component properties. Thus, the designer can quantify the suitability of the design. The best known DFA methods are the Boothroyd–Dewhurst System, the Lucas DFA Methodology and the Hitachi Assembly Ability Evaluation Method [21–23]. The method proposed in this article was based on assembly sequence determination using artificial neural networks. It is related to manual assembly and is largely based on the assessment of transitions between individual assembly states using a score taken from the following DFA rating factors charts (these data are used to train the neural network):

• The stability of the assembly unit (Table 1).

The stability criterion determines the assembly sequences with the fewest unstable states. A stable unit is one that remains in the assembled position regardless of the force applied. The acting forces can be the force of gravity or the force associated with the movement of the part/assembly.


**Table 1.** Assembly unit stability index based on the DFA (own study based on [22,23]).

• The movement and orientation index.

This is the criterion of the ability to move and orientate. Handling difficulties from the Boothroyd–Dewhurst table are rated from "easy to deliver and orientate—9" to "Requires gripping tools—0". Details of the assessment are given in Table 2.

**Table 2.** The criterion of the ability to move and orientate based on the DFA (own study based on [22,23]).


•The ease of joining index.

The ease of connection criterion evaluates whether the part is easy to grasp and assemble in a given process state (Table 3).

**Table 3.** The ease of joining index (own study based on [22,23]).


• The accessibility of the joining location index (Table 4). The accessibility determines whether it is sufficient to secure the part. Accessibilitydepends on the location of the parts in the product or its subassembly.


**Table 4.** The accessibility of the joining location based on the DFA (own study based on [22,23]).

#### **3. Artificial Neural Networks in Assembly Sequence Planning Problem**

An assembly sequence planning problem belongs to a class of optimization problems known as NP-complete. For such a class of problems, in order to find an optimal solution with regard to specific criteria, it is necessary to search the entire set of feasible solutions to ensure finding the optimal assembly sequence. With high complexity and the interdisciplinary nature of assembly problems, this strategy is very time-consuming and not practical in most industrial applications. For this reason, other heuristic techniques are often used in this case, allowing to find a solution that is close to the optimal one. The authors of this paper proposed a new approach to the problem of planning the assembly sequences. For this purpose, they used artificial neural networks to predict the assembly time determined by the assessment of transitions between its individual states according to the selected DFA criteria. This solution is an information-processing model inspired by a natural biological system, in which the interconnected processing elements (neurons) xn perform an input operation that will ultimately solve the problem of assessing the assembly sequence, whose optimization criterion comes down to determining the assembly time. Neurons are associated with synapses that are assigned weights with specific values, wk. Additionally, the intercept w0–bias is defined as the sensitivity threshold. A neuron is in an active state when the sum of the weighted input signals is greater than or equal to the sensitivity threshold w0. The vector of input data introduced into the network is multiplied by the synaptic weights according to the function of the postsynaptic potential to obtain the signal y, which is the output value of the network. Figure 2 shows the model of signal processing between neurons. The artificial neural network processes information by activating the neurons of the hidden and output layers with activation functions (transfer functions). The literature gives the highest efficiency of the logistic, exponential, linear, sine and hyperbolic tangent functions, which were used in the prediction model developed in this publication.

**Figure 2.** Schematic representation of the artificial neural network.

#### **4. The Concept of the System Supporting the Determination of the Assembly Sequence Based on Artificial Neural Networks and DFA Rating Factors for Manual Assembly**

The proposed method was based on artificial neural networks and selected DFA rating factors and aimed to assist in determining the order of manual assembly. It was assumed

that, at the current stage of research, it is used in a specific mechanical production company, where the conditions of the assembly process are relatively constant for the introduced new products, which undergo the ASP process, and the assembly processes carried out in it were used to teach the network. Such a system, together with an increase in the number of analyzed assembly processes, improves the efficiency of new process time estimation. The stability of the conditions are related to a specific production company and its concerns, for example, the available machine park, production organization, process control and control, and the level of training of employees (these are constant elements for the network). As input data for the process, the previously discussed assessments of transition between individual assembly states were used based on selected DFA rating factors. The operation of the method is, therefore, aimed at estimating the time for all sequences permissible due to the constraints of a constructional nature (feasible), and thus enables the selection of the most favorable one, due to the analyzed criteria in relation to a specific assembly process.
