**2. Literature Review**

Various methods of assembly support called DFX—design for X—has been developed and spread across industry methods such as QFD (quality function deployment) [4–6] used in the processes of implementing product customer requirements, FMEA (failure mode and effect analysis) [7]—related to the prediction and prevention of problems at the product design stage, DFA (design for assembly) [8–10]—e.g., design for manufacturing (DFM) regarding the shaping of the design process of components and the product itself [9,11–15]. Decisions made at the product design stage have a significant impact on production costs, efficiency and quality of production.

In the process of implementing the product, the impact of design on the cost of its implementation is very significant. The share of design costs varies around 5% of the costs of starting production of a new product but affects about 70% of the cost of the product after its implementation into production. While the rationalization activities of the production process at the production stage of the product (direct labor costs and indirect production costs often account for around 40% of the cost of the final product) affect only about 10% on the cost of production of the product [1,14,16]. This information is based on the US market cost structure. This means that the actions related to the changes in the production project (with relatively low costs incurred) in the right time have the greatest real impact on the production costs of the final product [1,15,17]. Swift [15] and others [1,7,18] analyzed the percentage of problems that occur in companies that have not performed and DFA activity during the product design development. For example, 35.9% had problems with the assembly of individual parts of the evaluated design. DFA methods have been selected to evaluate how their effects provide the greatest impact on manufacturing cost [1,14,15,19,20]. The DFA methods described in the literature and used in manufacturing practice were aimed at serial and mass product production [8–10,12,15,17,21]. There has been described in the latest scientific studies some of new DFA methods connected with CAD/CAM systems or Life Cycle assessment, some attempts to use fuzzy logic were also done however complete fuzzy DFA method for different kinds of production hasn't been recognized in literature [18,21–25] thus the proposed method also opens to small-lot and unit production.

Considering the methods described in the literature, we chose to focus on the analysis of unit and small-series production. We also point out that a method is needed to evaluate the production process in a comprehensive way that takes care of machining, assembly and production organizations. The use of fuzzy logic is justified by the need to estimate data due to the lower availability of complex analytical tools in small businesses, development budgets and product testing are limited there and there may be no accurate data for the reasons listed above. The method developed is open and other or additional criteria may be considered according to the production conditions of the company concerned. It may also have been interesting to develop DFA methods in medical procedures and the medical industry [26–30]. Another area of interest is the processes of electrical and electronic components that have been rapidly developing in recent years [28,31,32].

#### **3. Design Manufacturability Assessment in Terms of Unit and Low-Volume Production**

#### *3.1. Assumptions for the New Method Design for Manufacturability*

The justification for the emergence of a new fuzzy method to assess the technology of the structure resulted from the observed lack of flexibility of the described methods of Boothroyd-Dewhurst and Lucas. These methods were created in the 1980s where there was demand in the economy and was focused on serial and mass production. The current development of the economy and technology means that the modern economic system is characterized by a much greater need for flexibility in terms of production methods: high volume, low volume and in units. The need to create a more flexible method which is adaptable to the type of production was noticeable [33].

The design process should be determined from the point of view of various usability criteria—Figure 1. The assessment should consider many other various factors, sales, service, spare parts availability, production series, types of equipment, available assembly techniques, level of automation, cooperative services, possibilities of application commercial components, crew technical culture, etc. In small-lot and serial production conditions, the design process for new product production was based on simplified production documentation. Due to the low production series, production data

results from the project were rarely verified at the production stage, while the experience gained from this stage was used in the production projects of new products. Concerning mass production, particular attention from the point of view of cost criterion was paid to the possibility of using unified and standardized elements included in the final product, the use of work stations and workshop aids for processing and assembly of various elements included in the products making up the program production and introduction of group machining processes, process phases, group operations for various elements [34–36]. The newly proposed method using fuzzy inference was characterized by such flexibility. In the literature cited in the study [15,19,37–40] there has been lack of studies enabling in the absence or uncertain data to estimate the times of assembly operations. The developed method has the features of novelty and meets the needs of production practice.

**Figure 1.** Modified design and development process to produce a new product.

#### *3.2. The Course of the New Method Design for Manufacturability*

In this study, there was a proposed new model of the product design analysis process which was carried out by experts representing: product design, machining process design, assembly process design, quality assurance, product cost analysis, OHS and environmental protection. Their inputs were assessed with the help of fuzzy sets methods following Figure 2. The assessment of the design manufacturability from the point of view of the assembly process was the first step followed by the machining process and production organization. According to experts, the order of assessment results from the size of the impact of the assessed design manufacturability on production efficiency. The feedback in the assessment activities results from the impact of decisions made in one stage on the other stage assessments.

**Figure 2.** Structural analysis of the structure's technology in the proposed method.

Due to the costs of accurate analyses, there was less possibility to determine the performance parameters of the designed process in a unit, small-lot production due to unique, unstable and non-rhythmic production in the form of values, in the form of deterministic assessments. Therefore, fuzzy logic may be useful in such production conditions. Experts determine the fuzzy marks based on their own experience in the order given in Figure 2. The assessment was made on a scale of 0 to 100. Triangular symmetrical distributions were used for the assessments. The assessment method was presented below: when assessing, experts can be guided by their own production experience, they can also use data tables in the Boothroyd & Dewhurst and Lucas methods.

The assessment was related to the set of linguistic variables Vi = {V1, ... , Vn} and ∈N—{0}, defining the input and output criteria of technology. The linguistic variable Vi was described by a quadruple:

$$\begin{bmatrix} \Pi\_{\text{i}} & T\_{\text{i}}(\text{L}) \end{bmatrix} \begin{bmatrix} \Omega\_{\text{i}} \ \mathbf{M}\_{\text{i}} \end{bmatrix} \tag{1}$$

where: Li = {L1, ... , Ln}, i ∈ N—{0}—set of linguistic variable names, Ti(Li) = {T1(L1), ... , Tn(Ln)}, i ∈ N—{0}—set of countable determinations of linguistic variables, tij = {t11, t12, ... , tnm}, i, j ∈ N—{0}, tij - Ti (Li)—set of linguistic values of linguistic variables, Ω<sup>i</sup> = {Ω1, ... , Ωn}, i ∈ N—{0}—set of linguistic ranges of variables Vi, Mi = {Mi, ... , Mn}, i ∈ N—{0}—set of semantic rules, mij = {m11, m12, ... , mmn}, and, j ∈ N—{0}, mij - Mi—range of variation in linguistic value tij with an assessment of belonging from 0 to 1 [41].

The assessment of the assembly process capability followed by the assessment of assembly technology and production organization corresponds to the stage of developing the project documentation of the product design. The applied variables V1, V2, V3, V4, V5, V6 in the scope of machining technologies, assembly, production organization are shown in Figure 2. The assessment, depending on the scope of information obtained, can be carried out for individual components of the product, groups of elements, its assemblies or also in a holistic way [41]. Sets of Vi variables can be modified and changed depending on the nature of the target process for which we design the product. This gives the fuzzy method a significant advantage in terms of flexibility. In the example presented, the set of variables Vi was prepared for medium-sized plant and small-lot production. It is illustrated by an example of one stage of the developed method to better illustrate the course of proceedings.

Variables that, in addition to deterministic values, can assume imprecise values—fuzzy. The triangular membership function can be defined using the following formula.

$$\mu\_A(\mathbf{x}) = \begin{cases} 0 & dla \quad \mathbf{x} \le a \, lub \, \mathbf{x} \ge c \\ \underset{\begin{subarray}{c} \frac{\mathbf{x} - \mathbf{a}}{c - a} \, \, dla \\ \frac{c - \mathbf{x}}{c - b} \, \, dla \end{subarray}}{\, dla} & a \le \mathbf{x} \le b \\ b \le \mathbf{x} \le c \end{cases} \tag{2}$$

where *a*, *b* and *c* are parameters meeting the condition *a* < *b* < *c*.

Figure 3: presents a graph of the membership function of a given Formula (1).

**Figure 3.** Graph of the triangular membership function described by the Formula (1).

It was assumed that two input variables (×1 and ×2) and a single output variable (y) are related, respectively: {small, medium, large}, {short, medium, long} and {bad, medium, good}. What can be presented in the form of language rules:

R1W IF X1 is small and X2 is short, THEN Y is bad; also R2W IF X1 is small and X2 is medium, then Y is bad, too R3W IF X1 is medium and X2 is short, THEN Y is medium; also R4W IF X1 is large and X2 is medium, THEN Y is medium; also R5W IF X1 is large and X2 is long and Y is good

The rules can be presented in the decision table (Table 1) whereas, an example of a fuzzy partition is shown in Figure 4.


**Table 1.** Sample decision table.

**Figure 4.** Example of a fuzzy partition where vs. -, S-, M-, L-, VL-.

We perform calculations for the values of V1, Vi + 1 by reading the values from the graphs in Figures 3 and 4, according to the inference rule "min" specific rules were activated on the basis of which we set conclusions for the selected component that we evaluate. The next stage was the aggregation of conclusions, we should activate the selected rules for the selected component. In Mamdani's inference, which we use, there was a maximum operation as an operator of the aggregation of inference results obtained based on individual rules. For low average technology (range <0; 60>), the conclusion assumes a min value (0.67; medium technology)—lower value 0.67 or the value of the function, medium low technology. Fuzzy logic means that in the process of fuzzification, each rule was given a certain fuzzy value and must then be converted back to the real value, for this purpose we have defuzzification. In the work for defuzzification, a center of gravity method was proposed, which serves to sharpen the resulting fuzzy set and consists in determining the value of y \*, which was the center of gravity of the area under the curve μ*wyn* (y).

The Mamdani processing structure of fuzzy set inference methods consist of the following five elements:


The reference model of the project was of the type: multiple inputs—multiple outputs MIMO. To compile results according to the above MATLAB software was used for the model.

The proposed DFA model of conduct based on fuzzy logic and the use of multiple entries—fuzzy rules (Figure 5) and multiple outputs enables efficient operation also in small-lot production conditions when there was no data from design verification by building many versions of prototypes and testing subsequent assumptions and design effects.

**Figure 5.** Reference model of the project is of the type: multiple entries—multiple outputs.

#### **4. Implementation**

#### *4.1. Input Assumptions*

Based on the analysis of the above methods of assessing the product's producibility, an improved proprietary approach was proposed in the process to shape the product's productiveness. The illustration of the presented proposals is presented on the example of a single-stage gear in Figure 6. General purpose gearboxes are designed in the form of a series of types from the point of view of market demand, production costs and delivery time to the customer. The gearbox is shown in Figure 6 was designed in a traditional way (welded body, many bolted joints, etc.).

**Figure 6.** Diagram of the analyzed gearbox. 2—body; 5, 6, 7, 8—bearing caps; 10—shaft; 11—pinion; 12—tooth gear; 14—spacing rings; 17; 16—bearings; 18, 19; —seals; 21, 22, 23—keys; 25, 26—washers; screws.

A manufacturability analysis of the design was carried out for the adopted criteria presented in Figure 7. To illustrate the progress of the procedure in the method, the method of assessing the technological efficiency of the structure is more widely presented, on the example of the assembly of two elements—the gear housing and cover.

**Figure 7.** Model of the new method design for manufacturability based on three successive stages and substages of fuzzy inference (without feedback).

### *4.2. Assembly Manufacturability Fuzzy Assessment*

The assessment was carried out in three substages—substage 1 (access, number of workshop aids), substage 2 (orientation, maneuverability), substage 3 (assemblability, processes).
