4.2.1. Assembly Manufacturability Assessment—Substage 1

It was assumed that the assembly technology of the considered elements depends on two factors, which were: accessibility, number of workshop aids. The experts determined the ocean for the parameter "Access" = 20, "number of workshop aids" = 55. The functions of belonging linguistic variables for the given factors are given in Tables 2 and 3, the bases of rules for them are presented in Tables 4 and 5.



**Table 3.** Membership functions in tabular form of linguistic variables for "number of workshop aids".


**Table 4.** Rules database for "access".


**Table 5.** Rules database for "number of workshop aids".


The "Access" factor is described by formulas:

$$\mu\_{VERY\ HARD}(\mathbf{x}) = \begin{cases} \frac{\Im 0 - \mathbf{x}}{30 - 0} \, dla \, 0 < \mathbf{x} < 30\\ \mathbf{x} = 0 \, dla \, 30 \le \mathbf{x} \le 100 \end{cases} \tag{3}$$

$$\mu\_{RESTRED}(\mathbf{x}) = \begin{cases} \frac{\mathbf{x}}{30 - 0} \, dla \, 0 < \mathbf{x} < 30\\ \frac{60 - \mathbf{x}}{60 - 30} \, dla \, 30 < \mathbf{x} < 60\\ \mathbf{x} = 0 \, \, dla \, 60 \le \mathbf{x} \le 100 \end{cases} \tag{4}$$

$$\text{If } \text{ $^\mu$ } \text{MEDIM } \text{RESTRICED} \left( \mathbf{x} \right) = \begin{cases} \mathbf{x} = \mathbf{0} \text{ } \text{dla x} \le \mathbf{3} \mathbf{0} \\\begin{array}{c} \frac{\mathbf{x} - \mathbf{3} \mathbf{0}}{60 - \mathbf{3} \mathbf{0}} \text{ dla } \mathbf{3} \mathbf{0} < \mathbf{x} < \mathbf{6} \mathbf{0} \\\frac{100 - \mathbf{x}}{100 - \mathbf{6} \mathbf{0}} \text{ dla } \mathbf{6} \mathbf{0} < \mathbf{x} < 100 \end{array} \tag{5}$$

$$\mu\_{EASY}(\mathbf{x}) = \begin{cases} \quad \mathbf{x} = 0 \text{ dla} & \mathbf{x} \le 60\\ \quad \frac{\mathbf{x} - 60}{100 - 60} \text{ dla} & 60 < \mathbf{x} < 100 \end{cases} \tag{6}$$

The fuzzy rules for assembly technology are presented in Table 6.

**Table 6.** Fuzzy rules table for assembly technology—substep 1.


In order to make the method compared with traditional methods transparent, the evaluations and results were scaled. The best theoretical value for the design feasibility of the structure maybe 100. After scaling, this rating may have a maximum value of 1.00. The assessments of the efficiency according to the new method will be equal to x/100 where x was the given assessment of the structure's efficiency. For the body, for the values "access" = 20 and "number of workshop aids" = 55 based on Figure 8, according to the above-mentioned inference rule "min", the following rules were active:



**Figure 8.** Aggregation of rules for assembly technology Substep 1.

The practical approach of aggregation of rules was that for example for Rule 11 for values 20 and 55 calculated value 0.67 defines surface area under function "medium on Figure 9. After taking into account rules 10, 11, 14 and 15, in Mamdani's inference there was a maximum operation as an operator of the aggregation of inference results obtained on the basis of individual rules, therefore rules 10 and 15 which have the same "medium low" rating, we choose MAX so we activate rule 15. Hence, activated were rules 11,14,15. Complete aggregated values for assembly technology in substep 1 are given in Figure 8.

**Figure 9.** Aggregation of rules for assembly technology 2.

The next step was defuzzification (sharpening) of the parameter value to provide the predicted factor value. The basis of this step was the resulting membership function represented in a fuzzy form, while the inference should end with providing a specific numerical value, hence the need to sharpen. Various methods can be used to carry out this process: center of gravity, average maximum, first maximum, last maximum. The center of gravity method was selected:

$$\begin{cases} \ y = \frac{\mathbf{x}}{20} \\ \ y = 0.17 \end{cases}; \ 0.17 = \frac{\mathbf{x}}{20}; \ \mathbf{x} = 20 \cdot 0.17 = 3.4 \tag{7}$$

$$\begin{cases} \ y = \frac{\mathbf{x}}{20} \\ \ y = 0.33 \end{cases}; \ 0.33 = \frac{\mathbf{x}}{20}; \ \mathbf{x} = 20 \cdot 0.33 = 6.6 \tag{8}$$

$$\begin{cases} \quad y = \frac{80-\chi}{80-60} \\ \quad y = 0.67 \end{cases}; \ 0.67 = \frac{80-\chi}{20}; \ \mathbf{x} = 80-13.4 = 66.6 \tag{9}$$

$$\begin{cases} \quad y = \frac{x - 20}{40 - 20} \\ \quad y = 0.33 \end{cases}; \ 0.33 = \frac{x - 20}{20}; \ \mathbf{x} = 20 \cdot 0.33 + 20 = 26.6 \tag{10}$$

$$\begin{cases} \quad y = \frac{x - 20}{40 - 20} \\ \quad y = 0.67 \end{cases}; 0.67 = \frac{x - 20}{20}; \text{ x} = 20 \cdot 0.67 + 20 = 33.4 \tag{11}$$

Defuzzied center of gravity value:

$$r = \frac{r\_1}{r\_2} = \frac{\int\_0^{80} y \cdot \mu\_{B'}(y) dy}{\int\_0^{80} \mu\_{B'}(y) dy} \tag{12}$$

$$\begin{array}{l} r = \neq \stackrel{\text{f}}{\int\_{3.4}^{6.6} y \cdot \frac{y}{20} \, dy} \\ \quad \quad \quad \quad \quad \quad \begin{array}{l} \quad \stackrel{\text{33.4}}{\text{ }} y \cdot \frac{y - 20}{20} \, dy + \stackrel{\text{26.6}}{\int} y \cdot 0.33 \, dy \\ \quad \quad \quad \quad \quad \quad \quad \begin{array}{l} \quad \stackrel{\text{26.6}}{\text{ }} y \cdot 0.33 \, dy \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array} \tag{13}$$

where:

$$r\_1 = \left[\frac{y^2}{12}\right]\_0^{3.4} + \left[\frac{y^3}{60}\right]\_{3.4}^{6.6} + \left[\frac{y^2}{6}\right]\_{6.6}^{26.6} + \left[\frac{1}{20} \cdot \left(\frac{y^2}{3} - 10y^2\right)\right]\_{26.6}^{33.4} + \left[\frac{y^2}{3}\right]\_{33.4}^{66.6} + \left[\frac{1}{20} \cdot \left(40y^2 - \frac{y^2}{3}\right)\right]\_{66.6}^{80} \tag{14}$$

$$r\_1 = 0.96 + 4.14 \times 110.67 + 102.31 \times 1106.67 + 219.02 = 1644.76$$

$$\begin{aligned} r\_1 &= 0.96 + 4.14 + 110.67 + 103.31 + 1106.67 + 319.02 = 1644.76\\ r\_2 &= \int \mu\_{\mathcal{V}}(y) dy = P\_1 + P\_2 + P\_3 \end{aligned} \tag{15}$$

$$\begin{aligned} P\_1 &= (6.6 - 0) \cdot 0.17 = 1.1; \; P\_2 = (20 - 0) \cdot 0.33 = 6.6; \; P\_3 = \frac{[60 + 33.2] \cdot 0.67}{2} = 31.22\\ r\_2 &= \int \mu\_{\mathcal{B}^\*}(y) dy = 1.1 + 6.6 + 31.22 = 38.9 \end{aligned} \tag{16}$$

$$r = \frac{1644.76}{38.9} = 42.2\tag{17}$$

The assessment of technology for the 1st stage assumes for the adopted access assessment-20 and the number of workshop aids-55. The value of ~42.20 was determined.

#### 4.2.2. Assembly Manufacturability Assessment—Substage 2

The component's technology is determined, assuming that it depends on two factors, which were: orientation, maneuverability. The functions of belonging linguistic variables for the given factors are given in Tables 7 and 8, the bases of rules for them are presented in Tables 9 and 10. The expert group made the following assessment: orientation—10, maneuverability—35.

**Table 7.** Membership functions in tabular form of linguistic variables for orientation.



**Table 8.** Membership functions in tabular form of linguistic variables for maneuverability.

**Table 9.** Rule base for orientation.


**Table 10.** Rule base for maneuverability.


Aggregation of rules for assembly technology 2 is shown in Figure 9.

The technological assessment for the 2nd stage assumes for the adopted assessment of orientation—10 and maneuverability—35. The value equal to −31.0 was determined.

4.2.3. Assembly Manufacturability Assessment—Substage 3

The technology of the 3rd component was determined, assuming that it depends on two factors, which were: assembly, processes. The functions of belonging linguistic variables for the given factors are given in Tables 11 and 12. The expert group made the following assessment: assemblability = 20, joining processes = 35.

**Table 11.** Membership functions in tabular form of linguistic variables for assemblability.



**Table 12.** Membership functions in tabular form of linguistic variables for processes.

Aggregation of rules for assembly technology 3 is shown in Figure 10.

**Figure 10.** Aggregation of rules for assembly technology 3.

The technological assessment for the 3rd stage was assumed for the accepted assessment of assemblability—70 and joining processes—10. The value equal to −36.0 was determined.

#### *4.3. Fuzzy Assessment of Design for Machinability—for Example*

To decrease the number of components of a product may increase its complexity and increase its manufacturing costs. The final product can be easy to assemble and expensive to process its components.

The condition for the correct determination of the cost-related factors involved in the production process of a given element was information about the characteristics that this element has from the point of view of construction, production and organization of production. The main task that must be performed was to determine the value of the costs of implementing individual operations. The cost of product processing and organization of production includes material costs, costs of cooperation and processing of a given operation. Classification of elements should include its type, e.g., shaft, sleeve, specify dimensions, the accuracy of workmanship, etc. Based on technological similarity, the costs of individual operations can be determined in accordance with the data in the database of costs of operation of technologically closest components [36,39,42].

The assessment was based on a multi-level classification of elements, assemblies made in the enterprise, etc. (Figure 11). The new element was assigned to a given shape representative based on the designer's decision—Figures 12 and 13. Based on the shape and design parameters from the manufacturing processes database, the process of the element with the same shape code and parameters most like the parameters of the new element was searched. Having the process of manufacturing the nearest element at your disposal and data on the value of cost factors in connection with the time and then cost calculation system, you can specify the production costs of the designed element [34,36,42].

**Figure 11.** Example of the first level of the production item classifier–restrictions on unnecessary diversity.

**Figure 12.** Graphical representation membership functions for linguistic variables for technological capabilities.

**Figure 13.** Graphical representation membership functions for linguistic variables for Software Capability.

Result of aggregation of rules for design for machining manufacturability 1 calculated as center of gravity of the surface under the curve presented in Figure 14—design for machining manufacturability 1 assessment for Substep 1 was 29.9.

**Figure 14.** Aggregation of rules for machining technology 1.

The design for machining processing 2 assessment was determined in a similar procedure: tool machining capability V3 = 10, compliance requirements V4 = 35. Results of aggregation of rules for design for machining manufacturability 2 calculated as center of gravity of the surface under the curve are presented in Figure 15—the design for machining technology—machining processing assessment for substep 2 was 30.6.

**Figure 15.** Aggregation of rules for machining technology 2.

The design for machining processing 3 assessment was determined in the same procedure: Energy consumption V5 = 70, waste, environmental aspects V6 = 10. Aggregation of rules for design for machining manufacturability 3 calculated as center of gravity of the surface under the curve presented in Figure 16—design for machining technology—machining processing assessment for substep 3 was 36.

**Figure 16.** Aggregation of rules for design for machining technology 3.

#### *4.4. Fuzzy Assessment of the Design for a Manufacturing Organization*

In the next step called assessment of design for manufacturing organization 1 we perform calculations for the values of V1 and V2 by reading the values from the relevant graphs as Figures 14 and 15, 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. It depends on two factors, which were: number of components and the possibility of group processing, experts have determined the rating as Number of components V1 = 20, the possibility of group processing V2 = 20. The result of aggregation of rules for design for manufacturing organization 1 calculated as center of gravity of the surface under the curve design for manufacturing organization 1 was 40 (Figure 17).

**Figure 17.** Aggregation of rules for design for manufacturing organization 1.

The design for manufacturing organization 2 assessment was determined in similar procedure. It depends on two factors, which were: component normalization V3 = 20, target cost V4 = 55.

Aggregation of rules for design for manufacturing organization 2 calculated as center of gravity of the surface under curve presented in Figure 18—design for manufacturing organization 2 was 31. design for manufacturing organization 3 assessment was determined in a similar procedure. It depends on two factors: quality of assembly V5 = 70, reuse components V6 = 10.

**Figure 18.** Aggregation of rules for design for manufacturing organization 2.

Aggregation of rules for design for manufacturing organization 3 calculated as center of gravity of the surface under curve presented in Figure 19—design for manufacturing organization 3 was 36.

**Figure 19.** Aggregation of rules for assessment of design for manufacturing organization 3.

#### *4.5. A Fuzzy Assessment of Design for Technology*

A complete fuzzy analysis of the technology was carried out in the same stages as for the sample "body" component presented in previous chapters using MATLAB software. Each component of the analyzed transmission was assessed by a group of experts according to their best knowledge in the field of technology, organizational and cost options. Expert assessments were entered in Table 13 and were used in subsequent stages as input to fuzzy analyses. The stages of the analysis were identical to those presented in Figure 3. Calculations of the fuzzy technology assessment method were made using the fuzzy logic toolbox package, which is an addition to the MATLAB program. In the MATLAB FIS editor window, the number of entries and exits is defined and given a name. In this case, two inputs were specified in each step, for example, technological capabilities V1, software capability V2 and one output—The design for machining manufacturability 1. The logical method I (And), logical OR (Or), type of implication, type of aggregation (Aggregation), sharpening method (Defuzzification) were also defined. The analysis selected a uniform representation of the membership function. It can be obtained by using a membership function with a uniform shape and parametric definition of the function. In the case of the assessment of technology, triangular and trapezoidal functions were used.

It should be added that the parametric description of the triangular membership function is the most economical, it only requires three parameters, which are important in practical applications of the method in small lot production industry. After determining the membership sets, one should proceed to the next step of fuzzy analysis of manufacturability. It is creating a set of linguistic rules representing the relationships between system variables. The rules are the heart of the entire regulator. In MATLAB, the next part of the FIS editor is used—Rule Editor. Entered rules can be edited in several different ways. Rule Editor uses the words if, and, or, then, which are the closest to natural language. The final step is to read sharp results using "Rule viewer" or graphical explanation using "Surface viewer".

"Surface viewer" and its graphs—surface charts are additional tools useful in the assessment of construction. With the help of such charts, we can quickly obtain the result of the Technology component without the need for complex calculations. In the chart below "Assembly technology 1" if, for example, the "Access" and the "number of workshop aids" rating would change from (0.55, 0.2)—point A to (0.75, 0.8)—point B, then from the surface chart in Figure 20 we can read the value of Assembly technology 1 at the level of 0.6. In the process of selecting variants to improve the technology, this is a very useful tool that allows you to quickly assess how potential product changes can affect the result of the technology.



**Figure 20.** Surface viewer—The surface of the dependence of the variable "installation technology 1" on the input variables for "gears".

For each of the components, calculations were made using the above scheme MATLAB R2017b—the "fuzzy logic designer" module. The development of an approximate representation of knowledge and fuzzy inference methods enables the construction of models for assessing technology to support decision making in conditions of uncertainty and lack of complete information about the problems being solved. Premises and conclusions in these systems were developed using fuzzy logic elements. The knowledge contained in the system should come mainly from a field expert and the effectiveness and efficiency of the system operation depend mainly on the ability to model this knowledge by the system designer. The elements of fuzzy logic presented in the article were used in solving tasks in the field of technological preparation of production. An important problem was the correct definition of fuzzy sets by determining for them the course of belonging functions. [source-fuzzy logic toolbox—user's guide]

Table 14 presents a summary of the results of individual components obtained in the fuzzy transmission analysis. An acceptability criterion of 0.55 was adopted for each component (modeled on the recommendations of the Lucas method and other DFA methods as well as the opinions of experts from industrial practice), elements of lower value should be redesigned. The following transmission components need to be redesigned as a result of the above assessment: body, cover, breather, oil level indicator, covers, additional processes, nameplate, shaft assembly and inlets. Elements rated as not requiring redesign were gears, bearings, washers, screws.

Expert assessments mentioned above are shown in Table 13—those values were used as input to fuzzy analyze conducted in MATLAB—"fuzzy logic designer" module. The results are shown in Table 14.

**Table 14.** A set of fuzzy method results for individual components for specified criteria.

#### *Appl. Sci.* **2020** , *10*, 3935

### **5. Results and Discussion**

In the study, the indicators of the assessment of the manufacturability of the structure were determined for the sample product presented in Figure 6. As a result of the analysis after the proposed changes, the new form of the gear structure change is illustrated in Figure 21.

**Figure 21.** Construction form of the gearbox after the changes were made.

The study cites a comparison of new and currently used in mass production methods of construction technology in Table 15 and Figure 22 which presents the values of the indicators according to the traditional methods and the newly proposed method.

**Figure 22.** Comparison of methods for gear groups.


**Table 15.** Comparison of methods design for manufacturability.
