**3. Methodology**

They base their assessments on knowledge and experience as well as data from records.

*3.1. Basic Definitions of Intuitionistic Fuzzy Sets*

**Definition 1.** *An intuitionistic fuzzy set A in the universe of discourse X is defined with the form* [15]:

$$\tilde{A} = (\mathbf{x}, \,\,\mu\_{\tilde{A}}(\mathbf{x}), \theta\_{\tilde{A}}(\mathbf{x}) \, | \, \mathbf{x} \in X), \tag{1}$$

*where:*

*The numbers μA*-(*x*) → [0, 1] *and <sup>ϑ</sup>A*-(*x*) → [0, 1] *denote the membership degree and nonmembership degree.*

*With the condition*

$$\begin{cases} 0 \le \mu\_{\bar{A}}(\mathbf{x}) + \theta\_{\bar{A}}(\mathbf{x}) \le \mathbf{1}, \forall \mathbf{x} \in X\\ \pi\_{\bar{A}}(\mathbf{x}) = 1 - \mu\_{\bar{A}}(\mathbf{x}) - \theta\_{\bar{A}}(\mathbf{x})\\ 0 \le \pi\_{\bar{A}}(\mathbf{x}) \le \mathbf{1}, \forall \mathbf{x} \in X \end{cases}$$


*The value of <sup>π</sup>A*-(*x*) *is called the degree of indeterminacy (or hesitation). The smaller <sup>π</sup>A*-(*x*)*, more certain A* - .

**Definition 2.** *An IFS A* = *x*, *μA*-(*x*), *<sup>ϑ</sup>A*-(*x*)*x* ∈ *X of the real line is called an intuitonistic fuzzy number (IFN) whose membership function and non-membership function are defined as follows* [40]:

$$
\mu\_{\bar{A}}(\mathbf{x}) = \begin{cases}
\frac{\mathbf{x} - a}{\bar{b} - a} \cdot \mu\_{\bar{A}}(\mathbf{x}) & \text{if} \quad a \le \mathbf{x} < b \\
& \mu\_{\bar{A}}(\mathbf{x}) \qquad \text{if} \qquad \mathbf{x} = b \\
& \frac{\mathbf{c} - \mathbf{x}}{\mathbf{c} - b} \cdot \mu\_{\bar{A}}(\mathbf{x}) & \text{if} \quad b < \mathbf{x} \le \mathbf{c} \\
& 0 & \text{else}
\end{cases}
\tag{2}
$$

*and*

$$\theta\_{\bar{A}}(\mathbf{x}) = \begin{cases} \frac{b - \mathbf{x} + (\mathbf{x} - \mathbf{a}) \cdot \theta\_{\bar{A}}(\mathbf{x})}{b - \mathbf{a}} & \text{if } \quad a \le \mathbf{x} < b \\\\ \frac{\theta\_{\bar{A}}(\mathbf{x})}{c - b} & \text{if } \quad \mathbf{x} = b \\\\ 1 & \text{else} \end{cases} \tag{3}$$


*where a*, *b*, *c are real numbers, and a* ≤ *b* ≤ *c*.

> TIFN can be denoted as

$$\dot{A} = \left( [a, b, c]; \mu\_{\bar{A}}(\mathbf{x}), \vartheta\_{\bar{A}}(\mathbf{x}) \right), \tag{4}$$

**Definition 3.** *In compliance with the Definition 2, let A* = [*<sup>a</sup>*1, *b*1, *<sup>c</sup>*1]; *μA*-(*x*), *<sup>ϑ</sup>A*-(*x*) *and B* - = [*<sup>a</sup>*2, *b*2, *<sup>c</sup>*2]; *μB*-(*x*), *ϑB*-(*x*) *be two positive TIFNs. Additionally, λ is the real number. The operations of these TIFNs are given by* [41]:

$$\tilde{A} + \tilde{B} = \left( [a\_1 + a\_2, b\_1 + b\_2, c\_1 + c\_2]; \min\left( \mu\_{\tilde{A}}(\mathbf{x}), \mu\_{\overline{B}}(\mathbf{x}) \right), \max\left( \vartheta\_{\overline{A}}(\mathbf{x}), \vartheta\_{\overline{B}}(\mathbf{x}) \right) \right), \tag{5}$$

$$
\overrightarrow{A} - \overrightarrow{B} = \left( [a\_1 - c\_2, b\_1 - b, c\_1 - a\_2]; \min\left( \mu\_{\overrightarrow{A}}(\mathbf{x}), \mu\_{\overrightarrow{B}}(\mathbf{x}) \right), \max\left( \vartheta\_{\overrightarrow{A}}(\mathbf{x}), \vartheta\_{\overrightarrow{B}}(\mathbf{x}) \right) \right), \tag{6}
$$

$$
\widetilde{A} \cdot \widetilde{B} = \left( [a\_1 \cdot a\_2, b\_1 \cdot b\_2, c\_1 \cdot c\_2]; \min\left( \mu\_{\widetilde{A}}(\mathbf{x}), \mu\_{\widetilde{B}}(\mathbf{x}) \right), \max\left( \vartheta\_{\widetilde{A}}(\mathbf{x}), \vartheta\_{\widetilde{B}}(\mathbf{x}) \right) \right), \tag{7}
$$

$$
\lambda \cdot \bar{A} = \left( [\lambda \cdot a\_1, \lambda \cdot b\_1, \lambda \cdot c\_1]; \mu\_{\bar{A}}(\mathbf{x}), \; \theta\_{\bar{A}}(\mathbf{x}) \right), \tag{8}
$$

$$\hat{\mathcal{A}}^{\lambda} = \left( \left[ a\_1^{\lambda}, b\_1^{\lambda}, c\_1^{\lambda} \right]; \mu\_{\bar{A}}^{\lambda} \left( \mathbf{x} \right), \left( 1 - \left( 1 - \theta\_{\bar{A}}(\mathbf{x}) \right)^{\lambda} \right) \right), \tag{9}$$

$$\left(\overrightarrow{A}\right)^{-1} = \left(\left[\frac{1}{d\_1}, \frac{1}{c\_1}, \frac{1}{b\_1}, \frac{1}{a\_1}\right]; \mu\_{\vec{A}}(\mathbf{x}), \theta\_{\vec{A}}(\mathbf{x})\right),\tag{10}$$

**Definition 4.** *A usual defuzzification method can be defined as mapping IFNs into scale value and taking the median* [42]*, so that:*

$$a = \frac{1}{12} \cdot (a\_1 + 4 \cdot b\_1 + c\_1) \cdot \left(1 - \theta\_{\bar{A}}(\mathbf{x}) + \mu\_{\bar{A}}(\mathbf{x})\right) \tag{11}$$

**Definition 5.** *Let A*- = [*<sup>a</sup>*1, *b*1, *<sup>c</sup>*1]; *μA*-(*x*), *<sup>ϑ</sup>A*-(*x*) *and B*- = [*<sup>a</sup>*2, *b*2, *<sup>c</sup>*2]; *μB*-(*x*), *ϑB*-(*x*) *be two positive TIFNs. The Hamming distance* [43] is:

$$d\_H = \frac{1}{3} \cdot \left( |a\_1 - a\_2| + |b\_1 - b\_2| + |c\_1 - c\_2| \right) + \max\left( |\mu\_{\tilde{A}}(\mathbf{x}) - \mu\_{\tilde{B}}(\mathbf{x})|, |\left( \theta\_{\tilde{A}}(\mathbf{x}) - \theta\_{\tilde{B}}(\mathbf{x}) \right)| \right) \tag{12}$$

#### *3.2. Definition of a Finite Set of Decision Makers*

In this manuscript, the term DM means the FMEA team of each company. It is common for the FMEA team at the level of each company to be composed of: production manager, FMEA leader, and quality manager. It should be emphasized that they make the decision by consensus. DMs can be presented by the set *e* = {1, ... , *e*, ... *E*} The total number of considered DMs is marked with E, DM index *e =* 1, . . . , *E.*

#### *3.3. Choice of Appropriate Linguistic Variables for Describing the Relative Importance of RFs*

Defining a set of RFs against which failures are evaluated can be formally presented as a set {1, . . . , *k*,..., *<sup>K</sup>*}. Total number of RFs is denoted as *K*, and *k*, *k =* 1, ..., *K* is index of RF. In conventional FMEA [44] three RFs are defined: severity of consequence (*k =* 1), frequency of failure (*k =* 2), and possibility of detecting failure (*k =* 3), as in this research.

In compliance with the evidence from literature [10,14], the RFs may have different relative importance. The fuzzy rating of the relative importance of RFs are based on the pre-defined linguistic expressions and their corresponding TIFNs are presented in the Table 1.

**Table 1.** The relative importance of RFs.


The domains of these TIFNs are defined into real line into interval [1–5]. The value 1 indicates the lowest and the value 5 the highest relative importance of the considered RFs. The overlap of TIFNs describing the relative importance of RFs is large. This indicates a lack of knowledge of DMs about the importance of the considered criteria in SMEs of the manufacturing sector.

#### *3.4. Choice of Appropriate Linguistic Variables for Describing the RF Values, Aplicability of Quality Methods, and Implementation Costs of Quality Methods*

In the production process, numerous failures can occur, which can be formally represented by the set of indices {1, . . . , *i*,..., *<sup>I</sup>*}, where I presents the total number of failures, and the index of each failure is denoted as *i*, *I =* 1, ... , *I.* In this research, failures that are identified at the level of each observed SME are considered.

Analysis and reduction of the identified failures can be performed by applying numerous quality methods that can be formally represented by the set of indices {1, . . . , *m*,..., *<sup>M</sup>*}. The total number of quality methods is denoted as *M*, and *m*, *m =* 1, ..., *M* is index of quality method. In this research, quality methods are selected according to [6].

The evaluation of the value of RFs, *vik* is performed by DM at the level of each SME. Applicability of quality method m for failure analysis *i*, *vmi*, *I =* 1, ..., *I*, are assessed by quality manager at the level of each considered SME and presented in Table 2.


**Table 2.** The RF values and degree of belief that quality methods are applicable.

The domains of these TIFNs are defined in the common measurement scale [1–9]. The value 1 indicates the lowest and the value 9 indicates the highest values of RFs.

Implementation costs of considered quality methods, *cm* were evaluated by the quality manager and presented in Table 3.



The domains of these TIFNs are defined in the common measurement scale [0, 1]. The value 0 indicates the lowest and the value 1 indicates the highest values of implementation costs.

## **4. The Proposed Algorithm**

Step 1. The relative importance of RF *k*, *k =* 1, ..., *K* is assessed by each DM *e*, *e =* 1, *..*, *E*:

$$
\stackrel{\text{g}}{k} \tag{13}
$$

Step 2. The aggregated relative importance of RF *k*, *W k* , *k =* 1, ..., *K*, is given by using fuzzy averaging operator:

*W*


$$
\tilde{\mathcal{W}}\_k = \frac{1}{E} \cdot \tilde{\mathcal{W}}\_k^c \tag{14}
$$


Step 3. The representative scalar of TIFN, *W k* , *Wk*, *k =* 1, ..., *K* is given by using the defuzzification procedure [42].


Step 4. Construct the weights vector of RFs, [*<sup>ω</sup>k*]<sup>1</sup>×*K*. The element of weights vector of RFs, *ωk* given by using the linear normalization procedure, so that:

$$
\omega\_k = \frac{\mathcal{W}\_k}{\sum\_{k=1,\dots,K} \mathcal{W}\_k} \tag{15}
$$

Step 5. The value of each RF, *k =* 1, ..., *K* for each failure *i*, *I =* 1, ..., *I* is assessed by DM and could be presented by TIFN, *vik*.

Step 6. Determine the priority index for each failure *i*, *I =* 1, ..., *I* by using fuzzy geometric mean:

$$
\overline{\hat{R}P\overline{N}\_i} = \prod\_{k=1,\ldots,K} \left(\hat{v}\_{ik}\right)^{\omega\_k} \tag{16}
$$

Step 7. Degree of beliefs that quality methods are applicable to the analysis of identified failure, *vmi* and costs of implementation of quality methods, *cm* are assessed by DM.

Step 8. Determine the applicability of the quality method, *zmi*, to eliminate failure i, *m =* 1, ..., *M; I =* 1, ..., *I:* -

$$
\widetilde{z}\_{mi} = \widetilde{\nu}\_{mi} \cdot \textit{RPN}\_{i} \tag{17}
$$

Step 9. The normalized value of applicability of the quality method m at the level of failure *i*, *m =* 1, ..., *M; I =* 1, ..., *I*, is: -

$$
\widetilde{\tau}\_{mi} = \frac{z\_{mi}}{\widehat{\mathbb{Z}}^\*} \tag{18}
$$

where, *z*<sup>∗</sup> is the maximum applicability of the method, so that:

$$
\hat{z}^\* = VHV \prod\_{k=1,\ldots,K} (VHV)^{\omega\_k} \tag{19}
$$

Step 10. The total applicability of the method *m*, *m =* 1, ..., *M* is obtained according to the expression:

$$
\widetilde{r}\_m = \max\_{i=1,\ldots,I} (\widetilde{r}\_{mi}) \tag{20}
$$

Ranking of uncertain values, *rmi* is performed according to crisp values *rmi*, so that:

$$r\_m = \max\_{i=1,\ldots,l} defuzz \ (\overleftarrow{r}\_{mi}) \tag{21}$$

Step 11. Let us set the KP problem: The fitness function:

$$\max\_{j=1,\ldots,J} \sum d(\tilde{r}\_{m\prime} \, \tilde{c}\_m)\_{j\prime} , \, j \in \{1,\ldots,m,\ldots,M\} \tag{22}$$

where, *d* (*rm*, *cm*) is calculated as the Hamming distance between two TIFNs [43].

The objective to:

$$\frac{1}{M - 1} \cdot \sum\_{j = 1, \dots, J} \left( d(\tilde{c}\_{m\prime} \tilde{c}) \right)^2 \le \frac{1}{M - 1} \cdot \sum\_{m = 1, \dots, M} \left( d(\tilde{c}\_{m\prime} \tilde{c}) \right)^2 \tag{23}$$

where:

$$
\widetilde{\mathcal{c}} = \frac{1}{M} \cdot \sum\_{m=1,\dots,M} \widetilde{c}\_m
$$

And

$$d(\stackrel{\sim}{r}\_{m}, \stackrel{\sim}{c}\_{m})\_{j} = \begin{cases} 1 & \text{if } object \text{ } j \text{ is selected} \\ 0 & \text{otherwise} \end{cases} \tag{24}$$

Step 12. The near optimal solution of the treated KP problem is found by using GA. The encoding schemes are differentiated according to the problem domain. The wellknown encoding schemes are binary, octal, hexadecimal, value-based, and tree. Binary encoding is the used encoding scheme in this paper. Each chromosome is represented using a binary string. In binary encoding, every chromosome is 0 or 1. In knapsack problem, binary encoding is used in GA to show the presence of items, 1 for the presence of an item and 0 for the absence of an item.

The initial parameter setting with a population of 100 individuals was gradually reduced to 30 without loss in solution quality, also a further increase in the number of interactions over 1000 was not significant.

The parameters of the applied GA are: generational GA, roulette wheel parent selection, elitism 0.05%, number of individuals in the population 30, selection 0.95, mutation 0.02, and number of iterations 1000. Furthermore, a part of the code of the implemented GA is given, which provides the condition that the chromosome has a precisely determined number of units. In this way, the generation of correct units is ensured without the need for subsequent rejection of defective ones. The following part of the code is given below:

public string Generate(int NumberOfAllMeasures, int NumberOfMeasures) { string ret = new String('0', NumberOfAllMeasures); int[] measuresRB = new int[NumberOfAllMeasures]; double[] measuresRand = new double[b NumberOfAllMeasures]; for (int i = 1; i <= NumberOfAllMeasures; i++) { measuresRB[i-1] = i-1; measuresRand[i-1] = random.NextDouble(); } Array.Sort(measuresRand, measuresRB); char[] ch = ret.ToCharArray(); for (int i = 1; i <= NumberOfMeasures; i++) { ch[measuresRB[i-1]] = '1'; } return new string(ch);
