**5. Case Study**

In this research, the research is realized in 24 manufacturing SMEs located in Bosnia and Herzegovina (B&H). Respecting the official data, it can be said that: (i) account of considered SMEs for more than 15% of total gross domestic product (GDP), (ii) about 20% of total employment in B&H, and (iii) almost 90% of exports, such as they have a significant impact on economic growth in B&H. Experiences of good practice show that one of the most important problems of an operational managemen<sup>t</sup> team is to maintain the reliability of the manufacturing process over a long period of time. In this way, the defined business goals of the company can be realized to a high degree. Improving the reliability of the manufacturing process is realized through the accurate identification of potential failures and through the selection of adequate quality methods by which the identified failures are reduced and/or eliminated.

As is known in the literature, there are no procedures or recommendations on how to identify failures in manufacturing processes or how to choose quality methods, which leads to increased reliability of the manufacturing process. Experiences of best practice in the process industry show that failure identification is always based on evidence data and experience and knowledge of decision makers, as in this research. The set of quality methods that can potentially be applied is defined according to the recommendations from the relevant literature.

Assessment of the relative importance of RFs, their values, applicability of treated quality methods, as well as the costs of implementation were obtained using the interview method. Questionnaires were sent to the FMEA team asking them to express all estimates using the given pre-defied linguistic expressions. They also returned the completed questionnaires by e-mail.

#### *The Illustration of the Proposed Model*

Fuzzy assessment of the relative importance of RFs is performed by DM at the level of each SME (Step 1 of the proposed Algorithm). This is presented in Table 4.


The proposed aggregation and defuzzification procedures (Step 2 to Step 3 of the proposed Algorithm) is illustrated for example (*k =* 1):

$$\tilde{W}\_1 = \frac{1}{24} \cdot (18 \cdot H + 6 \cdot M) = \frac{1}{24} \cdot ([51, 111, 120]; 0.6, 0.3) = ([2.13, 4.63, 5]; 0.6, 0.3)$$

The precise number of TIF *W* 1 , *W*1 is:


$$W\_1 = \frac{1}{12} \cdot (2.13 + 4 \cdot 4.63 + 5) \cdot (1 - 0.3 + 0.6) = 2.776$$

Similarity, the relative importance of the rest RFs are given:


*W* 2 = ([1.375, 3.666, 4.875]; 0.6, 0.3) and *W*2 = 2.266 *W* - 3 = ([1.312, 2.771, 4.375]; 0.6, 0.3) and *W*3 = 1.817

The weights vector (Step 4 of the proposed Algorithm) is:

$$\omega\_k = \frac{W\_k}{\sum\_{k=1,\dots,K} W\_k} \\ 0.40, \quad 0.33, \quad 0.26$$

The assessment values of RFs for SME (*e =* 1) are given in a Table 5 (Step 5 of the proposed Algorithm):




**Table 5.** *Cont.*

The proposed procedure (Step 6 of the proposed Algorithm) is illustrated for the failure (*I =* 1):

*RPN* - 1 = ([6, 7.5, 9]; 0.75, 0.2)0.40·([6, 7.5, 9]; 0.75, 0.2)0.33·([1, 1, 2.5]; 0.65, 0.3)0.26 *RPN* - 1 = ([2.07, 2.26, 2.43]; 0.89, 0.09)·([1.81, 1.95, 2.07]; 0.91, 0.07)·([1, 1, 1.27]; 0.89, 0.09) = ([3.73, 4.40, 6.41]; 0.89, 0.09)

> RPN values for all other failures were calculated in a similar way and shown in Appendix A.

> According to the proposed procedure (Step 7 of the proposed Algorithm), the degree of beliefs is calculated.

> Applicability of methods with respects to priority failures as well as calculation of normalized values (Step 8 to Step 9 of the proposed Algorithm) is illustrated in the following example:

$$\begin{array}{rcl} \widetilde{z}\_{61} = \widetilde{v}\_{61} \cdot \overline{\widetilde{RPN}}\_{1} &=& ([6, 7.5, 9]; 0.75, 0.2) \cdot ([3.73, 4.40, 6.41]; 0.89, 0.09) \\ &=& ([27.41, 47.08, 66.70]; 0.75, 0.20) \end{array}$$

Let the maximum applicability of the method be given by using the expression (4.7):

$$\hat{z}^\* = \left( [7.5, 9, 9]; 0.8, 0.15 \right) \prod\_{k=1, \dots, 3} \left( [7.5, 9, 9]; 0.8, 0.15 \right)^{0.4} \cdot \left( [7.5, 9, 9]; 0.8, 0.15 \right)^{0.35}$$

$$\cdot \left( [7.5, 9, 9]; 0.8, 0.15 \right)^{0.26} = \left( [55.12, 79.24, 79.24]; 0.8, 0.15 \right)$$

So that, the normalized assessment of the applicability of the method (*m =* 6) at the level of failure (*i =* 1) is:

$$\widetilde{r}\_{61} = \frac{([27.41, 47.08, 66.70]; 0.75, 0.20)}{([55.12, 79.24, 79.24]; 0.8, 0.15)} = ([0.35, 0.59, 1.21]; 0.75, 0.20)$$

To apply the proposed procedure (Step 10 of the proposed Algorithm) it is necessary to determine representative scalars, so that:

$$r\_{61} = \frac{1}{12} \cdot (0.35 + 4 \cdot 0.59 + 1.21) \cdot (1 - 0.2 + 0.75) = 0.506$$

Ranking of uncertain values, *r*61 is performed according to crisp values *r*61, so that:

*r*6 = max *<sup>i</sup>*=1,...,*Idef uzz* 

⎛⎜⎜⎝

0.178; 0.169; 0.413; 0.076; 0.120; 0.506; 0.105; 0.078; 0.230; 0.137; 0.274; 0.066; 0.258; 0.080; 0.101; 0.043; 0.413; 0.418; 0.486; 0.243; 0.268; 0.156; 0.080; 0.204; 0.487; 0.117; 0.089; 0.133; 0.208; 0.191; 0.191; 0.204; 0.227; 0.289; 0.340; 0.149; 0.199; 0.129; 0.120; 0.167; 0.156; 0.191; 0.191 ⎞⎟⎟⎠ = 0.506

> Therefore:

$$\vec{r}\_{61} = ([0.35, 0.59, 1.21]; 0.55, 0.4)$$

In the same way, each quality method *m*, *m =* 1 ,.., *M* is accompanied by the normalized total applicability shown in Table 6. The same table shows the other input data.

**Table 6.** Input data.




Determine the mean value of implementation costs:

$$
\widetilde{\mathcal{E}} = \frac{1}{49} \cdot \sum\_{m=1,\dots,49} \widetilde{\mathcal{E}}\_{m} = ([0.14, 0.35, 0.62]; 0.55, 0.40)
$$

And the variance of implementation costs:

$$\frac{1}{49 - 1} \cdot \sum\_{m = 1, \dots, 49} d \left( d(\tilde{c}\_{m\prime} \tilde{c}) \right)^2 = \frac{1}{48} \cdot 2.1117 = 0.044$$

Assumed that the reliability manager should select 10 quality methods. This assumption was introduced based on best practice experience. Under this assumption, the KP problem (Step 11 of the proposed Algorithm) is:

The objective function

$$\max\_{j=1,\dots,10} d(\hat{r}\_{\mathfrak{m}\prime} \, \hat{c}\_{\mathfrak{m}})\_{j\prime} \; j \in \{1,\dots,m,\dots,M\}$$

The objective to:

$$\frac{1}{9} \cdot \sum\_{j=1,\dots10} \left( d(\tilde{c}\_{m\prime}\tilde{c}) \right)^2 \le 0.04$$

GA (Step 12 of the proposed Algorithm) was applied to find a near-optimal solution. The stop criterion is defined to have a number of iterations equal to 1000.

It has been shown that about 300 iterations are achieved near the optimal solution (see Figure 1).

**Figure 1.** Value of fitness function by iterations.

For the near-optimal solution, the solution obtained in the last iteration was adopted: By applying GA, the quality methods that the reliability manager in the considered company should implement in order to eliminate failures, or increase the reliability of the manufacturing process is:

solution 1: {*m* = 7, *m* = 19, *m* = 21, *m* = 29, *m* = 35, *m* = 38, *m* = 43, *m* = 45, *m* = 46, *m* = 49} solution 2: {*m* = 7, *m* = 19, *m* = 21, *m* = 22, *m* = 29, *m* = 35, *m* = 43, *m* = 45, *m* = 46, *m*=49}

It is worth to mention that the treated problem could be solved by the branch and bound algorithm [45] but it has a significant limitation since it is applicable to the lower size problems. In the presented case study, the output of the GA and branch and bound algorithm does not indicate output results deviation. As the number of quality methods in practice is increasing, the application of GA seems to be more adequate due to the nonpalatability of the branch and bound algorithm.

Based on the results of the survey, it can be concluded that most quality tools can be successfully applied to the analysis and identification of failures, which will ultimately lead to the elimination of failures, and thus lean waste. The obtained results show 10 methods that need to be applied in order to finally eliminate failures and increase the reliability of the manufacturing process. These methods can be implemented simultaneously. The period of implementation and education of employees for the implementation of these methods is not long. Another important fact is that the implementation of the above methods is not faced with the challenge of reorganization. The organizational culture of different SMEs will affect the effectiveness of the application of methods, so the effectiveness of the application should be observed at the level of each individual enterprise. In SMEs that have implemented the total quality managemen<sup>t</sup> (TQM) and quality system according to the ISO 9001 standard, most of these tools are already in use and do not require additional investment and additional costs for the application of these methods.

This new problem-solving approach sets the standard for those who are just applying the problem-solving approach, as well as for those who are interested in continuously improving existing problem-solving methodologies.
