1. Introduction

2. Continuous Parts of the Combined System

3. Dependability

4. Materials and Methods

4.1. Development Fuzzy Model

The first step when creating a fuzzy model is the definition of linguistic variables that refer to the partial indicators of dependability, namely:

Reliability represents the probability, at a certain level of confidence, that the system (machine) will successfully perform the function for which it is intended, without failure and within the specified performance limits, taking into account the previous time of system use, during the specified duration of a task. When it is used in the prescribed manner and for the purpose for which it is intended, under the specified load levels. [11]

Maintainability as a set of structural characteristics that affect the time to eliminate failures or the time of performing other maintenance procedures, is an internal property of the observed technical system, therefore it is called structural maintainability. The following parameters affect the maintainability: t - technology, e - tools and equipment, u- unification, d - diagnostics, m - manipulativeness, s - standardization [11,38].

For the technical system to successfully perform the set tasks, it is necessary to provide logistical support and numerous conditions. The logistic support combines the management process with appropriate technical measures to define the necessary support and create conditions for the realization of the given function of the technical system goal.

Logistical support performances according to the ISO-IEC Standard are defined as: "The ability of maintenance system, i.e. the organization that performs maintenance, to provide under given conditions the required maintenance of the technical system in accordance with the maintenance policy [35,36] the Standards of the IEC 300 series deal with the concept of logistic support for maintenance [37].

Figure 5. Presentation of partial indicators of dependability.

Figure 5. Presentation of partial indicators of dependability.

In terms of the number of linguistic variables, it can be inferred that seven is the maximum count of variables that a person can rationally recognize simultaneously while retaining the same meaning [39].

Taking this statement into account, the five ratings (linguistic variables), defined as follows: excellent (exc), good (good), average (aver), adequate (adeq), and poor (poor), will be considered in this paper. Linguistic variables (ratings) are given in the form of triangular fuzzy numbers, and their graphic representation is presented in the following figure.

Corresponding fuzzy numbers of the mentioned linguistic variables are defined by (according to Figure 6):

$${\mu }_{poor}=\left(1,0.25,0,0,0\right),$$

$${\mu }_{adeq}=\left(0.25,1,0.25,0,0\right),$$

$${\mu }_{aver}=\left(0,0.25,1,0.25,0\right),$$

(1)

$${\mu }_{good}=\left(0,0,0.25,1,0.25\right),$$

$${\mu }_{exc}=\left(0,0,0,0.25,1\right).$$

Partial indicators t, e, u, d, m, s more closely determines the partial indicator M- maintenance convenience, while M – maintainability together with R-reliability and F- logistic support determine the D- dependability of the system. It will be shown in the following how the dependability D is determined based on indicators of maintainability M, reliability R and logistical support F, while the maintenance convenience M is obtained similarly based on partial indicators t, e, u, d, m, s.

The idea of this work is to obtain a more accurate assessment of the dependability of CCS systems at the open pit Gacko. This assessment was identified as the best possible among the worst expected ratings of the partial availability indicators (R, M , and F). Let the partial indicators R, M, F be shown in the form of the following triangular fuzzy numbers

$${\mu }_R\left({\mu }_R^1,{\mu }_R^2,{\mu }_{R,}^3{\mu }_R^4,{\mu }_R^5\right),{\mu }_M\left({\mu }_M^1,{\mu }_M^2,{\mu }_{M,}^3{\mu }_M^4,{\mu }_M^5\right), { \mu }_F\left({\mu }_F^1,{\mu }_F^2,{\mu }_{F,}^3{\mu }_F^4,{\mu }_F^5\right).$$

(2)

In the next step, the max-min composition is performed on them. If the mentioned partial indicators R, M, F are observed, it is possible to make $C=5^3=125$ combinations of corresponding membership functions, which will be further denoted with

$${\mu }^{ijk}\left({\mu }_R^i{,\mu }_M^j, {\mu }_F^k\right), i,j,k\in \left\{1,2,3,4,5\right\}$$

(3)

Each of these combinations represents one possible assessment of the dependability and the following two values can be associate with it

$${\mathsf{\Omega }}_{ijk}={\displaystyle \frac{\left[i+j+k\right]}{3}}$$

(4)

and

$$m^{ijk}=\min \left\{{\mu }_R^i,{\mu }_M^j,{\mu }_F^k\right\}$$

(5)

${\mathsf{\Omega }}_{ijk}$ takes values from the set $\left\{1, 2, 3, 4, 5\right\}$, and each of the mentioned values can be associated with the number ${\mu }^l$ which represents the maximum value of $m^{ijk} \mathrm{o}\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}{\mathsf{\Omega }}_{ijk}$ is equal to $l$, za $l\in \left\{1, 2, 3, 4, 5\right\}$, that is

$${\mu }^l=\mathrm{m}\mathrm{i}\mathrm{n}\left\{m^{ijk}:{\mathsf{\Omega }}_{ijk}=l\right\}.$$

(6)

In this way, a rating for the dependability of $D$ is obtained

$$\mu =\left({\mu }^1,{\mu }^2,{\mu }^3,{\mu }^4,{\mu }^5\right)$$

(7)

Using the best-fit method, see [11], to transform the obtained ratings into belonging to the fuzzy set, determined by (2), the distance is used that is defined by

$$d_i=d\left(\mu ,{\mu }_i\right)=\sqrt{\sum _{j=1}^5{({\mu }^j-{\mu }_{i,j})}^2},{\mu }_i=\left({\mu }_{i,1},{\mu }_{i,2},{\mu }_{i,3},{\mu }_{i,4},{\mu }_{i,5}\right),$$

(8)

For ${\mu }_i\in \left\{{\mu }_{poor},{\mu }_{adeq} , {\mu }_{aver},{\mu }_{good} ,{\mu }_{exc} \right\}$. Small values $d_i$ indicate proximity to the linguistic variable ${\mu }_{ i}$. Accordingly, let $d_{min}$ be the minimum value of the obtained distances $d_1,d_2, d_3, d_4, d_5$, then the reciprocal value of the relative distances can be associated to each of them that is determined by:

$${\alpha }_i={\displaystyle \frac{d_{min}}{d_i}},i\in \left\{1,2,3,4,5\right\}.$$

(9)

If for some i the distance value$d_i$ is equal to 0, then the corresponding value of the reciprocal value of relative distance is ${\alpha }_i=1,$while the other values of the reciprocal relative distances are equal to 0. The normalized values of thus obtained reciprocal values of the relative distances are determined by:

$${\beta }_i={\displaystyle \frac{{\alpha }_i}{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4+{\alpha }_5}},i\in \left\{1,2,3,4,5\right\}$$

(10)

and present belonging to the appropriate rating:

$$A=\left\{\left({\beta }_1,"\mathrm{exc}\text{"}\right),\left({\beta }_2,"\mathrm{good}\text{"}\right),\left({\beta }_3,"\mathrm{aver}\text{"}\right),\left({\beta }_4,"\mathrm{adeq}\text{"}\right),\left({\beta }_5,"\mathrm{poor}\text{"}\right)\right\}$$

(11)

In the end, the appropriate linguistic rating is obtained as follows:

$$Z={\displaystyle \frac{{5\beta }_1+{4\beta }_2+{3\beta }_3+{2\beta }_4+{1\beta }_5}{{\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4+{\beta }_5}}$$

(12)

In a similar way, a rating for maintainability M is obtained from the linguistic variables t, e, u, d, m, s which is later used to determine the dependability D.

Figure 7 shows the fuzzy model algorithm for dependability evaluation.

4.2. Case Study – Open Pit Gacko

4.2.1. Results of Expert Evaluation

Determination of the dependability of the system and its partial indicators was processed through the results obtained through questionnaires related to the expert evaluation of the partial indicators of operational safety. The questionnaire contained detailed descriptions of the partial indicators themselves. In the expert evaluation, 10 experts were surveyed. The first 5 experts are representatives of the Gacko Mine (experts from this field with many years of work on these systems - minimum 10 years of work experience), and the other 5 experts are external experts with many years of experience in the field of open pit mining. Ratings are expressed using membership functions representing predefined linguistic variables ranging from 'poor' to 'excellent' within a scale of 0 to 1. Additionally, a parameter can be associated with multiple linguistic variables simultaneously, ensuring that the total sum of ratings equals 1. The following tables give the results of expert evaluation for each part of the CCS system. The layout of one questionnaire is given in Figure 8.

Table 1. Results of expert evaluation for Crusher SB 1515.

Table 1. Results of expert evaluation for Crusher SB 1515.

Expert	Type	poor	adeq	aver	good	exc	Expert	Type	poor	adeq	aver	good	exc
1	R	0	0.5	0.5	0	0	6	R	0	0	0	0.6	0.4
	t	0	0.2	0.8	0	0		t	0	0.4	0.6	0	0
	e	0	0.6	0.4	0	0		e	0	0.2	0.8	0	0
	u	0.2	0.8	0	0	0		u	0	0.1	0.9	0	0
	d	0	0.5	0.5	0	0		d	0	0.4	0.6	0	0
	m	0	0.3	0.7	0	0		m	0	0.3	0.7	0	0
	s	0	0.1	0.9	0	0		s	0	0.7	0.3	0	0
	F	0.4	0.6	0	0	0		F	0.2	0.8	0	0	0
2	R	0	0	0.3	0.7	0	7	R	0	0	0.3	0.7	0
	t	0	0	0.35	0.65	0		t	0	0	0.2	0.8	0
	e	0	0	0.2	0.8	0		e	0	0	0	0.4	0.6
	u	0	0	0.6	0.4	0		u	0	0.4	0.6	0	0
	d	0.2	0.8	0	0	0		d	0	0	0	0.6	0.4
	m	0.5	0.5	0	0	0		m	0	0	0.2	0.8	0
	s	0	0.7	0.3	0	0		s	0	0	0.3	0.7	0
	F	0	0.3	0.7	0	0		F	0	0.4	0.6	0	0
3	R	0	0	0.3	0.7	0	8	R	0	0	0	0.8	0.2
	t	0	0	0.4	0.6	0		t	0	0.4	0.6	0	0
	e	0	0	0.45	0.55	0		e	0	0.3	0.7	0	0
	u	0	0.25	0.75	0	0		u	0	0.6	0.4	0	0
	d	0	0.1	0.9	0	0		d	0	0	0.2	0.8	0
	m	0.4	0.6	0	0	0		m	0	0	0.7	0.3	0
	s	0	0.6	0.4	0	0		s	0	0.4	0.6	0	0
	F	0	0.3	0.7	0	0		F	0	0.1	0.9	0	0
4	R	0	0	0.1	0.9	0	9	R	0	0	0	0.9	0.1
	t	0	0	0.2	0.8	0		t	0	0.3	0.7	0	0
	e	0	0.5	0.5	0	0		e	0	0	0.2	0.8	0
	u	0.4	0.6	0	0	0		u	0	0	0.4	0.6	0
	d	0	0.4	0.6	0	0		d	0	0	0.2	0.8	0
	m	0.3	0.7	0	0	0		m	0	0	0.7	0.3	0
	s	0.25	0.75	0	0	0		s	0	0.3	0.7	0	0
	F	0.15	0.85	0	0	0		F	0.2	0.8	0	0	0
5	R	0	0	0.3	0.7	0	10	R	0	0	0.6	0.4	0
	t	0	0.3	0.7	0	0		t	0	0.3	0.7	0	0
	e	0	0.7	0.3	0	0		e	0	0.4	0.6	0	0
	u	0	0.4	0.6	0	0		u	0	0.6	0.4	0	0
	d	0.3	0.7	0	0	0		d	0	0.1	0.9	0	0
	m	0	0.6	0.4	0	0		m	0	0.7	0.3	0	0
	s	0.3	0.7	0	0	0		s	0	0.2	0.8	0	0
	F	0	0.3	0.7	0	0		F	0	0.4	0.6	0	0

Table 2. Results of expert evaluation for Crusher SB 1315.

Table 2. Results of expert evaluation for Crusher SB 1315.

Expert	Type	poor	adeq	aver	good	exc	Expert	Type	poor	adeq	aver	good	exc
1	R	0	0	0	0.6	0.4	6	R	0	0	0	0.6	0.4
	t	0	0	0.2	0.8	0		t	0	0.4	0.6	0	0
	e	0	0	0	0.7	0.3		e	0	0.2	0.8	0	0
	u	0.45	0.55	0	0	0		u	0	0.1	0.9	0	0
	d	0	0	0.1	0.9	0		d	0	0.4	0.6	0	0
	m	0	0.3	0.7	0	0		m	0	0.3	0.7	0	0
	s	0	0.2	0.8	0	0		s	0	0.7	0.3	0	0
	F	0	0	0.4	0.6	0		F	0.2	0.8	0	0	0
2	R	0	0	0.3	0.7	0	7	R	0	0	0.3	0.7	0
	t	0	0	0	0.3	0.7		t	0	0	0.2	0.8	0
	e	0	0	0	0.4	0.6		e	0	0	0	0.4	0.6
	u	0	0	0.2	0.8	0		u	0	0.4	0.6	0	0
	d	0	0	0.4	0.6	0		d	0	0	0	0.6	0.4
	m	0.5	0.5	0	0	0		m	0	0	0.2	0.8	0
	s	0	0	0.8	0.2	0		s	0	0	0.3	0.7	0
	F	0	0	0.2	0.8	0		F	0	0.4	0.6	0	0
3	R	0	0	0.3	0.7	0	8	R	0	0	0	0.8	0.2
	t	0	0	0.4	0.6	0		t	0	0.4	0.6	0	0
	e	0	0	0.45	0.55	0		e	0	0.3	0.7	0	0
	u	0	0.25	0.75	0	0		u	0	0.6	0.4	0	0
	d	0	0.1	0.9	0	0		d	0	0	0.2	0.8	0
	m	0.4	0.6	0	0	0		m	0	0	0.7	0.3	0
	s	0	0.6	0.4	0	0		s	0	0.4	0.6	0	0
	F	0	0.3	0.7	0	0		F	0	0.1	0.9	0	0
4	R	0	0	0.1	0.9	0	9	R	0	0	0	0.9	0.1
	t	0	0	0.2	0.8	0		t	0	0.3	0.7	0	0
	e	0	0.5	0.5	0	0		e	0	0	0.2	0.8	0
	u	0.4	0.6	0	0	0		u	0	0	0.4	0.6	0
	d	0	0.4	0.6	0	0		d	0	0	0.7	0.3	0
	m	0.3	0.7	0	0	0		m	0	0	0.4	0.6	0
	s	0.25	0.75	0	0	0		s	0	0.3	0.7	0	0
	F	0.15	0.85	0	0	0		F	0.2	0.8	0	0	0
5	R	0	0	0.3	0.7	0	10	R	0	0	0.6	0.4	0
	t	0	0.3	0.7	0	0		t	0	0.3	0.7	0	0
	e	0	0.7	0.3	0	0		e	0	0.4	0.6	0	0
	u	0	0.4	0.6	0	0		u	0	0.6	0.4	0	0
	d	0.3	0.7	0	0	0		d	0	0.1	0.9	0	0
	m	0	0.6	0.4	0	0		m	0	0.7	0.3	0	0
	s	0.3	0.7	0	0	0		s	0	0.2	0.8	0	0
	F	0	0.3	0.7	0	0		F	0	0.4	0.6	0	0

Table 3. Results of expert evaluation for belt conveyors.

Table 3. Results of expert evaluation for belt conveyors.

Expert	Type	poor	adeq	aver	good	exc	Expert	Type	poor	adeq	aver	good	exc
1	R	0	0.5	0.5	0	0	6	R	0	0	0	0.6	0.4
	t	0	0.2	0.8	0	0		t	0	0.4	0.6	0	0
	e	0	0.6	0.4	0	0		e	0	0.2	0.8	0	0
	u	0.2	0.8	0	0	0		u	0	0.1	0.9	0	0
	d	0	0.5	0.5	0	0		d	0	0.4	0.6	0	0
	m	0	0.3	0.7	0	0		m	0	0.3	0.7	0	0
	s	0	0.1	0.9	0	0		s	0	0.7	0.3	0	0
	F	0.4	0.6	0	0	0		F	0.2	0.8	0	0	0
2	R	0	0	0.3	0.7	0	7	R	0	0	0.3	0.7	0
	t	0	0	0.35	0.65	0		t	0	0	0.2	0.8	0
	e	0	0	0.2	0.8	0		e	0	0	0	0.4	0.6
	u	0	0	0.6	0.4	0		u	0	0.4	0.6	0	0
	d	0.2	0.8	0	0	0		d	0	0	0	0.6	0.4
	m	0.5	0.5	0	0	0		m	0	0	0.2	0.8	0
	s	0	0.7	0.3	0	0		s	0	0	0.3	0.7	0
	F	0	0.3	0.7	0	0		F	0	0.4	0.6	0	0
3	R	0	0	0.3	0.7	0	8	R	0	0	0	0.8	0.2
	t	0	0	0.4	0.6	0		t	0	0.4	0.6	0	0
	e	0	0	0.45	0.55	0		e	0	0.3	0.7	0	0
	u	0	0.25	0.75	0	0		u	0	0.6	0.4	0	0
	d	0	0.1	0.9	0	0		d	0	0	0.2	0.8	0
	m	0.4	0.6	0	0	0		m	0	0	0.7	0.3	0
	s	0	0.6	0.4	0	0		s	0	0.4	0.6	0	0
	F	0	0.3	0.7	0	0		F	0	0.1	0.9	0	0
4	R	0	0	0.1	0.9	0	9	R	0	0	0	0.9	0.1
	t	0	0	0.2	0.8	0		t	0	0.3	0.7	0	0
	e	0	0.5	0.5	0	0		e	0	0	0.2	0.8	0
	u	0.4	0.6	0	0	0		u	0	0	0.4	0.6	0
	d	0	0.4	0.6	0	0		d	0	0	0.7	0.3	0
	m	0.3	0.7	0	0	0		m	0	0	0.4	0.6	0
	s	0.25	0.75	0	0	0		s	0	0.3	0.7	0	0
	F	0.15	0.85	0	0	0		F	0.2	0.8	0	0	0
5	R	0	0	0.3	0.7	0	10	R	0	0	0.6	0.4	0
	t	0	0.3	0.7	0	0		t	0	0.3	0.7	0	0
	e	0	0.7	0.3	0	0		e	0	0.4	0.6	0	0
	u	0	0.4	0.6	0	0		u	0	0.6	0.4	0	0
	d	0.3	0.7	0	0	0		d	0	0.1	0.9	0	0
	m	0	0.6	0.4	0	0		m	0	0.7	0.3	0	0
	s	0.3	0.7	0	0	0		s	0	0.2	0.8	0	0
	F	0	0.3	0.7	0	0		F	0	0.4	0.6	0	0

4.3. Determination the Partial Indicator of Maintainability M

On the basis of the submitted results, the following estimates were obtained for each analyzed part of the CCS system.

Table 4. Ratings of maintainability indicators for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

Table 4. Ratings of maintainability indicators for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

	Crusher SB 1515					Crusher SB 1315					Belt conveyors
	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc
t	0.0000	0.2850	0.5250	0.1900	0.0000	0.1300	0.4800	0.3200	0.0700	0.0000	0.0700	0.3550	0.4450	0.1300	0.0000
e	0.0600	0.2550	0.4150	0.2700	0.0000	0.3400	0.4300	0.2000	0.0300	0.0000	0.0800	0.3850	0.4150	0.1200	0.0000
u	0.0000	0.1000	0.4650	0.3750	0.0600	0.0000	0.3600	0.3500	0.2450	0.0450	0.0000	0.1400	0.5300	0.3300	0.0000
d	0.0400	0.1700	0.4400	0.3000	0.0500	0.1200	0.4300	0.3500	0.1000	0.0000	0.0040	0.2200	0.4900	0.2500	0.0000
m	0.0000	0.1700	0.3400	0.3700	0.1200	0.0700	0.3000	0.3300	0.2100	0.0900	0.0070	0.1800	0.2700	0.3700	0.1100
s	0.0000	0.0700	0.4300	0.4450	0.0055	0.0000	0.1700	0.4500	0.3250	0.0550	0.0000	0.0800	0.4300	0.4600	0.0300

Table 5. Final rating for partial indicators t, e, u d, m, s, for Crusher SB 1515, Crusher SB 1315 and belt conveyors in the form of fuzzy number.

Table 5. Final rating for partial indicators t, e, u d, m, s, for Crusher SB 1515, Crusher SB 1315 and belt conveyors in the form of fuzzy number.

	Crusher SB 1515					Crusher SB 1315					Belt conveyors
	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc
t	0.0713	0.4163	0.6438	0.3213	0.0475	0.2500	0.5925	0.4575	0.1500	0.0175	0.1588	0.4838	0.5663	0.2413	0.0325
e	0.1238	0.3738	0.5463	0.3738	0.0675	0.4475	0.5650	0.3150	0.0800	0.0075	0.1763	0.5088	0.5413	0.2238	0.0300
u	0.0250	0.2163	0.5838	0.5063	0.1538	0.0900	0.4475	0.5013	0.3438	0.1063	0.0350	0.2725	0.6475	0.4625	0.0825
d	0.8250	0.2900	0.5575	0.4255	0.1250	0.2275	0.5475	0.4825	0.1875	0.0250	0.0950	0.3525	0.6075	0.3725	0.0625
m	0.0425	0.2550	0.4750	0.4850	0.2125	0.1450	0.4000	0.4575	0.3150	0.1425	0.1150	0.2650	0.4075	0.4650	0.2025
s	0.0175	0.1775	0.5588	0.5663	0.1663	0.0425	0.2825	0.5738	0.4513	0.1363	0.0200	0.1875	0.5650	0.5750	0.1450

On the basis of the obtained ratings in the form of fuzzy numbers, the ratings obtained using the max-min composition for the specified parts of the system are shown in the following table.

Table 6. Ratings obtained for the partial indicator of maintainability using the max-min composition.

Table 6. Ratings obtained for the partial indicator of maintainability using the max-min composition.

M- maintainability	poor	adeq	aver	good	exc
Crusher SB 1515	0.1250	0.4845	0.4850	0.4163	0.0713
Crusher SB 1315	0.0250	0.3150	0.4575	0.4575	0.2275
Belt conveyors	0.0625	0.4650	0.4650	0.4650	0.0950

Figure 9. Distribution of output values for indicator M maintainability for Crusher SB 1515 using max-min composition.

Figure 9. Distribution of output values for indicator M maintainability for Crusher SB 1515 using max-min composition.

Figure 10. Distribution of output values for indicator M maintainability for Crusher SB 1315 using max-min composition.

Figure 10. Distribution of output values for indicator M maintainability for Crusher SB 1315 using max-min composition.

Figure 11. Distribution of output values for indicator M maintainability for belt conveyors using max-min composition.

Figure 11. Distribution of output values for indicator M maintainability for belt conveyors using max-min composition.

4.4. Determination of Partial Indicators of Reliability and Logistical Support of Parts of the Continuous Prats of Combined System - R and F

On the basis of the submitted results, the following estimates were obtained for each analyzed part of the continuous system when these two partial indicators are concerned.

Table 9. Ratings of partial reliability and logistical support for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

Table 9. Ratings of partial reliability and logistical support for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

	Crusher SB 1515					Crusher SB 1315					Belt conveyors
	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc
R	0.0700	0.6400	0.2400	0.0500	0.0000	0.2700	0.6200	0.1100	0.0000	0.0000	0.0700	0.6500	0.2400	0.0400	0.0000
F	0.0000	0.0000	0.4200	0.4850	0.0950	0.0000	0.2600	0.3700	0.3250	0.0450	0.0000	0.0000	0.4600	0.5000	0.0400

Using the estimated values for the partial indicators for system maintainability, the final ratings for the partial indicators R, M , and F for Crusher SB 1515, Crusher SB 1315 and belt conveyors in the fuzzy number form.

Table 10. Corresponding fuzzy numbers for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

Table 10. Corresponding fuzzy numbers for Crusher SB 1515, Crusher SB 1315 and belt conveyors.

	Crusher SB 1515					Crusher SB 1315					Belt conveyors
	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc	poor	adeq	aver	good	exc
R	0.2300	0.7175	0.4125	0.1100	0.1250	0.4250	0.7150	0.2650	0.0275	0.0000	0.2325	0.7275	0.4125	0.1000	0.0100
M	0.1250	0.4845	0.4850	0.4163	0.0713	0.0250	0.3150	0.4575	0.4575	0.2275	0.0625	0.4650	0.4650	0.4650	0.0950
F	0.0000	0.1050	0.5413	0.6138	0.2163	0.0650	0.3525	0.5163	0.4288	0.1263	0.0000	0.1150	0.5850	0.6250	0.1650

Based on the obtained ratings in the form of fuzzy number, the ratings obtained using the max-min composition for the specified parts of the CCS system are shown in the following table.

Table 11. Obtained ratings for operational safety using the max-min composition.

Table 11. Obtained ratings for operational safety using the max-min composition.

D-dependability	poor	adeq	aver	good	exc
Crusher SB 1515	0.0713	0.4125	0.4850	0.4850	0.1050
Crusher SB 1315	0.0275	0.2650	0.3033	0.3033	0.2478
Belt conveyors	0.0950	0.4125	0.4650	0.4650	0.6250

Figure 12. Distribution of output values for D dependability for Crusher SB 1515 using max-min composition.

Figure 12. Distribution of output values for D dependability for Crusher SB 1515 using max-min composition.

Figure 13. Distribution of output values for D dependability for Crusher SB 1315 using max-min composition.

Figure 13. Distribution of output values for D dependability for Crusher SB 1315 using max-min composition.

Figure 14. Distribution of output values for D dependability for belt conveyors using max-min composition.

Figure 14. Distribution of output values for D dependability for belt conveyors using max-min composition.

4.5. Dependability of the CCS System at the Open Pit Gacko

On the basis of obtained ratings for dependability of the system parts, the overall rating of dependability was obtained using the max-min composition and reads as follows:

(0.07125, 0.3033, 0.30333, 0.30333, 0.105).

Further analysis shows that the corresponding values obtained by the best fit method are equal to:

(1.02977, 0.78942, 0.71215, 0.77866, 0.99645).

The corresponding reciprocal values of the relative distances ${\alpha }_i$ are:

(0.69156, 0.90211, 1, 0.91458, 0.71468),

while the values of normalized reciprocal values ${\beta }_i$ are equal to:

(0.16376, 0.21362, 0.23680, 0.21657, 0.16923).

Appropriate linguistic evaluation $Z={\displaystyle \frac{{5\beta }_1+{4\beta }_2+{3\beta }_3+{2\beta }_4+{1\beta }_5}{{\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4+{\beta }_5}}=2.9860.$

On a scale of 1-5, the mentioned system in operation has a center of gravity of linguistic assessment for the max-min composition of 2.9860. Dependability is 59%.

Figure 15. Distribution of output values for D dependability for CCS system using max-min composition.

Figure 15. Distribution of output values for D dependability for CCS system using max-min composition.

5. Verification of Fuzzy Model

Figure 16. The authors of the article with representatives of the Gacko mine during a tour of the field (private archive).

Figure 16. The authors of the article with representatives of the Gacko mine during a tour of the field (private archive).

Figure 17. The percentage of failures of the CCS system in the period from 2018-2023. years.

Figure 17. The percentage of failures of the CCS system in the period from 2018-2023. years.

Figure 18. Dependability of the continuous part of the CCS system in the period from 2018-2023. years.

Figure 18. Dependability of the continuous part of the CCS system in the period from 2018-2023. years.

6. Conclusions

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