# Development of the Availability Concept by Using Fuzzy Theory with AHP Correction, a Case Study: Bulldozers in the Open-Pit Lignite Mine

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Availability Concept in the Technical Systems of Maintenance Engineering

_{1}is uptime, and t

_{2}is downtime. The time when the system is in its proper state can be divided into inactive time or the standby time (t

_{11}) and the time when the system is in operation (t

_{12}). The time of failure is divided into: organizational time (t

_{21}), logistical time (t

_{22}) and active repair time (t

_{23}), which can be time for corrective repairs (t

_{231}) and time for preventive repairs (t

_{232}). The times t

_{21}and t

_{22}refer to: Defects, design interventions, administrative work, spare parts, tools, skilled labor, etc. The active repair time includes repair, assembly, disassembly, replacement, etc. The timetable is not always the same type. Figure 1 is just one of the possible examples.

_{o}(t). If, when determining the availability in time of failure, only the active time of corrective and preventive maintenance is taken into account, we are referring to achieved availability:

^{–λ·t}and convenience of maintenance M(t) = 1 − e

^{−μ t}, it is known as:

- failure intensity: $\lambda =\frac{1}{MTBF}=const.$
- maintenance intensity: $\mu =\frac{1}{MDT}=const.$

_{A}value is called the coefficient of availability and is obtained when A(t) is calculated for t→ ∞, or when the availability value becomes stationary (Figure 2).

_{i}is required. In real terms related to complex machine engineering systems, this condition is usually not fulfilled or only partially fulfilled. Another problem is that we cannot infer the availability structure by knowing the availability coefficient. In other words, we do not know the impact of partial indicators such as reliability, the convenience of maintenance, and the level of support.

#### 2.2. Expert Fuzzy-AHP Synthesis Model Availability

#### 2.2.1. Fuzzy Inference in the Synthesis Model

- ‘A‘(R)—No sudden, unplanned failures were recorded.
- ‘B‘(R)—There are some interruptions in work. Negligible impact on the time state picture of the technical system.
- ‘C’(R)—Failures occur. In most cases, they are expected, and therefore, in some way they can be planned. Failures can be eliminated on the spot.
- ‘D’(R)—Occurrence of failure is frequent. The reliability of the machine is low. Efficiency is reduced.
- ‘E’(R)—Constant breakdowns occur. The machine is not at the required working level.

- ‘A’(M)—Any intervention can be fully planned in terms of time and work organization. Diagnosis is simple. Repairs are quick. No corrosion. Defective parts are not of a large mass. It is possible to plan time and work organization.
- ‘B’(M)—Quick identification of weaknesses is possible (errors, faults …). It is constructively easy to repair. There may be some minor interference errors.
- ‘C’(M)—Possible difficulties during preventive and service maintenance, for reasons of constructive nature, inaccessibility of parts, due to the appearance of corrosion, the mass of the element, and the like.
- ‘D’(M)—It is not possible to plan the duration of the intervention and the organization of work. There are a number of complications during dismantling and assembly.
- ‘E’(M)—The breakdown cannot be remedied in an acceptable time. It is necessary to disconnect the machine from the operating unit for a longer period of time.

- ‘A’(S)—Any work with the machine can be fully planned in terms of time and organization. There are spare parts and tools. There are trained repairmen. The workshop is close. There are no administrative difficulties.
- ‘B’(S)—Administrative and logistical support is at a satisfactory level. Supply of spare parts is fast. Workshop is at a short distance. Possible purchase of necessary paperwork.
- ‘C’(S)—All activities related to maintenance support (spare parts, tools, workshops, employee training, etc.) are at a satisfactory level. Utilization of the machine is correct in most cases.
- ‘D’(S)—There are difficulties in purchasing spare parts. Additional training is necessary. There are administrative difficulties. Utilization of the machine is a little bit harder than expected.
- ‘E’(S)—There are no spare parts. The workers are not trained. There are administrative problems. The workshop is remote. Every utilization of the machine is full of unpredictability due to inadequate training, logistical support, etc. It is not possible to plan activities in the context of time and organization.

_{i}, M

_{i}и S

_{i}are defined through membership function μ and class j = 1 to n:

^{3}combinations among themselves. Each combination A

_{c}represents practically one possible assessment A

_{i}(8).

_{R,M,S}

_{(j = 1,…, n)}≠ 0, are taken into account, then we obtain the outcomes (o = 1 to O, where O ≤ C). Each outcome has the corresponding values (iv) and (v) that further identify it for the estimate.

_{c}value is calculated and rounded as an integer, in the following way:

- w
_{i}is the influential factor of the corresponding partial indicator on availability obtained on the basis of mutual ranking of partial indicators, where w_{Ri}+ w_{Mi}+ w_{Si}= 1 (Equation (17)); - j
_{c}is a class to which the corresponding fuzzy number (9) belongs for the observed membership function and the given combination c, where j_{c}= 1, …, n;

_{R}, μ

_{M}, μ

_{S}in the vector A

_{c}(10) is requested as follows:

_{c}value. The number of such groups can be 0 to n.

#### 2.2.2. AHP Ranking Model

_{max}is the weighted mean of coefficient λ

_{i}which calculations are given in Equation (20),

## 3. Results: Case Study Availability of Bulldozers

_{(BN)}> A

_{(BO)}).

#### 3.1. Preparation of Questionnaires, Statistical Processing and Fuzzification of Expert Opinions

- with ‘A‘, three out of four analysts (experts): $\frac{((1\cdot 0.7)+(1\cdot 0.6)+(1\cdot 0.5))}{4}=0.450$
- with ‘B‘, all four analysts (experts): $\frac{((1\cdot 0.3)+(1\cdot 0.4)+(1\cdot 0.9)+(1\cdot 0.5))}{4}=0.525$
- with ‘C‘, only one analyst (expert): $\frac{(1\cdot 0.1)}{4}=0.025$

#### 3.2. AHP Ranking

_{R}= 0.1630, W

_{M}= 0.2968, W

_{S}= 0.5401. Calculated weighting coefficients are further implemented into the fuzzy model with the results shown in Table 7.

#### 3.3. Max–Min Composition

_{RB1-N}, μ

_{MB1-N,}and μ

_{SB1-N}(9).

^{3}= 1000 combinations. The combinations carry the code mark in the general record: j

_{R}-j

_{M}-j

_{S}, for j = 1 … 10. In this example, the following combinations are possible: 1-1-1; 1-1-2; 1-1-3; …; 10-10-8; 10-10-9; 10-10-10.

_{R B}

_{1}= 0.164; W

_{M B}

_{1}= 0.297; W

_{S B}

_{1}= 0.539.

- For J
_{c}= 6, 14 combinations were recorded: 4-6-6, …, 9-6-6; - For J
_{c}= 7, 51 combinations were recorded: 4-6-8, …, 10-8-6; - For J
_{c}= 8, 65 combinations were recorded: 4-6-10, …, 10-10-7; - For J
_{c}= 9, 39 combinations were recorded: 4-6-8, …, 10-10-9; - For J
_{c}= 10, 6 combinations were recorded: 7-10-10, …, 10-10-10;

_{c}.

#### 3.4. Identification

_{A(B1-N)}according to (22) and μ

_{A}according to (2)

_{min}= d

_{2}:

_{2}= 0.28970, μ

_{3}= 0.17108, μ

_{4}= 0.15354, μ

_{5}= 0.15645

_{(B1-N)}= (0.22922/’A’, 0.28970/’B’, 0.17108/’C’, 0.15354/’D’, 0.15645/’E’)

#### 3.5. Results Discussion

_{B}

_{2-N}= 3.38, Z

_{B}

_{3-N}= 3.36, Z

_{B}

_{1-O}= 3.07, Z

_{B}

_{2-O}= 3.25, Z

_{B}

_{3-O}= 3.17,

_{B1}= 3.18, Z

_{B2}= 3.32, Z

_{B}

_{3}= 3.26

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time state picture. t

_{1,}uptime; t

_{11}, standby time; t

_{12}, time in operation; t

_{2}, downtime; t

_{21}, organizational time; t

_{22}, logistic time; t

_{23}, repair time (t

_{231}, corrective; t

_{232}, preventive repair).

**Figure 4.**Comparative analysis of all bulldozers availability and analysis according to different models. (

**a**) for all of the analysed machines; (

**b**) for machines according to manufacturer.

**Figure 5.**Comparative analysis of bulldozer availability according to different grades. (

**a**) for all of the analysed machines; (

**b**) for machines according to manufacturer.

The Level of Importance | Numerical Value | Reciprocal Value |
---|---|---|

Extreme importance | 9 | 1/9 (0.111) |

Very strong to extreme importance | 8 | 1/8 (0.125) |

Very strong importance | 7 | 1/7 (0.143) |

Strong to very strong importance | 6 | 1/6 (0.167) |

Strong importance | 5 | 1/5 (0.200) |

Moderate to strong importance | 4 | 1/4 (0.250) |

Moderate importance | 3 | 1/3 (0.333) |

Equal to moderate importance | 2 | 1/2 (0.500) |

Equal importance | 1 | 1 (1.000) |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

RI | 0.00 | 0.00 | 0.52 | 0.89 | 1.11 | 1.25 | 1.35 | 1.40 | 1.45 | 1.49 | 1.51 | 1.53 | 1.56 | 1.57 | 1.59 |

Years of Operation | B1 | B2 | B3 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

t_{1}, h | t_{2}, h | A(t) | t_{1}, h | t_{2}, h | A(t) | t_{1}, h | t_{2}, h | A(t) | |||||

N | 1 | 519 | 20 | 0.96 | 0.96 | 934 | 25 | 0.97 | 0.97 | 753 | 37 | 0.95 | 0.94 |

2 | 1893 | 92 | 0.95 | 3004 | 128 | 0.96 | 3741 | 290 | 0.93 | ||||

O | 3 | 3372 | 334 | 0.91 | 0.89 | 3415 | 262 | 0.93 | 0.90 | 3476 | 384 | 0.90 | 0.84 |

4 | 4100 | 498 | 0.89 | 3631 | 367 | 0.91 | 3102 | 572 | 0.84 | ||||

5 | 4325 | 431 | 0.91 | 4296 | 494 | 0.90 | 2635 | 622 | 0.81 | ||||

6 | 3601 | 449 | 0.89 | 4127 | 445 | 0.90 | 2757 | 664 | 0.81 | ||||

7 | 1438 | 234 | 0.86 | 2894 | 387 | 0.88 | 2008 | 343 | 0.85 |

Analyst | B1-N | B2-N | B3-N | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ||

1. | R | 0.7 | 0.3 | 0.8 | 0.2 | 0.6 | 0.4 | |||||||||

M | 0.4 | 0.6 | 0.7 | 0.3 | 0.5 | 0.5 | ||||||||||

S | 0.3 | 0.7 | 0.6 | 0.4 | 0.6 | 0.4 | ||||||||||

2. | R | 0.6 | 0.4 | 0.6 | 0.4 | 0.3 | 0.7 | |||||||||

M | 0.6 | 0.4 | 0.8 | 0.2 | 0.6 | 0.4 | ||||||||||

S | 0.5 | 0.5 | 0.4 | 0.6 | 0.5 | 0.5 | ||||||||||

3. | R | 0.9 | 0.1 | 1 | 1 | |||||||||||

M | 0.4 | 0.6 | 0.6 | 0.4 | 0.7 | 0.3 | ||||||||||

S | 0.2 | 0.8 | 0.6 | 0.4 | 0.7 | 0.3 | ||||||||||

4. | R | 0.5 | 0.5 | 0.7 | 0.3 | 0.4 | 0.6 | |||||||||

M | 0.7 | 0.3 | 0.7 | 0.3 | 0.7 | 0.3 | ||||||||||

S | 1 | 1 | 0.3 | 0.7 | ||||||||||||

Σ | R | 0.450 | 0.525 | 0.025 | 0 | 0 | 0.525 | 0.475 | 0 | 0 | 0 | 0.325 | 0.675 | 0 | 0 | 0 |

M | 0.525 | 0.475 | 0 | 0 | 0 | 0.700 | 0.300 | 0 | 0 | 0 | 0.625 | 0.375 | 0 | 0 | 0 | |

S | 0.250 | 0.750 | 0 | 0 | 0 | 0.650 | 0.350 | 0 | 0 | 0 | 0.525 | 0.475 | 0 | 0 | 0 |

Analyst | B1-O | B2-O | B3-O | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ‘A’ | ‘B’ | ‘C’ | ‘D’ | ‘E’ | ||

1. | R | 0.6 | 0.4 | 0.7 | 0.3 | 0.8 | 0.2 | |||||||||

M | 0.6 | 0.4 | 0.6 | 0.4 | 0.4 | 0.6 | ||||||||||

S | 0.3 | 0.7 | 0.5 | 0.5 | 0.4 | 0.6 | ||||||||||

2. | R | 0.9 | 0.1 | 0.5 | 0.5 | 0.2 | 0.8 | |||||||||

M | 0.8 | 0.2 | 0.8 | 0.2 | 0.5 | 0.5 | ||||||||||

S | 0.4 | 0.6 | 0.1 | 0.9 | 0.4 | 0.6 | ||||||||||

3. | R | 0.1 | 0.9 | 0.3 | 0.7 | 0.7 | 0.3 | |||||||||

M | 0.6 | 0.4 | 0.9 | 0.1 | 0.8 | 0.2 | ||||||||||

S | 0.9 | 0.1 | 0.8 | 0.2 | 1 | |||||||||||

4. | R | 0.5 | 0.5 | 0.2 | 0.8 | 1 | ||||||||||

M | 0.3 | 0.7 | 0.5 | 0.5 | 0.1 | 0.9 | ||||||||||

S | 0.8 | 0.2 | 0.4 | 0.6 | 0.2 | 0.8 | ||||||||||

Σ | R | 0 | 0.300 | 0.675 | 0.025 | 0 | 0 | 0.425 | 0.575 | 0 | 0 | 0 | 0.050 | 0.825 | 0.125 | 0 |

M | 0.150 | 0.375 | 0.375 | 0.100 | 0 | 0.150 | 0.650 | 0.200 | 0 | 0 | 0.150 | 0.650 | 0.200 | 0 | 0 | |

S | 0.075 | 0.275 | 0.575 | 0.075 | 0 | 0.025 | 0.650 | 0.325 | 0 | 0 | 0.150 | 0.700 | 0.150 | 0 | 0 |

AHP Preferences | B1-N (B2-N, B3-N) | B1-O | B2-O | B3-O | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R | M | S | R | M | S | R | M | S | R | M | S | |

R | 1 | 1/2 | 1/3 | 1 | 1 | 1 | 1 | 1/3 | 1/3 | 1 | 2 | 2 |

M | 2 | 1 | 1/2 | 1 | 1 | 1 | 3 | 1 | 1 | 1/2 | 1 | 1 |

S | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1/2 | 1 | 1 |

AHP Ranking | B1-N | B1-O | B2-N | B2-O | B3-N | B3-O |
---|---|---|---|---|---|---|

W_{R} | 0.1630 | 0.3333 | 0.1630 | 0.1428 | 0.1630 | 0.5000 |

W_{M} | 0.2968 | 0.3333 | 0.2968 | 0.4286 | 0.2968 | 0.2500 |

W_{S} | 0.5401 | 0.3333 | 0.5401 | 0.4286 | 0.5401 | 0.2500 |

λ_{max} | 3.00921 | 3 | 3.00921 | 3 | 3.00921 | 3 |

CI | 0.00460 | 0 | 0.00460 | 0 | 0.00460 | 0 |

CR | 0.00885 | 0 | 0.00885 | 0 | 0.00885 | 0 |

Machine | ‘A’—Excellent | ‘B’—Good | ‘C’—Average | ‘D’—Adequate | ‘E’—Poor |
---|---|---|---|---|---|

B1-N | 0.22922 | 0.28970 | 0.17108 | 0.15354 | 0.15645 |

B2-N | 0.31801 | 0.21294 | 0.15800 | 0.15400 | 0.15705 |

B3-N | 0.29108 | 0.23916 | 0.16110 | 0.15290 | 0.15576 |

B1-O | 0.14472 | 0.22777 | 0.31916 | 0.17073 | 0.13762 |

B2-O | 0.13784 | 0.37904 | 0.21289 | 0.13971 | 0.13052 |

B3-O | 0.14204 | 0.29836 | 0.27904 | 0.14466 | 0.13591 |

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**MDPI and ACS Style**

Djenadic, S.; Ignjatovic, D.; Tanasijevic, M.; Bugaric, U.; Jankovic, I.; Subaranovic, T.
Development of the Availability Concept by Using Fuzzy Theory with AHP Correction, a Case Study: Bulldozers in the Open-Pit Lignite Mine. *Energies* **2019**, *12*, 4044.
https://doi.org/10.3390/en12214044

**AMA Style**

Djenadic S, Ignjatovic D, Tanasijevic M, Bugaric U, Jankovic I, Subaranovic T.
Development of the Availability Concept by Using Fuzzy Theory with AHP Correction, a Case Study: Bulldozers in the Open-Pit Lignite Mine. *Energies*. 2019; 12(21):4044.
https://doi.org/10.3390/en12214044

**Chicago/Turabian Style**

Djenadic, Stevan, Dragan Ignjatovic, Milos Tanasijevic, Ugljesa Bugaric, Ivan Jankovic, and Tomislav Subaranovic.
2019. "Development of the Availability Concept by Using Fuzzy Theory with AHP Correction, a Case Study: Bulldozers in the Open-Pit Lignite Mine" *Energies* 12, no. 21: 4044.
https://doi.org/10.3390/en12214044