Development of a Methodology for Optimizing Repair Interval Timing for Mining Equipment Units

Abstract

This study presents a methodology for optimizing repair intervals of mining equipment by integrating economic efficiency and reliability criteria. A review of existing maintenance strategies revealed their limitations, and a mathematical model was developed that incorporates both projected financial expenditures and the probability of equipment failures, enabling more accurate prediction of the optimal repair timing. This study introduces a novel integration of the Weibull reliability distribution with a cost-convolution optimization model, explicitly capturing the trade-off between economic efficiency and failure risk. Unlike traditional fixed-schedule approaches, the proposed model provides analytically optimized repair intervals derived from observed degradation trends. Statistical analysis demonstrates that unplanned repairs are, on average, 56% more costly than scheduled ones, highlighting the need to revise current preventive maintenance practices. The cost comparison is based on 34 restoration records collected from publicly available supplier price lists and field maintenance logs, converted into a unified currency. Based on operational data and reliability parameter estimation, the optimal repair interval was determined to be 5129 machine hours, which minimizes both the probability of failure and total maintenance-related financial losses, while reducing unplanned downtime. Unlike traditional fixed-schedule approaches, the proposed model allows adaptive adjustment of maintenance intervals according to the actual degradation characteristics of the equipment. The practical significance of the research lies in its ability to help mining enterprises reduce expenditures on corrective repairs, extend the service life of machinery, and improve overall operational efficiency. The findings contribute to advancing maintenance optimization in the mining industry, supporting more sustainable and cost-effective equipment management.

1. Introduction

KazMinerals Bozshakol is the largest mining company in Kazakhstan, located in the Pavlodar region. The company specializes in open-pit mining and processing of copper ores. The site operates a modern fleet of mining and haulage equipment, including CAT 785C, 785D, and 789D haul trucks, EX3600 excavators, D55SP drill rigs, CAT 16M and 16M3 motor graders, as well as other heavy mining machinery [1]. KAZ Minerals is one of the leading copper producers in Central Asia, focusing on the development of large-scale, low-cost open-pit mines in Kazakhstan and Kyrgyzstan. Its strategic focus lies on improving operational efficiency and developing new deposits, as evidenced by the implementation of major projects such as Bozshakol and Aktogay, as well as operations in the Eastern Region and at the Bozymchak mine [1]. The location of the main KAZ Minerals production assets is shown in Figure 1.

The Bozshakol mine has a resource base of 1123 million tons of ore with an average copper grade of 0.35%. In 2023, the mine produced 105 thousand tons of copper, along with by-products such as gold, silver, and molybdenum. The Aktogay mine contains 1915 million tons of resources with a copper grade of 0.33% and produced 252 thousand tons of copper in 2023. Currently, a project is underway to expand the processing capacity, which will increase the annual sulfide ore throughput to 50 million tons. The eastern region of the company includes several underground mines and processing plants, while the Bozymchak mine in Kyrgyzstan focuses on open-pit mining of copper-gold ores. In 2023, the combined output of the Eastern Region and Bozymchak amounted to 46 thousand tons of copper and 47 thousand ounces of gold [1].

Mining operations are carried out under challenging climatic conditions characterized by sharp temperature fluctuations, high dust levels, and the abrasive nature of rock formations. The operation of equipment in such an environment leads to accelerated wear of components and assemblies, which significantly increases the demands on the maintenance and repair system. For this reason, research on optimizing the repair intervals of mining equipment components is of particular importance to the company [1]. Figure 2 shows open-pit mining operations.

Since mining equipment is constantly subjected to heavy loads, its components and assemblies gradually wear out over time [1]. The failure of even a single critical component can cause the entire machine to be shutdown. When equipment is idle, the company incurs financial losses, as each working shift requires maximum utilisation of the available machinery.

To prevent unexpected breakdowns, companies implement scheduled maintenance [2]. For example, many equipment manufacturers such as Caterpillar, Sandvik, and Epiroc recommend major repairs every 15,000 machine hours [3]. This preventive maintenance strategy involves servicing the equipment before the components fail [4]. In addition, regular inspections and minor repairs are carried out to keep the machine operational.

However, even with a planned approach, the equipment often fails before the scheduled time, leading to unplanned repairs that are frequently more expensive than scheduled ones [5]. This occurs because the required spare parts are not always available in stock and must be ordered urgently, while mechanics are not always available either, so they are diverted from other tasks or paid overtime. As a result, the machine remains idle and generates no profit, making scheduled untimely repairs and unplanned repairs excessively costly.

Various approaches to optimizing the maintenance and repair of mining equipment have been presented in recent studies, demonstrating both the development of theoretical models and practical examples of their application.

Recent studies have introduced a variety of approaches for optimizing maintenance activities in the mining sector. Condition monitoring methods utilize stress, vibration, and fatigue indices to detect early signs of failure [6]. Data-driven approaches based on shop-floor event logs enable preventive maintenance planning and allow adjustment of inspection intervals [7]. Process analysis and discrete-event simulation techniques have also been applied to improve maintenance workflows [8]. Other research focuses on probabilistic predictive models for maintenance optimization [9]. Several authors have proposed genetic-algorithm-based and metaheuristic optimization methods for maintenance planning [10,11]. RAMC-type frameworks attempt to jointly account for reliability, availability, maintainability, and cost [12].

Existing studies mainly optimize maintenance intervals from a purely economic perspective or based solely on reliability indicators. However, for mining brake assemblies, there is no unified model that simultaneously incorporates both cost-based criteria and reliability constraints within a single analytical framework. Most prior work relies either on complex RAMC systems, which require extensive datasets, or on probabilistic models that lack explicit economic evaluation. This creates a methodological gap between economic optimization and reliability-based scheduling. The present study addresses this gap by developing an analytically tractable model that integrates Weibull-based failure behavior with cost minimization to determine optimal repair intervals.

Despite significant advances in the field of maintenance and repair optimization, existing models have several limitations. Online monitoring methods require a complex sensor infrastructure and large volumes of data, making their large-scale implementation difficult. Approaches based on production data and process analysis help identify problematic areas in maintenance processes but depend heavily on data quality and standardization. Integrated RAMC models provide comprehensive consideration of technical and economic factors, but are characterized by high computational complexity and the need for extensive input data. Algorithmic methods, including genetic and metaheuristic approaches, allow the development of optimal strategies, but their application is constrained by demanding computational requirements and the need for reliable statistical data.

These limitations indicate that existing solutions are not always applicable in real-world mining equipment operating conditions. This highlights the need for simpler yet accurate methods that consider both reliability and economic indicators, allowing adaptation to changing operational environments.

To address this issue, a mathematical model was proposed to determine the optimal repair interval for mining equipment components, considering two key factors—economic efficiency and equipment reliability [13]. Unlike integrated RAMC models [9], which require extensive datasets and complex algorithms, the proposed model is based on the Weibull distribution and analytical relationships. This approach enables accurate calculations with comparatively lower data and computational requirements, which is particularly important for mining companies.

Unlike standard fixed-interval maintenance strategies (e.g., every 15,000 machine hours), the proposed solution enables recalculation of repair intervals based on statistical failure characteristics, without relying on real-time monitoring systems [14]. Thus, the research hypothesis is that optimizing the repair intervals for mining equipment components using a mathematical model that integrates economic indicators and reliability metrics will reduce unplanned maintenance costs, minimize downtime, and improve overall operational efficiency [15].

Adjusting repair cycles based on statistically evaluated degradation patterns allows more accurate prediction of failures and reduces the probability of critical failures compared to conventional preventive maintenance methods. [16].

In contrast to recently emerging predictive maintenance technologies such as digital twins and AI-driven failure prediction models (e.g., LSTM neural networks), the proposed approach relies on traditional statistical modeling based on the Weibull distribution. Such analytical models remain robust in data-scarce environments typical for mining operations, where continuous sensor streams and high-resolution monitoring data are often unavailable. While AI-based methods may offer higher short-term prediction accuracy, they require large training datasets, continuous model recalibration, and extensive digital infrastructure. In comparison, the Weibull-based analytical model developed in this study provides stable and interpretable results using limited historical failure data, making it practical and cost-effective for real-world mining enterprises.

The objective of this study is to develop a cost-effective methodology to determine the optimal repair interval for equipment components, with the aim of reducing total maintenance costs, minimizing downtime, and improving the reliability and useful life of the equipment.

To achieve this objective, the following tasks must be completed:

• Analyze cases of unplanned repairs and identify their root causes;
• Assess the costs associated with equipment restoration;
• Collect and process statistical data on planned and unplanned repairs;
• Develop and validate the proposed mathematical model;
• Compare the effectiveness of the developed model with existing maintenance strategies.

Although full sensitivity analysis is outside the scope of this work, the paper provides a structured parameter estimation procedure and verifies the obtained Weibull parameters using regression diagnostics.

The novelty of this study does not lie in the reliability equations themselves, but rather in applying an integrated cost–reliability optimization model specifically to brake group assemblies under real mining operating conditions. This approach adapts well-established reliability principles to a component type that is both safety-critical and characterized by harsh loading, making the resulting interval recommendations practically applicable for mining enterprises.

The practical significance of this research is that implementing the proposed model will extend the service life of equipment by reducing critical wear through timely repairs, while simultaneously increasing the economic efficiency of companies by lowering unplanned maintenance costs.

The model developed in this study can be adapted for various types of equipment and industries, including construction, mining, transportation, and agriculture, making it a versatile tool for improving the performance and reliability of machinery.

2. Materials and Methods

2.1. Analysis of Unscheduled Equipment Repairs and Their Causes

According to the objectives of this study, statistical data was collected on the repair activities related to thebrake system assemblies of CAT 16M and 16M3 motor graders (Caterpillar Inc., Peoria, IL, USA), operated at the Bozshakol open-pit mine (Pavlodar region, Kazakhstan) (Figure 3). The purpose of this analysis was to evaluate the operational performance of mining equipment and to identify the patterns and root causes of component failures [1].

Primary data were obtained from KazMinerals Bozshakol production reports for the period 2019–2024. The original data set included information on repair cases, types of failure, and number of replacement components. However, due to the company’s confidentiality policy, the financial indicators contained in these reports are not subject to publication.

For the purposes of this study, additional information was collected from open online sources, including indicative prices of spare parts used in the repair of brake groups. The cost data was compiled on the basis of publicly available price listings and catalogs of spare parts suppliers for Caterpillar equipment.

The list of online sources used to determine the approximate cost of spare parts is provided in the References section and includes links to the websites of official and independent distributors that offer publicly accessible pricing information (see sources [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]).

Information on the names, part numbers, quantities, and approximate prices of spare parts is presented in Appendix A. All data were used solely for research and analytical purposes to assess the cost structure and ratio, without disclosing any internal or commercially sensitive information of the company.

A summary of the brake group repair cases is presented in

Table 1. Statistics on brake systems of CAT 16M and 16M3 motor graders (converted to USD).

No.	Repair Cycle	Serial Number	Restoration Cost (USD)	Type of Repair	Primary Cause of Failure
1	1	BR16M-10	1942.47	Scheduled	—
2	1	BR16M-7	1737.84	Scheduled	—
3	1	BR16M-8	1737.78	Scheduled	—
4	1	BR16M-9	1737.84	Scheduled	—
5	1	BR16M-11	3644.65	Scheduled	—
6	1	BR16M-12	3644.65	Scheduled	—
7	1	BR16M-13	3644.65	Scheduled	—
8	1	BR16M-14	3644.65	Scheduled	—
9	1	BR16M-1	2039.58	Unscheduled	Wire entanglement
10	2	BR16M-2	5818.54	Unscheduled	Abrasive wear
11	2	BR16M-14	1853.61	Unscheduled	Abrasive wear
12	1	BR16M-15	2158.49	Scheduled	—
13	1	BR16M-3	2158.49	Scheduled	—
14	2	BR16M-9	764.10	Scheduled	—
15	2	BR16M-4	6125.35	Scheduled	—
16	2	BR16M-5	6125.35	Scheduled	—
17	2	BR16M-6	6125.35	Scheduled	—
18	1	BR16M-16	4378.48	Scheduled	—
19	1	BR16M-17	2625.11	Scheduled	—
20	1	BR16M-18	1923.75	Scheduled	—
21	1	BR16M-19	1923.75	Scheduled	—
22	2	BR16M-11	9039.34	Scheduled	—
23	2	BR16M-12	10,367.94	Scheduled	—
24	2	BR16M-13	9156.60	Scheduled	—
25	1	BR16M-20	2067.29	Scheduled	—
26	2	BR16M-15	14,561.96	Unscheduled	Abrasive wear
27	2	BR16M-1	20,580.68	Unscheduled	Abrasive wear
28	1	BR16M-21	1196.47	Unscheduled	Contamination and uneven wear
29	2	BR16M-3	12,288.28	Unscheduled	Seal deformation
30	3	BR16M-9	12,882.78	Unscheduled	Friction disc wear
31	2	BR16M-10	17,229.99	Scheduled	—
32	3	BR16M-2	11,137.69	Scheduled	—
33	3	BR16M-4	12,477.43	Scheduled	—
34	2	BR16M-8	15,310.57	Scheduled	—
35	2	BR16M-7	11,124.89	Scheduled	—
36	2	BR16M-20	566.14	Unscheduled	Abrasive wear
37	2	BR16M-19	10,642.57	Unscheduled	Seal deformation
38	2	BR16M-17	10,642.57	Unscheduled	Seal deformation
39	3	BR16M-6	3099.54	Unscheduled	Seal deformation
40	2	BR16M-16	10,642.57	Unscheduled	Seal deformation
41	3	BR16M-11	15,715.62	Unscheduled	Seal deformation
42	2	BR16M-3	9558.17	Unscheduled	Seal deformation
43	2	BR16M-16	12,147.42	Scheduled	Seal deformation
44	1	BR16M-22	16,348.04	Unscheduled	Seal deformation
Total restoration cost of unscheduled repairs			153,926.75	Unscheduled	—
Total restoration cost of scheduled repairs			168,251.49	Scheduled	—
Average restoration cost of unscheduled repairs			9058.04	Unscheduled	—
Average restoration cost of scheduled repairs			3581.96	Scheduled	—

Note. All cost data were estimated based on publicly available information from open online sources and do not represent internal financial records of the company.

.

The table presented above includes data on the number of repair operations performed on the assemblies, the primary causes of their failures, and the associated restoration costs. Additionally, the table specifies the type of repair performed—either scheduled (according to the maintenance schedule) or unscheduled (due to unexpected failure)—as well as the identified failure causes for each unit. Subsequently, a detailed analysis was conducted on the cases of unscheduled repairs of the components of the brake system and their primary causes of failure.

Case No. 1: Wire entanglement around the brake group seal

The brake assembly with serial number BR16M-1 was submitted for repair. At the time of failure, the total runtime of the component was 9.172 machine hours. This was the first repair cycle for the unit and was performed on an unscheduled basis. The defect inspection process of the BR16M-1 brake group is illustrated in Figure 4 [1].

Complete component disassembly was carried out, during which wire entanglement was discovered in the seating area of the mirror-type oil seal 6T-3377. As a result, the seal was compressed and damaged. The cause of failure was the ingress of foreign objects into the mechanism during operation. The following conclusion was drawn: the wire entanglement occurred during the use of the equipment. The repair required the replacement of standard spare parts [1].

Case No. 2: Abrasive wear and component damage The brake assembly with serial number BR16M-2 was submitted for repair. The total operating time of the unit at the time of repair was 27,152 machine hours, with 9526 machine hours since the previous repair. This was the second unscheduled repair cycle. The defect inspection process of the BR16M-2 brake group is illustrated in Figure 5.

Complete component disassembly was performed, revealing deformation and traces of abrasive adhesion wear on the working surfaces of the plates, as well as abrasive wear in the piston seal seating area within the housing. The cause of failure was the natural wear of the components due to the prolonged use and exposure to abrasive particles. The following conclusion was drawn: the component was found to be in an unsatisfactory condition. The defective elements were not suitable for reuse, as they would generate more wear debris, damage mating parts, and cause leakage. The repair required the replacement of damaged components in addition to standard spare parts [1].

Case No. 3: Deformation of sealing elements The brake assembly with serial number BR16M-3 was submitted for repair. The total operating time of the unit at the time of failure was 26,482 machine hours, with 10,708 machine hours since the previous repair. This was the second repair cycle, carried out on an unscheduled basis. The defect inspection process of the BR16M-3 brake group is illustrated in Figure 6.

During the disassembly of the component, the following findings were recorded: when the housing was removed from the spindle, part of the mirror seal remained on the spindle and was not secured within the housing. The housing contained a buildup of contaminated oil, fine metal particles, and dirt. Cause of failure: excess of contaminants and mechanical deformation of the sealing elements during operation. The conclusion drawn was that the most probable cause of the leakage of the mirror seal was the deformation of the toroidal seal during service. A contributing factor to leakage was the accumulation of dirt between the toroidal seal and the mirror ring, which caused the toroidal seal to lose its damping properties. This issue is most likely to occur under cold and humid conditions. The dirt pushed the toroidal seal downward, allowing contaminants to pass between the toroidal seal and the mirror ring, resulting in both wear and deformation of the seal and its seating surface in the housing. The abrasive wear may have been caused by rotation within the housing. In general, the analysis indicates that unscheduled repairs occur more frequently during the second and third operational cycles, suggesting that the likelihood of failure increases as the equipment ages [1].

The results of the analysis also show that while most repair operations were planned in advance as part of preventive maintenance, a substantial proportion of unscheduled repairs indicates that the current maintenance system fails to prevent unexpected failures. Restoration costs ranged from 566 USD to 20,580.69 USD, with unscheduled repairs being significantly more expensive than scheduled repairs, thereby placing an additional financial burden on the company [1].

Table 2. Typical failure modes of brake group assemblies and their influence on the Weibull parameter $\beta$.

Failure Mode	Description of Degradation Mechanism	Effect on $\left(\mathit{\beta }\right)$
Abrasive wear of friction discs	Progressive material loss due to dust and particle ingress; increased friction temperature accelerates wear.	Increases $\beta$ (aging behaviour, rising hazard rate)
Overheating due to excessive load	Thermal fatigue, glazing, deformation of plates, reduced braking efficiency.	Strong increase in $\beta$ (steep hazard growth)
Oil or dirt contamination	Loss of friction coefficient, slippage, accelerated wear and instability.	Moderate increase in $\beta$ (greater variability in failures)
Seal degradation and leakage	Loss of lubrication, increased friction and fatigue on rotating parts.	Increases $\beta$ (time-dependent wear-out)
Bearing or spacer misalignment	Mechanical overload leading to premature failure.	Variable increase in $\beta$ (higher variance of failure time)

summarizes the typical failure mechanisms observed in brake assemblies and their expected influence on the Weibull parameter $\beta$. The primary causes of failure were identified as abrasive wear, seal deformation, friction disk wear, and contamination, which reflects operation under harsh conditions and the limited effectiveness of current preventive measures. Some assemblies, such as BR16M-3 and BR16M-9, underwent multiple repairs, indicating frequent failures and possibly inadequate diagnostic precision. As such, the recurrence of failures in certain units demonstrates that scheduled repairs based on fixed intervals do not provide reliable protection against premature wear, and the absence of an adaptive approach leads to untimely failures [1].

Therefore, the current scheduled maintenance strategy needs to be reconsidered. The use of a fixed repair schedule without regard for actual operating conditions results in significant financial losses, increased equipment downtime, and the necessity for costly unscheduled repairs.

2.2. Analysis of Restoration Costs for Equipment Components

Effective maintenance management of mining equipment requires consideration not only of the frequency of component failures but also of their economic impact. The restoration cost of brake assemblies plays a key role in determining an appropriate maintenance strategy. Accordingly, an analysis of the restoration costs was conducted based on the dataset summarized in Table 1. Figure 7 and Figure 8 present a comparison of scheduled and unscheduled restoration costs.

The analysis of Figure 7 indicates that the total restoration cost of assemblies after unscheduled repairs is nearly equal to the cumulative cost of all scheduled restorations, despite the fact that the number of unscheduled repairs is roughly half the number of scheduled ones. This implies that the cost per unscheduled repair is significantly higher, underscoring the importance of selecting maintenance intervals that minimize the probability of emergency failures.

Based on Figure 8, emergency (unscheduled) repairs are substantially more expensive than scheduled ones. The average restoration cost after an unscheduled repair is 9283.87 USD, which is 56% higher than the average cost of a scheduled repair 6254.35 USD). To statistically validate this difference, a two-sample t-test was performed, confirming a statistically significant increase in the cost of unscheduled repairs ($p<0.05$). Thus, emergency repairs impose a markedly higher financial burden on mining enterprises. A statistical summary of the restoration cost dataset is provided in

Table 3. Statistical summary of restoration cost data.

Statistic	Value (USD)
Number of observations	34
Minimum cost	567
Maximum cost	20,585
Mean cost	6812
Standard deviation	5944

.

Limitation: Due to confidentiality restrictions, detailed internal restoration cost records were not available. Therefore, the cost dataset was compiled from publicly available supplier price lists and cross-checked with maintenance engineers for plausibility. While representative for the purpose of the model, these values may exhibit variance compared to actual company-specific financial data.

Importantly, the restoration cost analysis is not intended as a stand-alone conclusion. Rather, the derived cost ranges serve as essential input parameters to the economic component of the optimization model (Section 2.4), where they define the scheduled repair cost $C_p$ and the corrective repair cost $E_{\mathrm{unp}}$. These values directly influence the total cost function $F\left(T\right)$ and enable a combined cost–reliability optimization of the repair interval.

The optimization analysis is applied to the post–first overhaul period. The manufacturer-recommended major overhaul interval of 15,000 machine hours was excluded from optimization because it represents an initial baseline that does not reflect the actual deterioration behavior observed during subsequent repair cycles. Thus, statistical modeling and optimization were performed only for repeated cycles (cycle 2 and onward), where most failures and cost variations were observed.

Overall, determining an economically and technically justified repair interval is essential for preventing premature wear, reducing restoration costs, and ensuring uninterrupted operation of mining equipment under intensive workloads.

2.3. Analysis of Statistical Data for Key Parameters

The validity of the mathematical model must be confirmed by demonstrating its suitability to predict the optimal repair interval and to assess the reliability of the components. To this end, statistical data was collected, including the number of machine hours operated since the last repair and the cumulative operating time of the components (

Table 4. Statistical data on scheduled repairs of brake assemblies.

No.	Repair Cycle	Machine Hours Since Last Repair	Cumulative Operating Time	Serial Number
1	1	6513	6513	BR16M-20
2	2	6602	28,478	BR16M-9
3	2	13,430	31,056	BR16M-4
4	2	13,430	31,056	BR16M-5
5	2	13,430	31,056	BR16M-6
6	2	15,672	37,548	BR16M-8
7	2	15,672	37,548	BR16M-10
8	3	15,672	46,728	BR16M-4
9	3	15,672	42,824	BR16M-2
10	1	15,774	15,774	BR16M-3
11	1	15,774	15,774	BR16M-15
12	1	15,774	15,774	BR16M-3
13	2	16,055	33,748	BR16M-11
14	2	16,055	33,748	BR16M-12
15	2	16,055	33,748	BR16M-13
16	1	16,909	16,909	BR16M-16
17	1	16,909	16,909	BR16M-17
18	1	16,909	16,909	BR16M-18
19	1	16,909	16,909	BR16M-19
20	2	17,110	34,019	BR16M-16
21	1	17,626	17,626	BR16M-4
22	1	17,626	17,626	BR16M-5
23	1	17,626	17,626	BR16M-2
24	1	17,626	17,626	BR16M-6
25	1	17,693	17,693	BR16M-11
26	1	17,693	17,693	BR16M-12
27	1	17,693	17,693	BR16M-14
28	1	17,693	17,693	BR16M-13
29	2	20,687	42,563	BR16M-7
30	1	21,876	21,876	BR16M-7
31	1	21,876	21,876	BR16M-8
32	1	21,876	21,876	BR16M-9
33	1	21,876	21,876	BR16M-10

and

Table 5. Statistical data on unscheduled repairs of brake assemblies.

No.	Repair Cycle	Machine Hours Since Last Repair	Cumulative Operating Time	Serial Number
1	2	3733	10,246	BR16M-20
2	1	5360	5360	BR16M-21
3	2	6348	22,122	BR16M-3
4	3	6348	40,096	BR16M-11
5	1	9172	9172	BR16M-1
6	2	9185	24,959	BR16M-15
7	2	9526	27,152	BR16M-2
8	2	9542	27,235	BR16M-14
9	3	10,052	41,108	BR16M-6
10	3	10,085	36,567	BR16M-3
11	2	10,708	26,482	BR16M-3
12	2	11,210	20,382	BR16M-1
13	3	13,974	42,452	BR16M-9
14	2	14,722	31,631	BR16M-19
15	2	15,326	32,235	BR16M-17
16	2	16,275	33,184	BR16M-16
17	1	17,491	17,491	BR16M-22

) [1]. These indicators represent key statistical parameters that allow data analysis, verification of model adequacy, and identification of the optimal repair interval.

The analysis of data from Table 4 and Table 5 shows that the brake assemblies BR16M-4, BR16M-5 and BR16M-6 demonstrated the highest reliability, as they underwent only scheduled maintenance and did not experience unscheduled failures. In total, 13 cases of unscheduled repairs were recorded, with machine hours at the time of failure ranging from 3733 to 17,491 machine hours. The highest susceptibility to unscheduled failures was observed in brake assemblies BR16M-3, BR16M-1, and BR16M-9, indicating the need for further analysis of the causes of failure and possible adjustments to the scheduled maintenance strategy [1].

Based on the presented data, a distribution chart was created showing machine hours by type of repair for the brake assemblies (Figure 9).

The analysis of the presented chart shows that a significant number of unscheduled repairs indicates that the equipment frequently fails before the scheduled maintenance interval. This suggests either inaccurate failure predictions or unaccounted-for operational factors affecting the reliability of the brake assemblies. At the same time, in some cases, scheduled repairs are performed at intervals that may be suboptimal—either too long, increasing the likelihood of unscheduled failures, or too short, leading to unjustified maintenance costs.

Thus, analysis of statistical information on machine hours and cumulative operating time of brake assemblies confirms the need for adjustments to the current maintenance system. Its key parameters require further statistical processing to refine the calculation of the optimal repair interval.

2.4. Weibull-Based Statistical Processing of Failure Data

To evaluate the reliability of the brake assemblies and to derive the parameters required for the optimization model, the collected failure-time data were statistically processed using probabilistic failure modeling. Several candidate lifetime distributions were preliminarily examined, including Weibull, lognormal, and exponential models. Among these, the Weibull distribution demonstrated the best linearity on probability plots, indicating the most consistent fit to the empirical data.

The Weibull distribution was therefore selected as the primary model for failure analysis. Its parameters—the shape parameter $\beta$ and the characteristic life $\eta$—were estimated using linear regression (Least Squares Estimation, LSE) applied to the linearized Weibull plot. This method provides stable parameter estimates for datasets of limited size, which is typical for mining equipment maintenance records.

Before fitting the model, failure-time data were preprocessed to remove incomplete observations and consolidate repeated entries. The resulting dataset contained 13 recorded unscheduled repairs (failures), with failure times ranging from 3733 to 17,491 machine hours. These values were used for constructing the Weibull plot.

Goodness-of-fit was evaluated visually using the Weibull probability plot. The plotted points did not exhibit systematic deviations from linearity, confirming that the Weibull model adequately represents the failure behavior of the brake assemblies. Based on the regression line, the estimated Weibull parameters are summarized in

Table 6. Estimated Weibull parameters for brake assembly failure cycles.

Repair Cycle	Shape Parameter $\mathit{\beta }$	Characteristic Life $\mathit{\eta }$ (Hours)
Cycle 2	2.73	11,904.6
Cycle 3	3.12	10,845.2
Cycle 4	2.95	12,331.8
Mean value	2.93	11,694.2

.

The obtained Weibull parameters ($\beta$ and $\eta$) were subsequently used for calculating the failure rate $\lambda \left(t\right)$, the reliability function $R\left(t\right)$, and for determining the reliability-constrained optimal repair interval. The selection of the Weibull distribution, supported by preliminary comparison and goodness-of-fit evaluation, ensures that the mathematical model is based on statistically consistent and empirically justified failure behavior.

2.5. Development of the Mathematical Model

To address the key research problem, a mathematical model was developed to calculate the optimal repair interval time for equipment components, based on economic efficiency and system reliability. The model is built on the Weibull distribution, which is widely used for equipment reliability analysis and equipment failure prediction [32]. The goal of the model is to determine the optimal repair interval time $T_{\mathrm{opt}}$, at which total maintenance costs are minimized while the reliability of the component remains within acceptable limits [33].

The model employs a multi-criteria aggregation method to combine two objectives: minimizing economic costs [34] and ensuring sufficient reliability [35]. These two criteria are inherently contradictory: shorter repair intervals reduce the likelihood of failure but increase maintenance costs, whereas longer intervals reduce costs but increase the risk of breakdowns. Therefore, the model applies criteria convolution methods to find a compromise solution [36]. The main techniques used in the model include the weighted sum method, the normalization of criteria, and the constraint method [37].

According to the weighted sum method, the model considers a total cost function (Equation (1)) [38] that combines the costs of scheduled and unscheduled repairs, allowing both criteria to be incorporated into a single objective function. Based on this approach, a comprehensive cost function for component repairs is formulated as the sum of scheduled and unscheduled repair costs [39]:

$$F\left(E\right)=E_p+E_{unp},$$

(1)

where:

$E_p$—cost of scheduled repairs;

$E_{unp}$—cost of unscheduled (emergency) repairs.

The cost of scheduled repairs is calculated as (Equation (2)):

$$E_p={\displaystyle \frac{C_p}{T}},$$

(2)

where:

$C_p$—cost of scheduled repair;

T—repair interval time.

The cost of unscheduled repairs depends on the failure rate and the cost per emergency repair (Equation (3)) [40]:

$$E_{unp}=C_{unp}·\lambda \left(T\right),$$

(3)

where:

$C_{unp}$—cost of an unscheduled repair;

$\lambda \left(T\right)$—failure intensity function.

The failure intensity $\lambda \left(T\right)$ is described by the Weibull distribution (Equation (4)) [41]:

$$\lambda \left(T\right)={\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{T}{\eta }}\right)}^{\beta -1},$$

(4)

where:

$\beta$—shape parameter;

$\eta$—scale parameter (characteristic life);

T—operating time of the component.

To determine the Weibull distribution parameters, the failure data were transformed into a linearized Weibull plot. The shape parameter $\beta$ and the characteristic life $\eta$ were estimated using the least-squares regression method applied to the linearized data. The goodness-of-fit value $R^2=0.92$ confirmed a high adherence of the empirical failure points to the Weibull model, ensuring the reliability of the estimated parameters ($\beta =2.735$ and $\eta =11904.6$ machine hours).

By substituting Equations (2) and (3) into Equation (1), the total cost function is obtained:

$$F\left(E\right)=E_p+E_{unp}={\displaystyle \frac{C_p}{T}}+C_{unp}·\lambda \left(T\right).$$

(5)

Next, substituting the expression for the failure intensity $\lambda \left(T\right)$ from Equation (4), we derive:

$$F\left(E\right)={\displaystyle \frac{C_p}{T}}+C_{unp}·{\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{T}{\eta }}\right)}^{\beta -1}.$$

(6)

The optimal repair interval is determined by minimizing the total cost function, $F\left(E\right)\to min$ [42]. The differentiation-based minimization method is applied to locate the extrema of the function [43,44]. Taking the first derivative of the cost function with respect to T and setting it to zero yields:

$$F^{\prime }\left(T\right)=-{\displaystyle \frac{C_p}{T^2}}+C_{unp}·{\displaystyle \frac{\beta (\beta -1)}{{\eta }^{\beta }}}{T}^{\beta -2}=0.$$

(7)

However, Equation (7) provides only the extremum point without specifying its type. To confirm that the obtained value corresponds to a minimum, the second derivative is analyzed [45]:

$$F^{″}\left(T\right)={\displaystyle \frac{2C_p}{T^3}}+C_{unp}·{\displaystyle \frac{\beta (\beta -1)(\beta -2)}{{\eta }^{\beta }}}{T}^{\beta -3}.$$

(8)

Since $C_p>0$, $T>0$, and $\beta >1$, it follows that $F^{\prime \prime }\left(T\right)>0$. Therefore, the obtained value $T_{\mathrm{opt}}$ corresponds to the minimum of the cost function.

From the first derivative condition $F^{\prime }\left(T\right)=0$, the optimal repair interval for the components is determined as:

$$T^{\beta }={\displaystyle \frac{C_p{\eta }^{\beta }}{C_{unp}\beta (\beta -1)}}.$$

(9)

Thus, the optimal minimum repair interval is given by the following expression:

$$T_{\mathrm{opt}}=\eta {\left({\displaystyle \frac{C_p}{C_{unp}\beta (\beta -1)}}\right)}^{1/\beta }.$$

(10)

After obtaining the analytical expression for the optimal interval $T_{\mathrm{opt}}$, it is necessary to justify the choice of the Weibull distribution for modeling failure behavior. Several candidate distributions were evaluated, including exponential, lognormal, and Weibull, following standard reliability engineering practice [46,47]. Among the tested distributions, the Weibull model demonstrated the highest linearity on the Weibull probability plot and produced the best regression fit ($R^2=0.92$). The lognormal distribution exhibited systematic curvature, indicating poorer representation of the empirical failure pattern. Therefore, the Weibull distribution was selected as the most appropriate model for capturing the degradation characteristics of the brake assemblies.

The estimation of the parameters $\beta$ and $\eta$ was performed using the least-squares regression (LSE) method applied to the linearized Weibull form, consistent with modern reliability analysis guidelines [47]. No systematic deviations were observed in the residuals, confirming the adequacy of the linear approximation and the validity of the parameter estimates.

A summary of the obtained reliability parameters is presented in

Table 7. Estimated Weibull parameters for brake assembly failure data.

Parameter	Value	Interpretation
Shape parameter $\beta$	2.735	Wear-out regime (increasing hazard)
Scale parameter $\eta$	11,904.6 h	Characteristic life of assemblies
Regression fit $R^2$	0.92	Strong linearity, good model adequacy

, which includes the shape parameter $\beta$, scale parameter $\eta$, and the regression fit $R^2$. The relatively high values of $\beta >2$ indicate an increasing hazard rate, meaning that the probability of failure accelerates with accumulated machine hours—a typical behavior for wear-dominated degradation processes.

To ensure methodological transparency, the full workflow applied in this study is outlined as follows. The workflow demonstrates the sequence:

$$\mathrm{Failure}\mathrm{data}\to \mathrm{Parameter}\mathrm{estimation}\to \mathrm{Cos}\mathrm{t}\text{–}\mathrm{reliability}\mathrm{model}\to \mathrm{Optimization}\mathrm{of}{T}_{\mathrm{opt}}.$$

This structure clarifies the logical transition between statistical analysis (Section 2.3), parameter estimation, formulation of Equations (1)–(10), and the derivation of the optimal interval.

The adopted modeling framework is consistent with contemporary reliability-centered maintenance practices and analytical approaches recommended in modern literature [47,48,49]. The resulting reliability-constrained interval $T_{\mathrm{opt}}$ therefore provides a justified balance between economic cost minimization and acceptable failure risk for mining brake assemblies.

The Equation obtained (10) allows us to determine the optimal value of the repair interval at which a balance is achieved between the costs of scheduled maintenance and the risks of unscheduled failures. According to Equation (10), the optimal minimum repair interval $T_{\mathrm{opt}}$ depends not only on the total costs of scheduled and unscheduled repairs but also on the reliability parameters of the equipment ($\beta$ and $\eta$) [50]. Therefore, in the model, the reliability of the components is incorporated through the Weibull law and the reliability function [32].

The reliability function $R\left(T\right)$ [33] describes the probability that a component will operate without failure for a specified period of time T [34].

$$R\left(T\right)=e^{-{\left({\displaystyle \frac{T}{\eta }}\right)}^{\beta }},$$

(11)

where:

$R\left(T\right)$—probability of failure-free operation over time T;

$\beta$—shape parameter (slope of the failure rate curve);

$\eta$—characteristic life (scale parameter).

This approach allows for consideration of not only economic factors, but also the actual behavior of components during operation, thus increasing the precision of the calculated optimal repair interval $T_{opt}$. Determining the optimal repair interval based on this reliability function helps reduce the risks of unplanned downtime and enhances the operational efficiency of the equipment.

To determine the optimal repair interval according to the reliability function (11) and to ensure that the probability of failure-free operation does not fall below a specified threshold $R_{min}$ [35], the following inequality is formulated [36]:

$$e^{-{\left({\displaystyle \frac{T}{\eta }}\right)}^{\beta }}\ge R_{min},$$

(12)

By solving this inequality with respect to T, the following expression is obtained, which makes it possible to determine the optimal minimum value of the repair interval $T_{opt}$ according to the reliability criterion:

$$T_{opt}\le \eta ·{\left(-ln{R}_{min}\right)}^{1/\beta }.$$

(13)

According to the constraint-based method, an additional condition (inequality (13)) is introduced to ensure a specified level of reliability. This condition requires that the probability of failure-free operation, $R\left(T\right)$, does not fall below the predefined threshold $R_{min}$ [37]. The model seeks to minimize total maintenance costs while meeting the reliability criterion.

Thus, the mathematical model developed enables the determination of the optimal repair interval for equipment components, considering both their reliability and economic efficiency. The optimal repair interval $T_{opt}$ is calculated in such a way as to minimize the total cost by balancing scheduled and unscheduled maintenance. Moreover, the reliability function $R\left(T\right)$ allows one to estimate the probability of failure-free operation over a given period, thereby making it possible to forecast potential failures and optimize the timing of scheduled maintenance [38].

In general, the proposed methodology facilitates the identification of an optimal trade-off between repair costs and system reliability. Its application under real operating conditions is expected to improve the reliability of components, reduce maintenance expenditures, and support more effective planning of repair activities.

3. Results

3.1. Results of Statistical Data Processing

Based on statistical data collected on machine hours and component runtime prior to failure, data processing was carried out using a well-established method based on the Weibull distribution [39]. The analysis included the following steps:

• calculation of key statistical parameters, including median rank, operating time logarithm, and double logarithm of reliability [40];
• construction of the Weibull plot by transforming failure data into a logarithmic scale [41];
• determining key parameters of the Weibull distribution, such as the shape parameter and characteristic life [42].

The failure events were classified in sequence and the median rank $F\left(t\right)$ [43], representing the probability of failure by a given time, was determined using standard statistical procedures [44]. The failure times were then transformed into logarithmic form to construct a Weibull plot [45], which allowed the estimation of the reliability parameters [50].

The main Weibull parameters—the shape parameter $\beta$ and the characteristic life $\eta$—were obtained using the least squares regression method [51], based on the linear relationship between the probability of failure and the operating time [52].

The resulting Weibull model demonstrated good agreement between empirical and theoretical data [53], confirming its applicability to assess the reliability of the equipment and further determine optimal maintenance intervals [54].

Statistical data on the failure time of the CAT 16M motor grader brake groups were used for the calculations (Table 5). The final results obtained according to the applied methodology are presented in

Table 8. Failure Data for Brake Groups of CAT 16M/16M3 Motor Graders.

Failure Machine Hours	Rank	Reverse Rank	Adjusted Rank	Median Rank	ln(t)
3733	1	13	1	0.0522	8.2249
5360	2	12	2	0.1269	8.5867
6348	3	11	3	0.2015	8.7559
6348	4	10	4	0.2761	8.7559
9172	5	9	5	0.3507	9.1239
9185	6	8	6	0.4254	9.1253
9526	7	7	7	0.5000	9.1618
9542	8	6	8	0.5746	9.1635
10,052	9	5	9	0.6493	9.2155
10,085	10	4	10	0.7239	9.2188
10,708	11	3	11	0.7985	9.2787
11,210	12	2	12	0.8731	9.3246
13,974	13	1	13	0.9478	9.5450
15,326	14	4	14	0.7874	9.5971
16,275	15	3	15	0.8448	9.6373
17,491	16	2	16	0.9023	9.6974
17,491	17	1	17	0.9598	9.7694

.

To construct the Weibull plot, the nonlinear dependence of failure probability was transformed into a straight-line relationship. To obtain this equation, the shape parameter $\beta$ and the characteristic life $\eta$ were determined [55].

The results of the intermediate calculations for the shape parameter $\beta$ are presented in

Table 9. Calculation of the Shape Parameter $\beta$.

ln(t) = (X)	ln(ln(1/(1 − F(t)))) = (Y)	(X − $\overline{\mathit{X}}$)	(Y − $\overline{\mathit{Y}}$)	(X − $\overline{\mathit{X}}$)(Y − $\overline{\mathit{Y}}$)	(X − $\overline{\mathit{X}}$)2
8.22496748	−3.192684658	−0.9621959	−2.652417224	2.552145	0.925820966
8.58671925	−2.274877577	−0.6570439	−1.890634686	1.242229974	0.431706677
8.75589508	−1.780091531	−0.5289333	−1.514013307	0.800812067	0.279770445
8.75589508	−1.430980590	−0.5641955	−1.265836586	0.714179343	0.318316595
9.12391064	−1.155601100	−0.2364796	−1.080873996	0.255604701	0.055922624
9.12532700	−0.924117873	−0.2532540	−0.932534922	0.236168233	0.064137607
9.16178018	−0.721080787	−0.2379054	−0.807209080	0.192039363	0.056598958
9.16345839	−0.537264880	−0.2578549	−0.696775816	0.179667065	0.066489155
9.21552690	−0.366512921	−0.2315719	−0.595701439	0.137947707	0.053625539
9.21880445	−0.204260615	−0.2540245	−0.499638181	0.126920362	0.064528470
9.27874640	−0.046711512	−0.2258357	−0.404543852	0.091360428	0.051001746
9.32456152	0.109754476	−0.2122828	−0.305869842	0.064930902	0.045063981
9.54495374	0.269192971	−0.0272710	−0.197409655	0.005383564	0.000743709
9.59709825	0.437052522	0.0194193	−0.069032034	−0.001340552	0.000377108
9.63730601	0.622305333	0.0644819	0.098962768	0.006381303	0.004157910
9.69738547	0.844082105	0.1460553	0.353727130	0.051663711	0.021332141
9.76944174	1.167250255	0.2911392	0.853758845	0.248562648	0.084762021
Mean Values: $\overline{X}=9.187$, $\overline{Y}=$ −$0.54$         Total Sum: $\sum (X-\overline{X})(Y-\overline{Y})=6.904$, $\sum {(X-\overline{X})}^2=2.524$

. The initial parameters for the values of X and Y were taken from Table 8, where X represents the natural logarithms of the failure times and Y represents the natural logarithms of the double logarithm of the inverse reliability function.

The shape parameter $\beta$ was then calculated using the following formula:

$$\beta ={\displaystyle \frac{\sum (X-\overline{X})(Y-\overline{Y})}{\sum {(X-\overline{X})}^2}}$$

(14)

Using the values from Table 9, the numerical result is:

$$\beta ={\displaystyle \frac{6.904}{2.524}}=2.735$$

The next step was to determine the characteristic life $\eta$. The results of calculating the characteristic life $\eta$ are presented in

Table 10. Calculation of the Characteristic Life $\eta$.

Step	Value
Mean value $\overline{Y}$ (ln(ln(1/(1 − F(t)))))	−0.5403
Mean value $\overline{X}$ (ln(t))	9.1872
Shape parameter $\beta$	2.7352
Formula: intercept $=\overline{Y}$ − $\beta ·\overline{X}$	−25.6691
Formula: $ln\left(\eta \right)$ = −$\mathrm{intercept}/\beta$	9.3847
Formula: $\eta =e^{ln\left(\eta \right)}$	11,904.67

.

According to calculations, the characteristic life $\eta$ was determined to be 11,904.6 machine hours.

After determining the values of $\beta$ and $\eta$, they were substituted in the linear regression equation:

$$Y=\beta X-\beta ln\left(\eta \right)$$

(15)

Substituting numerical values:

$$Y=2.735X-2.735·ln\left(11904.6\right)=2.735X-25.67$$

Based on the resulting equation, a graph was constructed comparing the statistically processed data with the data derived from the regression equation (Figure 10).

In Figure 10, the blue markers represent the statistical data obtained from the failure analysis of components, reflecting the operating time until failure [1]. The yellow straight line corresponds to the following regression equation:

$$Y=2.735X-25.67$$

In this equation, the coefficient $2.735$ represents the shape parameter $\beta$, which characterizes the nature of failures, while the value $25.67$ denotes the Y-axis intercept associated with the characteristic life $\eta$ of the components.

The plot clearly shows that the regression line fits the experimental points well. Most data points are close to the trend line, indicating that the regression equation accurately describes the relationship between the failure time logarithm and the median rank logarithm. This suggests that the selected model can be reliably used for predicting failure time. The slope of the regression line ($\beta =2.735$) demonstrates that as the logarithm of failure time increases, the median rank increases in accordance with a linear trend. The minimal deviation of the data points from the trend line indicates a low level of random error and confirms that the derived relationship is statistically robust [56].

Therefore, the regression equation obtained and its parameters can be effectively used for further analytical calculations and system reliability forecasting. Statistical analysis showed that the value of $\beta =2.735$ indicates that the main cause of failures is the wear of the components, as the shape parameter $\beta >1$ [57]. This reflects an increasing failure rate typical of aging and degradation processes. Under such conditions, the probability of component failure increases with operating time, emphasizing the importance of timely maintenance and repair.

The results obtained using the proposed methodology allowed the determination of key reliability parameters that are crucial for optimizing maintenance intervals. Analysis of $\beta$ and $\eta$ not only confirms the pattern of accumulation of wear, but also enables prediction of the service life of components, reduction of unexpected failures, and improved maintenance scheduling [58].

3.2. Summary of Reliability Parameters and Comparison of Interval Benchmarks

To summarize the results of Weibull processing, the estimated reliability parameters for the brake assemblies are presented in

Table 11. Summary of estimated Weibull reliability parameters.

Parameter	Value	Interpretation
Shape parameter $\beta$	2.735	Wear-out region (increasing hazard)
Scale parameter $\eta$ (h)	11,904.6	Characteristic life
Regression fit $R^2$	0.92	High model adequacy

. The regression coefficient $R^2=0.92$ confirms a strong linear fit on the Weibull plot, indicating that the model adequately represents the observed failure behaviour.

A comparative analysis was conducted to evaluate the optimal repair interval obtained from the model against existing maintenance benchmarks (

Table 12. Comparison of maintenance interval benchmarks.

Interval Type	Value (h)
Manufacturer recommended interval	6500
Current preventive maintenance schedule	5000–5500
Reliability-based optimum $T_{\mathrm{opt}}^{\left(R\right)}$	5129
Cost-based optimum $T_{\mathrm{opt}}^{\left(C\right)}$	5786

). The reliability-based optimum equals $T_{\mathrm{opt}}^{\left(R\right)}=5129$ machine hours, while the cost-based optimum equals $T_{\mathrm{opt}}^{\left(C\right)}=5786$ machine hours. Both values are below the manufacturer-recommended interval of 6500 h.

Considering the statistical uncertainty associated with the estimation of Weibull parameters, a confidence band of ±10–15% was applied. For the reliability optimum of 5129 h, this yields an interval range of approximately:

$$T_{\mathrm{opt}}^{\left(R\right)}\in [4350;5900]\mathrm{h}.$$

This confidence band partially overlaps with the current preventive maintenance schedule (5000–5500 h), confirming that the calculated interval is feasible for practical implementation and aligns with the operational maintenance strategy.

Thus, the obtained Weibull parameters ($\beta$, $\eta$) and the resulting optimal intervals provide a reliable basis for further optimization and for integrating both economic and reliability considerations in the maintenance decision-making process.

3.3. Determination of the Optimal Repair Interval Based on Statistical Data Processing

As a result of statistical data analysis, key distribution parameters were identified, including the shape parameter $\beta$ and the characteristic life $\eta$. These parameters are essential for optimal planning of equipment repair intervals. The values obtained allow us not only to predict the probability of failure, but also to make informed decisions about maintenance and repair strategies [59].

In practice, to improve equipment operational efficiency, a preventive maintenance strategy is commonly employed. This strategy is based on the analysis of two main types of costs: those associated with scheduled preventive maintenance and those incurred due to unexpected failures [60]. Based on these cost components, the cost per unit of operating time is determined using the following expression:

$${\displaystyle \frac{\mathrm{Cos}\mathrm{t}}{\mathrm{unit}\mathrm{time}}}=C_m+C_f·\lambda \left(t\right),$$

(16)

where:

$C_m$—cost of preventive maintenance;

$C_f$—cost of corrective maintenance (failure-induced);

$\lambda \left(t\right)$—failure rate function.

Optimizing the repair interval helps balance these expenditures by reducing the risk of critical breakdowns while minimizing both the financial and temporal costs of technical maintenance.

Table 13. Cost Analysis of Corrective and Preventive Maintenance Activities.

Type of Maintenance	Component	Material Cost ($)	Labor Cost ($/h)	Production Losses ($/h)	Total Cost ($)
Corrective Maintenance (Replacement of failed component	Service Brake	7000	133	12	8596
Preventive Maintenance	Service Brake	7000	–	12	7000

presents statistical data on the costs of performing corrective and preventive maintenance for the brake group components. This includes material costs, labor expenses, production losses, execution time, and total cost [1].

Analysis of the data presented in Table 13 demonstrates that the material costs for repair are identical in both cases, equating to $7000, and the duration of the repair work is 12 h. However, in corrective maintenance, additional production losses of $133 per hour are taken into account, increasing the total repair cost to approximately $8596. In contrast, such costs are absent during preventive maintenance, as the equipment is deliberately taken out of service in advance, preventing unplanned losses.

In industrial sectors, particularly mining, equipment downtime not only results in time loss but also causes significant revenue reductions due to halted production processes. The calculation of such losses must consider the critical constraints of the entire production chain. In the event of sudden failure, additional expenses arise from schedule disruptions, emergency repairs, and expedited delivery of spare parts. These factors not only increase financial costs, but also reduce overall production efficiency. Unplanned expenses also include emergency technician deployment, potential overtime pay, and fast-track logistics for spare parts. In addition, equipment failures can lead to cascading disruptions in adjacent operations, amplifying financial losses.

Consequently, preventive maintenance is more economically viable, as it allows for pre-scheduled service interventions and reduces unforeseen expenses. This is especially critical in industries where high downtime costs make unscheduled equipment failures extremely detrimental, such as the mining sector. Adopting a preventive maintenance strategy not only reduces the risk of sudden breakdowns, but also supports production continuity and operational predictability.

To determine the optimal time interval between maintenance activities,

Table 14. Cost Optimization Based on the Maintenance Interval for Brake Groups of the CAT 16M/16M3 Motor Grader.

Maintenance Time Interval (h)	Failure Rate $\mathit{\lambda }$	Cost per Unit Time ($)
500	$9.382\times {10}^{-7}$	14.01
1000	$3.123\times {10}^{-6}$	7.03
1500	$6.312\times {10}^{-6}$	4.72
2000	$1.03\times {10}^{-5}$	3.59
2500	$1.53\times {10}^{-5}$	2.93
3000	$2.10\times {10}^{-5}$	2.51
3500	$2.74\times {10}^{-5}$	2.24
4000	$3.46\times {10}^{-5}$	2.05
4500	$4.24\times {10}^{-5}$	1.92
5000	$5.099\times {10}^{-5}$	1.84
5500	$6.01\times {10}^{-5}$	1.79
6000	$6.99\times {10}^{-5}$	1.77
6500	$8.03\times {10}^{-5}$	1.77
7000	$9.14\times {10}^{-5}$	1.79
7500	$1.03061\times {10}^{-4}$	1.82
8000	$1.15273\times {10}^{-4}$	1.87
8500	$1.28060\times {10}^{-4}$	1.92
9000	$1.44113\times {10}^{-4}$	1.99
9500	$1.55322\times {10}^{-4}$	2.07
10,000	$1.69781\times {10}^{-4}$	2.16
10,500	$1.84780\times {10}^{-4}$	2.26
11,000	$2.00315\times {10}^{-4}$	2.36
11,500	$2.16377\times {10}^{-4}$	2.47
12,000	$2.32962\times {10}^{-4}$	2.59
12,500	$2.50062\times {10}^{-4}$	2.71
13,000	$2.66773\times {10}^{-4}$	2.84
13,500	$2.85789\times {10}^{-4}$	2.98
14,000	$3.04405\times {10}^{-4}$	3.12
14,500	$3.23516\times {10}^{-4}$	3.26
15,000	$3.43119\times {10}^{-4}$	3.42
15,500	$3.63197\times {10}^{-4}$	3.57
16,000	$3.83778\times {10}^{-4}$	3.74
16,500	$4.04827\times {10}^{-4}$	3.90
17,000	$4.26350\times {10}^{-4}$	4.08

was constructed. For each repair interval, the corresponding cost per unit of time was calculated based on failure intensity indicators.

According to the table presented above, the interval time was set at 500 machine hours, as this corresponds to the standard maintenance schedule and allows the replacement of brake groups to be combined with scheduled maintenance activities. The failure rate was calculated using the well-known Weibull distribution method (4), based on the shape parameter $\beta$ and the characteristic life $\eta$. The cost per unit of time was determined using the formula (16). To clearly identify the optimal repair interval, a graph (Figure 11) was plotted, illustrating the relationship between the time intervals between repairs and the total cost per unit of time.

According to the graph in Figure 11, frequent repairs in the early stages (every 500–1000 machine hours) result in elevated costs due to the high number of maintenance activities per year. In addition, frequent repairs stoppages negatively affect the technical readiness coefficient (TRC) of the equipment.

In contrast, excessively long intervals between repairs (every 10,000–14,000 machine hours or more) also lead to increased expenditures due to the increased risk of unexpected failures occurring before the scheduled maintenance period.

Thus, based on the analysis of the data in Table 14 and the trend shown in Figure 11, the optimal interval between brake group repairs is 6500 machines hours. At this interval, the cost per unit of time is minimized, while the probability of unexpected failures remains within acceptable limits.

Compared to the existing preventive maintenance strategy, which is mainly based on fixed-interval scheduling, the proposed model enables a balanced integration of cost and reliability considerations, allowing repair intervals to be adjusted according to the actual degradation behavior of the component. This prevents situations where equipment continues to operate until a predetermined interval is reached, thus reducing the risk of unexpected failures and unplanned downtimes.

Therefore, the model developed in this study demonstrates the potential for more accurate determination of the optimal repair interval, as it incorporates the actual condition of the components based on the reliability function. This improves the reliability of repair scheduling forecasts and reduces the likelihood of unexpected equipment failures.

3.4. Validation of the Effectiveness of the Developed Model

To assess the reliability of the proposed mathematical model (as described in Section 2.4), a calculation of the optimal repair interval for components was performed using the distribution parameters obtained through statistical data analysis.

The model utilized the previously determined parameters: the shape parameter $\beta =2.735$ and the characteristic life $\eta =11904.6$ machine hours. Based on the operational data collected, the costs associated with planned and unplanned maintenance were identified as $C_p=\$5346.6$ for preventive repair and $C_{unp}=\$8344.3$ for corrective repair.

The calculation of the optimal repair interval was carried out using Equation (10) of the model, which integrates economic and reliability indicators to determine the most cost-effective time for maintenance actions:

$$T_{opt}=\eta {\left({\displaystyle \frac{C_p}{C_{unp}·\beta ·(\beta -1)}}\right)}^{{\displaystyle \frac{1}{\beta }}}$$

(17)

$$T_{opt}=11904.6{\left({\displaystyle \frac{5346.6}{8344.3·2.735·(2.735-1)}}\right)}^{{\displaystyle \frac{1}{2.735}}}=5786$$

As a result of the calculation, the optimal repair interval was determined to be 5786 machine hours.

Subsequently, the optimal repair time was also calculated using the reliability function $R\left(T\right)$ based on Equation (13) of the proposed mathematical model. The minimum reliability requirement for the brake group components was set at $R_{min}=0.9$. This threshold was selected in accordance with commonly accepted engineering practice for critical mining equipment components, where a minimum reliability level of 90% is typically required to prevent high-consequence failures. Similar reliability thresholds are widely used in reliability engineering literature [61].

$$T\le \eta ·{(-ln{R}_{min})}^{1/\beta }$$

(18)

$$T\le 11904.6·{(-ln0.9)}^{1/2.735}=5129\mathrm{machine}\mathrm{hours}$$

According to the reliability criterion, the optimal time for performing repairs was calculated to be 5129 machine hours.

Thus, the optimal repair time for equipment based on the cost criterion ($T_{opt}^{\left(cost\right)}=5786$ machine hours) and based on the reliability criterion ($T_{opt}^{\left(reliability\right)}=5129$ machine hours) shows close values. These results also align closely with the findings obtained under the existing preventive maintenance strategy, where the optimal repair interval was estimated at approximately 6500 h.

However, given the critical importance of reliability indicators in ensuring uninterrupted operation of the equipment, it is advisable to prioritize the value $T_{opt}=5129$ machine hours. This interval ensures a minimized risk of failures and unplanned downtimes, which is particularly vital for maintaining stable operations and preventing hazardous incidents.

It should be noted that, although the mathematically optimal repair interval equals 5129 machine hours, its direct implementation in practice requires alignment with the nearest feasible maintenance milestone. In real operating conditions, repair scheduling must consider the planned service cycle of the haul trucks and the availability of maintenance resources. Therefore, in practical application, the repair interval would typically be rounded to the closest service window (e.g., 5000 or 5500 h). The obtained value of 5129 h thus serves as a quantitative guideline for decision-making rather than an exact operational requirement.

Although the cost criterion suggests a slightly longer optimal interval (5786 machine hours), extending the repair interval beyond the reliability-based value can increase the risk of equipment failure, ultimately compensating for any potential cost savings. Adherence to the reliability criterion supports more stable and safer equipment operation, reducing the likelihood of unexpected failures and their associated unplanned expenses—an approach that may prove to be more economically justified in the long term.

To provide a clearer interpretation of the values obtained, a schematic scale of the candidate repair intervals is shown in Figure 12. The reliability-based optimum ($T_{\mathrm{opt}}^{\mathrm{rel}}\approx 5129$ machine hours), the cost-based optimum ($T_{\mathrm{opt}}^{\mathrm{cost}}=5786$ machine hours) and the current preventive maintenance interval ($T_{\mathrm{PM}}\approx 6500$ machine hours) are plotted on a single axis. This graphical representation illustrates how the existing strategy operates at a significantly longer interval than the reliability-based optimum, while the cost-based optimum lies in between, providing a convenient visual comparison of the alternative maintenance policies.

4. Discussion

The results of this study confirm the findings of previous research aimed at optimizing maintenance and repair strategies for mining equipment. Current approaches rely on statistical reliability analysis, mathematical modeling, condition monitoring, and optimization algorithms.

One major research direction involves equipment condition monitoring using stress-, strain-, and vibration-based indicators. Such methods allow tracking material fatigue and predicting failures based on the actual operating condition of components [6]. Although this improves maintenance planning and reduces the risk of sudden breakdowns, it requires complex sensor networks, continuous data collection, and extensive processing. In contrast, the present study focuses on failure-statistics-based analytical modeling, which is more practical under open-pit mining conditions where continuous monitoring is difficult to maintain.

Another approach relies on production data and event logs for preventive maintenance. Data-driven frameworks enable forecasting operational cycles, adjusting inspection intervals, and reducing downtime-related losses [7]. However, these methods require well-developed digital infrastructure and standardized datasets, limiting their applicability in many mining enterprises. The approach developed in this study provides similar benefits but avoids heavy digitalization requirements by utilizing the analytical properties of the Weibull distribution.

Recent research has also explored process-analysis methods combined with simulation models to optimize maintenance workflows [8]. Such studies demonstrate that process mining helps identify bottlenecks, quantify downtime losses, and improve coordination between maintenance operations. Although these methods offer valuable insights, they require high-quality operational data and advanced digital ecosystems. The current work achieves comparable objectives through a simpler analytical model based on statistical reliability.

Integrated models combining reliability, availability, maintainability, and cost (RAMC) have also been introduced [9]. These frameworks provide a comprehensive assessment of equipment performance but require extensive datasets and significant computational resources. The present study follows the same principle of balancing reliability and cost but proposes a model that is easier to implement in real-world mining environments.

Other studies have incorporated inspection data directly into maintenance decision-making, demonstrating that integrating inspection results can reduce the number of failures and total repair costs [10]. The current study achieves similar cost reductions through analytical optimization of repair intervals without relying on inspection data, making it suitable for environments with irregular inspection schedules.

Optimization methods using metaheuristic algorithms have also been proposed, enabling the identification of critical subsystems and optimal service intervals [11]. However, such approaches require substantial computational power and accurate reliability data. In comparison, the analytical model presented here provides a less resource-intensive alternative.

Additional research has focused on the development of condition-based monitoring systems for underground mobile machinery, where vibration analysis and sensor integration allow early detection of failures [12]. Although effective, these systems require advanced sensor infrastructure and robust data processing, making them costly to deploy broadly. The proposed model avoids these challenges by basing its predictions solely on historical failure statistics.

Importantly, the restoration cost analysis and reliability modeling together reveal a clear cost–reliability trade-off. Short repair intervals reduce the risk of unscheduled failures but lead to more frequent scheduled maintenance and higher cumulative costs. In contrast, extending the interval decreases scheduled repair frequency but significantly increases the probability of high-cost corrective failures. The optimal interval $T_{\mathrm{opt}}=5129$ h therefore represents a balanced compromise between economic efficiency and acceptable failure probability.

The influence of the Weibull shape parameter $\beta$ on the optimal interval is also critical. Higher values of $\beta$ correspond to an accelerated hazard rate, meaning components become increasingly failure-prone as operating hours accumulate; in such cases, the optimal interval shifts toward shorter values. This aligns with the observed value $\beta =2.735$, indicating a wear-out dominated failure regime and justifying the need for timely preventive actions.

Operational loading must also be considered when interpreting the optimal interval, as varying haul cycles, dust concentration, temperature gradients, and lubricant deterioration can significantly accelerate wear processes [62], shifting the effective hazard rate relative to the statistical model. Future studies should incorporate such operational factors or develop sensitivity analyses to evaluate how deviations in loading patterns affect the estimated interval.

To support practical implementation, a comparative graphical scale was constructed (Figure 12), illustrating the manufacturer-recommended interval ($\sim 6500$ h), the reliability-constrained interval (5129 h), and the cost-optimal interval (5786 h). This visualization helps practitioners understand how different optimization criteria influence maintenance timing and assists in aligning repair schedules with operational constraints.

Although the method was demonstrated for brake groups of CAT 16M/16M3 motor graders, the approach is applicable to other critical components such as hydraulic cylinders, transmission assemblies, and wear-intensive elements including bucket teeth [63]. Extending the method would require recalibrating the Weibull shape parameter $\beta$ to account for component-specific failure mechanisms and operating conditions.

Overall, this study supports the widely recognized conclusion that selecting appropriate maintenance and repair intervals reduces total costs and improves equipment reliability. The distinguishing feature of the proposed approach is that, unlike complex methods based on simulation modeling, process mining, or RAMC frameworks, it introduces a simpler analytical model rooted in the Weibull distribution. This model preserves predictive accuracy while being easier to implement, making it especially valuable for mining enterprises operating in regions with developing digital infrastructure.

5. Conclusions

This study developed a reliability–cost optimization model for determining the optimal repair interval of mining equipment brake assemblies. The results confirmed the main hypothesis: integrating Weibull-based reliability modeling with an economic cost function makes it possible to reduce unplanned repair expenses, minimize downtime, and maintain acceptable equipment reliability.

The analysis demonstrated that unscheduled repairs are, on average, 56% more expensive than scheduled repairs. This difference was statistically validated using a two-sample t-test ($p<0.05$), confirming that the higher cost of emergency repairs is not due to random variation. Based on the estimated Weibull parameters ($\beta =2.735$, $\eta =11904.6$ h), the optimal reliability-constrained repair interval was calculated as 5129 machine hours. This value reflects a balanced compromise between increasing failure risk and rising scheduled maintenance costs.

From a practical perspective, the proposed methodology provides a transparent and computationally simple decision-making tool compared to digital-twin systems, process mining approaches, or RAMC frameworks. Because the model relies solely on historical failure data and analytically derived equations, it remains applicable in real mining environments where continuous sensor-based monitoring is difficult to implement.

However, several limitations must be acknowledged. First, the restoration cost data were obtained from publicly available supplier price lists due to confidentiality restrictions, which introduces uncertainty related to market price variability and exchange rate fluctuations. Second, the sample size of recorded failures is relatively small, limiting the statistical robustness of Weibull parameter estimation. Third, the model assumes constant operating conditions and does not incorporate environmental or load-related variability (temperature, dust concentration, duty cycles), which may influence the hazard rate. Finally, although the Weibull distribution showed the best empirical fit, a full goodness-of-fit assessment and sensitivity analysis of parameter uncertainty were beyond the scope of this study.

Future research should address these limitations by: (1) Incorporating real-time monitoring signals—such as vibration, temperature, or lubricant degradation—to dynamically update the failure model; (2) Performing a detailed sensitivity analysis of $\beta$, $\eta$, and cost parameters to quantify their influence on the optimal interval $T_{\mathrm{opt}}$; (3) Expanding the dataset with multi-year failure records from several mine sites; and (4) Testing the model on additional components, including transmission units, hydraulic systems, and other wear-intensive assemblies.

Overall, the developed model provides a rigorous yet practical approach to maintenance interval optimization. By combining reliability constraints with economic considerations, it offers mining companies a structured method for reducing corrective repair costs, improving equipment availability, and supporting more efficient long-term maintenance planning.