Equipment failures and associated maintenance have an impact on the profitability of mines. Implementing maintenance at suitable time intervals can save money and improve the reliability and maintainability of mining equipment. This paper discusses aspects of maintainability prediction for mining machinery. For this purpose, a software tool, called GenRel, is developed. In GenRel, it is assumed that failures of mining equipment caused by an array of factors follow the biological evolution process. GenRel then simulates the failure occurrences during a time period of interest using Genetic Algorithms (GAs) coupled with a statistical methodology. Two case studies on maintainability analysis and prediction of a mine’s hoist system in two different time intervals, three months and six months are discussed. The data are collected from a typical underground mine in the Sudbury area in Ontario, Canada. In each case study, a statistical test is carried out to examine the similarity between the predicted data set and the real-life data set in the same time period. The objectives include an assessment of the applicability of GenRel using real-life data and an investigation of the relationship between data size and prediction results. Discrete and continuous probability distribution functions are applied to the input data.
To maximize profit;
To reduce the cost of poor reliability;
To reduce unplanned maintenance costs and frequency;
To provide more accurate short-term forecasts for equipment operating hours;
To overcome challenges imposed by global competition;
To take advantage of lessons learned from other industrial sectors such as aerospace, defense, and nuclear power generation [
Maintainability is defined as a characteristic of design and installation expressed as a probability that an item will be retained in or restored to specified conditions within a given period of time, when maintenance action is performed in accordance with prescribed procedures and resources [
A system with better maintainability would inherently provide the benefit of lower maintenance costs, less time to recover with lower breakdown frequency, less complexity of maintenance tasks and relatively reduced man-hours [
or, the maintainability function is defined as:
where
Genetic Algorithms Initialization
Generate initial data list A from random numbers
Fitness evaluation
Find suitable fitness function for this data list
Selection
Select a set of promising candidates B from the data list
Crossover
Apply crossover to the candidate set B, and get an offspring set C
Mutation
Some mutations happen in the offspring set C, then obtain the new offspring set C'
Replacement
Use the new offspring set C to replace initial data list A
Termination
If the termination criteria are not met, go to “Fitness evaluation”. In this thesis, termination criteria are the convergence limit and the maximum number of iterations.
Research on GAs has a wide spectra of applications from computer science [
Reliability and maintainability models with or without co-variances are based on the use of rigorous and complicated statistical techniques which include, for instance, theoretical probability distribution fitting, trend and serial correlation tests, and require assumptions of homogeneous or non-homogeneous Poisson process or assumptions of proportionality of the hazard rate. The assumptions and statistical constraints of probabilistic reliability and maintainability models limit the ability of these models to accurately represent and fit all real life mining conditions [
GenRel is a computer model developed in MS-Excel using Visual Basic for reliability and maintainability assessment of mining equipment based on Genetic Algorithms (GAs).
In a typical genetic algorithm, variables of interest are coded. Afterwards, the processes of mate selection, crossover, and mutation are repeated until the fitness function yields desired values [
In GenRel, the user can define the maximum number of iterations, the convergence limit, and the probability of mutation, see
The raw data used in GenRel is derived from historical records in terms of Time Between Failures (TBF) in the case of reliability predictions and in terms of Time To Repair (TTR) in the case of maintainability studies. The distribution fitting process in GenRel selects the best fitted probability distribution function
where
and
where
1From Equation (5), an inverse of the function can be expressed as
ted probability distribution functions for the Raw Evaluation Data set and for the Raw Input Data set, respectively. In order to generate new data from the Raw Input Data, we use the Inverse Transform Technique, or ITT [
buted random variable R ~ U (0,1). Then, six sets of generated data can be yielded by
The generated data is then used for mate selection and crossover in which a random number determines the positions and total number of crossovers, yielding offspring data. Afterwards, mutation is performed at a specified rate defined by the user as mutation probability. Six sets of new offspring data follow the respective best fitted Exponential probability distribution functions, denoted by parameter pairs (
is designed to measure the fitness of each individual offspring data set, for detailed discussion of this procedure, see [
If the smallest value of the six fitness function values is not greater than the user-defined convergence limit, then the iterative process is terminated and GenRel is considered applicable for prediction of future data; otherwise another iteration will be implemented as long as the pre-set maximum number of iterations is not exceeded and the smallest fitness function value is within a user defined convergence limit. The convergence limit is the upper limit of deviation between the probability distribution function of the generated data set and the probability distribution function of the Raw Evaluation Data set.
Overall, after the above described algorithmic process is successful, GenRel can be applied to predict future data. Otherwise, GenRel is considered not acceptable for prediction of future data based on the raw data under study. For details of the algorithmic procedure applied in GenRel, see [
There are four main procedures in GenRel.
Input data of GenRel can be either Time Between Failures (TBF) (for reliability assessment) or Time To Repair (TTR) data (for maintainability predictions), field data from mine sites must be entered as TBF or TTR data. For discrete distribution fitting, the requirement of input data is to be integer, thus TBF or TTR data must be transformed to integer to meet the statistical fitting requirements.
Prior to statistical analysis and probability distribution fitting, the data should be tested for trends and serial correlations. The purpose of these tests is to verify the assumption that the data is Independent and Identical Distribution (IID), [
If there is a specific probability distribution that can fit the Raw Input Data, then we can use the inverse transform technique to generate new data sets.
Convergence criteria include the convergence limit, the probability of mutation, and the maximum number of iterations. If the smallest fitness function value falls within the convergence limit, then the validation process of convergence is considered successful. Otherwise, we run the crossover and mutation processes, iterating the initially generated data sets, achieving an acceptable value and not exceeding the maximum number of iteration.
After the successful completion of the verification process for the input data, GenRel then can be applied to predict future data. GenRel is run to generate the initial offspring data set of the same size as the Raw Input Data set with the parameters values obtained from the Raw Input Data distribution fitting process. In GenRel, six data sets are considered sufficient. In these six data sets, selection, cross-over, and mutation are performed until either of the following two criteria is satisfied: (1) the fitness value of the best offspring data set falls within the user-defined convergence limit; (2) total iterations run in GenRel reaches the user-defined maximum number of iterations. Upon completion of the prediction, the t-test is used to examine the statistical similarity of the predicted data set with the Raw Evaluation Data set [
Increases in depth of underground mines, requirements of increased productivity, expectations of reduction in energy consumption, together with better safety requirements provoke challenges to mine extraction systems. In deep underground mines, hoist systems play a strategic role to turn mineral resources into profits.
Data was gathered from a typical underground mine (named for simplicity in this paper as Mine A) in the Sudbury mining area in Ontario, Canada. Since the case studies are related to the maintainability characteristics of hoists, Time To Repair (TTR) data is entered.
In this paper, we consider the case study with prediction of TTR data for the period from April 1st to June 30th 2007 based on historical data from January 1st to March 31st 2007. Continuous and discrete distribution fitting will be tested based on the input data. With preliminary manipulations,
Even though the data collection period is not as recent, this has no impact on the development work to apply GenRel and examine the proposed methodology.
Type of TTR | TTR (Hours) | TTR Frequency | Percent of Total TTR |
---|---|---|---|
MP | 320.25 | 65 | 45.57% |
OI | 294.25 | 36 | 41.87% |
MM | 21.75 | 14 | 3.09% |
MO | 16.5 | 7 | 2.35% |
ME | 50 | 26 | 7.11% |
Total | 702.75 | 148 | 100.00% |
Before GenRel is run, the validity of the Independent and Identical Distribution (IID) assumption for the data sets should be examined. Graphical tests show that the data set under study is independent and identically distributed. Then, the verification process aims to provide a prescreening result of GenRel’s applicability to the data set under study. In the case study, TTR data for the period of January 1st to March 31st 2007 is the data set of interest, which is to be used as Raw Input Data set to predict repair data for the period of April 1st to June 30th 2007, if GenRel is to be found applicable through the verification process.
As described in Chapter 3, TTR data for the period of January 1st to March 31st 2007 is divided chronologically into two sets: (Raw Input Data Set and Raw Evaluation Data Set) with equal number of data entries, 74.
In the verification stage, continuous and discrete probability distribution fitting will be used as fitting distribution functions to find the best fit.
For discrete distribution function, the best fitting distribution function, calculated by @Risk® [
In Microsoft® Excel® with Visual Basic, there isn’t a built-in Poisson inverse function that can directly be called to generate random variables from the best fitted normal distribution with location parameter Lambda. A Poisson inverse function has been programmed in GenRel to complete this procedure, see [
Next, the initial population of six sets of generated data starts to cross over and mutate at a rate defined by the user (e.g. mutation probability of 0.05). Elite individuals which possess smaller cost function values have better chance to survive and to be selected to reproduce offspring through selection, cross-over, and mutation. These genetic operations iterate until either the maximum number of iterations, (which is set to 15) or the convergence limit, (which is set to 0.05), has been reached. In this case, after one iteration, the fitness function value falls within the user-defined convergence limit, as shown in
Therefore, it is concluded that GenRel is applicable to analyze the data set from January 1st to March 31st 2007.
For continuous probability distribution function, the best fitting distribution function, calculated by @Risk® [
In the case of discrete probability distribution fitting, after one iteration, GenRel returns a set of TTR data as the final prediction of failures for the hoist system at Mine A during the time period from April 1st to June 30th, 2007. Results from @Risk® show that the Poisson probability distribution best fits the predicted data set. The parameter of the Poisson probability distribution is Lambda = 4.5203.
Iteration | Sum of Raw Evaluation Data Parameters | Upper Evaluation Limit Based on Convergence Limit of GA | Lower Evaluation Limit Based on Convergence Limit of GA | Parameter Deviation of Generated Data | Accepted |
---|---|---|---|---|---|
Convergence? | |||||
1 | 5.0135 | 0.250675 | 0 | 0.027 | yes |
Iteration | Sum of Raw Evaluation Data Parameters | Upper Evaluation Limit Based on Convergence Limit of GA | Lower Evaluation Limit Based on Convergence Limit of GA | Parameter Deviation of Generated Data | Accepted |
---|---|---|---|---|---|
Convergence? | |||||
1 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
2 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
3 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
4 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
5 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
6 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
7 | 908.09027 | 45.4045135 | 0 | 332.4346 | no |
8 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
9 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
10 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
11 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
12 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
13 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
14 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
15 | 908.09027 | 45.4045135 | 0 | 169.1355 | no |
As in the case study of a three-month period, a case study is considered: for the period from May 1st to October 31st, 2007 based on historical data from November 1st, 2006 to April 30th, 2007.
As mentioned earlier, similar graphical testing processes are implemented, and show that the data sets under study are independent and identically distributed.
Type of Failure | TTR (Hours) | TTR Frequency | Percent of TTR |
---|---|---|---|
MP | 679 | 151 | 43.37% |
OI | 551.25 | 72 | 35.21% |
OP | 159.25 | 78 | 10.17% |
MM | 58.25 | 31 | 3.72% |
ME | 63.5 | 39 | 4.06% |
MO | 54.5 | 25 | 3.48% |
Total | 1565.75 | 396 | 100.00% |
Iteration | Sum of Raw Evaluation Data Parameters | Upper Evaluation Limit Based on Convergence Limit of GA | Lower Evaluation Limit Based on Convergence Limit of GA | Parameter Deviation of Generated Data | Accepted |
---|---|---|---|---|---|
Convergence? | |||||
1 | 3.5 | 0.175 | 0 | 0.0606 | yes |
Iteration | Sum of Raw Evaluation Data Parameters | Upper Evaluation Limit Based on Convergence Limit of GA | Lower Evaluation Limit Based on Convergence Limit of GA | Parameter Deviation of Generated Data | Accepted |
---|---|---|---|---|---|
Convergence? | |||||
1 | 3.71471 | 0.1857355 | 0 | 0.0501 | yes |
In the case of discrete probability distribution fitting, with one iteration, GenRel returns a set of TTR data as the final prediction of failures of the hoist system at Mine A during the time period from May 1st to October 31st, 2007. Results from @Risk® show that the Poisson probability distribution is the best fit. The parameter of the Poisson probability distribution is Lambda = 3.7348.
Thus, the maintainability function is shown as follows, (for details, see [
Following table presents maintainability values based on the TTR data:table_table_tableλ = 3.7348.
In the case of continuous probability distribution fitting, after one iteration, GenRel returns a set of TTR data as the final prediction of failures of the hoist system at Mine A during the time period from May 1st to October 31st, 2007. Results from @Risk® show that the Lognormal probability distribution fits the predicted data set best. Parameters of Lognormal probability distribution are μ = 4.6514, σ = 14.048 and shift = 0.18658.
Thus, the maintainability function is shown as follows, (for details, see [
with
where,
θ: is the shift;
σ: is the mean of the associated Normal distribution;
μ: is the standard deviation of the associated Normal distribution.
Following are some maintainability values based on the TTR data:
μ = 4.6514, σ = 14.048 and θ = 0.18658.
The case studies discussed in this paper indicate the potential of a genetic algorithms based on software, called GenRel, to predict maintenance characteristics of a mine’s hoist system, expressed in terms of maintainability. A case study at time intervals of three months is successful with discrete probability distribution fitting but not with continuous probability fitting. A case study at time intervals of six months shows acceptable predictions at a given level of confidence, 5%. However, the results with discrete distribution fitting show less significance difference than the results with continuous distribution fitting. Further investigation is required to examine the reason behind these unsuccessful predictions. Research should be carried out to examine whether the reasons are time based or dependent on the failure characteristics of the hoist system.
The authors wish to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for the funding support to this research.
Chao Xu,Nick Vayenas, (2015) Maintainability Analysis Software of Mine’s Hoist System Based on Genetic Algorithms for Data Collection Periods of Three and Six Months. Open Access Library Journal,02,1-13. doi: 10.4236/oalib.1102022