Application of a Selected Pseudorandom Number Generator for the Reliability of Farm Tractors

Abstract

Knowledge of the use-to-failure periods of process equipment, including agricultural vehicles, is essential for the determination of their durability and reliability. Obtaining any empirical data on this issue is difficult and sometimes impossible. Experimental studies are costly and time-consuming. Manufacturers are usually reluctant to share such data, claiming that the information is classified for the sake of their companies. The purpose of this study was to compare empirical data with data generated using adequate statistical tools. The newly generated and very similar in value pseudorandom numbers were obtained by simulations using the Monte Carlo, Latin hypercube sampling and Iman-Conover methods. Reliability function graphs obtained from the generated time-series (use-to-failure periods) with matching Weibull distribution had very similar shape and scale parameters. They were are also comparable to parameters from experimental data extracted from a Polish Zetor agricultural tractor service station. The validation of the applied methods was limited as it was carried out only on the basis of the available data. Analysis of line graphs of cumulative deviations of the values of use-to-failure periods (times-to-fail) generated against empirical times-to-fail indicated that the best method in the studied case was the Monte Carlo method.

1. Introduction

Agricultural equipment is a specific group due to its demanding working environment and specific usage [1]. Agricultural machines are characterized by great diversity, as they are operated in particularly harsh conditions (they are exposed to aggressive media, such as manure, fertilizers, pesticides or abrasion and cohesion; their effectiveness in the field is determined, for example, by the adhesion of the surfaces of tools working in the soil). They work mainly outdoors (rarely in farm buildings), sometimes on slopes, and their use is seasonal. Agricultural machinery is also characterized by low utilization of their potential working time, while requiring expensive storage, which is logistically difficult. Long operating cycles require regular maintenance according to manufacturer guidelines. Users require that such machines have high reliability, that technological processes are not disrupted by service interruptions, and that they guarantee low future value loss.

The operation of agricultural appliances and vehicles is a continuous process, which is discrete in states.

Research fields dealing with the reliability of farm machinery including tractors [2,3,4,5] as well as other appliances are addressed more frequently in the scientific literature [6,7].

Reliability is a measurable and quantitative parameter, which represents the probability that failure will not occur within the assumed period of time.

Various techniques are used in order to obtain demanded data and publish reliability rankings [8]. The most popular are graphical methods [9,10,11,12], fault tree analysis [13,14,15,16,17], probability distributions and models [18], as well as applications of machine learning and reliability estimations [8]. The multistate approach to the reliability of agricultural devices can be based on Markov chains, which allow for intermediate states such as a state of partial operability or a state of incomplete task operability [19,20,21]. The reliability of farm tractors can also be estimated with the Bernoulli competing risks or Kaplan-Meier methods [3,22].

The classical theory of reliability has developed a wide range of functional and numerical characteristics, which require a priori data to be defined.

Due to the long life of agricultural machinery including tractors, it is difficult to determine the actual times-to-failure. By definition, these are renewable objects that can be repaired more than once. That is why, for example, agricultural tractors tend to be used longer than predicted by various methodologies for their operating time. According to the IBMER methodology, the operation time of farm tractors in Poland is assumed [23] to be 20 years. On the other hand, according to Agroscope—the Swiss Confederation’s center of excellence for agricultural research—the impairment period of a farm tractor is 15 years [24].

A full service history enables an a posteriori quantification of the reliability of agricultural machinery and tractors with dedicated methods and techniques based on assumptions of the classical theory of reliability. It distinguishes only two states describing the condition of a technological device: able or unable to perform the tasks for which it was designed [25,26].

Due to the fact that manufacturers consider data on the failure rate of agricultural equipment to be sensitive, it is difficult to access them. Without actual, credible and reliable data, it is impossible to create reliability rankings, which are so important for agricultural practice.

In order to overcome the problem of a posteriori data availability, data generation procedures using a number of generators are applied, and to check their usability, their quality is compared to incomplete data.

The software-generated numbers are not truly random because the computer uses an algorithm based on distribution, and they are not secure because they rely on deterministic, predictable algorithms. Random number generators (in our case—pseudorandom number generators) have been known for several decades and are constantly being developed [27,28,29,30]. If the algorithm is known, successive elements of the sequence can be predicted, so there is no randomness. The generated sequence of numbers has a discrete rather than continuous distribution.

The first extensive use of the Monte Carlo (MC) method by von Neumann in 1944 allowed for understanding of the fundamentals of the chain reaction, the basis for construction of the atomic bomb. The MC simulation method is intuitive and universal. It is used for solving both random and nonrandom tasks, such as complex integrals, differential equations, nonlinear equations and optimization (minimization), as well as simulation of dynamical systems in physics [31,32,33,34,35,36]. It can be used for issues where analytical methods fail, e.g., to assess the reliability of systems with a complex arrangement of elements or the analysis of systems with dependent operating times and device failures [37]. The accuracy of the result obtained with the MC method depends primarily on the number of randomizations (checks) and the quality of the pseudorandom number generator. It usually increases with the number of trials, although this is not always the case. The pseudorandom number generator has a finite quantity of random numbers in a cycle. For example, this method is used for integration when the time taken to obtain the result is more important than its accuracy. The classic Monte Carlo method has the following advantages: intuitive nature, ease of implementation, insensitivity to the shape of the failure area and the ability to make a simulation in an original space [31,32,34,36,38,39].

The MC method is inextricably linked to the development of numerical methods and computer programs using random simulation techniques to mimic the statistical population. In conventional statistical analysis software packages, the computer constructs the community according to the user’s recipe. Then, a random sample from the community for each MC repetition is simulated, analyzed and stored. After multiple repetitions, the stored results will imitate the distribution of the statistical sample.

LHS is a statistical method of making a sample of probable sets of parameter values with a multivariate probability distribution. The LHS method has been used since 1980, after publication of the study by McKay et al. [40] and the development of computer codes in the following years. The difference between the LHS method and standard randomization consists in the fact that it takes previously generated samples into account. Moreover, it is necessary to assume in advance how many sampling points are needed. In the standard MC method, samples are generated without taking the previously randomized ones into account, and the number of samples is unknown. In the LHS method, the places where the samples were taken from are remembered in the matrix notation (row and column). The optimal hypercube is generated in the orthogonal method, in which the probabilistic space is divided into equally probable subspaces. All orthogonal sampling points are selected simultaneously. Each subspace is sampled from the same density. In this way, orthogonal sampling ensures that the set of random numbers is a very good representation of actual random sample variability, whereas traditional (extensive) sampling is simply a set of random numbers without any guarantee of representativeness. Usually, the latter method is enough to achieve sufficient accuracy for frequent problems, on condition that the assumptions of the generation of the optimal hypercube are met. In the classic MC method, several hundred times more randomizations would normally be needed in order to achieve the same accuracy.

The purpose of this study was to compare empirical defect/reliability data on Zetor tractors collected over the course of a year with data generated by Monte Carlo techniques or Latin hypercube sampling. Additionally, correlation-based methods such as Iman-Conover and a combination of Latin hypercube sampling with Iman-Conover were tested.

2. Materials and Methods

The service history of 74 agricultural tractors (4 Zetor models, 45–90 kW power) serviced at an authorized service center in 2012 was used as an empirical base (the actual data). The faults were not divided either by Zetor tractor model or by individual vehicle componentry. It was assumed that each defect eliminated the tractor from use and was equally important. It did not matter which defect occurred.

The 200 records (as in

Table 1. Data from 200 exemplary failures for different models of Zetor tractors [41], Poznań University of Life Sciences.

No.	Model	VIN *	EOH **	Symptoms of Failure	Broken Part	Cause of Failure
1	2	3	4	5	6	7
1.	Forterra 95	1337	330	air escapes	quick coupler	damaged sealing flange
2.	Forterra 115	6708	323	wrong sensor indication	air pressure sensor	short circuit
3.	Forterra 125	2534	221	worn out mounts	wheel disc	inaccurate processing
4.	Forterra 105	1286	130	no light	headlight	short circuit
5.	Forterra 125	2408	220	worn out mounts	wheels	wheel disc
…	…	…	…	…	…	…
196.	Proxima 85	2657	610	shock absorption failure	cabin	gas spring
197.	Proxima 85	3028	487	shock absorption failure	cabin	gas spring
198.	Proxima 85	3028	487	engine hour meter failure	panel	short circuit
199.	Proxima Plus 85	1690	4	voltage drop	battery	faulty battery cell
200.	Forterra 125	2743	304	shock absorption failure	gas spring	damaged surface of sealing flange

* VIN chassis number; ** engine operating hours.

) of tractor use-to-failure periods (time-to-failure), expressed as engine operating hours (EOH), were used to simulate 1000 consecutive use-to-failure periods with the commercial Statistica package (v13.3, StatSoft, Cracow, Poland).

In the first step, the empirical data were analyzed through a Weibull distribution because it is optimal for the failure times of tractors recorded since the beginning of their operation [7,42]. The offset threshold was assumed to be zero, which meant that only values greater than zero could occur. A 95% confidence interval was assumed. The results were ranked using the Kolmogorov-Smirnoff algorithm.

Simulated numerical data were generated with the ‘Distributions and Simulation’ module in the ‘Statistics’ tab of the Statistica software. This module enables ‘experiments’ by simulating multivariate data with a defined distribution while maintaining covariance between variables. In our study, two-parameter Weibull distribution was used (as for empirical data).

In the second step, Monte Carlo (MC) and Latin hypercube sampling (LHS) methods were selected. For both methods, data generation is based on the best-fit distributions, whereas the correlations between variables are ignored.

Latin hypercube sampling involves dividing the range of each theoretical distribution into N parts so that the probability of selecting each part is the same. N is the sample size. The values of each variable are randomized so that each interval for each variable occurs only once. For example, if there are two variables, X and Y, and the sample size is 3, the first sampling may produce X from interval no. 1 and Y from interval no. 2. In the second sampling, interval no. 1 for X and interval no. 2 for Y are disregarded. This ensures coverage of the full range of variability of all variables.

There are four stages of sampling:

• The theoretical distributions are divided into N intervals of equal probability.
• The value of the cumulative distribution function is randomized from each interval.
• The cumulative distribution function is inverted to obtain the value of the variable for the randomized value of the cumulative distribution function.
• The values of individual variables are randomly combined to obtain a multivariate random variable. In this procedure, the relationships between variables are ignored.

The next step is selection of the simulation method from the correlation-based group where data generation is based on the best-fit of the distribution while maintaining the correlation between variables [43]. The Iman-Conover (IC) and LHS combined with the Iman-Conover (LHS + IC) methods were used. The number of attempts generated during the simulation is entered in the ‘Options’ field (the standard number is 100; in this study, it was classically set at 1000), and the initial value for the random number generator (the Generator kernel) is entered. Empirical data, i.e., failure times, were recorded with an accuracy of 1 EOH (engine operating hour). By default, data generated in the used software were rounded to three decimal places.

3. Results

The 200 records of tractor use-to-failure periods (time-to-fail) expressed as engine operating hours (EOH) were used to simulate 1000 consecutive times in the program. All four random number generators available in Statistica were checked. The results are shown in

Table 2. The empirical data (EOH) and data generated by the four estimation methods.

No.	Empirical DataEOH	New Random Data (EOH) Generated with Method
		MC	LHS	IC	LHS + IC
1	2	3	4	5	6
1.	2	15	13	16	13
2.	3	15	15	17	15
3.	3	15	17	20	17
4.	4	15	19	22	19
5.	4	15	20	24	19
…	…	…	…	…	…
196.	1041	164	164	166	164
197.	1474	165	165	166	165
198.	1474	167	165	167	166
199.	1474	168	166	169	166
200.	1498	168	167	169	166
…	…	…	…	…	…
996.	-	1197	1281	1305	1284
997.	-	1209	1304	1318	1309
998.	-	1291	1319	1337	1313
999.	-	1317	1349	1361	1356
1000.	-	1357	1383	1389	1381

.

The graph generated in the Statistica software (option ‘Weibull analysis and reliability/failure time analysis’) was used for further analysis and formulation of conclusions concerning the usefulness of a specific dedicated method for the creation of sets of new numerical data. The graph in Figure 1 shows the obtained course of the reliability function fitted with the Weibull distribution for 200 empirical data items from Table 1, column 2.

The following plots (Figure 2a–d) show the reliability functions R(t) for 1000 data items, which were randomly generated by the pseudorandom simulation methods.

4. Discussion

Data deficiency is a problem in reliability research. Without a database of specific damage times (time-to-fail), it is not possible to calculate reliability functions and create user-relevant reliability rankings. The use of pseudorandom number generators has made it possible to enlarge the database.

The assumed confidence interval results in the fact that is very likely (degree of probability—95%) that the plots and, in consequence, the reliability functions R(t) based on the empirical use-to-failure periods (times-to-fail) are almost identical to those generated randomly.

Both the shape parameters α and scale parameters λ for the two-parameter Weibull distribution R(t) = exp(−λ·t^α) are similar.

The shape parameter α is always greater than 1 (1.49, 1.53, 1.55, 1.54 and again 1.55). This means that the data well describe the third stage of machine wear on the Lorentz wear curve. The curve illustrates the inoperability of machinery as a result of the accumulation of irreversible changes, continuous ageing of materials, wear and tear, deformation of the structure or a gradual change in the value of device parameters beyond acceptable limits. For shape parameters α > 1, the initial decrease in reliability R(t) is small, but later it increases considerably (as in the exponential distribution). It is a classic course of the reliability function for technological devices, including agricultural machinery and farm tractors.

The values of scale factor λ, and thus the expected value of failure occurrence (454, 450, 452, 455 and again 452 EOH), enable calculation of the failure intensity ratio. With the arithmetic mean of 453 EOH, as much as 63.2% (characteristic operability) of Zetor tractors will have failed to operate by that time, with an intensity of 0.0022 EOH⁻¹. In the graphs (Figure 1 and Figure 2), it will always be the value of R(t) = 36.8% (exactly e⁻¹) on the ordinate axes. It is difficult to clearly indicate which of the random number generators is better for this task.

In order to estimate which generator can be considered the best in the sense of the most accurate match, a 100% linear cumulative plot was constructed (Figure 3), which includes the deviation of the 200 time-to-fail data generated by the four methods from values obtained empirically at the Zetor brand service station.

As shown in Figure 3, the actual state is best represented by the times-to-fail generated by the Monte Carlo (MC) method. The spread around the zero value is smallest among all the used methods. The sum of deviations is also the smallest as it amounts only to 510 EOH, compared to 1707 EOH (for the combined LHS and IC method) and 1821 EOH (for the LHS method only) in absolute terms. The least favorable results come from the IC method (as much as 2980 EOH difference).

The efficiency of agricultural production largely depends on the performance of agricultural machinery in various operating conditions [5]. There are more than 100 different types of technological devices related to plant and animal production in a modern farm (or rather an agricultural enterprise). This is a wide spectrum of machinery ranging from simple devices (or rather tools), such as cultivators, to complex ones with several thousand (e.g., about 4000 individual parts in a farm tractor) or even more than a dozen thousand parts (e.g., about 18,000 parts in a combine harvester, and even 25,000 parts in a self-propelled forage harvester). As far as the improvement of durability and reliability is concerned, such systems (usually serial structures) can be broken very easily. This was already noticed by Einstein, who said “Everything should be made as simple as possible, but not simpler”, thus confirming the KISS (Keep it simple, stupid) rule, which is well-known to computer programmers.

Due to the characteristic traits of agricultural devices, their application is very narrow, and, in consequence, they are expensive. In order to minimize the probability of wrong purchase decisions, it is recommended to follow the example of the automotive industry [44] and use rankings, including rankings of reliability and failure rates. Due to difficult access to actual data on failures (time and type) of agricultural machinery and vehicles, it is necessary to use various statistical estimators for their valuation. There are computer techniques enabling estimation of how the statistics will behave in repeated randomization.

The ‘Distributions and Simulations’ module of the Statistica program enables the fitting of theoretical distributions to data, assessment of the goodness of fit and generation of multivariate data from the fitted distributions with the possibility of maintaining correlations between them.

Instead of waiting until the appropriate amount of data on reliability is obtained, theoretical distributions can be fitted to data already collected, and then the course of events can be simulated. The simulation results enable evaluation of the phenomenon under analysis—in our case, evaluation of the reliability of Zetor farm tractors.

5. Conclusions

The simulation research led to the following final conclusions:

• Empirical (sampling) data which allow for historical events give reliable results of the assessment of the reliability of technological devices such as farm tractors. However, it is difficult to obtain such data from reliable sources such as manufacturers and service technicians of agricultural equipment. Therefore, it is helpful to identify alternative ways of obtaining these data, e.g., available statistical methods such as random/pseudorandom number generators.
• Simulation methods are a separate group of methods of analysis of the reliability of technological devices, which is important for practice. The unquestionable advantages of simulation methods are ease of implementation, the possibility of obtaining results with any accuracy, and insensitivity to nondifferentiability of the limit function or the existence of multiple design points.
• The verification and validation of failure times of the working units obtained from four dedicated random number generators in the Statistica program showed that they could be successfully used in agricultural practice to estimate the failure probability value. It is impossible to indicate the best adaptive method due to the small dataset of only 200 elements, i.e., the failure times of Zetor tractors which were repaired at an authorized service station in Poland.
• Each of the random number generators tested in our study can be regarded as dedicated because it enables the estimation of failure times of technological devices in various units of durability, i.e., CTU (conventional time units). For farm tractors, it is an hour of engine operation, which depends on the engine load, i.e., an engine-operating hour. For agricultural machinery, these may also be a clock hour (e.g., for seed drills, combine harvesters or potato harvesting machines), hectare (e.g., for ploughs and sprayers) or year (e.g., for slurry tankers). Other conventional time units are kilometers travelled (e.g., for transport sets) and tones or kilograms of capacity (e.g., for forage harvesters and collecting presses). The durability of relays and contactors in mechatronic systems can be measured with the number of their correct operations. However, as such data are not publicly available, there are no rankings of machinery and vehicle reliability relevant to farmers.
• There are only two states describing the condition of a technological device in the classical theory of reliability. Proven random number generators are also based only on this assumption. All identified failures (their recorded times) are treated as equally important, which does not reflect reality. Therefore, further improvement of the quality of random number generators is necessary.
• The future is likely to be quantum generators of pseudorandom numbers. Such generators are already finding applications in encryption devices to enhance the security of distributed systems.