A Multistate Model of Reliability of Farming Machinery

The article describes a multistate model of reliability of farming machinery as a deductive stochastic model of the process of changes in the technical conditions observed during operation. These conditions determine the capacity of machinery to fulfil functions, simultaneously keeping safety and maintaining acceptable costs of possible repairs. The theory of semi-Markov processes was used to solve the problem. After detailed analysis of the symptoms of damage to exemplary groups of farming machinery (rotary mowers, rotary harrows and harvesting presses) we obligatorily and arbitrarily proposed an optimal four-state reliability model to describe changes in technical conditions. In contrast to the classic reliability theory, which allows only two states of technical usability (either a machine is fit to function or not), we also allowed intermediate states, because not all types of damage affect the functionality of machinery. This approach increases the probability of technical usability of machinery and rationally delays the moment of premature repair.


Introduction
Reliability is the ability of a technical device to perform its required functions under stated conditions for a specified time.According to the classic reliability theory, there are only two states of usability of a technical device, i.e. the state of being usable and the state of being unusable.A device is technically usable if it is able to perform its required functions at a specified moment or interval of time.The state of unusability is the opposite.The two-value approach to reliability, without intermediate states, simplifies reality, because there are repairable devices, composed of numerous parts, which form systems. Farming machinery is an example of such devices [1, 2, 3 and 4].
Contemporary machines, which are used at each farm, must be reliable due to the specific conditions of their work.Their periods of operation are usually long (ranging from a dozen years to several decades).They work together with other machines in simple or complex units (e.g. a tractor and plough unit, a sowing and cultivation unit) during short agrotechnical terms (ranging from a few hours while spraying orchards to a few weeks during harvesting) and under difficult field and weather conditions.Due to the specific character of their work an agrotechnical procedure can be conducted even if the machine is not fully technically usable.Therefore, the traditional reliability theory should be modified by introducing additional, acceptable states of reliability.
A four-state stochastic model was proposed to model the reliability of cars [5].Apart from the state of full (complete) usability and the state of full (complete) unusability, the authors also assumed two intermediate states, i.e. the state of partial (incomplete) usability and the state of incomplete task-related usability.This approach seems to be justified and it can also be applied for the quantification of reliability of tractors and farming machinery.Hypothetically, there could be a different number of intermediate states between the state of usability and the state of unusability, so it requires an individual approach, depending on the group of machinery under consideration.
The aim of the study was to make methodological assumptions of a multistate probabilistic reliability model, which will allow for actual changes in the technical condition of farming machinery during field work.The theory of semi-Markov processes was used to solve the problem, as it is successfully used in many branches of business activity and fields of science [6, 7, 8, 9, 10, 11, 12, 13 and 14].

Material and methods
Many methods, which differ in precision and range of application, were developed to calculate indicators of the reliability of technical devices with a complex structure.There are analytical methods, simulations and mixed methods.Analytical methods of calculation enable detailed analysis of random events and processes.The Markov process method and, increasingly often, the semi-Markov process method are methods specialising in the analysis of random processes [15, 16, 17 and 18].If the Markov process method is to be applied, we need an exponential distribution of the probability of damage, which causes some limitations.This distribution is characterised by constant intensity of damage λ = const.
In practice, semi-Markov processes are proposed for the modelling of actual operation processes.They are characterised by the possibility to use various distributions of probability freely, which are generalisations of the exponential distribution, e.g. the Weibull distribution and the gamma distribution.
At the beginning it is necessary to make a very precise definition of complete usability of farming machinery.Only a brand-new machine which is released from a factory and has not worked even one hour (t = 0) is in the state of complete usability (CU) and its reliability amounts to R(0) = 1.As the machinery uptime increases, its reliability decreases.If the state of technical usability is not restored regularly, the state of complete unusability (CUn) will result.In practice, this situation should not occur and the state of usability should be restored regularly upon the observation of symptoms (organoleptic tests and diagnostic systems).A used machine will never achieve the 100% usability of a brand-new product.However, we made some simplifications of material reality in this study and the machine will achieve the CU state after being repaired.
According to the definition of reliability presented in the Introduction, a farming machine is in the state of usability only if is able to perform its required functions.The basic utility functions of farming machinery are listed in Polish standards -General Requirements and Tests, in the chapter: Operating Requirements.For example, the basic utility functions of tractor-drawn mouldboard ploughs are: stubble-field skimming and shallow pre-sowing ploughing on light and medium soils for skimming ploughs as well as medium and deep ploughing (pre-sowing and pre-winter ploughing) on all soil types for medium and deep ploughs (PN-89/R-36598).These requirements are also specified for other groups of farming machinery, e.g.rotary mowers (PN-90/R-36506).Their basic utility function is to mow meadow grass and low-stem green forage and to arrange it into windrows.A harvesting press is a tractor-drawn or self-propelled machine for collecting hay and straw in windrows or raked into rolls.Other utility functions are pressing (piston press) and baling (baler) (PN-91/R-55026).
According to the Lorenz curve, even the simplest machines lose their properties during operation and pass into intermediate states (IN).If the symptoms indicating partial usability of a machine are ignored or overlooked, the usability of the whole machine may be reduced.In consequence, it may generate costs, which are particularly significant in the case of expensive farming machinery.Delays in sowing, spraying or harvesting due to an unexpected downtime and a major repair cause serious economic problems in agricultural enterprises.
The authors of this study proposed the graph method and the theory of semi-Markov processes to solve the problem of farming machinery passing from the CU through the IN to the CUn state.
The graph theory enables pictorial presentation of the functioning scheme of a particular process.Semi-Markov processes are stochastic processes with specific properties.In order to model the process of change in the technical condition of farming machinery and the states of its reliability the semi-Markov process {W(t): t ≥ 0} can be defined by means of the homogeneous Markov renewal process [19].The process is fully defined if its function matrix is known: This function matrix is an adjacency matrix whose elements are the probabilities of transition from state s i to state s j over a period of time which is not longer than t (i ≠ j; i, j = 1, 2,..., k).
Another condition is to know the initial distribution of this process: The construction of the semi-Markov model {W(t): t ≥ 0} of the actual process of change in the technical condition of farming machinery as the states of its reliability is the sine qua non to apply the theory of semi-Markov processes.These models require that the following conditions be met:  the Markov condition, i.e. the future evolution of the state of any technical device must only depend on its present condition;  random variables: T i (the duration of state s i , no matter which state follows) and T ij (the duration of state s i on condition that it is followed by state s j ) must have non-exponential distributions.

Results
The operation of farming machinery under natural conditions enables distinction of a few characteristic changes in the technical condition.However, not all of these states make further effective and high quality operation of the machinery impossible.The long-term experience gained during research on operation let us observe several characteristic states of partial usability of farming machinery.Table 1 lists typical symptoms of the states of unusability of exemplary groups of farming machinery, which can be categorised as four universal classes.
For further considerations concerning the reliability of farming machinery a set of four states was arbitrarily assumed S = {s 1 , s 2 , s 3 , s 4 }, as is shown in Figure 1.The machine is operated within the entire range of its capacity according to specifications, without any symptoms of damage.
s 2state of partial usability The machine can be operated like in state s 1 , but the quality of work is worse (blades are worn out or minimally damagedlower quality of work, but the machine is still functional) s 3state of incomplete taskrelated usability Blades are damaged considerablylesser weight of the blade or one blade is missing because the fixing element has been broken.The machine is still basically functional, but the quality of work is very low, because it does not mow evenly.Apart from that, the missing blade causes uneven load of the rotating parts and possible damage to the propulsion system.s 4state of complete unusability (CUn) The device does not function adequately (blades are missing, the propulsion system is broken and it is impossible to operate the device).

Rotary harrow (active)
s 1state of complete usability (CU) The machine is operated within the entire range of its capacity according to specifications, without any symptoms of damage.
s 2state of partial usability The coupling in the propulsion system is not adjusted.Higher load at high revolutions.s 3state of incomplete taskrelated usability The coupling in the propulsion system does not work.It may be maladjusted or broken.If the machine is overloaded, the working elements and racks and pinions in the rotor propulsion system may be badly damaged.s 4state of complete unusability (CUn) The propulsion system is brokenit is impossible to operate the machine, especially at high revolutions.
Harvesting press (baler) s 1state of complete usability (CU) The machine is operated within the entire range of its capacity according to specifications, without any symptoms of damage.
s 2state of partial usability The terrain copy system is broken or the pick-up is maladjusted.The machine collects biomass, so it is functional, but the malfunctioning system of pick-up level adjustment (the work level is too low) may cause premature wear of working parts, such as pick-up fingers, the propulsion system and pick-up cover.They may also let foreign bodies, such as stones enter the chamber.When pick-up fingers are too close to the ground, they wear out and become damaged soon.They may be broken off.In consequence, there are gaps, which may cause inadequate picking up of the mass.The operator makes a decision when to qualify a particular degree of wear as terminal.When the pick-up working position is too low, soil may enter the press chamber and cause premature wear of rotating metal elements of the pick-up, the rotor, blades and press chamber.The working elements of the pick-up, the rotor, blades and press chamber may become badly damaged.In consequence, the cutting system may become damaged (the feeding rotor and blades).The machine still makes bales although the system which should shred the collected material is out of order.s 4state of complete unusability (CUn) A foreign body enters the machine and causes permanent damage to the working elements and the propulsion system, which renders the machine unusable.

Compiled by the authors
The states are marked with nodes.The arcs with arrows show the relations, i.e. possible changes in the technical condition.Two values were allocated to the arcs, i.e. probability (p) and the time (T) of transition between individual states.4. The multistate approach is a reliable reflection of the material reality.The article provides the method which can be adapted and used according to one's heuristic knowledge by making individual adjustments concerning intermediate states and the probable consequences of ignoring them.It is the user that makes an arbitrary decision about the acceptable quality of work, safety and costs of repair of machinery.

Fig. 1 .
Fig. 1.A directed graph of the four-state model of the actual process of farming machinery operation.

s 3 -
state of incomplete taskrelated usability

Table 1 .
The states of reliability of selected groups of farming machinery and possible symptoms of damage