PROCEEDINGS OF THE 8TH INTERNATIONAL CONFERENCE
AND
29TH ANNUAL GENERAL MEETING
(YOLA 2008)
THEME:
AGRICULTURAL ENGINEERING AND THE CHALLENGES OF THE
MILLENNIUM DEVELOPMENT GOALS
VENUE:
MULTI-PURPOSE HALL, FEDERAL UNIVERSITY OF TECHNOLOGY
YOLA, ADAMAWA STATE, NIGERIA
DATE:
JANUARY 28 – 1 FEBRUARY 2008
PROC, NIAE: Volume 29, 2008
© NIAE, 2008
THE NIGERIAN INSTITUTION OF AGRICULTURAL ENGINEERS
ISBN 0794-8387
FUZZY LOGIC MODELLING OF FARM TRACTOR RELIABILITY IN KWARA STATE, NIGERIA
A.O. Ogunlela and T. A. Ishola
Agricultural Engineering Department,University of Ilorin, Ilorin.
ABSTRACT
Preliminary works of data collection and analysis were carried out to examine the reliability of farm tractors.
Sugeno-type Fuzzy Logic model was developed to estimate the reliabilities of farm tractors and their systems. The
results show that Fuzzy Logic model is reasonably accurate in predicting reliability of farm tractors. The fuel system
was observed to be the most reliable of the tractor systems. The electrical system had the greatest tendency to failure
on the farm tractors surveyed.
Keywords: Reliability, Fuzzy Logic model, Farm tractors.
1.
INTRODUCTION
As more and more capital in the form of machinery
replaces manual labour on the farm, the reliability of
the equipment used assumes greater importance.
Indeed, deeper insight into failures and their
prevention is to be gained by comparing and
contrasting the reliability characteristics of systems
that make up the tractor. Reliability is defined as the
probability that the equipment or system will
complete a specific task under specified conditions
for a stated period of time (Amjad and Chaudhary,
1988). Hence, reliability is a mathematical expression
of the likelihood of satisfactory operation. A failure
may be referred to as any condition which prevents
operation of a machine or which causes or results in a
level of performance below expectation.
Owing to the importance of timeliness of operations
in obtaining high yields, machinery breakdown
especially at busy period such as sowing or
harvesting can lead to large losses of revenue apart
from the cost of repairing the equipment. If estimates
could be made of when equipment is likely to fail,
this would assist in planning machinery purchases
and spare parts inventories and reduce costs.
The concept of Fuzzy Logic was conceived by Lotfi
Zadeh in 1965, and he presented it as a way of
processing data by allowing partial set membership
rather than crisp set membership or non-membership
(Kaehler, 2005).In Fuzzy Logic, an assertion is
allowed to be more or less true. A true value in Fuzzy
Logic is a real number in the interval [0, 1] rather
than the set of two true values {0,1} of Classical
Logic. Areas of application of Fuzzy Logic include
cement kiln control, information retrieval systems,
laboratory water level controller, etc (Jaya et al.,
2004).
Fuzzy Logic provides a simple way to arrive at a
definite conclusion based upon vague, ambiguous,
imprecise, noisy or missing input information. A
Fuzzy Logic approach control problem mimics how a
person would make decisions, only much faster. It
uses an imprecise but very descriptive language to
deal with input data more like a human operator
(Kaehler, 2005).
Fuzzy Logic is an important decision making tool for
comparing a developed product with a similar
product available in the market. This technique helps
to transfer the opinion of experts as well as objective
information, into numerical values which can be used
for decision making (Jaya et al., 2004).
Fuzzy Inference system has been
successfully used in predicting many biological
parameters. The use of Fuzzy Inference System is a
means of modelling that allows apt description of
mapping inputs to outputs (Brown-Brandl et al.,
2003).
Fuzzy models use qualitative relationships
and can include intuitive understanding. They may be
developed significantly faster than analytical models.
Fuzzy models also eliminate the need of coefficients
in relationships to account for uncertainty between
variables (Center and Verma, 1997). Applications of
Fuzzy Inference system to model complex physical
and biological systems are well documented and
growing (Brown-Brandl et al., 2003).
The main objective of this work
was to use a Fuzzy Logic model to estimate
reliability of farm tractors during their
operating life. This will help in assessing the
suitability of imported tractors to Nigeria.
Necessary steps can then be taken to make
these imported tractors more reliable.
2.
MATERIALS AND METHODS
2.1
Theoretical Development
R(t )  1  F (t )
Amjad and Chaudhary (1988) used
Weibull failure model to apply reliability
theory to farm machinery with special
reference to mechanical reaper. It was
claimed that the Weibull failure model is
very basic and applicable to all farm
machines such as combines, tractors etc. The
Weibull cumulative distribution function
(cdf) or failure function is given as
(3)
F t   1  exp  [(t   )  ]
(1)
where
logarithm.
exp = base of Naperian
α
=
scale
R(t )  exp  [ t  ]
(4)
For the estimation of α and β (Weibull
parameters) simple regression analysis was
used. This method is based on the fact that
the reliability function of Weibull
distribution can be transformed into a linear
function of Ln t by means of double
logarithmic transformation. Taking the
natural logarithm of both sides of equation
(2) twice, gives:
parameter
(months).
β = shape parameter or
Weibull slope (ratio).
γ = location parameter or
lower bound of life (months).
t = time
successive failure (months).
between


















F t   1  exp  [ t  ]



















Ln t  1 Ln Ln











1
 Ln 
1 F (t)
(6)
This is of the form Y = Mx + C
where
(2)
M = 1/ β
(Slope of the linear
equation)
The reliability function R(t)
given as
is therefore







(5)

Amjad and Chaudhary (1988) also stated
that in the case of farm machinery, the first
failure can be expected as soon as the
machine is placed in service. Hence, the
lower bound of the life or location parameter
(γ) is zero. Thus γ = 0 and the Weibull
cumulative density function becomes







Ln Ln 1   Ln t  Ln 
1 F (t)
C = Ln α
(Intercept of the linear
equation)
The Weibull model provides
considerable flexibility in describing failure
distribution (Amjad and Chaudhary, 1988).
That is, upon setting β = 1, equation (2)
becomes exponential distribution with a
delay (which can be thought of as a
guarantee period within which no failure can
occur, or a minimum life). Thus, the
assumption of a constant failure rate
(exponential model) is also included as a
special case in this Weibull failure model.
Wingate-Hill (1981) and Amjad
and Chaudhary (1988) pointed out that
reliability of a machine is a product of the
individual component's reliability. If the
engine, transmission, tyre and steering have
reliabilities of R1, R2, R3, and R4
respectively, for a specified condition, then
the reliability, Rt , of the tractor as a whole
is obtained as
Rt = R1 R2 R3 R4
(7)
2.2
Data Collection and Analysis
A questionnaire was designed to
collect data on time between successive
failures of each of the systems of farm
tractor. The tractor systems considered were
engine, transmission, hydraulic, steering,
Table 1. Weibull Parameters (α and β)
of Tractor Systems
α
β
Engine
31.75
2.88
Hydraulic
25.87
4.36
Steering
19.79
2.91
Transmission
23.36
2.98
Fuel
39.31
3.39
Cooling
24.99
3.51
Traction
19.12
3.14
Electrical
18.41
3.19
Tractor
System
electrical, traction, cooling and fuel. A total
of sixty questionnaires were administered to
twenty-six organizations. The organizations
were well spread all over Kwara State,
Nigeria. Forty-five tractors were surveyed
and they were all still serviceable. They had
covered a period of operation ranging from
thirty to one hundred and thirty-six months.
The tractors were generally the medium
power rating tractors and were used for
tillage, haulage and stationary barn yard
operations. The data obtained from time
measurement were compiled and used to
quantify reliability. The data were grouped
into class intervals and analysed using
median rank and regression equations in
conjunction with the Weibull model. The
time between successive failures data for
each system of all tractors irrespective of the
make was analysed to obtain the overall
Weibull parameters, α and β (Table 1) and
reliability of each system. With the use of
equation (7), the overall reliability of the
whole tractor was obtained from the
individual reliability of the eight systems of
the tractor considered. This computation was
done for various times between successive
failures. The individual reliability of the
eight systems of the tractor constituted the
input data while the overall reliability of the
whole tractor obtained from them was the
output data. Hence, sets of input/output data
were generated.
2.3
Modelling
The reliability input/output data set
was randomly divided into two subsets. Set
1 consisted of 85 independent data points
and was used in the model development. Set
2 consisted of the remaining 30 data points,
and the model evaluation was completed
using this set. In a detailed analysis, a
random selection of 10 data points was
evaluated to ensure accurate model
predictions.
Fuzzy Inference Systems
The Sugeno-type Fuzzy Inference
system was developed in this research. For a
system with two input variables and one
output variable, Sugeno-type fuzzy logic
models have the form:
If x is A and y is B, then z = f (x, y)
(8)
where x and y are system input variables and
z is system output variable. A and B are
antecedent membership functions and f (x,
y) is a crisp function in the consequent.
Usually f (x, y) is a polynomial function of
the input variables x and y; but it could be
any function as long as outputs produced by
the
Fuzzy
Inference
system
can
appropriately simulate the system being
modelled (Mathworks, Inc., 2002).
Clustering
Clustering is used to establish
membership
functions
during
the
development of Fuzzy Inference system.
Usually, cluster definition is based on spatial
distances between data. In a cluster space of
numerical data, the distance between any
two points is less than the distance between
any point in the cluster and any point outside
the cluster (Mathworks, Inc., 2002). For
fuzzy clusters, the boundary between any
two clusters is fuzzy. Membership functions
are established by assigning a degree of
belonging for each data point to a given
cluster as the fulfillment of the membership
function. In this research, subtractive
clustering was used to establish initial
Sugeno-type membership functions.
Neural Network for Tuning Fuzzy Inference
Systems
A key characteristic of artificial
neural networks is their ability to learn, a
process by which a neural system acquires
the ability to carry out certain tasks by
adjusting its internal parameters according
to some learning algorithms. The initial
Sugeno-type Fuzzy Inference system was
optimized by ANFIS (Artificial NeuroFuzzy Inference System) (Mathworks, Inc.,
2002), in which the Fuzzy Inference System
is implemented into the framework of a
supervised feed-forward type artificial
neural network. The training, or learning,
algorithm of the artificial neural network is a
combination of back-propagation and the
least-squares methods. The synapses in
ANFIS are not given values (synaptic
weights). They only indicate flow direction
of data. Parameters that are adjusted are in
node transfer functions. In other words, the
membership function parameters are
adjusted, not synaptic weights.
Fuzzy Logic Model
This model, consisting of eight
inputs (individual reliabilities of the engine,
transmission, hydraulic, steering, electrical,
traction, cooling and fuel systems of the
tractors) and one output (overall reliability
of the tractors), was implemented using
MatLab(Mathworks, Inc., 2002), using a
Sugeno-type modelling approach. The
Fuzzy Logic Model was developed using the
aforementioned ‘dataset 1’ as a training set,
and ‘dataset 2’ as the testing set. The model
was developed using ‘genfis2’ (which
utilizes subtractive clustering) within Matlab
(Mathworks, Inc., 2002), and ANFIS as the
neural net training routine.
3.
RESULTS AND DISCUSSION
The description of the model was a
set of five rules dictating the mapping of the
inputs to the output. The five rules are
shown graphically in Figure 1. Note that the
inputs are combined as fuzzy clusters
described with Gaussian membership
functions. The model uses a logical ‘AND’
relationship to combine the data space into
fuzzy clusters. The degree of belonging of
an input vector to a particular cluster defines
the contribution of the associated rules. The
ultimate output is a weighted average of
each of the contributing rules. Examining
the rules presented in figure 1 reveals that
the overall reliability of the tractor is
sensitive to all the inputs (reliabilities of the
tractor systems). The projections of each
cluster on each of the tractor systems change
dimensions at every rule which indicates
that they all have some impacts on the
output (overall reliability of the tractor).
A plot of the Fuzzy Logic model
estimated reliability and the reliability
values obtained from reliability function
equation for farm tractors (Figure 2) shows
that the Fuzzy Logic model was able to
capture the relationship between the eight
inputs (individual reliabilities of the tractor
systems) and the output (overall reliability
of the tractor) with a good fit. The
coefficient of determination (R2) was 0.85,
the slope was 0.96 and the standard error of
the estimate was 0.15.
Fuzzy Logic model of each tractor
system was also developed using time
between failure as input and reliability of the
system as output. The values of Fuzzy Logic
model estimated reliabilities and the actual
(calculated) values of reliability were very
close. The coefficients of determination (R2)
were virtually unity (0.9999). However, in
order to compare the relative reliabilities of
the tractor systems, the Fuzzy Logic model
estimated reliabilities were plotted against
time between failures as shown in Figure 3.
With reference to Figure 3, the fuel system
was found to be the most reliable. It has the
least slope. The second most reliable system
REFERENCES
Amjad, S. I. and Chaudhary, A. P. 1988. Field
reliability of farm machinery. Journal of
Agricultural Mechanization in Asia,
Africa and Latin America. 10(1):73 - 78.
Brown-Brandl, T. M., Jones, D. D. and Woldt,
W. E. 2003. Evaluating Modelling
Techniques for Livestock Heat Stress
was the engine. This is followed by the
hydraulic, cooling, transmission, steering,
traction and the electrical systems in order
of decreasing reliability. The highest
reliability value of the fuel system could be
explained by the fact that routine
maintenance of changing the fuel filters on
the farm tractors reduces the frequency of
breakdown of the system. The low reliability
value of the traction system is
understandable from the fact that the farm
tractors in the area of study operate on rough
terrains where tree stumps and other
damaging materials are not properly
removed. Likewise, the relatively low
reliability value of the steering system is due
to the fact that the tractors surveyed are
made to work on rocky, difficult and rough
fields.
4.
CONCLUSION
This
paper
reported
an
investigation to assess the reliability
functions from the breakdown records of
farm tractors. Sugeno-type Fuzzy Logic
model was developed to estimate the
reliabilities of the various systems of the
farm tractor and the overall reliability of the
tractors. The results show that the Fuzzy
Logic model estimated reliability is
reasonably accurate and is comparable to the
reliability values obtained from reliability
functions. Comparisons of the Fuzzy Logic
model estimated reliabilities of the tractor
systems showed that steering, traction, and
electrical systems have higher tendency to
failure than other systems of the tractor. The
fuel system was observed to be the most
reliable of the tractor systems.
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ASAE Annual International Meeting,
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Center, B. and Verma, B. P. 1997. A Fuzzy
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pp 1 - 8.
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(accessed 2006).
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Wingate-Hill, R. 1981. The application of
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Figure 1. MatLab (Mathworks, Inc., 2002) interactive interface describing “Fuzzy Logic Model.” Each row
of membership functions represents a rule and consists of eight membership functions, corresponding to
each input. The first eight columns represent the eight inputs, (reliabilities of the tractor systems). The last
column represents the weighted output (overall reliability of the tractor) of each of the five rules.