Preventive Maintenance and Replacement Model for Mechanically
Repairable Systems with Linearly Increasing Hazard Rate
NSE UDOH
Department of Statistics
University of Uyo
Nwaniba Road, Akwa Ibom State
NIGERIA
INIOBONG UKO
Department of Statistics
University of Uyo
Nwaniba Road, Akwa Ibom State
NIGERIA
Abstract: - Most machines fail due to lack of appropriate preventive maintenance (PM) and replacement
schedule, and this failure leads to higher cost of repair maintenance, distortion of production schedule, elongated
downtime period and reduced productivity. These could however be avoided by the utilization of optimal PM
and replacement models suited for the specific kind of system. It is on this premise that this work develops an
optimal PM and replacement model for mechanically repairable systems with linearly increasing hazard rate
which failure distribution of the system is characterized by the Rayleigh distribution. The failure times of a
Rolls Royce dredging machine was used as real-time data to obtain the PM and replacement schedule for the
machine at respective cost ratios. The results showed that the model provided an effective maintenance schedule
for the machine and ensures optimal performance.
Key-Words: - preventive maintenance, replacement model, linearly increasing hazard rate, Rayleigh
distribution, dredging machine, repairable systems
Received: May 9, 2022. Revised: September 5, 2023. Accepted: October 7, 2023. Published: November 3, 2023.
1 Introduction
Preventive maintenance (PM) is an effective method
of enhancing the condition of a machine’s
functionality. It aids in minimizing cost of
maintenance and unexpected failure of machines. A
PM policy outlines the scheduling requirements for
PM activities. It may be periodic, which calls for
machine maintenance at integer multiples of a
predetermined period or sequential which keeps the
system running at a series of intervals that may have
different lengths. Both of these PM plans share the
same presumption that the machine will only need
minimal maintenance if it breaks down in between
PM actions. When a machine breaks down, minimal
repair just gets it back to working condition (as-
good-as old); it does not get the machine healthier
overall. In other words, minor repairs have no impact
on the machine’s age or hazard rate. Authors in [1]
and [2] defined Preventive maintenance (PM) as a
set of activities to be performed before system
failures, aimed at keeping the system in a good
working state and reducing its operational
expenditure. A sequential PM policy with failure
rate threshold for lease items with Weibull lifetime
distribution was developed by [3] and applied it to
leased equipment. The work showed that any
product failure that occurs during the term of the
lease is fixed with minimum repairs, and if a
minimal repair takes longer than expected, the lessee
may be charged a fee. Furthermore, additional PM
actions were carried out in an effort to reduce
product failures. The best threshold value and
accompanying maintenance degrees were
determined using this maintenance scheme and a
mathematical model of the predicted total cost in
order to reduce cost.
In 2009, [4] reviewed maintenance policies with
emphasis on replacement, imperfect PM, and
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inspection policies in a broader context. The three
common models of block replacement, simple
replacement, and periodic replacement with
minimum maintenance were changed into finite
replacement models. Subsequently, an ideal
preventive maintenance and replacement plan for a
system was determined using a novel mathematical
model by [5] with three potential actions; keeping
the system up to date, replacing it, and doing nothing
for each of the discrete, equally spaced intervals that
made up the maintenance planning horizon, noting
that every choice has a price and it influences the
system's failure pattern. In a later development, [6]
integrated dynamic programming with the branch
and bound method to find the best PM plan for a
sequence of repairable and maintainable system
which components have increasing rate of failure. In
order to meet the system's objectives, the best
choices for each component during each period were
examined. These were all based on the general
increasing hazard rate condition of mechanically
repairable systems. A repairable machine is a
machine in which damaged parts can be serviced or
readjusted so as to get the system into proper
working condition. In this type of machine, a failure
to one part of the machine does not necessary mean
a failure to the entire machine. Also, damaged parts
are not completely replaced regularly.
In 1880, [7] proposed the Rayleigh distribution to
solve a problem in the field of acoustics. Since then,
a lot of researches in various branches of science and
technology has been done in relation to this
distribution. The generalized Rayleigh distribution
was put forth by [8], and various estimating
techniques have been used to determine its
parameters. Authors in [9] investigated the
estimation of the Rayleigh distribution's parameter
in the presence of various censoring sampling
strategies, including type-I, type-II, and progressive
type-II censored sampling. The Weibull distribution
with scale parameter 2 is a particular instance of the
Rayleigh distribution which is of interest in this
work. When the shape parameter is adjusted to 1, the
Rayleigh distribution changes into the chi square
distribution with two degrees of freedom. The failure
rate or hazard function of the Rayleigh distribution,
according to [10], is a crucial property since it rises
over time. This suggests that when failure time is
dispersed in accordance with the Rayleigh model,
excessive aging/piece occurs. The Rayleigh
distribution's hazard rate increases linearly over
time. It has several uses, including reliability
analysis, clinical investigations, life testing
experiments and applied statistics. It is frequently
used to simulate the behaviour of systems with rising
failure rates. The two-parameter Rayleigh
distribution provides a simple but nevertheless
useful model for the analysis of lifetimes, especially
when investigating reliability of technical
equipment.
Several replacement maintenance models and
policies abounds in the literature. For instance, [11]
proposed that the first and last triggering event
approaches for replacement with minimal repairs of
whichever occurs last should be used in
optimizations for policy consideration when
replacement times could be scheduled at a planned
time, T of operation and at a number, N of minimal
repairs to compare with the traditional approach of
whichever occurs first. The long-run average cost
rates was minimized by [12] to estimate the best
scheduled replacement instants. In order to examine
the effects of such variations, cost-rate minimizing
models were created, presuming that the real PM
time and the scheduled PM time varied from one
another in a probabilistic manner. Also, [13]
reviewed general maintenance policies under the key
areas of maintenance: holistic review, concept
planning, development planning and optimization
planning. Furthermore, a non-periodic preventive
maintenance schedule for repairable systems using
failure rate threshold was developed by [14].
Over time, many combined PM and replacement
models have been proposed to improve system
maintenance. Of particular interest is [15] who
proposed a PM and replacement schedule based on
the age and hazard models. The result showed that
the age model outperformed the hazard model. In a
later development, [16] formulated a hybridized PM
model on the assumption that PM is imperfect by
combining the age reduction model and the hazard
rate adjustment model of [15] for improved decision.
Also, [17] developed a geometric imperfect
preventive maintenance and replacement model for
ageing repairable systems with higher degrees of
deterioration. The model has three phases: the
average life span, beyond the average life span, and
beyond the initial replacement age of the system.
The model was a generalization of [16] to produce a
PM and replacement timeline for ageing
mechanically repairable systems at various phases of
deterioration.
It is however noted that these models were
developed for the general case of repairable systems
with increasing hazard rate mostly characterized by
the Weibull failure function. In this work, we
consider a special class of repairable systems with
linear increasing hazard rate and therefore propose
an optimal PM and replacement model for this class
of system which were not considered in previous
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works. The Rayleigh distribution is utilized as the
failure distribution characterizing the failure rate of
this class of system, [10]. The proposed model is
implemented on the Roll Royce dredging machine to
obtain optimal PM and replacement schedule.
2 Problem Formulation
The Rayleigh distribution is the proposed failure
distribution for this work because of it has the
property of linear increasing hazard rate (LIHR). It
is derived from the Weibull distribution when the
shape parameter is 2. The 2-parameter Weibull
distribution is given by;
0,0,0;
1

tettf t
The hazard and cumulative hazard functions are
respectively given as:

ttHandtth 1
For
2
, we have the Rayleigh distribution as
follows;
And its respective hazard and cumulative hazard
functions are:
222
2ttHandtth
2.1 Formulation of imperfect preventive and
replacement model for repairable systems
with linearly increasing hazard rate (LIHR)
In [16] a hybrid model which is a combination of the
hazard rate adjustment model and the age reduction
model of [15] was formulated. The model is given
by;
xutdhxt 11
where d is the hazard
rate adjustment factor, u is the age reduction factor
and x is the operating time before the next PM;
,,0;10;1 12 ttxud
th
is the
failure rate function for
1
,0 tt
.
The PM activity at time,
1
t
generates a new failure
rate function, λ(t) for
12 ,ttt
with
xdh
as the
failure rate function in the subsequent PM interval
which solely depends only on h(x) and the associated
PM activity. In other words, λ(t) is dependent on
both h(t) for
1
,0 tt
and u, the magnitude of the
PM activity in time
.
1
t
2.2 The average cost of running the system
per unit time
The associated cost model to PM and replacement
model is often used to evaluate the performance of
the repairable system and also to determine expected
time for safe and appropriate maintenance. The aim
is to minimize the expected cost of maintenance.
Hence, the expected cost rate model is;
Nk
N
kk
kkk
N
kkmpr
Nyyu
yuHyHDQQNQ
yyyCC
1
1
11
1
21
1
1
,...,
(1)
where
r
Q
,
mp QandQ
are respectively the cost of
replacement maintenance, preventive maintenance
and minimal repair of the machine,
,1...0, 210
1
1
uuudD k
iik
k
D
and
k
yH
are the product of the hazard rate
adjustment factor and the cumulative hazard
function occurring within the interval
,,
1kk tt
which is between the time of
PMkth
1
and the
kth PM respectively and
11 kk yu
is the effective
age of the system right after
th
k1
PM.
3 Problem Solution
3.1 Minimizing expected cost per unit time
To generate optimal PM and replacement plan for
mechanically repairable systems with linearly
increasing hazard rate, we shall determine optimal
PM intervals by finding the optimal values of
k
y
(k
=1,2,3,…,N) and at replacement point, N as decision
variables to minimize the expected cost rate in (1);
see [18], [15] and [16]. Let
CyyyC N,...,21
; In
order to minimized the cost function, we take the
partial derivative of (1) with respect to
k
y
and
equate the obtained derivative to zero as follows;
0
1
11
1
2
1
1
11
1
1
1
1
k
N
kkN
kkkk
N
kkmpr
kkkkkkmk
N
kkN
kyuy
uyuHyHDQQNQ
yuhDuyhDQyuy
y
C
01
1 kkkkkkkm uCyuhDuyhDC
kkkkkkkm uCyuhDuyhDQ 1
1
(2)
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1,...,3,2,1 Nk
Where;
k
yh
is the hazard function and
k
yH
is
the cumulative hazard function,
kk yuh
is the
adjusted hazard function of the machine after
th
k
PM where
.1,...,3,2,1 Nk
Similarly, at replacement point, N;
1,...,3,2,1
)3(
0
0
1
1
1
1
1
1
Nk
CyuhDuyhDQ
CyuhDryhDQ
yuy
CyuhPDyhDQ
y
C
NNkNNNm
NNkNNNm
k
N
kkN
NNkNNNm
N
0
1
k
D
since replacement occurs at the
th
k
PM.
NNm yhDQC
(4)
By substituting (4) into (2) we have;
kNNmkkkkkkm uyhDQyuhDuyhDQ 1
111
1,...,3,2,11
111 NkuyhDyuhDuyhD kNNkkkkkk
(5)
where
12
16
k
k
dk
is the hazard rate adjustment
factor and
12
k
k
uk
is the age improvement
factor, [16]
Also, from (1);
11
1
1
1
11
kkk
N
kkmprNk
N
kkyuHyHDQQNQyyuC
prkkk
N
kkmNk
N
kkQNQyuHyHDQyyuC 11 11
1
1
1
(6)
Substituting (4) into (6), we obtain;
prkkk
N
kkmNk
N
kkNNm QNQyuHyHDQyyuyhDQ 11 11
1
1
1
m
pr
kkk
N
kkNk
N
kkNN Q
QNQ
yuHyHDyyuyhD 1
111
1
1
1
(7)
3.2 Algorithm for generating PM and
Replacement Schedule
Based on the preceding results, the following
computational algorithm would be used;
Step1: solve for
k
y
as a function of
N
y
Step2: Substitute
k
y
into (7)
Step3: Choose N to minimize
NN yhP
Step4: Obtain
k
y
from the expression in step 1
Step5: obtain
Nkyuyx kkkk ,...,3,2,1,
11
The input parameters are the cost
r
Q
,
p
Q
and
m
Q
with ratios
,
p
m
p
r
Q
Q
and
Q
Q
the Weibull parameters
are
and
, and the adjustment factors are
k
d
and
k
u
.
3.3 Implementation of the optimal PM and
replacement algorithm
Step 1: Substituting the Rayleigh hazard function in
(5), we have;
kNNkkkkkk uyDyuDuyD 1222 22
1
2
kNNkkkk uyDuDDy 1
2
1
N
kkk
kN
ky
uDD
uD
y
2
1
1
(8)
Step 2:
From (7) we have;
m
pr
kkk
N
kkNk
N
kkNN Q
QNQ
yuHyHDyyuyhD 1
111
1
1
1
By substituting Equation (8) into (7) we obtain;
11
1
2
1
1
1
1
1
kkk
N
kkNN
kkk
kN
N
kkNN yuHyHDyy
uDD
uD
uyhD
m
pr
Q
QNQ 1
m
pr
kkk
N
kkNN
N
kN
kkk
k
NN
Q
QNQ
yuHyHDyyD
uDD
u
yhD
1
1
11
1
1
1
2
1
2
2
1
2
1
kkk
k
krDD
u
let
m
pr
k
N
kkNNN
N
kkNN Q
QNQ
yHDyyPyhD 1
1
1
1
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Substituting the Rayleigh hazard and cumulative
hazard functions, we have;
m
pr
k
N
kkNN
N
kkNNN Q
QNQ
yDyyDyD 1
2
2
1
1
1
2

m
pr
k
N
kk
N
kkNkNN Q
QNQ
yDDyyD 2
2
1
1
1
1
12
Note:
22
1
1
2
1
NNk
N
kNk
N
kkyDyDyD
Hence,
m
pr
NNk
N
kN
N
kkNkNN Q
QNQ
yDyDDyyD 2
22
1
1
1
1
1
12
1
1
2
1
1
1
2
N
N
kkNm
pr
N
DDQ
QNQ
y
(9)
Step 3: To obtain optimal N, we seek optimal
number
*
N
which minimizes
NN yhD
Let
NNNN yDyhDNB 2

2
1
1
2
2
2
2
1
1
1
2
2
12
1
1
2
N
N
kkNm
prN
N
N
kkNm
pr
N
DDQ
QNQD
DDQ
QNQ
DNB
Let
2
2
m
Q
A
2
1
1
1
2
1
N
kkN
pr
D
QNQA
A necessary condition for the existence of a finite
*
N
which minimizes B(N) is that
*
N
satisfies the
inequalities;
11 NBNBandNBNB
. This
follows;
2
1
1
1
2
2
1
1
1
21
N
kkN
pr
N
kkN
pr
D
QNQA
D
QNQA
p
r
NNN
N
kkN
NNN
NNN Q
Q
DD
D
DD
DDN
11
1
1
1
1
11
1
11
1
1
p
r
NNN
N
kkN
Q
Q
N
DD
D
NB
1
12
1
1
1
1
1
1
*
(10)
Where
NkdDand
uDD
uk
iik
kkk
k
k,...,3,2,1,
11
1
2
1
2
Similarly,
p
r
Q
Q
NB 1
*
3.4 Application of the proposed model
The inter failure times of Rolls Royce - RB211
Engine Dredging machine was studied and found to
follow a Rayleigh distribution with rank 1 and scale
parameter
5.2
with the help of Easyfit (5.6)
software. Rolls Royce is a dredging machine that is
used to suck out accumulated sediment from the
bottom or banks of bodies of water, rivers, lakes or
streams. See Fig.1 in the appendix.
Recall:
12
16
k
k
dk
is the hazard rate adjustment factor
and
12
k
k
uk
is the age improvement factor.
NkdD k
iik ,...,3,2,1,
1
1
is the cumulative
hazard rate adjustment factor
The optimal N, cost ratios
p
r
Q
Q
and
p
m
Q
Q
were
obtained from equation (10) as shown in Table 1.
Also, the effective age
k
y
and optimal preventive
maintenance and replacement schedule
k
x
was
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Volume 3, 2023
obtained for the dredging machine from (8) and Step
5 of the algorithm respectively.
Table 1: Optimal PM and replacement Schedule
for Rolls Royce Dredging Machine with
LIHR (‘0000)
*
N
1
3
5
7
9
11
13
p
rQ
Q
2
5
10
20
30
50
80
1
x
0.2
502
0.4
225
0.5
594
0.7
911
0.9
689
1.2
508
1.5
822
2
x
0.2
305
0.2
356
0.3
194
0.3
856
0.5
742
0.6
179
3
x
0.2
315
0.1
899
0.2
525
0.3
023
0.4
918
0.4
796
4
x
0.1
567
0.2
047
0.2
434
0.3
065
0.3
821
5
x
0.2
405
0.1
809
0.2
140
0.2
680
0.3
331
6
x
0.2
004
0.1
906
0.2
375
0.2
942
7
x
0.2
111
0.1
708
0.2
118
0.2
615
8
x
0.1
542
0.1
904
0.2
344
9
x
0.1
812
0.1
718
0.2
110
10
x
0.1
557
0.1
906
11
x
0.1
955
0.1
728
12
x
0.1
550
13
x
0.2
000
3.5 Discussion of results
Table 1 shows the optimal number, N* of PM and
replacement at the last point in row 1. The cost ratios
are contained in row 2 while rows x1 … x13 are the
operating times of the machine under different cost
ratios. The decreasing pattern of the operating times
in columns 1 to 13 except the last one which is the
replacement point shows shorter operating times
before next PM. This calls for frequent PM due to
usage and aging which is in line with the result
obtained in the works by [15], [16], [19] and [17].
For instance, under the cost ratio of 20,000 in the
first column of Table 1, replacement should be
carried out on the machine after about 2,502hours of
operation. If the company chooses to continue with
the use of the machine, then it moves to the next
column with a higher cost ratio of 50,000. In this
column, the first PM is carried out on the machine
after about 4,225hours of operation, the next PM is
carried out after about 2,305hours of operation,
being the second cycle and finally replacement is
recommended in the third cycle after about
2,315hours of operation. If the operator still chooses
to continue with the use of the machine, it moves to
the next column with next higher cost ratio and so
on.
4 Conclusion
A PM and replacement model has been
developed in this work for a special class of
mechanically repairable systems with linearly
increasing hazard rate (LIHR) which failure rate
is characterized by the Rayleigh distribution.
The proposed model is shown to provide
optimal PM and replacement schedule for this
class of systems which were not provided for in
earlier models. This model was applied to the
Rolls Royce dredging machine which failure
times was found to follow the Rayleigh
distribution with scale parameter 2.5 to obtain
optimal PM and replacement schedule. The
machine has a linearly increasing hazard rate
(LIHR), which means the machine deteriorates
linearly with time. It is found that for a system
with LIHR, PM is carried out more often at
different costs levels which guarantees safe
operation and of course, conforms to earlier
results by [15], [16] and others. The frequent
PM schedule obtained in this work will reduce
the effective age and downtime of the machine
as well as avoiding unplanned failures thereby
increasing the uptime of a machine.
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The authors equally contributed in the present
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problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
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Conflict of Interest
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that are relevant to the content of this article.
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n_US
DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2023.3.19
Nse Udoh, Iniobong Uko
E-ISSN: 2732-9984
214
Volume 3, 2023
Appendix
Figure 1: Rolls Royce dredging machine connected to a suction pipe
DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2023.3.19
Nse Udoh, Iniobong Uko
E-ISSN: 2732-9984
215
Volume 3, 2023