An Efficient Harmony Search Optimization for Maintenance Planning to the Telecommunication Systems

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An Efficient Harmony Search Optimization
for Maintenance Planning to the
Telecommunication Systems
Fouzi Harrou and Abdelkader Zeblah
1University of Bourgogne, Institute of Automotive and Transport Engineering (ISAT),
49rue Mademoiselle Bourgeois, 58000 Nevers,
2University Of Sidi Bel Abbes, Engineering Faculty,
1France
2Algeria
1. Introduction
A necessary precondition for high production is availability of the technical equipment. In
addition, reliability engineers have to build a reliable and efficient production system. The
system reliability affects essentially the reliability of its equipments. This characteristic is a
function of equipment age on system’s operation life. In this work, we consider series-
parallel systems. To keep the desired levels of availability, strongly performs a preventive
maintenance actions to components are best than breakdown maintenance. This suggestion
is supported by a number of case studies demonstrating the benefits of PM in (Gude et al,
1993). In this case, the task is to specify how PM activity should be scheduled. One of the
commonly used PM policies is called periodic PM, which specifies that systems are
maintained at integer multiple of some fixed period. Another PM is called sequential PM, in
which the system is maintained at a sequence of interval that have unequal lengths. The first
kind of PM is more convenient to schedule. Contrary the second is more realistic when the
system require more frequent maintenance at it age. A common assumption used in both
these PM is that minimal repair is conducted on system if it fails between successive PM
activities. In other words, minimal repairs do not change the hazard function or the effective
age of the system.
Traditionally PM models assume that the system after PM is either as good as new state in
this case is called perfect PM or simply replacement, as bad as old state the same as minimal
repair, where he only restores the function of the system, this concept is well understood in
the literature (Brown et al, 1983). The more realistic assumption is that the system after PM
not return at zero age and remains between as good as new and as bed as old. This kind of
PM is called imperfect PM. The case when equipment fails (damage), a corrective
maintenance (CM) is performed which returns equipment to operating condition, in fact
specially, the task of preventive maintenance actions served to adjust the virtual age of
equipment. Our particular interest is under investigation to present an harmony search
algorithm which determines the optimal intervals of PM actions to minimize maintenance-
cost rate or maximize mission reliability.

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1.1 Summuray of previous work
Several years ago, much work was reported on policy optimization of preliminary planned
PM actions with minimal repair as in (Zhao, 2003), (Borgonovo et al, 2000). Most of these
researches are based on two popular approaches to determine the optimal intervals for a PM
sequence. The first is reliability-based method and the second is optimization method.
In the first one the PM is performed whenever the system availability or the hazard function
of the system reaches a predetermined level and the optimal PM intervals will be selected.
The second is finding the optimal intervals as a decision variable in the optimization
problem. (Lin et al 2000) presents an algorithm to determine the optimal intervals based on
the reliability-based method and in there models the effective age reduction and hazard
function are combined. (Levitin et al, 2000) present a genetic algorithm which determine a
minimal cost plan of the selecting PM actions which provides the required levels of power
system reliability. A list of possible PM actions available for each MSS, are used. Each PM
action is associated with cost and reduction age coefficient of its implementation.
1.2 Approach and outlines
The proposed approach is based on the optimization method using harmony search
algorithm, which determines the intervals sequence of PM actions to minimize the
maintenance-cost subject to availability or (reliability) constraints. The goal of the proposed
approach is to know when, where, to which component and what kind of available PM
actions among the set of available PM actions should be implemented. To evaluate the
reliability and the effect of PM actions of series-parallel MSS, UGF method is applied. It’s
proved to be effective at solving problem of MSS redundancy and maintenance in (Monga et
al, 1997), (Levitin et al, 1999) and (Ushakov et al, 2002).
2. Preventive maintenance
It has been shown that the incorporation of the preventive maintenance has a benefit and
success. Also it was observed that the impact of the decrease of component failure rate and
improvement of component reliability is vital to maintain efficiency of production. The


Fig. 1. Series-parallel Power System

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major subject of maintenance is focused on the planning maintenance service of the power
system. Such as cleaning, adjustment and inspection performed on operation’s lifetime are
classed as a preventive maintenance policy. However, all actions of PM not capable to
reduce age component to zero age is imperfect. There are two main alternatives for
modeling an imperfect PM activity. The first one assumes that PM is equivalent to minimal
repair with probability p and 1p
−
is the equivalent to replacement in (Nakagawa, 1999).
The second model where the imperfect PM directly analyzes how the hazard function or the
effective age change after PM as in (Lin et al, 2000). The proposed model is based on
reduction age concept. Let consider the series-parallel MSS system shown in figures 1.
If the component j undergoes on PM actions calendar at chronological times as follows:
(
1, ...,
j jn
tt ) (1)

Based on the second model description, the effective age after i-th PM actions may be
written as:
( )
t
(
i
( ) (
t
)
jj ji
tt
ττ+
=+−
for
()
1, 1
ji ji
tttin
+
< <≤ ≤
(2)
and
immediately after the i–th PM action which ranges in the interval [0, 1]. By definition, we
assume that(0)0
=
,
0
0
=
and
actions is, where
1
and
0
. In the first case the component at least be restored to “as
bed as old” state which assumes that PM does not affect the effective age. In the second case
the model reduce the component age to “as good as new”, which means that the component
age reaches zero age (replacement). In fact, all PM actions which improve the component
age are imperfect. As it be mentioned and demonstrated in (Lin et al, 2000), the hazard
function of component j , as function of its actual age, can be calculated as
)
11
()()( ) (
+
)
j jii j jij ji ji ji
ttttt
τ ε τε τ
++
−−
==−
where ()
j ji t
τ+
is the age of component
j τ
jt
iε is the age reduction coefficient. Two limits for PM
iε =
iε =

()
*
j
0
( )t
jjj
hhh
τ=+
(3)

where
and
interval between PM actions i and
( )
correspond to the initial age of equipment. The reliability of the equipment j in the
1i + can be written as:
jh t is the hazard function is defined when equipment does not undergo PM actions
0 jh

( )t
()
*
j
( )exp( )
j
jji
j
t
r th x dx
τ
τ+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=− ∫


()
exp(( ))( ( ))
j
τ
jj jij
HtHt
τ
+
=−
(4)

( )
j
H τ represents the accumulative hazard function. Clearly if
reliability reaches the maximum and is equal to 1.
The Minimal repairs are performed if MSS equipment fails between PM actions, and there
cost expected in interval [0, t] can be given as
ji
tt
=
in equation (4) the

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0
( )
t
Mjjj
Cc h x dx
= ∫
(5)
Possible equipment j , undergoes PM actions at each chronological time
case, the total minimal repair cost is the sum of all cost can be written as :
1, ...,
j
jjn
tt
, in this

()
()
1
1
00
()
( )( (
τ
))(())
jj
t
n
∑
n
∑
j ji
∫
Mjjjjjjijj ji
ii
t
jji
Cch x dxcHtHt
τ
τ
τ
+
+
+
==
+
==−
(6)
where
0
0
jt
=
and
1jnj
tT
+=
where T represents the lifetime.
3. Optimization problem
Let consider a power system organized with components connected in series arrangement.
Each component contains different component put in parallel. Components are
characterized by their nominal performance rate
minimal repair cost
j
C . The system is composed of a number of failure prone components,
such that the failure of some components leads only to a degradation of the system
performance. This system is considered to have a range of performance levels from perfect
working to complete failure. In fact, the system failure can lead to decreased capability to
accomplish a given task, but not to complete failure. An important MSS measure is related
to the ability of the system to satisfy a given demand.
When applied to electric power systems, reliability is considered as a measure of the ability
of the system to meet the load demand (W), i.e., to provide an adequate supply of electrical
energy (Ξ ). This definition of the reliability index is widely used for power systems: see
e.g., (Ross, 1993), (Murchland, 1975), (Levitin et al, 1996), (Levitin et al, 1997) and (Levitin et
al, 1998). The Loss of Load Probability index (LOLP) is usually used to estimate the
reliability index (Billinton et al, 1990). This index is the overall probability that the load
demand will not be met. Thus, we can write R = Probab(ΞMSS≥ W) or R = 1-LOLP with
LOLP = Probab(ΞMSS<W). This reliability index depends on consumer demand W.
For reparable MSS, a multi-state steady-state instantaneous availability A is used as
Probab(ΞMSS ≥ W). While the multi-state instantaneous availability is formulated by
equation (7):
j
Ξ , hazard function
j h (t) and associated

{}
,( )
j
MSS
A
j
D
t WP t
Σ ≥
= ∑
(7)
where ΞMSS (t) is the output performance of MSS at time t . To keep system reliability at
desired level, preventive and curative maintenance can be realized on each MSS. PM actions
modify components reliability and CM actions does not affect it. The effectiveness of each
PM actions is defined by the age reduction coefficient ε ranging from 0 to 1. As in (Levitin
et al, 2000), the structure of the system as defined by an available list of possible PM actions
( ) v for a given MSS. In this list each PM actions ( ) v is associated with the cost of its
implementation
( )
p
C v , and
( )M v is the number of equipment affected corresponding to
their age reduction ( ) v
ε
. Commonly the system lifetime T is divided into y unequal

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lengths, and each interval have duration
is performed. This action will be performed if the MSS reliability
than the desirable level
0
R .
Let us remark that the increase in the number of intervals increases solution precision. On
the other hand, the number of intervals can be limited for technical reasons. All the PM
actions performed to maintain the MSS reliability are arranged and presented by a vector
V as they appear on the PM list. Each time the PM is necessary to improve the system
reliability; the performed following action is defined by the next number from this vector.
When the scheduled PM action
iv+ action should be performed at the same time and so on. For a given vector V , the total
number
component j 1
jJ
≤ ≤
. For all scheduled PM actions
be expressed as
y
θ 1
yY
≤≤
, at each end of this latter an PM action
( , )
R t w becomes lower
iv was insufficient to improve reliability, automatically the
1
j n and chronological times of PM action in equation (1) are determined for each
ivV
∈
. The total cost of PM actions can

( )
V
( )
v
1
N
ppi
i
CC
=
= ∑
(8)
and the cost of minimal repair can be calculated as

( )
V
()
1
10
( (
τ
))(())
j n
∑
J
Mjj jij ji
ji
CcHtHt
τ+
+
==
=−
∑
(9)
The optimization problem can be formulated as follows: find the optimal sequence of the
PM actions chosen from the list of available actions which minimizes the total maintenance
cost while providing the desired MSS availability. That is,
Minimize:

( )
f V
( )
V
( )
V
pM
CCC
→=+
(10)
Subject To:

Aθ (V, D, t) ≥ R0
(11)
To solve this combinatorial optimization problem, it is important to have an effective and
fast procedure to evaluate the availability index. Thus, a method is developed in the
following section to estimate the system availability.
4. Reliability estimation based on Ushakov’s method
The last few years have seen the appearance of a number of works presenting various methods
of quantitative estimation of systems consisting of devices that have a range of working levels
in (Reinschke, 1985) and (El-Neweihi, 1984). Usually one considers reducible systems. In
general forms the series connection, the level of working is determined by the worst state
observed for any one of the devices, while for parallel connection is determined by the best
state. However, such the approach is not applicable for the majority of real systems.
In this paper the procedure used is based on the universal z-transform, which is a modern
mathematical technique introduced in (Ushakov, 1986). This method, convenient for
numerical implementation, is proved to be very effective for high dimension combinatorial