Summary
Preventive replacement is applied to improve device availability or increase the profit per unit time. In this paper, we study the 3-state model of age-replacement. We investigate the problem of maximization of profit per unit time for non-decreasing and unimodal the failure rate function. The purpose of this paper is to obtain conditions under which the profit per unit time approaches a maximum. Finally, two numerical examples to evaluate an optimal replacement age are presented.
Keywords: maintenance, replacement, profit per unit time, availability, optimization 1. Introduction
One important research area in reliability engineering is the study of various maintenance poli-cies. Maintenance can be classified by two major categories: corrective and preventive case, e.g. Pham and Wang [18] or Wu and Clements-Croome [19]. Corrective maintenance is any maintenance that occurs when the system fails. Some authors refer to corrective maintenance as a repair; we will use this approach in this paper. Preventive maintenance is any maintenance that occurs when system is not failed. In this paper, we will consider a preventive maintenance as an age-replacement and system consisting of one operating unit. One of the important areas, where the role of economic optimization will grow, is the maintenance and replacement of the technical unit. Maintenance and replacement is not only a question of technical matters, but indeed also an economic one. Using the repair facilities can repair the failed unit. It is not assumed that after a repair and after a preventive replacement a repair unit is operating perfectly. An additional preventive replacement may be rea-sonable to expand the unit profit of the system. The unit can be in three states: perfect operation, repair and preventive replacement. After Barlow & Prochan [2] have systematically studied the age-replacement policy and various situations, many analytical results have been obtained in Berg [4], Berg and Epstein [5], Ingram and Scheaffer [11] as well as Osaki and Nakagawa [17].
Many reliability systems can be modeled by using the semi-Markov process. An assumption of exponential of all due to these process distributions is rather unrealistic. A semi-Markov process {X (t): t 0} is a stochastic process in which the time interval between two successive transitions is a random variable whose distribution depends on the state from which the transition takes places as well as the state to which the next transition occurs.
This paper contains the analysis of introduction efficiency of the additional preventive replace-ment services after x time units of the system work. We formulate the criteria for the existence of the profit maximum per unit time and the availability for differently-aging classes of the distribu-tions of the time of the proper work. The values of the objective function depend on the mean value of the preventive replacement time, mean repair time and the transition probabilities matrix of the
Markov chain embedded in the semi-Markov process. For a unit with an increasing failure rate func-tion, we show that there exists a unique optimal replacement age such that the objective function (profit per unit time) is maximized.
In this paper, we formulate these criteria in a more general case than the IFR (Increasing Failure Rate function) class and for the definition of a new aging class (see [14, 15]). This class is between the IFRA (Increasing Failure Rate Average) and the NBUE (New Better than Used in Expectation). For definition of the classes IFRA and NBUE, see Barlow and Prochan [3]. The considered class contains some distributions with unimodal failure rate function. We give the corresponding criterion of a distribution with the unimodal failure rate function to a defined class. In this paper, we formulate the criteria of existence of a profit maximum per unit time, when the distribution of the time of the proper work has a unimodal failure rate function.
Other applications of the distributions with the unimodal failure rate function are considered in the papers Chang [6], Jiang et al [12,13]. In this paper, we also consider the particular case of the objective function and the availability, and we then formulate the criteria of existence of a maximum.
In the last part of this paper, we analyse two examples of the application method presented herein. This example contains the numerical calculation. In the Example 1, the time to a failure has the inverse Gaussian distribution. The inverse Gaussian distribution has the unimodal failure rate function. In Example 2, we consider the modification of the failure rate function presented in papers [1] and [9], and for different values of its parameters, we calculate the values of the objective func-tion.
2. Preliminaries and assumptions
In this paper we will present a semi-Markov model of preventive age-replacement. We consider a three–state semi-Markov process, with the state space S = {1, 2, 3}, and if X (t) = i, then the considered unit at moment t is in state i. We assume that 1 is the state of perfect operation, 2 is the state of repair and 3 is the state of preventive replacement. By zi, i = 1, 2, 3, we denote profit per
unit time for the state i. In this paper we assume that z1 > 0, z2 < 0, z3 < 0. If the unit is in state 1,
then it brings a profit and if the unit is state 2 or 3, then it generates a loss. We assume that
0 < 1 < 2 < … < n<…
one jump times and vn = n – n-1 for n 1, v0 = 0 are the sojourn times of states. A
semi-Markov process X (t) can be completely determined, if we know its the initial distribution P{ X (0) = i } = pi(0), i = 1, 2, 3
and its semi-Markov kernel defined by Q(t) = [ Qij(t) ], i,j = 1, 2, 3,
where
The sequence X( n) n N of the random variables is the Markov chain with the transition
probabilities matrix
P = [ pij = Qij() ] i,j = 1, 2, 3
called embedded Markov chain. The random variables Ti, i = 1, 2, 3 denotes the sojourn time
in state i, we have the distribution function Fi(t) = P{ Ti < t } = P ( n+1– n < t | X( n) = j }
or otherwise, we have
, i = 1, 2, 3 (1)
Let us also consider the distribution function associated to sojourn time in state I, before going to state j, defined by
(2)
It is known that
Fij(t) = P{ n+1– n < t | X( n+1) = j, X( n) = i }
In this paper we use from the limit theorem for the finite semi-Markov process (see Howard [10]). We assume that the mean values ETi, i = 1, 2, 3 are finite and positive and Markov chain
X( n), n = 0, 1, 2, … has one ergodic class. From these assumptions, we have the limit distribution
in form
Pj = for i = 1, 2, 3
and
, (3)
where , j = 1, 2, 3 is the limit distribution of the embedded Markov chain X( n), n = 0, 1, 2,
3,... with the probability transition matrix P = [ pij ], where pij = Qij(), i, j = 1, 2, 3. Distribution
, j = 1, 2, 3 is the solution of the system of linear equations
∈
¦
= = 3 1 j ij i(t) Q (t) F ° ¯ ° ® > = otherwise 0 , 0 p if p ) t ( Q ) t ( F ij ij ij ij}
i
)
0
(
X
|
j
)
t
(
X
{
P
lim
}
j
)
t
(
X
{
P
lim
t t→∞=
=
→∞=
=
¦
= = 3 1 k k * k j * j j ET p ET p P * jp
* jp
, i, j = 1, 2, 3, with the condition
3. Objective function
Let X(t) be a semi-Markov process with the continuous kernel Q(t). In the sequel, we define the process
Kj(t) = ,
where
I { X(u) = j } =
denotes a summarized time of the sojourn of process X(t) in state i and in the interval time < 0,t >. The value
L(t) =
denotes an average profit for the unit in the interval time < 0, t >. The limit
is the average profit per unit time for the infinity interval of time. The value L is the base in construction of the objective function. By the definition of process Kj(t), j = 1, 2, 3, we have
, j=1, 2, 3 Hence, According to (3), we obtain
¦
= = 3 1 i * j ij * ip p p¦
= = 3 1 i * i 1 p³
= t 0 du } j ) u ( X { I ¯ ® ≠ = j ) u ( X for 0 , j ) u ( X for 1¦
= 3 1 i i iEK (t) z t ) t ( L lim L t→∞ = j j t t P ) t ( EK lim = ∞ →¦
= = 3 1 i i iP z LL = (4)
Let us consider a random variable T1(x), defined by
(5)
According to the definition (5), if the sojourn time T1 in state 1 is less (smaller) than x, then unit
change state 1 to 2 or 3. If T1 x then the unit changes state from 1 to 3. Using the formula (5), we
obtain a new semi-Markov process with the transition matrix Px of the embedded Markov chain. The
elements of the first row of matrix Px depend to x. For p13(x), we have
p13(x) = P{ X( n+1) = 3 | X( n) = 1 } =
P{ X( n+1) = 3 | X( n) = 1, T1 < x } P{ T1< x | X( n) = 1 } +
P{ X( n+1) = 3 | X( n) = 1, T1 x } P{ T1 x | X( n) = 1 }
From (5), we have
P { X( n+1) = 3 | X( n) = 1, T1 x } = 1
Using the properties of the conditional probability, we obtain P{ X( n+1) = 3 | X( n) = 1, T1 < x }= P{X( n+1) = 3, T1 < x | X( n) = 1 }/ P{ T1< x | X( n) = 1} = Q13(x) / F1(x) Hence, p13(x) = Q13(x) + R1(x), where R1(x) = 1 – F1(x) By (2), we have p13(x) = p13F13(x) + R1(x)
Similarly, for the probability p12(x), we obtain
p12(x) = P{ X( n+1) = 2 | X( n) = 1 } =
P{ X( n+1) = 2 | X( n) = 1, T1 < x } P{ T1< x | X( n) = 1 } +
P{ X( n+1) = 2 | X( n) = 1, T1 x } P{ T1 x | X( n) = 1 }
From definition (5), we conclude
¦
¦
= = 3 1 i i * i 3 1 i i * i i ET p ET p z ¯ ® ≥ < = x T if , x , x T if , T ) x ( T 1 1 1 1P{ X( n+1) = 2 | X( n) = 1, T1 x } = 0
Hence,
P{ X( n+1) = 2 | X( n) = 1, T1 < x } = Q12(x) / F1(x)
and
p12(x) = Q12(x) = p12F12(x)
Now, the probability transition matrix Px is following
Px =
The objective function can be written in the form
g(x) = (6)
where , i = 1, 2, 3 are the limit probability for the Markov chain with the probability transition matrix Px, ET1(x) is the mean value of the random variable T1(x). For the mean value
ET1(x), we see that
ET1(x) =
Integrating by parts, we obtain
ET1(x) = (7)
The limit probability , i = 1, 2, 3 satisfy the linear system of equations
(8)
Using Cramer’s rule, we calculate the solution of the system (8). = (p21 p32 + p31) / W(x) » » ¼ º « « ¬ ª + 0 p p p 0 p ) x ( R ) x ( F p ) x ( F p 0 32 31 23 21 1 13 13 12 12 ) x ( p ET ) x ( p ET ) x ( p ) x ( ET ) x ( p z ET ) x ( p z ET ) x ( p z ) x ( ET * 3 3 * 2 2 * 1 1 * 3 3 3 * 2 2 2 * 1 1 1 + + + +
)
x
(
p
*i³
+ ≥ x 0 1 1(t) xP{T x} dF³
x 0 1(t)dt R)
x
(
p
*i » » ¼ º « « ¬ ª = » » » ¼ º « « « ¬ ª » » ¼ º « « ¬ ª − − 1 0 0 ) x ( p ) x ( p ) x ( p 1 1 1 p 1 ) x ( F p p p 1 * 3 * 2 * 1 32 12 12 31 21)
x
(
p
*1= (p32 + p12 p31F12(x)) / W(x) (9)
= (1 – p21 p12 F12(x)) / W(x)
where W(x) is the determinant of the linear system equations (8). Using solution (9), the objec-tive function (6) can be denoted by the form
g(x) = (10)
where the coefficients A, B, B1, C, C1 can be written as
A = p21 p32 + p31,
B1 = p12 (z2 ET2 p31 – z3 ET3 p21),
B = p12 (ET2 p31 – ET3 p21),
C1 = z2 ET2 p32 + z3 ET3,
C = ET2 p32 + ET3
The first derivative of the objective function (10) with respect to x can be written as
g’(x) = , (11)
where F1(x), F12(x) are the distribution functions of the random variables T1 and T12
respec-tively, and f1(x), f12(x) are the density functions of the random variables T1, T12 respectively. M(x)
is the nominative of the formula (10). The coefficients , and can be written as = A p12 [ ET2 p31 (z2 – z1) + ET3 p21 (z1 – z3)]
= A [ ET2 p32 (z1 – z2) + ET3(z1 – z3) ] (12)
= A ET2 ET3 (z2 – z3)
Next, we analyse M(x). If ET2 p31 ET3 p32, then the first derivative
M’(x) = A R1(x) + B f12(x) 0 for x 0
)
x
(
p
*2)
x
(
p
*3 C ) x ( F B ) x ( AET C ) x ( F B ) x ( ET Az 12 1 1 12 1 1 1 + + + + )} x ( f ) x ( R )) x ( R ) x ( F ) x ( ET ) x ( f ( { ) x ( M 1 12 1 1 12 1 12 2 α − +β +γM(x) is non-decreasing and M(x) M(0) = C > 0. Hence, M(x) > 0 for x 0. In the rest of this paper, we will assume that
(a) z1 > 0, z2 < 0, z3 < 0
(b) z2 < z3 (13)
(c) ET2 p31 ET3 p32
From these assumptions, we conclude that
< 0, > 0, < 0 (14)
4. Conditions for a maximum of the profit per unit time
In this chapter, we consider the conditions for a maximum of the objective function g(x). We will formulate these conditions in a particular case. Now, we assume that
F12(x) = F1(x). (16)
By (1) and (2), we conclude that
F13(x) = F1(x). (17)
The first derivate of the objective function g(x) can be written as
(18)
where r(x) = f1(x)ET1(x) – F1(x)R1(x),
M(x) = AET1 + BF1(x) + C.
The first derivate g’(x) can be written as
where r1(x) = 1(x)ET1(x) – F1(x),
1(x) = f1(x)/R1(x)
If z1 = 1, z2 = 0, z3 = 0, then g(x) is the limit availability (steady-state availability). We will
formulate the conditions for the maximum profit per unit time for the three cases of the failure rate function (for general cases see [16]).
) x ( f ) x ( R ) x ( r { ) x ( M 1 ) x ( ' g = α +β 1 +γ1 )} x ( ) x ( r { ) x ( M ) x ( R ) x ( ' g 1 1 2 1 α +β+γλ =
Proposition 1. If the failure rate function 1(t) is continuous, increasing to +, + f1(x) > 0
and the assumptions (13) are true, then exactly one point exists x0 such that g(x0) is the maximum
value of g(x).
Proposition 2. If the failure rate function 1(t) is continuous and increasing to 1, and
+ f1(0) > 0 and the assumptions (13) are true and
then exactly one point exists t0 such that g(x0) is the maximum value of g(x).
Proposition 3. If the failure rate function 1(t) is continuous and unimodal with the maximum
point x1, and the assumptions are true and + 1(0) > 0 and (ET11 – 1) + + 1 < 0, where
1 () = 1, then exactly one point exists t0 such that g(x0) is the maximum value of g(x).
5. Numerical examples
For the considered examples, we will assume that the probability transition matrix P is
P =
The mean values of the time T2 and T3 are ET2 = 0.2, ET3 = 0.05, and the profit per unit time
z1 = 1, z2 = – 1, z3 = 0.02.
Example 1.
Let IG(m, a) denote the inverse-Gaussian distribution with parameters m > 0, a > 0,which has the density
f(x) = , for x > 0
The mean value and variance of IG(m, a) are m and m3/a respectively. Chhikara and Folks [7,
8] show that the failure rate function (x) increases from zero at time t = 0 and then decreases to the nonzero asymptotic value a/2m2. For numerical calculation, we assume that () = d = a/2m2 = 4.
We analyze three cases of calculation m {0.3, 0.4, 0.5}. Fig. 1 shows the plots (x) for three cases. γ + α β − α > λ 1 1 ET » » ¼ º « « ¬ ª 0 2 . 0 8 . 0 2 . 0 0 8 . 0 2 . 0 8 . 0 0 ) x m 2 ) m x ( a exp( x 2 a 2 2 2 3 − − π −
∈
Figure 1. Plots of the failure rate function (x) Source: own work
Fig 2 shows the plots of the objective function g(x) for three cases.
Fig 2. Plots of the objective function g(x) for Example 1 Source: own work
Example 2. We assume that he unimodal failure rate function (x) has the form:
(x) = d – where b, d > 0 (19)
This function is a modification of the failure rate function discussed in [1] and [9]. For the failure rate function presented in paper [1] and [11], we have () = 0. It is clear that, if () = 0 then the objective function g(x) does not approach a maximum. For (19), it is easy to see that (0) = 0, () = d and 0 1 2 3 4 5 6 0 0,2 0,4 0,6 0,8 m=0.3 m=0.4 m=0.5 1 1,5 2 2,5 1 21 41 61 81 m=0.3 m=0.4 m=0.5 ) b ln( ) b x ( ) b x ln( db + +
= (e – b) = d for 0 < b < 1
The reliability function R(x) = exp(– (x)), where
(x) = d x – Let
w(x) = for b (0, 1)
The function w(x) has the minimum at x = 1 – b and w(1 – b) = d b ln(b) / 2 < 0
It is easy to see, that (x)/x = d for x = 1 / b – b and
Since, conclude that T IFRA. In this example we assume that d = 4. In fig. 3 we showe we assume that d = 4. Fig. 3 shows three plots (x) for b {0.82, 0.86, 0.90}.
Figure 3. Plots of the failure rate function (x) Source: own work.
) x ( Max 0 x≥ λ (1 eln(b)) b −
))
b
(
ln
)
b
x
(
(ln
b
ln
2
db
2 2−
+
) b ln ) b x ( (ln b ln 2 db 2 + − 2∈
d x ) x ( lim x = Λ ∞ →∈
∈
0
5
10
15
20
0
1
2
3
4
5
b=0.82
b=0.86
b=0.90
Fig. 4 shows the plots of the objective function g(x) for the different values of b.
Figure 4. Plots of the objective function g(x) for Example 2 Source: own work.
6. Closing remarks
The decision support models must be easy to use and integrated with other tools that the utility is using for maintenance and replacement planning purposes. It is important to establish a strategy for the use of the models, which are in accordance with the overall maintenance and replacement planning strategy in the utility. In this paper, we consider the method of maximization of the profit per unit time for the 3-state model of preventive maintenance. We give the existing criteria of the maximum of the objective function for different failure rate functions. To verify the validity of the maximization method of the objective function, we analyze two numerical examples.
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OPTYMALNE DECYZJE DLA OPTYMALNYCH POLITYK UTRZYMANIA Streszczenie
Profilaktyczna wymiana jest stosowana do podniesienia gotowoci maszyn lub podniesienia zysku na jednostk czasu. W pracy studiujemy 3-stanowy model wymian elementów według wieku. Badamy problem maksymalizacji zysku przypadajcego na jednostk czasu dla niemalejcych i jednomodalnych funkcji intensywnoci uszkodze. Celem tej pracy jest sformułowanie warunków, przy których zysk na jednostk czasu osiga maksimum. W kocu pracy rozwaano dwa przykłady numeryczne oceniajce optymalny wiek wymiany.
Słowa kluczowe: utrzymanie, wymiana, zysk na jednostk czasu, gotowo, optymalizacja
Leszek Knopik Wydział Zarzdzania
Uniwersytet Technologiczno-Przyrodniczy w Bydgoszczy ul. Fordoska 430, 87-791 Bydgoszcz
