Abstract - This paper represents a new development on the basic classical age replacement policy, heuristically using the reliability function. First, an applied mathematical model of reliability function is obtained, and then based on this model; optimal preventive age replacement policy will be determined to maximize the reliability of the system. For the special systems which meet exponential density function with constant hazard function, a different simple decision making model will be proposed to choose optimal replacement policy based on the reliability function. A real case study will be reviewed on the maintenance data collection of two different industrial machines that one of them meets the Weibull density and other one has Exponential density function as life time pattern.
Keywords - Hazard rate, maintenance, reliability function, replacement.
I. INTRODUCTION
Reliability is one of the most important scientific indexes to depend on systems performance. It’s more considerable real practical conditions of many systems that any little unpredicted failure causes unusual damage or costs [1].
Maintenance planning which has considerable role in reliability level of systems usually is classified into Corrective Maintenance (CM) and Preventive Maintenance (PM). The former corresponds to the actions that occur after the system break down. The later corresponds to the actions that come about when the system is operating [2].
Two basic classical replacement models, Age replacement policy and Block replacement policy, were quoted by Barlow and Hunter [3].
In the Age replacement policy, an operating system is replaced at age T, or at failure, whichever occurs first. In the Block replacement policy the system is replaced by a new one at times KT, k=1 ,2,… and at failures [4].
Several developed models based on two basic policies for solving replacement problems of different systems have been presented; such as systems subject to shock [5, 6, 7, 8, 9, 10], k-out-of-n systems [11, 12], different kinds of multi states systems involve minimal repair to improve the system conditions [13, 14, 15, 16, 17], warranted and guaranteed systems [18, 19, 20, 1] and the systems working with spare parts and maintenance tools safety stock [21, 22, 23, 24].
All mentioned models are almost based on total cost function or objectives functions of cost-benefit variables.
One of the most recognized is average cost per unit time which is denoted by Jiang and Ji [25]. They defined T as preventive replacement age, considering Cp, cost of preventive replacement and Ct cost of a failure which includes all costs resulting from the failure and its replacement.
However, in real situation there are some systems operating in high level of risk that unpredicted failures cause unusual costs.
Ki Mun et al. [1] has pointed to such systems and presented a multi-attribute model for determining an optimal replacement policy decreasing down time and system cost simultaneously.
This paper tries to present a model for obtaining an optimal age replacement policy based on reliability function. Thus the necessity of having a new model of reliability function has investigated mathematically in section II. The offering model according to Cook and Paulsen [26] and Arts et al. [27] in determining performance evaluation indexes have been presented in section III. Model solving and its considerations are discussed parametrically in section IV. The model is illustrated by a numerical example in Section V. Finally, the paper is concluded with a brief summary in SectionVI.
II. RELIABILITY FUNCTION BEHAVIOUR, OPTIMAL FEASIBLE SOLUTION
The reliability function for a system with life distribution function f(t) is defined as
∫
∞= () )
(t t f t dt
R and according to R(t)=1-F(t) is always decreasing in [0,∞). The function's maximized value occurs in t=0 and by R(0)=1 which acceding time variable to infinite, causes the function's value tends to zero. Also dtd R(t)≤0 is always true [28].
Accordingly, maximization of reliability factor is not applicable through the general function R(t) because replacement in t =0 is impossible. Fig.1 shows samples for reliability curves of some general probability density which clarifies the above concept.
So a mathematical model of reliability function is necessary to substitute R(t) that its maximized value occurs in a reasonable and unique point not being zero.
Mohamad Mahdavi
1, Mojtaba Mahdavi
21 Department of Industrial Engineering, Islamic Azad University Najafabad Branch, Isfahan, Iran
2 Young Researchers Club, Islamic Azad University Najafabad Branch, Isfahan, Iran
III. PROPOSED MODEL
Arts et al. [27] suggested performance indicators for evaluation of the maintenance activities. They propose the ratio of the preventive maintenance hours and total maintenance hours, is appropriate for evaluating the system dependability.
According to Cooke and Paulsen [26] the most reliable situation for a system is minimizing both corrective maintenance and preventive maintenance.
Considering the basic age replacement policy suggested by Barlow and Hunter [3] and Cooke and Paulsen's evaluation of dependability concept, (which implies Arts's discussion as well), a heuristic mathematical model could be defined. This model is composed by two basic parameters each of them an indicator to provide one of two conditions in mentioned definition (the least possible number of corrective maintenance and preventive maintenance).
A. Parameters definition
A.1. Minimizing the corrective replacement rate
For first condition, minimizing the corrective maintenance, Jiang and Ji [25] suggested the expected ratio of number of preventive maintenance events and total number of corrective and preventive maintenance events which obtained by:
) (1) ) (
( ) (
)
( RT
T F T R
T
R =
+
Defining the basic parameters of the model by Φi then Φ1 is obtained by:
) (2) ( )
1(T =RT
Φ
A.2. Minimizing the preventive replacement rate
The ideal state of second condition, minimizing the preventive maintenance, Realized when the preventive replacement occurs just before the equipment fails.
Although, in age replacement policy it is not possible, but it is applicable to utilize methods to minimized the time between preventive replacement and actual time of failure occurrence in long term. Mean Square Errors (MSE) is one of the mentioned methods. Supposing the model's second parameter is Φ2 then this parameter is defined by:
2 2
0 2 2 2
) ( )
(
) ( ) (
T dt
T t
T t E T
− +
=
−
=
−
= Φ
∫
∞ σ μ (3)where μ and
σ
are mean and standard deviation of life density of system accordingly.B. The model design
In order to combine the parameters Φ1 and Φ2 the following model is suggested:
) (4) ( / ) ( )
(T =Φ1s T Φ2 T Φ
where s, the positive constant value, is considered the decision making parameter which causes the following situations:
1) When s tends to zero, 1( ) ) 2
(T →Φ T
Φ . This implies the first factor is much less important than the second.
2) When s tends to 1, Φ(T)=
[
Φ1(T)/Φ12/s(T)]
s→Φ1s(T).This implies the fist factor is much more important than the second.
3) When s=1, which generally is supposed, two parameters is of same importance.
IV. ANALYTIC DISCUSSION AND PARAMETRIC SOLVING THE MODEL
A. For s≠1.
When s is not 1, equation 4 is like the following form:
2 (5)
2 ( )
) ) (
( T
T T R
s
−
= +
Φ σ μ
Solving this model is possible through mathematical methods and corresponds a unique positive solution which determines the optimal replacement polices.
B. For s=1.
In this case the equation 4 is like the following form:
(6)
2
2 ( )
) ( ) 1
( T
T T F
− +
= −
Φ σ μ
B: Exponential A: Normal
D: Weibull C: Hyper Exponential
Fig. 1. Reliability curves for four various distributions;
A: Exponential with λ=0.1. B: Normal with μ=10,σ =2. C: Weibull with α=2,.β=10 D: Hyper Exponential with
1 .
=0
λ ,k=0.25.
Solving the Equ. 6 by differential arithmetic lead to the following results.
[ ] [ ]
[
( )( ( ))]
( ) 01 ) ( 2
0 ) (
2 2 2
2 2
− = +
− +
−
−
−
= Φ
T
T T
f T F T dT T
d
μ σ
μ σ μ
2 (7)
2 ( )
) ( ) 2
( T
T T
h + −
= −
⇒ σ μμ
where h(t) is hazard function. Above equation results in the unique positive Topt which reflects optimal age replacement policy.
C. The especial state of exponential distribution
Fig. 2 shows hazard function (h(t)) curves of a Normal and Exponential distribution which are increasing and constant, respectively. It is also decreasing for Hyper Exponential distribution.
Economically conducting the preventive replacement for equipment with none-decreasing hazard function actually is considered waste of resources and budget. In such systems the failure occurrence rate in future, in spite of conducting the preventive replacement, either continues as former state (the constant hazard function) or naturally takes a decreasing routine (the decreasing hazard function). So implementing the preventive replacement means replacement of the equipment currently operating faultlessly, is not reasonable in economic point of view [29].
From the other side, based on the proposed model, suppose f(t) is Exponential with λ parameter. Then
σ
λ λ1 1
)
(t = = =
h and Equ. 7 turns to the following form:
2 (8)
2 1 1
1
) ( ) (
) ( 2
T T
− +
= −
λ λ
λ λ
Solving the above equation results T =0.
We propose a new simple heuristic conceptual model to solve the replacement problem for Exponential situations, using the Mean Time To Failure (MTTF) factor. Equ. 9 represent this model.
σ (9) p MTTF
T = −
where σ is the standard deviation of f(t) and P (0≤p<1) is the decision making parameter to control R(t). Optimal value of T can be found easily by a quick trial and error operation and precise interpolation using the bellow equations:
λ (10) MTTF p p
T= − =1−
) 1 (
1 (11)
1
)
( ⎟⎠ −
⎜ ⎞
⎝
⎛ −
− =
= p
p
e e
T
R λ λ
B: Exponential A: Normal
Fig. 2. Two different hazard functions.
A: Normal withμ=10, σ =2. B: Exponential with λ=0.1 It’s clear that R(t) stands in the interval
(
e−1=0.36787 ,1)
independently to λ . Some different obtained values of R(t) for several P are collected in TableI.V. A NUMERIC CASE STUDY
Sets of performance data for two electro mechanics equipments which named A and B, were collected and Table II shows the results of basic statistical analysis for determining the life probability distributions with parameters estimating (using the Anderson-Darling method). We use the contents of Table II for the next computations.
A. Determining optimal age replacement policy of A based on the reliability function
Fig. 3 illustrates the probability sheet, life density, reliability and hazard curves of equipment A.
Supposings=1, Equ. 7 can be configured parametrically.
It has been represented by Equ. 12 and Equ. 13 for a general Wiebull distribution with
α
andβ , shape and scale parameters, respectively.TABLE I
DIFFERENT R(t)VALUES FOR SEVERAL P ) (T Ri pi
) i (T Ri pi
i
0.54881 0.4
7 1
1 1
0.49658 0.3
8 0.90484 0.9
2
0.44932 0.2
9 0.81873 0.8
3
0.40657 0.1
10 0.74081 0.7
4
0.38674 0.05
11 0.67032 0.6
5
0.36787 0.00
12 0.60653 0.5
6
TABLE II
BASIC STATISTICAL ANALYSIS RESULTS FOR A AND B DATA SETS Coeff. of Variation Goodness
of Fit (AD) Parameters
St. D.
Mean Life System PDF
0.356 1.034
27 .
=3 α
3 .
=33 10.64 β 29.18 Wieb.
A
0.857 0.915
033 .
=0 λ 25.74 30.05 Exp.
B
Fig. 3. Probability sheet, life density, reliability and hazard function for A
(12)
1
) (
) 1 (
) ( ) ( 1
) ) (
(
−
−
− −
⎟⎟⎠
⎜⎜ ⎞
⎝
= ⎛
− =
=
α α β αβ
β β α
βα βα
t e
e t
F t t f
h t
t t
2 (13)
2 1
) (
) ( 2
T T T
− +
= −
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ −
μ σ
μ β
β
α α
108 . 29 ) (A =
Topt is resulted by solving Equ. 14, so we get Φopt(T)=0.00459 and then Ropt(T)=0.52804.
2 (14)
2 3
. 32
) 18 . 29 ( ) 64 . 10 (
) 18 . 29 ( 2 3
. 33 3 . 33
27 . 3
T T T
− +
= −
⎟⎠
⎜ ⎞
⎝
⎛
B. Determining optimal age replacement policy of B based on the reliability function
Fig. 4 illustrates the probability sheet, life density, reliability and hazard curves of equipment B. Equ. 15 is modified by supposing p=0.6 that results T =12.12 and
67032 . 0 ) (T =
R .
(15) 12
. 033 12 . 0
6 . 0 1− =
= T
VI. CONCLUSION
Unusual and irrecoverable costs which emanate any little failure in many systems, turns the reliability factor into the most important controlling and evaluating index for such systems. Replacement policies which are classified into preventive and corrective categories, have most important effect on the reliability function and hazard rate of a system during its operating time.
Fig. 4. Probability sheet, life density, reliability and hazard function for B
Constraints for optimization the general reliability function of the systems (R(t)) also, demonstrate the necessity of defining a new applied mathematical model for the reliability based decision making, specially replacement policy.
In this paper a conflated model of two operational procedures, decreasing corrective replacement occurrence rate and preventive replacement ratio reduction, was determined. Also in the special case, Exponential life density systems which have constant hazard function, a new simple conceptual model was discussed.
A real numeric case study was reviewed by applying the mentioned models.
In future works applying an appropriate algorithm to obtain the optimal result for Equ. 7 and 8 along with the possibility of sensitivity analyzing is suggested using mathematical decision making models.
Discussion to determine optimal value of S in Equ. 6 and P in Equ. 11 is another way to develop the result of the current research.
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