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11
Analysis of Some Proposed Replacement Policies
Tijjani A. Waziri
School of Continuing Education, Bayero University, Kano, Nigeria [email protected]
Bashir M. Yakasai
Department of Mathematical Sciences, Bayero University, Kano, Nigeria [email protected]
Rahama S. Abdullahi
School of Continuing Education, Bayero University, Kano, Nigeria [email protected]
Abstract
This paper is coming up with an age replacement cost model under the standard age replacement policy (SARP) for some multi-unit systems. Furthermore, some two other age replacement cost models will be constructed for the multi-unit systems under some proposed policies (policy A and policy B). For simple illustration of the proposed age replacement cost models under SARP, policy A and policy B, numerical example was provided, and the result obtained will be beneficial to engineers, maintenance managers and plant management, in selecting and applying the optimal preventive maintenance policies.
Keywords: failure rate, proposed policies, multi-unit systems
I. Introduction
Multi-components systems deteriorate and subsequently fail due to age and usage. To reduce the occurrence of system failures, management of organizations are always interested in selecting and implementing the optimal preventive replacement policy for normal system operation. Furthermore, in describing the reliability of a multi-unit system, it is necessary to specify how the units of the system are connected and provide the rule of the operation. The simplest form of the system configuration is the series configuration. Designing systems in parallel configuration is done with the intention to improve systems reliability. In most practical situations, a combination of both series and parallel configurations is inevitable.
Enogwe et al. [1] used the distribution of the probability of failure times and come up with a replacement model for items that fails un-notice. Fallahnezhad and Najafian [2] investigated the number of spare parts and installations for a unit and parallel systems, so as cut down the average
cost per unit time. Gertsbakh [3] described and presented some vital preventive maintenance models for some multi-component systems. Huang and Wang [4] constructed a time-replacement model for multistate systems, which can be used to determine the optimal time to replace the entire system, and this proposed approach provides further insight into the relationship between preventive maintenance policy setting and long-term system benefits. Jain et al. [5] developed Markov model for a multi-component system which is subjected to two types of failures, which are hardware failure and human error. Jain and Gupta [6] presented a preventive replacement model for a repairable system with multiple vacation and imperfect coverage. Lim et al. [7] presented some characteristics of some age substitution policies. Liu et al. [8] come up with mathematical models of uncertain reliability of some multi-component systems. Malki et al. [9] studied some age replacement policies of a parallel system with stochastic dependency. Murthy and Hwang [10] discussed that, the failures can be reduced through effective maintenance actions, and such maintenance actions can occur either at discrete time instants or continuously over time. Nakagawa [11] presented age replacement model for series and parallel system based on standard age replacement policy. Nakagawa et al. [12] presented the advantages of some proposed replacement policies. In an approach for analyzing the behavior of an industrial system under the cost free warranty policy, Niwas and Garg [13] developed a mathematical model of a system based on the Markov process, they also derived various parameters such as reliability, mean time to system failure, availability and expected profit for the system. Safaei et al. [14] studied the optimal preventive maintenance action for a system based on some conditions. Sudheesh et al. [15] studied age replacement model in discrete approach. Tsoukalas and Agrafiotis [16] presented a new replacement policy warrant for a system with correlated failure and usage time. Waziri and Yusuf [17] constructed an age replacement cost model for a parallel-series system based on some proposed policies, where they investigated the characteristics of the proposed policies. In trying to extending the optimal replacement time of multi-unit systems, Waziri et al. [18] come up with some proposed age replacement cost models involving discounting rate and minimal repair for a series system. Waziri [19] presented a discounted age replacement model for a unit based on discrete time. Wu et al. [20] proposed a new replacement policy and established corresponding replacement models for a deteriorating repairable system with multiple vacations of one repairman. Xie et al. [21] assessed the effects of safety barriers on the prevention of cascading failures. Zhao et al. [22] collected some recent results on age replacement policies and proposed some modified age replacement policies, such as optimal age replacement policy for a parallel system with a random number of units.
The literature review presented in this paper did not captured a way or strategy of extending the optimal replacement time of a multi-component system. This paper will proposed some proposed replacement cost models under some policies, so as to see the possibility of extending the optimal replacement time of some four multi-component systems, and this will be achieved through the following objectives:
1. By constructing age replacement cost model for series and parallel systems under the standard age replacement policy (SARP).
2. By constructing age replacement cost models for series and parallel systems under two proposed policies (policy A and policy B).
3. By providing a numerical example for simple illustration of the constructed replacement cost models.
II. Methods
Reliability measures namely reliability function and failure rates are used to obtain the expressions of replacement cost models for four systems under the standard age replacement policy (SARP) and under proposed two policies (policy A and policy B). A numerical example was given so to assess the three replacement policies.
III. Notations
• Tj (t): Level I failure rate of unit Dir for i = 1,2,3,4,5,6 .
• r*(t)\ Level II failure rate of unit Di, for i = 1,2,3,4,5,6 .
• R*(t): Reliability function of Level II failure of unit Bir for i = 1,2,3,4,5,6.
• SARP: Standard age replacement policy.
• Rsi(t): Reliability function of system Si due to Level II failure, for i = 1,2,3,4.
• CSi(T): Cost rate of system Si under SARP, for i = 1,2,3,4.
• CYSi(T): Cost rate of system Si under policy A, for i = 1,2,3,4.
• CZ(T): Cost rate of system SL under policy B, for i = 1,2,3,4.
• XOptimal replacement time of system Si under SARP, for i = 1,2,3,4.
• Kjj: Optimal replacement time of system Si under policy A, for i = 1,2,3,4.
• Z*si \ Optimal replacement time of system Si under policy B, for i = 1,2,3,4.
• Cir: Cost of unplanned replacement of failed Di due to Level II failure, for i = 1, 2, 3, 4, 5, 6.
• Cim: Cost of minimal repair of failed Di due to Level II failure, for i = 1, 2, 3, 4, 5, 6.
• Csp: Cost of planned replacement of system Si at planned replacement time T, for i = 1,2,3,4.
• Csr: Cost of un-planned replacement of system Si due to Level II failure, for i = 1, 2,3,4.
IV. Description of the Systems
Consider six units Dlr D2, D3, D4, D5 and D6, arranged in four different configurations, so as to formed four different systems, which are series-parallel system ( ), series-parallel system (S2), parallel-series system (S3) and parallel-series system (S4). All the six units are subjected to Level I and Level II failures, such that, Level I failure is repairable one, while the Level II failure is non-repairable failure. Since all the six units are subjected to Level I and Level II failures, then, this implies that, all the four systems are also subjected to Level I and Level II failures. See the Figure 1, Figure 2, Figure 3 and Figure 4 as the diagram of the four systems (Slr S2, S3 and S4).
Figure 1: Reliability block diagram of system S1
Figure 2: Reliability block diagram of system S2
Figure 3: Reliability block diagram of system S3
Figure 4: Reliability block diagram of system S4
V. Replacement Cost Models Under SARP
Some Assumptions for SARP:
1. If a system fails due to Level I failure, then the system is minimally repaired.
2. If a system fails due to Level II failure, then the whole system replaced completely with new one.
3. Both the two levels of failures (Level I and Level II) of the six units arrives according to non-homogeneous Poisson process.
4. Rate of Level I failure of the six units follows the order: r((t) > r*(t) > r5(t) > r2(t) > r+(t) > r-(t).
5. Rate of Level II failure of the six units follows the order: rl(t) > r*(t) > r,(t) > r)(t) > r;(t) > r-xt).
6. The cost of repair and replacement follows the order: Cim < CSp < CSr , for i = 1,2,3,4,5,6.
7. A system is replaced at a planned time T(T > 0) or at Level II failure, whichever occurs first.
8. The cost of planned replacement of a system is less than the cost of un-planned replacement.
9. The cost of repair of a failed unit is less than the cost of replacement of a unit.
10. All costs are positive numbers.
From the assumptions above, the probability that system S( will be replaced at planned replacement time T, before Level II failure occurs, is
R*S1(T) = (1- nU(i - RI(T))) x(1- nU(i - RI(T))) x(1- nU(i - RI(T))). (1)
From the assumptions above, the probability that system S2 will be replaced at planned replacement time T, before Level II failure occurs, is
R*S2(T) = (1- nh(1 - RI(T))) x(1- nh(1 - RI(T))) x R-(T) . (2)
From the assumptions above, the probability that system S* will be replaced at planned replacement time T, before Level II failure occurs, is
R#*(T) = 1-{1- R*(T)R-(T)G x(l- R)(T)R,(T)G x(l- Rl(T)RKT)) . (3)
From the assumptions above, the probability that system S4 will be replaced at planned replacement time T, before Level II failure occurs, is
R*S4(T) = 1-(1- Rl(T)R)(T)R*(T)) x (1 - R+ (T)R*5(T)R*6(T)) . (4)
The mean time of systems ^, S2 , S* and 5+ under SARS, is
J0r Rx(t)dt , for i = 1,2,3,4. (5)
The cost of un-planned replacement (failure due to Level II failure) of ^ and S2 in one replacement cycle under SARP, is
Csr{1- R*si(T)), for i = l,2,3,4. (6)
The cost of planned replacement at time T of ^, S2 , S* and 5+ in one replacement cycle under
SARS, is
CspR*siCO, for i = 1,2,3,4 . (7)
The cost of minimal repair of components D(, D2, D*, D4, D, and D6 due to Level I failure in one replacement cycle under SARP, is
JCT)= f Clmr1(t)Rtsi(t)dt+ f C2mr2Ct)R*SiCt)dt + f C3mr3Ct)R*si(t)dt
Jo Jo Jo
+ /0 Cm+CtWsiCt)dt + f0T C4%r5Ct)R^iCt)dt + f0T C5mr6Ct)Rtsi(t)dt . (8)
Using equations (5 ), (6 ), (7) and (8), the replacement cost rate of systems S(, S2 , S* and S4 under SARP is
-crfl-RcfCDV-™ RS; (T)+f' l(t) R# (t)cLt
CSi(T) = -8 ) 0 Sl() , i = 1,2,3,4, (9)
J0 R's'(t)ct
where
m (t) + L2m r2 (t) + L3mr3 (t) + C+m r+(t) + (t) + (t). (10)
Noting that, CSi (T) for i = 1,2,3,4, is adopted as an objective function of an optimization problem, and the main goal is to obtain an optimal replacement time T#'i that minimizes CSi (T), for i = 1,2,3,4.
VI. Replacement Cost Models Under Policy A
From assumption 4, observe that, Level II failure of units D1, D* and D, is higher than that of units D2, D4 or D6. Policy A is a preventive maintenance policy, in which the un-planned replacement of a system, which depends on the failure of units D(, D* and D, due to Level II. Noting that, the reliability function of a system due to policy A, depends on the location of units D(, D* and D, in a system. But when any of the units D2,D4 or D6 fails due to Level II failure, the failed unit is replace completely with new one and allow the system to continue operating from where it stopped.
Under policy A, we have the following descriptions:
1. System S( : the system is replace completely with new one when at least one of the components D(, D* or D, fails due to Level II failure. Now, the probability that system S( will be replaced at planned replacement time T, before Level II failure occurs due to policy A, is
R>1(T) = R1CT)R*CT)R, (T). (11)
2. System S2 : the system is replace completely with new one when all the three units D(,D3 and D, fails due to Level II failure. Now, the probability that system S2 will be replaced at planned replacement time T, before Level II failure occurs due to policy A, is
R£(T) = R((T)(l -(1- R*(T))(1 - RS(T))) .
(12)
3. System S* : the system is replace completely with new one when at least one of the units D(, D* or D, fails due to Level II failure. Now, the probability that system S* will be replaced at planned replacement time T, before Level II failure occurs due to policy A, is
Rg(T) = (1-C1- R*(T))(1 - R*(T))(1 - R,(T))G . (13)
4. System S4 : the system is replace completely with new one when any of the combination fails: D( and D5, or D* and D, fails. Now, the probability that system S* will be replaced at planned replacement time T, before Level II failure occurs due to policy A, is
R>*(T) = 1-(1- R((T)R*(T))(1 - R,(T)) . (14)
The mean time of systems of S(, S2 , S* and 5+ in one replacement cycle under policy A, is
J/ Rsi(t)dt , for i = 1,2,3,4. . (15)
The cost of un-planned replacement (failure due to Level II failure) of S(, S2 , S* and 5+ in one replacement cycle under policy A, is
Csr(1-R»*(T)), for i = 1,2,3,4 . (16)
The cost of planned replacement at time T of S(, S2 , S* and 5+ in one replacement cycle under policy A, is
CspR»i(T), for i = 1,2,3,4 . (17)
The cost of minimal repair of components D(, D2, D*, D4, D, and D- due to Level I failure in one replacement cycle under policy A, is
K Clmr1(t)R>*(t)dt+ K C)%r2(t)R>*(t)dt+ K C*mr*(t)R>*(t)dt , , ,
+ J0/ C+mT4 (t)R>* (t)dt + J^ C+mT, (t)R>* (t)dt + J, C,mV- (t)R>* (t)dt . (18)
The cost of replacement of components D2, D+ and D- due to Level II failure in one replacement cycle under policy A, is
J0 C)rr* (t)R#* (t)dt + JTQ C+rr* (t)R>* № Jj" C-rr-* №s? (№ (19)
Using equations (15), (16 ), (17), (18) and (19), the replacement cost rate of S(, S2 , S* and S+ under policy A, is
CYS1 (T) = -#'( 8 P #'j " #' -0 " #', for i = 1, 2,3,4 (20)
1 ( ) Jo'R§;(mt , , , , v '
where
K(t) =
Lim^i (t) + C2mr2 (t) + C*m r* (t) + (t) +
C5 m r5
(t) + (t), (21)
and
L(t) = C2$r* (t) + C+rr+* (t) + C-rr* (t) . (22)
Noting that, CYS1 (T) for i = 1,2,3,4, is adopted as an objective function of an optimization problem, and the main goal is to obtain an optimal replacement time Y#*1 that minimizes CYS1 (T), for i = 1, 2,3,4.
VII. Replacement Cost Models Under Policy B
Observe from assumption 4, that Level II failure of units D2, D4 and D6 is lower than that of units D(, D* or D,. Policy B is a preventive maintenance policy, in which the un-planned replacement of a whole system depends on the failure of units D2, D4 and D6 due to Level II. Noting that, the reliability function of a system due to policy B, depends on the location of units D2, D4 and D6 in a system. But when any of the units D1, D* or D, fails due to Level II failure, the failed unit is replace completely with new one and allow the system to continue operating from where it stopped.
Under policy B, we have the following descriptions:
1. System S( : the system is replace completely with new one when at least one of the units D2,D4 or D6 fails due to Level II failure. Now, the probability that system S( will be replaced at planned replacement time T, before Level II failure occurs due to policy B, is
R^CT) = R*1 CT)R4 CT)R*1CT). (23)
2. System S2 : the system is replace completely with new one when all the three units D2,D4 or D6 fails due to Level II failure. Now, the probability that system S2 will be replaced at planned replacement time T, before Level II failure occurs due to policy B, is
Rb*CT) = R1CT)RKT)R1 (T). (24)
3. System S* : the system is replace completely with new one when at least one of the components D2, D4 or D6 fails due to Level II failure. Now, the probability that system S* will be replaced at planned replacement time T, before Level II failure occurs due to policy B, is
R»*CT) = 1-C1- R2CT))C1 - R4CT))C1 - R-CD). (25)
4. System S4 : the system is replace completely with new one when any of the combination fails: D4 and D2, or D6 and D2 fails. Now, the probability that system S* will be replaced at planned replacement time T, before Level II failure occurs due to policy B, is
R»1CT) = 1-C1- RICT)R1 CT))C1 - R*CD). (26)
The mean time of systems of S(, S2 , S* and S4 in one replacement cycle under policy B, is
f0T RA* Ct)dt , for i = 1,2,3,4 . (27)
The cost of un-planned replacement (failure due to Level II failure) of S(, S2 , S* and S4 in one replacement cycle under policy B, is
Csr (i-RA* CT)), for i = 1,2,3,4 . (28)
The cost of planned replacement at time T of S(, S2 , S* and S4 in one replacement cycle under policy B, is
CspRAîCT), for i = 1,2,3,4 . (29)
Tijjani A. Waziri, Bashir M. Yakasai, Rahama S. Abdullahi RT&A, No 1 (67)
ANALYSIS OF SOME PROPOSED REPLACEMENT POLICIES_Volume 17, March 2022
The cost of minimal repair of components Di, D2, D*, D4, D, and D- due to Level I failure in one replacement cycle under policy B, is
f C(mriCt)RA*Ct)dt+ f C2mr2Ct)Rs1Ct)dt+ f C*mr*Ct)RA!Ct)dt
o o o
+ f0T C4mT4Ct)RAi*Ct)dt + f0T C4mr,Ct)RA*Ct)dt + / C,mr-Ct)Rbs* Wt . (30)
The cost of replacement of components Di, D* and D, due to Level II failure in one replacement cycle under policy B, is
f0T Ci$r* WA*Ct)dt + fT C*rr*Ct)RA*Ct)dt fT C5rr,Ct)RA*Ct)dt. (31)
Using equations (27), (28), (29), (30) and (31), the replacement cost rate of systems Si, S2 , S* and S4 under policy B, is
Csr{1-RH(T))+CspRH(T)+SÇ M(t)R#$(t)dt+i( N(t)R#$(t)dt
CZSi çr)=-i--' -, for 1 = 1,2,3,4, (32)
Jo Rsi (t>dt
where
MCt) = Lim^i Ct) + C2 m r2 Ct) + C*m r* Ct) + Ct) + C,m r,Ct) + C-mT-Ct), (33)
and
NCt) = CirrUCt) + C*rr**Ct) + C,rT^Ct) . (34)
Noting that, CZSi ÇT) for i = 1,2,3,4, is adopted as an objective function of an optimization problem, and the main goal is to obtain an optimal replacement time ZS^ that minimizes CZSi ÇT), for i = 1, 2,3,4.
VIII. Numerical Example
To illustrate the characteristics of the constructed replacement cost models under SARP, policies A and B. Let the time of Level I failure for the six units follows Weibull distribution:
r, (t) = Xi ^ tKi~(, for i = l,2,3,4,5,6, (35)
where > 1 and t > 0.
Also, let the time of Level II failure for the six units follows Weibull distribution:
r*(t) = X* k* tKi-(, for i = l,2,3,4,5,6, (36)
where > land t > 0.
Let the set of parameters and cost of repair and replacement be used throughout this particular example:
1. k(= 4, k2 = 3, k* = 3, k+= 3, 4 and k6= 2.
2. X( = 0.03, X2 = 0.002, X* = 0.03, X4 = 0.001 , X5 = 0.001 and Xb = 0.001 .
3. <= 4, k*2= 3.5, k*3= 4, k*4= 3.5, k*5= 4, and <= 3.5 .
4. X\ = 0.00033, r2 = 0.00025 , T3 = 0.00030, X\ = 0.00023, = 0.00025 and A*6 = 0.0002 .
5. Csr = 70 , Csp = 50 and Cim = 0.5, for i = 1, 2 , 3, 4, 5, 6 .
By substituting the parameters in equations (35) and (36), the following equations ( Level I and Level II failures ) below are obtained as follows:
r((t) = 0.12t3. (37)
r2(t) = 0.06t. (38)
r3(t) = 0.09t2. (39)
r4(t) = 0.003t2. (40)
r5(t) = 0.004t3. (41)
r6(t) = 0.002t. (42)
r* (t) = 0.00132t3. (43)
r.* (t) = 0.000875t2-5. (44)
r3*(t) = 0.00012t3. (45)
r4*(t) = 0.000805t2S. (46)
r5*(t) = 0.001t3. (47)
r-(t) = 0.0007t2,5. (48)
Tables 1, 2 and 3 below are obtained, by substituting the assumed cost of replacement/repair and rates of Level I and Level II failures obtained above ( equations (37) to (48) ) in the replacement cost models constructed above ( equations (9), (20) and (32) ), so as to determine the optimal replacement times of the four systems.
Table 1. Results obtained i from evaluating the replacement cost rates of systems Sv S2, S# and S$ under SARP.
T CS^T) CS2(T) CS3(T) CS4(T)
1 240.04 240.09 240.04 240.03
2 120.16 120.43 120.16 120.11
3 80.42 81.14 80.37 80.41
4 61.06 62.38 60.75 61.61
5 50.68 52.48 49.65 53.17
6 46.28 48.04 44.23 52.86
7 46.64 48.13 44.70 59.29
8 53.74 52.16 51.31 67.98
9 61.01 58.01 60.74 72.17
10 63.97 58.91 64.89 74.11
11 70.03 61.84 65.31 76.16
12 73.92 68.98 67.87 78.97
Table 2. Results obtained i from evaluating the replacement cost rates of systems Sv S2, S# and S$ under policy A.
T CYS1(T) CYS2(T) CYS3(T) CYS4(T)
1 240.78 240.65 240.57 240.57
2 122.87 121.82 121.18 121.19
3 87.58 84.10 81.96 82.03
4 76.01 68.08 62.89 63.41
5 76.62 62.52 52.13 54.13
6 83.71 64.18 46.27 51.36
7 89.44 70.76 44.88 53.95
8 90.24 77.06 48.34 59.77
9 92.73 77.01 55.49 64.92
10 94.99 73.65 61.79 66.28
11 95.45 76.44 62.77 67.90
12 98.00 79.00 65.57 69.72
Table 3. Results obtained i from evaluating the replacement cost rates of systems Sv S2, S# and S$ under policy B.
T CZSi(T) CZS2(T) CZS3(T) CZS^(T)
1 240.80 243.3 240.64 240.64
2 122.54 125.04 121.62 121.63
3 85.61 88.11 83.11 83.13
4 70.10 72.6 65.11 65.20
5 63.97 66.47 55.61 55.94
6 62.95 65.45 50.64 51.48
7 64.67 67.17 49.55 50.22
8 67.21 69.71 48.55 51.27
9 68.67 71.17 50.20 53.79
10 70.74 73.24 53.05 56.80
11 74.46 76.96 56.43 59.10
12 76.97 79.47 59.34 59.61
Table 4. The optimal replacement times of systems Sv S2, S# and S4 under SARP, policy A and policy B from tables 1, 2 and 3.
System Under SARP Under policy A Under policy B
s, X#( = 6.00 Y* = 4.00 Z#( = 6.00
s7 X#*2 = 6.00 Y#*2 = 5.00 Z#2 = 6.00
S, X#*3 = 6.00 Y#*3 = 7.00 Z#3 = 8.00
S4 X#*4 = 6.00 Y#*4 = 6.00 Z#4 = 7.00
3456789 10 11 PLANNED REPLACEMENT TIME T
Figure 5: The plot of cost rates of system S1 under SARP, policy A and policy B against planned replacement time T.
3 4 5 6 7 8 9 10 11 12 PLANNED REPLACEMENT TIME T
Figure 6: The plot of cost rates of system S2 under SARP, policy A and policy B against planned replacement time T.
4 5 6 7 8 9 10 PLANNED REPLACEMENT TIME T
Figure 7: The plot of cost rates of system S3 under SARP, policy A and policy B against planned replacement time T.
3456789 10 PLANNED REPLACEMENT TIME T
Figure 8: The plot of cost rates of system S4 under SARP, policy A and policy B against planned replacement time T.
Figure 9 : The plot of cost rates of the four systems under SARP against planned replacement time T.
1200
<
1000
I—I
hJ
O
Ph
PS № D Z
¡3
№
h
<
Pi H
« 200
O
U
800
600
400
•CYSi(T)
■CYS2(T)
CYS3(T)
■CYS4(T)
3456789 10 11 PLANNED REPLACEMENT TIME T
12
Figure 10 : The plot of cost rates of the four systems under SARP against planned replacement time T.
1200
pa
y 1000
Hj
o
^ 800 Pi
№ Q
Z 600 £ №
H 400 <
Pi
& 200
o
u
■CZSiCD
■CZS2(T)
123456789 10 11 12 PLANNED REPLACEMENT TIME T
Figure 11: The plot of cost rates of the four systems under SARP against planned replacement time T.
Some observations from the results obtained are as follows
1. From Table 4, observe that, optimal replacement time of systems S3 and S4 under policy B is higher than that of SARP and policy A.
2. From Table 4, observe that, optimal replacement time of systems S1 and S2 under SARP and policy B are the same.
3. From Table 4, observe that, optimal replacement time of all the four systems under SARP are the same.
4. From Figure 5, observe that:
CS^T) < CZSj (T) < CYSj (T). (49)
5. From Figure 6, observe that:
CS2(T) < CZS2(T) < CYS2(T). (50)
6. From Figure 7, observe that:
CZS3(T) < CYS3(T) < CS3(T). (51)
7. From Figure 8, observe that:
CZS4(T) < CYS4(T) < CS4(T). (52)
8. From Figure 9, observe that:
CS^T) < CS2(T) < CS3(T) < CS4(T). (53)
9. From Figure 10, observe that:
CYS1(T) < CYS2(T) < CYS3(T) < CYS+ÇT).
(54)
10. From Figure 11, observe that:
CZS((T) < CZS2(T) < CZS3(T) < CZS+(T).
(55)
IX. Discussion of Results Obtained
From the results obtained, we have the following observations:
1. It can be seen that, the optimal replacement time of the parallel - series systems (S* and S4) under policy B, is higher than that of SARP and policy A. Furthermore, the results also showed that, the cost rates of the parallel - series systems (S* and 5+) under policy B is lower than that of SARP and policy A. With these reasons, preventive maintenance of the parallel - series systems under policy B is optimal when compared to preventive maintenance of parallel - series systems under SARP and policy A.
2. It can be seen that, the optimal replacement time of series - parallel systems (S( and S2 ) under SARP and policy B are the same or very closed. While, the cost rates of the series - parallel systems (S( and S2) under SARP is lower than that of under policies A and B. With these reasons, preventive maintenance of the series-parallel systems under SARP is optimal when compared to preventive maintenance of the series-parallel systems under policies A and B.
Hence, from the observations above, we suggest maintenance managers and plant management to adopt policy B as an optimal preventive policy of maintaining multi-unit systems which are in parallel-series configuration. While for systems with series-parallel configuration, SARP should be adopted as an optimal preventive replacement policy.
In trying to come up with some modifications and extension of the age replacement policy, this paper presented some proposed age replacement cost models for multi-unit systems under standard age replacement policy (SARP), policy A and policy B. The results obtained, showed that, preventive replacement of parallel-series systems under policy B is optimal when compared to SARP and policy A. While, preventive replacement of series-parallel systems under SARP is optimal when compared to policies A and B. Thus, the results is beneficial to maintenance managers, in selecting the optimal preventive replacement policy. All the replacement cost models and the results presented in this paper are vital to engineers, maintenance managers and plant management for proper maintenance analysis, decision and safety for multi-unit systems.
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