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Comparisons of replacement policies with periodic times and repair numbers
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Reliability Engineering and System Safety 000 (2017) 1–10
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Comparisons of replacement policies with periodic times and repair numbers Xufeng Zhao a,b,∗, Cunhua Qian c, Toshio Nakagawa d a
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China Department of Mechanical and Industrial Engineering, Qatar University, Doha, 2713, Qatar c School of Economics and Management, Nanjing Tech University, Nanjing, 211800, China d Department of Business Administration, Aichi Institute of Technology, Toyota, 470-0392, Japan b
a r t i c l e
i n f o
Keywords: Periodic replacement Minimal repair Replacement delay Replacement last Unit failure
a b s t r a c t Periodic replacement policies modeled with the history of minimal repairs have been studied extensively. However, in the viewpoint of cost rate, there is no literature to compare replacement polices which are carried out at some periodic times and at a predetermined number of repairs. In this paper, we compare these two types of replacement policies analytically from the optimizations of the integrated models. It will be shown that there always exists a degradation model when any bivariate replacement policy is optimized and this is just the best choice of the comparisons. Not only that, the approaches of whichever occurs first and last are applied to model the above two types of policies, which are named as replacement first and replacement last, respectively, and their comparisons are also made. In addition, we delay the policy at repair to periodic time for easier replacement, and the modified replacement model, which is named as replacement overtime, is compared with the original ones. Numerical examples are also given and agree with all analytical discussions. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In order to be strong enough in production capacity, the total productive maintenance (TPM) has been indispensable in Japanese industry since 1970s, which involves an innovative concept of maintaining equipments by everyone in the organization [1,2]. TPM generally uses periodic and predictive maintenances to aim at maximizations of equipment utilization and production stability. Periodic maintenance [3] consists of periodically inspecting, cleaning and servicing equipment and replacing deteriorated parts to prevent serious breakdown and process problem, while predictive maintenance [4,5] is a method to operate the equipment to the limit of its service life, by measuring and analyzing data about deterioration at routine diagnosis and minor repairs. Normally, maintenances are more easily to be performed at periodic times in applications, e.g., a complete maintenance in TPM should be carried out on monthly holidays if the equipment must be stopped and a long time is required for maintenance [2]. Theoretical research works on periodic patterns in maintenance plans were studied extensively. Periodical inspection intervals were optimized to detect soft failures of a complex repairable unit, while hard failures create opportunities for additional inspections of all soft-type components [6,7]. Maintenance policies with periodic inspections were observed for the systems with
∗
several failures [8] and failure interactions [9]. Inspections are carried out periodically on the unit using the delay time concept during two failure state processes [10]. A k-out-of-n load-sharing unit that is periodically inspected to detect failed components was studied [11]. An optimal number of periodic inspections and its maintenance level to minimize the expected total warranty cost for the second-hand product during the warranty period were derived [12]. In order to achieve just-in-time (JIT) principle in TPM [1,2], repairs are included in maintenance schedules in response to all non breakdown deteriorations and resume quickly the operation of equipment after repairable failure. Not only that, the service life of important part can be predicted based on diagnosis at repairs for predictive maintenance decisions in TPM [3]. Repair models have been studied especially for large and complex systems, which consist of many kinds of units [13]. In recent works, models for repairable unit subjected to minimal repair [14], imperfect repair considering time-dependent repair effectiveness [15], age-based replacement with repair for shocks and degradation [16,17], inspection modeling for repairs [18], post-warranty maintenance with repair time threshold [19], random working models with replacement and minimal repair [20,21], etc., have been studied extensively. More recently, preventive maintenance should be planned jointly with the right type of repairs to achieve the best performance for a multi-state
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.ress.2017.05.015 Available online xxx 0951-8320/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: X. Zhao et al., Reliability Engineering and System Safety (2017), http://dx.doi.org/10.1016/j.ress.2017.05.015
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Reliability Engineering and System Safety 000 (2017) 1–10
unit has been studied [22]. A nonparametric estimation method for periodic replacement problem with minimal repairs has been proposed [23]. A case study of periodic maintenance policy under imperfect repair for off-road engines has been surveyed [24]. It has been well-known in reliability theory that the cost rate model for periodic replacement is formulated by using the cumulative hazard function H(t), where H(t) presents the expected number of failures during [0, t] when the failure rate of a repairable unit remains undisturbed by minimal repairs [25,26]. Recent periodic maintenance models based on the history of failures/repairs can be found extensively in above literatures. However, in the viewpoint of cost rate, there is no study until now to compare replacement polices that are carried out at some periodic times and at a predetermined number of repairs, which becomes the main problem to be discussed in this paper. Furthermore, in order to achieve the target of maximizations of equipment utilization and production stability in TPM [3], the approaches of modelings for replacement first [26], replacement last [27], and replacement overtime [28,29] are used to formulate our replacement models, that is, three approaches of whichever triggering event occurs first, whichever triggering event occurs last, and replacing over a planned measure will be taken into considerations in replacement policies, respectively. It has been shown in [21,27–29] that replacement last and replacement overtime could let the equipment operate as long as possible, which will also be discussed and compared for the models in this paper. Our comparisons start from formulating the integrated models with two types of maintenances that are planned at periodic times and at repair numbers. Obviously, when the optimal maintenance policies are obtained in separative models, it will be very easy to compare them with numerical examples in viewpoint of cost rates. However, we compare these two types of maintenances analytically from the optimizations of the integrated models. It will be shown that there always exists a degradation model when any bivariate replacement policy is optimized and it is just the best choice of the comparison, which agrees with the comparisons obtained in [21]. In addition, other comparative results among replacement policies are obtained analytically and numerically.
curs first. Then, the probability that the unit is replaced at time NT is 𝑃 𝐾 (𝑁𝑇 ), and the probability that it is replaced at failure K is PK (NT). Denoting 𝑃 𝑗 (𝑡) ≡ 1 − 𝑃𝑗 (𝑡), and noting that 𝑃𝑗 (𝑇 ) = 𝐾 ∑
𝑇
𝑃𝑗 (𝑇 ) =
𝑗=1 ∞ ∑ 𝑗=𝐾
𝑝𝑗−1 (𝑡)ℎ(𝑡)d𝑡,
∫0
𝑇
∫0
𝑃𝑗 (𝑇 ) =
𝑇
∫0
𝐾 ∑
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡,
𝑗=1
𝑝𝑗−1 (𝑡)ℎ(𝑡)d𝑡,
𝑃 𝑗 (𝑇 ) =
∞ ∑
𝑃𝐾−1 (𝑡)ℎ(𝑡)d𝑡,
∞
∫𝑇
𝑗=𝐾
∞
∫𝑇
𝑃 𝑗 (𝑇 ) =
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡, ∞
∫𝑇
𝑃𝐾−1 (𝑡)ℎ(𝑡)d𝑡,
the mean time to replacement is (𝑁 𝑇 )𝑃 𝐾 (𝑁 𝑇 ) +
𝑁𝑇
∫0
𝑁𝑇
𝑡d𝑃𝐾 (𝑡) =
∫0
𝑃 𝐾 (𝑡)d𝑡,
(1)
and the expected number of minimal repairs until replacement is 𝐾−1 ∑ 𝑗=0
=
𝑗𝑝𝑗 (𝑁𝑇 ) + 𝐾𝑃𝐾 (𝑁𝑇 ) = 𝐾 −
𝐾 ∑ 𝑗=1
𝑃𝑗 (𝑁𝑇 ) =
𝑁𝑇
∫0
𝐾 ∑ (𝐾 − 𝑗)𝑝𝑗 (𝑁𝑇 ) 𝑗=0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡,
(2)
Therefore, the expected repair and replacement cost rate is 𝑁𝑇
𝐶𝐹 (𝑁, 𝐾; 𝑇 ) =
𝑐 𝑅 + 𝑐 𝑚 ∫0
𝑁𝑇 ∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑃 𝐾 (𝑡)d𝑡
,
(3)
where 𝑐𝑅 = replacement cost at time NT or at repair K, and 𝑐𝑚 = cost of each minimal repair. The integrated model in (3) includes the following replacement policies: When the unit is replaced at time T (0 < T < ∞), 𝐶(𝑇 ) ≡ lim 𝐶𝐹 (1, 𝐾; 𝑇 ) = 𝐾→∞
𝑐𝑅 + 𝑐𝑚 𝐻(𝑇 ) , 𝑇
(4)
which agrees with the expected cost rate of the original periodic model with time T [25,26]. When the unit is replaced at time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞),
2. List of assumptions
𝐶(𝑁; 𝑇 ) ≡ lim 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) = 𝐾→∞
In this section, the following assumptions for failure, minimal repair and replacement are given:
𝑐𝑅 + 𝑐𝑚 𝐻(𝑁𝑇 ) 𝑁𝑇
(𝑁 = 1, 2, ⋯).
(5)
When the unit is replaced at repair 𝐾 (𝐾 = 1, 2, ⋯),
i. Failures of an operating unit occur at a nonhomogeneous Poisson 𝑡 process with mean value function 𝐻(𝑡) ≡ ∫0 ℎ(𝑢)d𝑢. Let pj (t) and Pj (t) be the respective probabilities of exact number j of failures and at least number j of failures occur in [0, t], then
𝐶 (𝐾 ) ≡ lim 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) = 𝑁→∞
𝑐𝑅 + 𝑐𝑚 𝐾 ∞
∫0 𝑃 𝐾 (𝑡)d𝑡
(𝐾 = 1, 2 ⋯).
(6)
When the unit is replaced at repair 𝐾 (𝐾 = 1, 2, ⋯) or at time T (0 < T < ∞), whichever occurs first,
∑ [𝐻(𝑡)]𝑗 −𝐻(𝑡) e and 𝑃𝑗 (𝑡) ≡ 𝑝𝑖 (𝑡) (𝑗 = 0, 1, 2, ⋯), 𝑗! 𝑖=𝑗 ∞
𝑝𝑗 (𝑡) =
𝑃 𝑗 (𝑇 ) =
𝑇
𝐶𝐹 (𝐾; 𝑇 ) ≡ 𝐶𝐹 (1, 𝐾; 𝑇 ) =
where 𝑃0 (𝑇 ) = 1 and 𝑃𝑗 (∞) = 1. ii. The unit undergoes minimal repairs at failure events, and begins to run again after repairs. It is also assumed that the failure rate h(t) remains undisturbed by minimal repairs, i.e., the unit after each minimal repair has the same failure rate as before failure. iii. A new unit is replaced at time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞), at repair number 𝐾 (𝐾 = 1, 2, ⋯), and at the next periodic time (𝑛 + 1)𝑇 when a number 𝐾 (𝐾 = 1, 2, ⋯) of repairs have been done during [𝑛𝑇 , (𝑛 + 1)𝑇 ]. iv. The times for minimal repair and replacement are negligible, and let cR be the cost of replacement and cm be the cost of minimal repair at each failure, where cR ≥ cm .
𝑐𝑅 + 𝑐𝑚 ∫0 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 𝑇
∫0 𝑃 𝐾 (𝑡)d𝑡
.
(7)
3.2. Optimum policies When the failure rate h(t) ≡ dH(t)/dt increases strictly with t to ℎ(∞) = ∞, we find optimum replacement policies of T, N and K for the above expected cost rates and comparisons among them are made in terms of cost rates. 3.2.1. Optimum T∗ , N∗ and K∗ We find optimum T∗ , N∗ and K∗ to minimize C(T) in (4), C(N; T) in (5) and C(K) in (6), respectively. (1) Optimum T∗ Optimum T∗ (0 < T∗ < ∞) to minimize C(T) satisfies [25,26] 𝑐 𝑇 ℎ(𝑇 ) − 𝐻(𝑇 ) = 𝑅 , 𝑐𝑚
3. Replacement first 3.1. Expected cost rate
(8)
and the resulting cost rate is
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at a repair number 𝐾 (𝐾 = 1, 2, ⋯), whichever oc-
𝐶(𝑇 ∗ ) = 𝑐𝑚 ℎ(𝑇 ∗ ). 2
(9)
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 1 Optimum T∗ in (8) and K∗ in (11) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 . cR /cm
1.0 2.0 5.0 10.0 20.0
𝑚 = 2.0
𝑚 = 3.0
T∗
C(T∗ )/cm
K∗
C(K∗ )/cm
T∗
C(T∗ )/cm
K∗
C(K∗ )/cm
1.00 1.41 2.24 3.16 4.47
2.00 2.83 4.47 6.32 8.94
2 2 5 11 20
2.26 3.01 4.59 6.40 9.00
0.79 1.00 1.36 1.71 2.15
1.89 3.00 5.53 8.77 13.92
1 2 3 6 10
2.24 3.36 5.76 8.97 14.08
(a) From (9) and (13), we obtain 𝐶𝐹 (𝐾; 𝑇𝐹∗ ) ≥ 𝐶(𝑇 ∗ ). That is, the original periodic replacement policy in (4) is more economical than that in (7) when a number K of minimal repairs is predefined. (b) Optimum policy to minimize CF (K; T) in (7) is (𝐾𝐹∗ = ∞, 𝑇𝐹∗ = 𝑇 ∗ ), in other words, a degradation model with T∗ is always found for the bivariate optimum policy of CF (K; T). That is, C(T∗ ) < C(K∗ ) in (4) and (6), which indicates the original periodic replacement policy in (4) is more economical than that in (6) when a number K of minimal repairs is optimized.
Table 2 Optimum 𝑇𝐹∗ in (12) and its cost rate when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
1.09 1.69 3.39 6.21 11.85
2.18 3.38 6.77 12.41 23.70
1.00 1.42 2.27 3.44 5.73
2.00 2.83 4.54 6.88 11.46
1.00 1.41 2.24 3.19 4.80
2.00 2.83 4.47 6.37 9.61
In addition, forming the inequality 𝐶𝐹 (𝐾 + 1; 𝑇 ) − 𝐶𝐹 (𝐾; 𝑇 ) ≥ 0 for given T,
(2) Optimum N∗ Forming the inequality 𝐶(𝑁 + 1; 𝑇 ) − 𝐶(𝑁; 𝑇 ) ≥ 0 for given T, 𝑐 𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ) ≥ 𝑅 . 𝑐𝑚
𝑇
∫0 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡 𝑇 ∫0
(10)
𝐿(𝑁 + 1) − 𝐿(𝑁) = (𝑁 + 1)
∫𝑁𝑇
[ℎ(𝑡 + 𝑇 ) − ℎ(𝑡)]d𝑡 > 0.
Therefore, there exists a finite and unique minimum N∗ (1 ≤ N∗ < ∞) to minimize C(N; T) which satisfies (10). If 𝐻(2𝑇 ) − 2𝐻(𝑇 ) ≥ 𝑐𝑅 ∕𝑐𝑚 , then 𝑁 ∗ = 1. Noting that
if T ≥ T∗ , where T∗ is given in (8), then 𝑁 ∗ = 1.
∞
∫0 𝑝𝐾 (𝑡)d𝑡
−𝐾 ≥
𝑐𝑅 , 𝑐𝑚
(11)
3.2.2. Optimum 𝑇𝐹∗ and 𝐾𝐹∗ In order to compare the optimum policies of C(T) and C(K) in (8) and (11), the optimizations of C(K; T) in (7) are discussed as follows. Differentiating CF (K; T) with respect to T and setting it equal to zero for given K, optimum 𝑇𝐹∗ satisfies ∫0
𝑃 𝐾 (𝑡)[ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 =
𝑐𝑅 , 𝑐𝑚
(𝑁+1)𝑇
∫𝑁𝑇
𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
(14)
∫0
𝑃 𝐾 (𝑡)d𝑡
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
𝑐𝑅 , 𝑐𝑚
(15)
and forming the inequality 𝐶𝐹 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
(12)
𝑁𝑇
∫(𝑁−1)𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
and the resulting cost rate is 𝐶𝐹 (𝐾; 𝑇𝐹∗ ) = 𝑐𝑚 ℎ(𝑇𝐹∗ ).
𝑐𝑅 , 𝑐𝑚
3.2.4. Optimum 𝑁𝐹∗ and 𝐾𝐹∗ In order to compare the optimum policies of N and K for C(N; T) in (5) and C(K; T) in (7), the optimizations of CF (N, K; T) in (3) are discussed as follows. We find optimum 𝑁𝐹∗ and 𝐾𝐹∗ to minimize CF (N, K; T) for given T. It has been shown above that if T > T∗ , then 𝑁𝐹∗ = 1 and 𝐾𝐹∗ is given in (14), and if 𝑇 = 𝑇 ∗ , then 𝑁𝐹∗ = 1 and 𝐾𝐹∗ = ∞. When T < T∗ , forming the inequality 𝐶𝐹 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
whose left-hand side increases strictly with K to ∞ [26]. Thus, there exists a finite and unique minimum K∗ (1 ≤ K∗ < ∞) to minimize C(K) which satisfies (11).
𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
3.2.3. Numerical example Tables 1 and 2 present optimum T∗ in (8), K∗ in (11), 𝑇𝐹∗ in (12) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 . It shows numerically in Table 1 that both T∗ and K∗ decrease with m and C(T∗ ) < C(K∗ ), and in Table 2 that 𝑇𝐹∗ decreases with K to T∗ and 𝐶(𝑇 ∗ ) ≤ 𝐶(𝐾; 𝑇𝐹∗ ) when 𝑚 = 2.0.
(3) Optimum K∗ Forming the inequality 𝐶(𝐾 + 1) − 𝐶(𝐾) ≥ 0, ∞
∫0
This result will be used to compare optimum policies of N and K for C(N; T) in (5) and C(K; T) in (7) in the following discussion when T ≤ T∗ and in Table 3 when T > T∗ .
𝐻(2𝑇 ) − 2𝐻(𝑇 ) > 𝑇 ℎ(𝑇 ) − 𝐻(𝑇 ),
∫0 𝑃 𝐾 (𝑡)d𝑡
𝑇
(a) If T > T∗ , then (19) > cR /cm , and there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) to minimize C(K; T) which satisfies (14) for given T. (b) If T ≤ T∗ , then 𝐾𝐹∗ = ∞, i.e., the policy with repair K becomes meaningless when a given T is small enough.
lim 𝐿(𝑁) = ℎ(∞) = ∞,
𝑁→∞ (𝑁+1)𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
whose left-hand side increases strictly with K to that of (8). Thus, we obtain
Denoting L(N) as the left-hand side of (10), 𝐿(1) = 𝐻(2𝑇 ) − 2𝐻(𝑇 ) > 0,
𝑝𝐾 (𝑡)d𝑡
𝑇
𝑁𝑇 ∫(𝑁−1)𝑇
(13)
𝑃 𝐾 (𝑡)d𝑡
𝑁𝑇
∫0
(𝑁+1)𝑇
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
(𝑁+1)𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 <
𝑐𝑅 . 𝑐𝑚
(16)
Noting that ∫𝑁𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑃 𝐾 (𝑡)d𝑡 increases strictly with N to ∞ and increases strictly with K to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 (Appendix A), the left-hand side of (15) increases strictly with K to that of (10). Thus, there exists a finite and unique minimum 𝑁𝐹∗ (1 ≤ 𝑁𝐹∗ < ∞) which satisfies (15), and 𝑁𝐹∗ decreases with K to N∗ given in (10) and 𝑁𝐹∗ ≥ 𝑁 ∗ .
Noting that the left-hand side of (12) increases strictly with T from 0 to ∞ and increases strictly with K to that of (8). Thus, there exists a finite and unique 𝑇𝐹∗ (0 < 𝑇𝐹∗ ≤ ∞) which satisfies (12), and 𝑇𝐹∗ decreases strictly with K to T∗ given in (8) and 𝑇𝐹∗ ≥ 𝑇 ∗ . Therefore, the following two comparative results are obtained: 3
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 3 Optimum N∗ in (10) and K∗ in (14) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. 𝑇 = 1.0
cR /cm
1.0 2.0 5.0 10.0 20.0
𝑇 = 5.0
N∗
C(N∗ ; T)/cm
K∗
CF (K∗ ; T)/cm
N∗
C(N∗ ; T)/cm
K∗
CF (K∗ ; T)/cm
1 1 2 3 4
2.00 3.00 4.50 6.33 9.00
∞ ∞ ∞ ∞ ∞
2.00 3.00 6.00 11.00 21.00
1 1 1 1 1
5.20 5.40 6.00 7.00 9.00
1 2 6 11 23
2.26 3.01 4.59 6.40 8.98
Table 4 Optimum 𝑁𝐹∗ in (15) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
1 2 4 ∞ ∞
2.19 3.38 6.77 12.41 23.70
1 2 2 4 6
2.00 3.00 4.56 6.88 11.46
1 2 2 3 5
2.00 3.00 4.50 6.38 9.61
Table 5 Optimum 𝑁𝐿∗ in (23) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
1 1 2 3 4
2.08 2.96 4.50 6.33 9.00
1 1 2 3 4
2.75 3.21 4.53 6.33 9.00
1 2 2 3 4
3.52 3.84 4.80 6.36 9.00
Forming the inequality 𝐶𝐹 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0, 𝑁𝑇
∫0
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾 (𝑡)d𝑡
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
𝑐𝑅 , 𝑐𝑚
which agrees with that of (11). Thus, 𝐾𝐹∗ increases strictly with N to K∗ and 𝐾𝐹∗ ≤ 𝐾 ∗ . Comparing (19) with the left-hand side of (10),
(17)
𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ) > 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 )
and forming the inequality 𝐶𝐹 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) > 0, 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑐 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < 𝑅 . 𝑐𝑚
>𝑁 [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] − 𝐻 (𝑁 𝑇 ). Thus, we obtain
(18)
(a) If N > N∗ , then there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) which satisfies (17). (b) If N ≤ N∗ , then 𝐾𝐹∗ = ∞, i.e., the model in (3) is degraded into the one in (5).
Substituting (15) for (18), 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
(𝑁+1)𝑇
<
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
,
It is easy to understand that when the approach of whichever occurs first is applied, replacement should be done as soon as possible when any policy is first triggered, which goes somewhat against the goal of TPM. This result will be used to compare the optimal policies of replacement last in the following section to decide which policy is more economical and also will be shown in Table 6.
which always holds for any N, as 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
(𝑁+1)𝑇
< ℎ(𝑁𝑇 ) <
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
.
This follows that 𝐶𝐹 (𝑁, 𝐾 − 1; 𝑇 ) > 𝐶𝐹 (𝑁, 𝐾; 𝑇 ), and optimum policy to minimize CF (N, K; T) is (𝑁𝐹∗ = 𝑁 ∗ , 𝐾𝐹∗ = ∞). That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of CF (N, K; T), which indicates replacement policy in (5) is more economical than that in (7). 𝑁𝑇 𝑁𝑇 Furthermore, noting that ∫0 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫0 𝑝𝐾 (𝑡)d𝑡 increases strictly with K to h(NT) and increases strictly with N (Appendix B), the left-hand side of (17) increases strictly with K to 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ).
3.2.5. Numerical example Table 3 presents optimum N∗ in (10) and K∗ in (14) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. It shows that when T ≤ T∗ in Table 1, 1 ≤ N∗ < ∞ and 𝐾 ∗ = ∞ and C(N∗ ; T) < CF (K∗ ; T), and when T > T∗ , 𝑁 ∗ = 1 and 1 ≤ K∗ < ∞ and C(N∗ ; T) > CF (K∗ ; T), which agree with the above analytical results. Table 4 presents optimum 𝑁𝐹∗ in (15) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. It shows that 𝑁𝐹∗ decreases with K to N∗ in (10). The comparison between Table 4 and Table 5 will be shown in the following section.
(19)
Thus, if 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) > 𝑐𝑅 ∕𝑐𝑚 , then there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) which satisfies (17). Furthermore, noting that the left-hand side of (17) increases strictly with N to
4. Replacement last
∞
∫0 𝑃 𝐾 (𝑡)d𝑡 ∞
∫0 𝑝𝐾 (𝑡)d𝑡
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 0, 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at a repair number 𝐾 (𝐾 = 0, 1, 2, ⋯), whichever
− 𝐾,
4
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Reliability Engineering and System Safety 000 (2017) 1–10
4.2. Optimum 𝐾𝐿∗
Table 6 Optimum 𝐾𝐹∗ in (17) and 𝐾𝐿∗ in (25) when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. 𝑁 =1
cR /cm
1.0 2.0 5.0 10.0 20.0
Forming the inequality 𝐶𝐿 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ 𝑁𝑇 + 𝑃 ( 𝑡 )d 𝑡 ∞ ∫𝑁𝑇 𝐾 ∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 [ ] ∞ 𝑐 − 𝐻 (𝑁 𝑇 ) + 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥ 𝑅 , ∫𝑁𝑇 𝑐𝑚
𝑁 =5
∞
𝐾𝐹∗
𝐾𝐿∗
𝐾𝐹∗
𝐾𝐿∗
∞ ∞ ∞ ∞ ∞
0 0 5 10 20
1 2 5 10 20
0 0 0 0 0
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
and forming the inequality 𝐶𝐿 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∞ ∫𝑁𝑇 𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡 𝑁𝑇 + 𝑃 ( 𝑡 )d 𝑡 𝐾 ∞ ∫𝑁𝑇 ∫𝑁𝑇 𝑝𝐾−1 (𝑡)d𝑡 [ ] ∞ 𝑐 − 𝐻 (𝑁 𝑇 ) + 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < 𝑀 . ∫𝑁𝑇 𝑐𝑚
occurs last. From the policy definition, it can be understood for replacement last that the unit could operate as long as until any replacement is last triggered, which is somewhat consistent with the goal of TPM. Then, the probability that the unit is replaced at time NT is PK (NT), and the probability that it is replaced at repair K is 𝑃 𝐾 (𝑁𝑇 ). The mean time to replacement is (𝑁 𝑇 )𝑃𝐾 (𝑁 𝑇 ) +
∞
∫𝑁𝑇
𝑡d𝑃𝐾 (𝑡) = 𝑁𝑇 +
∞
∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡,
𝑗𝑝𝑗 (𝑁𝑇 ) + 𝐾 𝑃 𝐾 (𝑁𝑇 ) = 𝐾 +
𝑗=𝐾
= 𝐻 (𝑁 𝑇 ) +
𝐾 ∑ 𝑗=1
∞ ∑ 𝑗=𝐾
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡
(20)
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡.
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡
∞
𝑁𝑇 + ∫𝑁𝑇 𝑃 𝐾 (𝑡)d𝑡
.
(21)
(22)
Clearly, comparing with the expected cost rates in (3)–(6), we obtain
𝐶𝐿 (𝑁, 0; 𝑇 ) = 𝐶𝐹 (𝑁, ∞; 𝑇 ) = 𝐶(𝑁; 𝑇 ),
𝑁𝐿∗
(𝑁+1)𝑇 ∞
−
∫𝑁𝑇
[ 𝑁𝑇 +
𝑃𝐾 (𝑡)d𝑡
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
∞
∫𝑁𝑇
] 𝑃 𝐾 (𝑡)d𝑡 − 𝐻 (𝑁 𝑇 )
𝑐𝑅 , 𝑐𝑚
𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡 ∞
−
∫𝑁𝑇
(27)
𝑁𝑇 e−𝐻 (𝑁 𝑇 ) 𝑁 𝑇 e−𝐻 ((𝑁 −1)𝑇 ) −𝐻 (𝑁 𝑇 )>𝑁 𝑇 ℎ(𝑁 𝑇 )−𝐻 (𝑁 𝑇 )> ∞ − 𝐻 (𝑁 𝑇 ). ∞ ∫𝑁𝑇 e−𝐻(𝑡) d𝑡 ∫(𝑁−1)𝑇 e−𝐻(𝑡) d𝑡
(23)
Thus, we obtain
and forming the inequality 𝐶𝐿 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) > 0, ∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡 [
.
then there exists a finite and unique minimum 𝐾𝐿∗ (1 ≤ 𝐾𝐿∗ < ∞) which satisfies (25). Furthermore, noting that the left-hand side of (25) increases strictly with N from that of (11). Thus, 𝐾𝐿∗ decreases strictly with N from K∗ and 𝐾𝐿∗ ≤ 𝐾 ∗ . Comparing (27) with the left-hand side of (10),
𝐾𝐿∗
We find optimum and to minimize CL (N, K; T) in (22) when T < T∗ . Forming the inequality 𝐶𝐿 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0,
∫𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡
𝑐 𝑁𝑇 e−𝐻 (𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) < 𝑅 , ∞ 𝑐𝑚 ∫𝑁𝑇 e−𝐻(𝑡) d𝑡
4.1. Optimum 𝑁𝐿∗
𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
to ∞. Thus, if
𝐶𝐿 (1, 0; 𝑇 ) = 𝐶(𝑇 ).
(𝑁+1)𝑇
,
𝑁𝑇
> ℎ(𝑁𝑇 ) >
𝑁𝑇 e−𝐻 (𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) ∞ ∫𝑁𝑇 e−𝐻(𝑡) d𝑡
𝐶𝐿 (0, 𝐾; 𝑇 ) = 𝐶𝐹 (∞, 𝐾; 𝑇 ) = 𝐶 (𝐾 ),
∫𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡
This follows that 𝐶𝐿 (𝑁, 𝐾 + 1; 𝑇 ) > 𝐶𝐿 (𝑁, 𝐾; 𝑇 ), and optimum policy to minimize CL (N, K; T) is (𝑁𝐿∗ = 𝑁 ∗ , 𝐾𝐿∗ = 0). That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of CL (N, K; T), which also indicates replacement policy in (5) is more economical than that in (7). ∞ ∞ Furthermore, noting that ∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 increases with ∞ − 𝐻 ( 𝑁 𝑇 ) − 𝐻( 𝑡 ) K from e ∕ ∫𝑁𝑇 e d𝑡 to ∞, and increases strictly with N from ∞ 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 (Appendix D), the left-hand side of (25) increases strictly with K from
∞
𝑐𝑅 + 𝑐𝑚 [𝐻(𝑁𝑇 ) + ∫𝑁𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡]
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
Therefore, the expected repair and replacement cost rate is 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) =
>
which always holds for any N, as
∞
𝑃 𝑗 (𝑁𝑇 ) = 𝐻 (𝑁 𝑇 ) +
𝑁𝑇
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑗 − 𝐾)𝑝𝑗 (𝑁𝑇 ) ∞
(26)
Substituting (24) for (25),
and the expected number of repairs until replacement is ∞ ∑
(25)
𝑁𝑇 +
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < (𝑁+1)𝑇
∞
∫𝑁𝑇
(a) If N < N∗ , then there exists a finite and unique minimum 𝐾𝐿∗ (1 ≤ 𝐾𝐿∗ < ∞) which satisfies (25). (b) If N ≥ N∗ , then 𝐾𝐿∗ = 0, i.e., the model in (22) is degraded into the one in (5).
] 𝑃 𝐾 (𝑡)d𝑡 − 𝐻 (𝑁 𝑇 )
𝑐𝑅 . 𝑐𝑚
(24)
Using the comparative method in [21], we conclude for replacement first and last that:
(𝑁+1)𝑇
Noting that ∫𝑁𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑃𝐾 (𝑡)d𝑡 increases strictly with N to ∞ and increases strictly with K from [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 (Appendix C), the left-hand side of (23) increases strictly with K from that of (10). Thus, from the assumption of T < T∗ , there exists a finite and unique minimum 𝑁𝐿∗ (0 ≤ 𝑁𝐿∗ < ∞) which satisfies (23), and 𝑁𝐿∗ increases with K from N∗ given in (10) and 𝑁𝐿∗ ≥ 𝑁 ∗ . Table 5 presents optimum 𝑁𝐿∗ in (23) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 4, it can be found that replacement last is more economical than replacement first when K is smaller or cM /cm is larger.
(a) If N < N∗ , then 1 ≤ 𝐾𝐿∗ < ∞ and 𝑁𝐹∗ = ∞, i.e., we should plan the policy of replacement last rather than replacement first. (b) If N > N∗ , then 1 ≤ 𝐾𝐹∗ < ∞ and 𝑁𝐿∗ = 0, i.e., we should plan the policy of replacement first rather than replacement last. (c) If 𝑁 = 𝑁 ∗ , then 𝐾𝐹∗ = ∞ and 𝐾𝐿∗ = 0, i.e., replacement should be planned only at a predefined time NT in (5). Table 6 presents optimum 𝐾𝐹∗ in (17) and 𝐾𝐿∗ in (25) when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Clearly, this numerical result agrees with the above analytical discussions. 5
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 7 Optimum 𝐾𝑂∗ in (32) and its cost rate when 𝑇 = 1.0 and 𝐻(𝑡) = 𝑡𝑚 .
5. Replacement overtime It has been indicated above that replacement policies are more easily to be performed at periodic times, e.g., monthly holidays. We next delay the replacement at repair K during [𝑛𝑇 , (𝑛 + 1)𝑇 ] to the next periodic time (𝑛 + 1)𝑇 for the above replacement first and last, which is also a possible plan during the busy production state for the JIT principle in TPM.
cR /cm
1.0 2.0 5.0 10.0 20.0
5.1. Replacement first Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at the next periodic time when a number 𝐾 (𝐾 = 1, 2, ⋯) of minimal repairs have occurred, whichever occurs first. Then, the probability that the unit is replaced at time NT is 𝑃 𝐾 (𝑁𝑇 ), and the probability that it is replaced over repair K is PK (NT). Letting 𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )] be the probability that i repairs occur in [𝑛𝑇 , (𝑛 + 1)𝑇 ], then the probability that the unit is replaced at time (𝑛 + 1)𝑇 (𝑛 = 0, 1, 2, ⋯) when more than K repairs have occurred in [𝑛𝑇 , (𝑛 + 1)𝑇 ] is 𝐾−1 ∑ 𝑗=0
=
𝑝𝑗 [𝐻 (𝑛𝑇 )]
∞ ∑
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}.
The mean time to replacement is (𝑁 𝑇 )
𝐾−1 ∑ 𝑗=0
=𝑇
𝑁−1 ∑ 𝑛=0
𝑝𝑗 [𝐻 (𝑛𝑇 )] +
𝑁−1 ∑
𝐾−1 ∑
𝑛=0
𝑗=0
[(𝑛 + 1)𝑇 ]
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}
(28)
−
𝑗=0
=
𝑛=0 𝑗=0
𝑝𝑗 [𝐻(𝑛𝑇 )]
∞ ∑ 𝑖=𝐾−𝑗
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ).
−
∞ ∑
(30)
𝑛=0
where
∑𝑁−1
𝑄(𝑁, 𝐾; 𝑇 ) =
𝑛=0
∞ ∑ 𝑐 [𝐻((𝑛 + 1)𝑇 − 𝐻(𝑛𝑇 ))]𝑃 𝐾 (𝑛𝑇 ) ≥ 𝑅 , 𝑐𝑚 𝑛=0
∑∞ 𝑁→∞
𝑁−1 ∑
𝑐𝑅 , 𝑐𝑚
(33)
𝑃 𝐾 (𝑛𝑇 )
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑁−1 ∑ 𝑛=0
𝑄(𝑁, 𝐾 − 1; 𝑇 )
(31)
𝑃 𝐾 (𝑛𝑇 ) −
𝑁−1 ∑ 𝑛=0
𝑐𝑅 . 𝑐𝑚
(34)
𝑁−1 ∑ 𝑛=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑐𝑅 , 𝑐𝑚
(35)
𝑃 𝐾 (𝑛𝑇 ) −
𝑁−1 ∑ 𝑛=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 . 𝑐𝑚 (36)
Substituting (33) for (36), 𝑄(𝑁, 𝐾 − 1; 𝑇 ) < [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )].
(32)
which always holds for any N, as 𝑄(𝑁, 𝐾 − 1; 𝑇 ) ≤ [𝐻(𝑁𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] < [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]. This follows that COF (N, K; T) decreases strictly with K, and the opti∗ = 𝑁 ∗ , 𝐾 ∗ = ∞), which mum policy to minimize COF (N, K; T) is (𝑁𝑂𝐹 𝑂𝐹 shows the same result as above in Section 3.2.4. That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of COF (N, K; T), which indicates replacement policy in (5) is more economical than that in (31).
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) , ∑𝑁−1 𝑛=0 𝑝𝐾 (𝑛𝑇 )
𝑄(𝐾; 𝑇 ) ≡ lim 𝑄(𝑁, 𝐾; 𝑇 ) =
𝑃 𝐾 (𝑛𝑇 )
and forming the inequality 𝐶𝑂𝐹 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
𝑛=0 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) . ∑ 𝑇 ∞ 𝑛=0 𝑃 𝐾 (𝑛𝑇 )
and
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑄(𝑁, 𝐾; 𝑇 )
∑∞
𝑃 𝐾 (𝑛𝑇 ) −
3.35 4.08 6.27 9.15 14.17
The left-hand side of (33) increases strictly with N to ∞ and increases strictly with K to that of (10). Thus, there exists a finite and unique ∗ (1 ≤ 𝑁 ∗ < ∞) which satisfies (33), and 𝑁 ∗ decreases minimum 𝑁𝑂𝐹 𝑂𝐹 𝑂𝐹 ∗ ≥ 𝑁 ∗. ∗ with K to N given in (10) and 𝑁𝑂𝐹 Forming the inequality 𝐶𝑂𝐹 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
We find optimum 𝐾𝑂∗ to minimize CO (K; T) in (31) when T < T∗ . Forming the inequality 𝐶𝑂 (𝐾 + 1; 𝑇 ) − 𝐶𝑂 (𝐾; 𝑇 ) ≥ 0, 𝑄(𝐾; 𝑇 )
𝑁−1 ∑ 𝑛=0
𝐶𝑂 (𝐾; 𝑇 ) = lim 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) 𝑐𝑅 + 𝑐𝑚
1 1 1 3 4
𝑛=0
5.1.1. Optimum 𝐾𝑂∗ When the unit is replaced at the next periodic time over repair K, the expected cost rate is
=
2.31 3.03 4.62 6.43 9.02
[𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )]
(29)
Therefore, the expected repair and replacement cost rate is ∑ 𝑐𝑅 + 𝑐𝑚 𝑁−1 𝑛=0 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) = . ∑ 𝑇 𝑁−1 𝑛=0 𝑃 𝐾 (𝑛𝑇 )
𝑁→∞
1 1 3 8 16
and forming the inequality 𝐶𝑂𝐹 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
(𝑖 + 𝑗)𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑁−1 ∑ 𝑛=0
𝑁−1 ∑ 𝑛=0
𝑗𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
𝐶𝑂 (𝐾𝑂∗ ; 𝑇 )∕𝑐𝑚
𝑛=0
𝑃 𝐾 (𝑛𝑇 ),
𝑁−1 ∑ 𝐾−1 ∑
𝐾𝑂∗
[𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]
and the expected number of repairs until replacement is 𝐾−1 ∑
𝐶𝑂 (𝐾𝑂∗ ; 𝑇 )∕𝑐𝑚
∗ and 𝐾 ∗ 5.1.2. Optimum 𝑁𝑂𝐹 𝑂𝐹 In order to compare the optimum policies of N and K for C(N; T) in (5) and CO (K; T) in (31), the optimizations of COF (N, K; T) in ∗ ∗ (30) are discussed as follows. We find optimum 𝑁𝑂𝐹 and 𝐾𝑂𝐹 to ∗ minimize COF (N, K; T) in (30) when T < T . Forming the inequality 𝐶𝑂𝐹 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
𝐾−1 ∑ 𝑗=0
𝑚 = 3.0
𝐾𝑂∗
which increases strictly with K to ∞ (Appendix E). Thus, there exists a finite and unique minimum 𝐾𝑂∗ (1 ≤ 𝐾𝑂∗ < ∞) which satisfies (32). Table 7 presents optimum 𝐾𝑂∗ in (32) and its cost rate when 𝑇 = 1.0 and 𝐻(𝑡) = 𝑡𝑚 . Comparing with Table 3 when 𝑇 = 1.0 and 𝑚 = 2, 0, 𝐾𝑂∗ < 𝐾 ∗ , and 𝐶𝑂 (𝐾𝑂∗ ; 𝑇 ) < 𝐶𝐹 (𝐾 ∗ ; 𝑇 ) when cR /cm becomes larger. It is easy to understand that repair cost would possibly increase due to replacement delay so that optimum replacement time should be earlier than the original one, and also, replacement overtime would be more economical than the original one if repair cost becomes smaller.
𝑝𝑖 [𝐻 ((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑖=𝐾−𝑗
𝑚 = 2.0
𝑛=0 [𝐻((𝑛 +
1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) , ∑∞ 𝑛=0 𝑝𝐾 (𝑛𝑇 ) 6
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X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10 Table 8 ∗ in (33) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Optimum 𝑁𝑂𝐹 𝐾=1
cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=5
𝐾 = 10
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
1 2 3 4 8
2.27 3.03 5.19 8.80 16.01
1 2 2 3 5
2.50 3.48 4.62 6.54 10.27
1 1 2 3 5
2.50 3.00 4.67 6.45 9.24
Table 9 ∗ in (40) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Optimum 𝑁𝑂𝐿 𝐾=1
cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=5
𝐾 = 10
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
1 2 2 3 5
2.31 3.02 4.50 6.33 9.00
2 2 2 3 5
3.18 3.55 4.67 6.35 9.00
2 2 3 3 5
3.99 4.27 5.10 6.47 9.00
Furthermore, noting that Q(N, K; T) increases strictly with K to [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] and increases strictly with N (Appendix E), the left-hand side of (35) increases strictly with K to
Therefore, the expected repair and replacement cost rate is ∑ 𝑐𝑅 + 𝑐𝑚 {𝐻(𝑁𝑇 ) + ∞ 𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 )} 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) = . ∑ 𝑁𝑇 + 𝑇 ∞ 𝑛=𝑁 𝑃 𝐾 (𝑛𝑇 ) (39)
𝑁 [𝐻 (𝑁 𝑇 )−𝐻 ((𝑁 − 1)𝑇 )]−𝐻 (𝑁 𝑇 )<𝑁 [𝐻 ((𝑁 + 1)𝑇 )−𝐻 (𝑁 𝑇 )]−𝐻 (𝑁 𝑇 ),
Clearly, comparing with the expected cost rates in (5), (30) and (31), we obtain
which agrees with (10). Thus, we obtain ∗ (1 ≤ (a) If N > N∗ , then there exists a finite and unique minimum 𝐾𝑂𝐹 ∗ < ∞) which satisfies (35) 𝐾𝑂𝐹 ∗ = ∞, i.e., the model in (30) is degraded into the (b) If N ≤ N∗ , then 𝐾𝑂𝐹 one in (5).
𝐶𝑂𝐿 (0, 𝐾; 𝑇 ) = 𝐶𝑂𝐹 (∞, 𝐾; 𝑇 ) = 𝐶𝑂 (𝐾; 𝑇 ), 𝐶𝑂𝐿 (𝑁, 0; 𝑇 ) = 𝐶𝑂𝐹 (𝑁, ∞; 𝑇 ) = 𝐶(𝑁; 𝑇 ). ∗ ∗ We find optimum 𝑁𝑂𝐿 and 𝐾𝑂𝐿 to minimize COL (N, K; T) in (39) when T < T∗ . Forming the inequality 𝐶𝑂𝐿 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ ∑ [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )] 𝑁 + 𝑃 𝐾 (𝑛𝑇 )
This result will be used to compare the optimal policies of replacement last in the following section to decide which policy is more economical and also will be shown in Table 9. ∗ Table 8 presents optimum 𝑁𝑂𝐹 in (33) and its cost rate when ∗ ≤ 𝑁 ∗ and 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 4, 𝑁𝑂𝐹 𝐹 ∗ ∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 ) < 𝐶𝐹 (𝑁𝐹 , 𝐾; 𝑇 ) when cM /cm becomes larger or given K becomes smaller.
𝑛=𝑁
− 𝐻 (𝑁 𝑇 ) −
𝑛=𝑁
𝑗=𝐾
𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
= 𝑁𝑇 + 𝑇
∞ ∑ 𝑛=𝑁
∞ ∑
[(𝑛 + 1)𝑇 ]
− 𝐻 (𝑁 𝑇 ) −
𝑗=0
− 𝐻 (𝑁 𝑇 ) −
𝑗=𝐾
𝑗𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
= 𝐻 (𝑁 𝑇 ) +
∞ ∑ 𝑛=𝑁
∞ 𝐾−1 ∑ ∑ 𝑛=𝑁 𝑗=0
𝑝𝑗 [𝐻(𝑛𝑇 )]
∞ ∑ 𝑖=𝐾−𝑗
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 . 𝑐𝑚
(41)
𝑛=𝑁
(37)
∞ ∑ 𝑛=𝑁
and the expected number of repairs until replacement is ∞ ∑
(40)
∗ (0 ≤ 𝑁 ∗ < ∞) Thus, there exists a finite and unique minimum 𝑁𝑂𝐿 𝑂𝐿 ∗ increases with K from N∗ given in (10) and which satisfies (40), and 𝑁𝑂𝐿 ∗ ≥ 𝑁 ∗. 𝑁𝑂𝐿 Forming the inequality 𝐶𝑂𝐿 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ ∑ ̃(𝑁, 𝐾; 𝑇 ) 𝑁 + 𝑄 𝑃 𝐾 (𝑛𝑇 )
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}
𝑃 𝐾 (𝑛𝑇 ),
∞ ∑ 𝑛=𝑁
𝐾−1 ∑
𝑛=𝑁
𝑐𝑅 , 𝑐𝑚
𝑛=𝑁
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 0, 1, 2, ⋯ ; 𝑇 > 0) or at the next periodic time when a number 𝐾 (𝐾 = 0, 1, 2, ⋯) of repairs have occurred, whichever occurs last. Then, the probability that the unit is replaced at time NT is PK (NT), and the probability that it is replaced over repair K is 𝑃 𝐾 (𝑁𝑇 ). The mean time to replacement is ∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
and forming the inequality 𝐶𝑂𝐿 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∑ [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] 𝑁 + 𝑃 𝐾 (𝑛𝑇 )
5.2. Replacement last
(𝑁 𝑇 )
∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑐𝑅 , 𝑐𝑚
(42)
and forming the inequality 𝐶𝑂𝐿 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∑ ̃(𝑁, 𝐾 − 1; 𝑇 ) 𝑁 + 𝑄 𝑃 𝐾 (𝑛𝑇 )
(𝑖 + 𝑗)𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑛=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ).
(38)
− 𝐻 (𝑁 𝑇 ) −
∞ ∑ 𝑛=𝑁
7
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 , 𝑐𝑚
(43)
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X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10 Table 10 ∗ ∗ in (33) and 𝐾𝑂𝐿 in (42) when Optimum 𝐾𝑂𝐹 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
where ̃(𝑁, 𝐾; 𝑇 ) ≡ 𝑄
𝑁 =1
It has been shown that replacement first includes: (i) The unit is replaced at time T, (ii) the unit is replaced at time NT, (iii) the unit is replaced at repair K, and (iv) the unit is replaced at repair K or at time T. We have obtained the comparative results as follows: Policy (i) is more economical than those (iii) and (iv). Policy (ii) is more economical than that (iv) only when T ≤ T∗ , which has also been obtained from the optimization of replacement last. When we compare the policies of replacement first and replacement last, i.e., (v) the unit is replaced at time NT or at repair K, whichever occurs first, and (vi) the unit is replaced at time NT or at repair K, whichever occurs last, it has been found that whether replacement last is more economical or not depends on the scale of the predefined policy and the cost ratio of replacement and repair. Finally, we delay replacement at repair K to periodic time, i.e., (vii) the unit is replaced at the next periodic time when a number K of repairs have occurred. Comparing with the original policy (iii), its optimum policy should be done early due to increasing repair cost. This modification of repair K has also been applied for replacement first and replacement last, i.e., (viii) the unit is replaced at time NT or at the next periodic time over repair K, whichever occurs first, and (ix) the unit is replaced at time NT or at the next periodic time over repair K, whichever occurs last. Similar comparisons have been obtained.
𝑁 = 10
∗ 𝐾𝑂𝐹
∗ 𝐾𝑂𝐿
∗ 𝐾𝑂𝐹
∗ 𝐾𝑂𝐿
∞ ∞ ∞ ∞ ∞
0 0 3 8 16
1 1 3 8 16
0 0 0 0 0
∑∞
𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) . ∑∞ 𝑛=𝑁 𝑝𝐾 (𝑛𝑇 )
Substituting (40) for (43), ̃(𝑁, 𝐾 − 1; 𝑇 ) < [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )], 𝑄 which does not hold for any N. This follows that COL (N, K; T) increases strictly with K, and the optimum policy to minimize COL (N, K; ∗ = 𝑁 ∗ , 𝐾 ∗ = 0), which shows the same result as above in T) is (𝑁𝑂𝐿 𝑂𝐿 Section 4.2. That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of COL (N, K; T), which indicates replacement policy in (5) is more economical than that in (39). ̃(𝑁, 𝐾; 𝑇 ) increases strictly with K Furthermore, noting that 𝑄 (Appendix F), the left-hand side of (42) increases strictly with K from ∑ −𝐻(𝑛𝑇 ) 𝑁 ∞ 𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]e − 𝐻 (𝑁 𝑇 ) ∑∞ −𝐻(𝑛𝑇 ) 𝑛=𝑁 e
Acknowledgement This paper was made possible by Qatar National Research Fund (No. PDRA1- 0116-14107) and National Natural Science Foundation of China (No. 71371097).
>𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ), Appendix A
which agrees with that of (10). Thus, we obtain ∗ (1 ≤ (a) If N < N∗ , then there exists a finite and unique minimum 𝐾𝑂𝐿 ∗ 𝐾𝑂𝐿 < ∞) which satisfies (42) ∗ = 0, i.e., the model in (39) is degraded into the (b) If N ≥ N∗ , then 𝐾𝑂𝐿 one in (5).
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 1, 2 ⋯ , (𝑁+1)𝑇
𝐻1 (𝑁, 𝐾; 𝑇 ) ≡
Finally, we conclude for replacement first and last that:
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇
∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
increases strictly with N from H1 (0, K; T) to h(∞) and increases strictly with K from H1 (N, 1; T) to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 .
∗ < ∞ and 𝑁 ∗ = ∞, i.e., we should plan the (a) If N < N∗ , then 1 ≤ 𝐾𝑂𝐿 𝐹 policy of replacement last rather than replacement first. ∗ < ∞ and 𝑁 ∗ = 0, i.e., we should plan the (b) If N > N∗ , then 1 ≤ 𝐾𝑂𝐹 𝑂𝐿 policy of replacement first rather than replacement last. ∗ = ∞ and 𝐾 ∗ = 0, i.e., replacement should be (c) If 𝑁 = 𝑁 ∗ , then 𝐾𝑂𝐹 𝐿 planned only at a predefined time NT in (5).
Proof. Note that ℎ(𝑁𝑇 ) < 𝐻1 (𝑁, 𝐾; 𝑇 ) < ℎ((𝑁 + 1)𝑇 ),
lim 𝐻1 (𝑁, 𝐾; 𝑇 ) = ℎ(∞),
𝑁→∞
𝐻((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 ) . 𝑇
lim 𝐻1 (𝑁, 𝐾; 𝑇 ) =
𝐾→∞
∗ in (40) and its cost rate when 𝑇 = 1.0, Table 9 presents optimum 𝑁𝑂𝐿 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 8, it can be found that replacement last is more economical than replacement first when K is smaller or cR /cm is larger. ∗ in (33) and 𝐾 ∗ in (42) when 𝑇 = Table 10 presents optimum 𝐾𝑂𝐹 𝑂𝐿 𝑚 1.0, 𝐻(𝑡) = 𝑡 and 𝑚 = 2.0, which also agrees with the above analytical ∗ ≤ 𝐾∗ discussions. Table 10 has the same property with Table 6, and 𝐾𝑂𝐹 𝐹 ∗ ≤ 𝐾∗ . and 𝐾𝑂𝐿 𝐿
Forming 𝐻1 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐻1 (𝑁, 𝐾; 𝑇 ),
6. Conclusions
which follows that H1 (N, K; T) increases strictly with N from H1 (0, K; T) to h(∞). Similarly, forming 𝐻1 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐻1 (𝑁, 𝐾; 𝑇 ),
(𝑁+2)𝑇
∫(𝑁+1)𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇
−
∫𝑁𝑇 (𝑁+2)𝑇
=
In the viewpoint of cost rate, replacement polices that are carried out at periodic times NT and at repair numbers K have been compared analytically from the optimizations of the integrated models. In addition, other comparative results among replacement policies, such as replacement first and replacement last, replacement overtime and its original model, have also been obtained analytically and numerically. Our comparisons have started from formulating the integrated models with two above replacement polices, which have been widely used in total productive replacement (TPM) management in Japanese industry.
∫(𝑁+1)𝑇
𝐾 ∑ 𝑗=0
𝑗=0
(𝑁+1)𝑇
8
∫𝑁𝑇
𝑝𝐾 (𝑡)
(𝑁+2)𝑇
∫𝑁𝑇
𝐾−1 ∑ 𝑗=0
𝑝𝑗 (𝑡)d𝑡 {𝐾−1 ∑ 𝑗=0
𝑃 𝐾 (𝑢)d𝑢
∫(𝑁+1)𝑇
(𝑁+1)𝑇
𝑝𝑗 (𝑡)ℎ(𝑡)d𝑡
∫𝑁𝑇
(𝑁+1)𝑇
=
𝑃 𝐾 (𝑢)ℎ(𝑢)d𝑢
𝑃 𝐾 (𝑡)
∫𝑁𝑇 𝐾 ∑
∫𝑁𝑇
{
(𝑁+1)𝑇
−
(𝑁+1)𝑇
∫𝑁𝑇
𝑗=0
d𝑡 > 0,
𝑝𝑗 (𝑢)d𝑢
(𝑁+1)𝑇
∫𝑁𝑇
(𝑁+1)𝑇
∫𝑁𝑇
}
𝑃 𝐾 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
(𝑁+1)𝑇
𝐾−1 ∑
𝑃 𝐾 (𝑡)d𝑡
𝑝𝑗 (𝑢)ℎ(𝑢)d𝑢 }
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
d𝑡
ARTICLE IN PRESS
JID: RESS X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10
(𝑁+1)𝑇
=
∫𝑁𝑇
𝑝𝐾 (𝑡)
(𝑁+1)𝑇
+
∫𝑡
(𝑁+1)𝑇
∫𝑁𝑇
𝑡
})
𝑝𝑗 (𝑡)
Proof. Note that
d𝑡.
̃ 1 (𝑁, 𝐾; 𝑇 ) < ℎ((𝑁 + 1)𝑇 ), ℎ(𝑁𝑇 ) < 𝐻
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
̃ 1 (𝑁, 𝐾; 𝑇 ) = ℎ((𝑁 + 1)𝑇 ). lim 𝐻
Thus, by using the similar method of Appendix A, Appendix C can be proved. □
d𝑡
} 𝑝𝐾 (𝑢)[ℎ(𝑢) − ℎ(𝑡)]d𝑡 d𝑢,
𝑡
∫𝑁𝑇
̃ 1 (𝑁, 𝐾; 𝑇 ) = ℎ(∞), lim 𝐻
𝑁→∞
𝐾→∞
}
(𝑁+1)𝑇
∫𝑡 {
̃ 1 (0, 𝐾; 𝑇 ) to h(∞) and increases strictly increases strictly with N from 𝐻 with K from [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 to ℎ((𝑁 + 1)𝑇 ).
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
∫𝑁𝑇
𝑗=0
{
𝑝𝐾 (𝑡)
(𝑁+1)𝑇
=
(𝐾−1 { ∑
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
Noting that ∫𝑁𝑇
[m5GeSdc;May 15, 2017;12:4]
Appendix D
the above equation is (𝐾−1 { }) (𝑁+1)𝑇 𝑡 ∑ 𝑝𝐾 (𝑡) 𝑝 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢 d𝑡 ∫𝑁𝑇 ∫𝑁𝑇 𝑗 𝑗=0 (𝐾−1 { }) (𝑁+1)𝑇 𝑡 ∑ − 𝑝𝑗 (𝑡) 𝑝𝐾 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢 d𝑡 ∫𝑁𝑇 ∫𝑁𝑇 𝑗=0 (𝐾−1 (𝑁+1)𝑇 ∑ 𝑡 = 𝑝𝐾 (𝑢)𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)] ∫𝑁𝑇 ∫ 𝑗=0 𝑁𝑇 {[ ]𝐾 [ ]𝑗 } ) 𝐻(𝑡) 𝐻(𝑡) × − d𝑢 d𝑡 > 0, 𝐻(𝑢) 𝐻(𝑢)
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , ∞
̃ 2 (𝑁, 𝐾; 𝑇 ) = 𝐻
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡 ∞
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 ∞
increases strictly with N from 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 to h(∞) and increases strictly ̃ 2 (𝑁, 0; 𝑇 ) to h(∞). with K from 𝐻 Proof. Note that ̃ 2 (𝑁, 𝐾; 𝑇 ) < ℎ(∞), ℎ(𝑁𝑇 ) < 𝐻 ̃ 2 (𝑁, 𝐾; 𝑇 ) = lim 𝐻 ̃ 2 (𝑁, 𝐾; 𝑇 ) = ℎ(∞). lim 𝐻
𝑁→∞
which follows that H1 (N, K; T) increases strictly with K from H1 (N, 1; T) to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 . □
𝐾→∞
Thus, by using the similar method of Appendix B, Appendix D can be proved. □
Appendix B Appendix E
For 0 < T < ∞, 𝑁 = 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , 𝑁𝑇
𝐻2 (𝑁, 𝐾; 𝑇 ) ≡
∫0
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , Q(N, K; T) increases strictly with N from H(T) to Q(∞, K; T) and increases strictly with K from Q(N, 0; T) to 𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 ).
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾 (𝑡)d𝑡 ∞
increases strictly with N from H2 (1, K; T) to 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 and increases strictly with K from H2 (N, 0; T) to h(NT).
Proof. Note that
Proof. Note that
𝐾→∞
𝐻2 (𝑁, 𝐾; 𝑇 ) < ℎ(𝑁𝑇 ),
lim 𝑄(𝑁, 𝐾; 𝑇 ) = 𝐻(𝑁𝑇 ) − 𝐻((𝑁 − 1)𝑇 ).
Forming 𝑄(𝑁 + 1, 𝐾; 𝑇 ) − 𝑄(𝑁, 𝐾; 𝑇 ),
1
lim 𝐻2 (𝑁, 𝐾; 𝑇 ) =
, ∞ ∫0 𝑝𝐾 (𝑡)d𝑡
𝑁→∞
𝑁 𝑁−1 ∑ ∑ [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) 𝑝𝐾 (𝑛𝑇 )
lim 𝐻2 (𝑁, 𝐾; 𝑇 ) = ℎ(𝑁𝑇 ).
𝑛=0
𝐾→∞
Differentiating H2 (1, K; T) with respect to T, 𝑝𝐾 (𝑇 )
𝑇
∫0
𝑛=0
−
𝑝𝐾 (𝑡)[ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 > 0,
𝑇
∫0
𝑝𝐾+1 (𝑡)ℎ(𝑡)d𝑡
𝑇
∫0
𝑝𝐾 (𝑡)d𝑡 −
𝑇
∫0
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑇
∫0
𝑛=0
𝑛=0
𝑁−1 ∑ 𝑛=0
𝑝𝐾 (𝑛𝑇 )
𝑝𝐾 (𝑛𝑇 )[𝐻 ((𝑁 + 1)𝑇 )−𝐻 (𝑁 𝑇 )−𝐻 ((𝑛 + 1)𝑇 ) + 𝐻(𝑛𝑇 )] > 0,
which follows that Q(N, K; T) increases strictly with N from H(T) to Q(∞, K; T). Similarly, forming 𝑄(𝑁, 𝐾 + 1; 𝑇 ) − 𝑄(𝑁, 𝐾; 𝑇 ),
𝑝𝐾+1 (𝑡)d𝑡,
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾+1 e−𝐻(𝑛𝑇 )
we have 𝐿1 (0; 𝐾) = 0 and
𝑛=0
𝑇 d𝐿1 (𝑇 ; 𝐾) = 𝑝𝐾 (𝑇 ) 𝑝 (𝑡)[𝐻(𝑇 ) − 𝐻(𝑡)][ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 > 0, ∫0 𝐾 d𝑇
−
which follows that H2 (N, K; T) increases strictly with K from H2 (N, 0; T) to h(NT). □
=
𝑗=0
𝑛=0
𝑗=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
{
(𝑁+1)𝑇
∫𝑁𝑇
(𝑁+1)𝑇
∫𝑁𝑇
𝑛 ∑ [𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )] 𝑗=0
𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
+
𝑃𝐾 (𝑡)d𝑡
𝑁−1 ∑
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑛
9
[𝐻(𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑁−1 ∑
×
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ ,
𝑁−1 ∑
𝑁−1 ∑
𝑛=0
Appendix C
̃ 1 (𝑁, 𝐾; 𝑇 ) ≡ 𝐻
𝑁 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
= 𝑝𝐾 (𝑁𝑇 )
which follows that H2 (N, K; T) increases strictly with N from H2 (1, K; ∞ T) to 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡. Similarly, denoting 𝐿1 (𝑇 ; 𝐾) ≡
𝑁−1 ∑
} .
[𝐻(𝑗𝑇 )]𝐾+1 e−𝐻(𝑗𝑇 )
ARTICLE IN PRESS
JID: RESS X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10
Noting that
=
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
=
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑁
̃(𝑁, 𝐾; 𝑇 ) increases strictly with K to h(∞). which follows that 𝑄
𝑗=𝑛
References
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
[𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑗=0
the above equation is 𝑁−1 ∑
𝑛 ∑ [𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑛=0
𝑗=0
[𝐻 (𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
× [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 ) − 𝐻((𝑗 + 1)𝑇 ) + 𝐻(𝑗𝑇 )][𝐻(𝑛𝑇 ) − 𝐻(𝑗𝑇 )] > 0, which follows that Q(N, K; T) increases strictly with K from Q(N, 0; T) to 𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 ). □
Appendix F ̃(𝑁, 𝐾; 𝑇 ) increases For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , 𝑄 ̃(0, 𝐾; 𝑇 ) to h(∞) and increases strictly with K from strictly with N from 𝑄 ̃(𝑁, 0; 𝑇 ) to h(∞). 𝑄 Proof. Note that ̃(𝑁, 𝐾; 𝑇 ) = ℎ(∞), lim 𝑄
𝑁→∞
̃(𝑁, 𝐾; 𝑇 ) = ℎ(∞). lim 𝑄
𝐾→∞
̃(𝑁 + 1, 𝐾; 𝑇 ) − 𝑄 ̃(𝑁, 𝐾; 𝑇 ), Forming 𝑄 ∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
𝑛=𝑁+1 ∞ ∑
−
𝑛=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
= 𝑝𝐾 (𝑁𝑇 )
∞ ∑ 𝑛=𝑁
∞ ∑
𝑝𝐾 (𝑛𝑇 )
𝑛=𝑁 ∞ ∑
𝑛=𝑁+1
𝑝𝐾 (𝑛𝑇 )
𝑝𝐾 (𝑛𝑇 )[𝐻((𝑛 + 1)𝑇 )−𝐻(𝑛𝑇 )−𝐻((𝑁 + 1)𝑇 )+𝐻 (𝑁 𝑇 )] > 0,
̃(𝑁, 𝐾; 𝑇 ) increases strictly with N from 𝑄 ̃(0, 𝐾; 𝑇 ) which follows that 𝑄 to h(∞). ̃(𝑁, 𝐾 + 1; 𝑇 ) − 𝑄 ̃(𝑁, 𝐾; 𝑇 ), Similarly, forming 𝑄 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾+1 e−𝐻(𝑛𝑇 )
𝑛=𝑁
∞ ∑
[𝐻(𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑗=𝑁
∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑛=𝑁 ∞ ∑
∞ ∑
[𝐻(𝑗𝑇 )]𝐾+1 e−𝐻(𝑗𝑇 )
𝑗=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑛=𝑁 ∞ ∑
×
□
[1] Yamashina H. Japanese manufacturing strategy and the role of total productive maintenance. J Quality Maint Eng 1995;1:27–38. [2] Wireman T. Total Productive Maintenance. Industrial Press; 2004. [3] Venkatesh J. An introduction to total productive maintenance (TPM). The Plant Maintenance Resource Center; 2007. p. 3–20. [4] Grall A, Dieulle L, Bérenguer C, Roussignol M. Continuous-time predictive-maintenance scheduling for a deteriorating unit. IEEE Trans Reliab 2002;51:141–50. [5] Zhou X, Xi L, Lee J. Reliability-centered predictive maintenance scheduling for a continuously monitored unit subject to degradation. Reliab Eng Unit Saf 2007;92:530–4. [6] Taghipour S, Banjevic D, Jardine A. Periodic inspection optimization model for a complex repairable unit. Reliab Eng Unit Saf 2010;95:944–52. [7] Taghipour S, Banjevic D. Optimum inspection interval for a unit under periodic and opportunistic inspections. IIE Trans 2012;44:932–48. [8] Huynh KT, Barros A, Benguer C, Castro IT. A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events. Reliab Eng Unit Saf 2011;96:497–508. [9] Golmakani H, Moakedi H. Periodic inspection optimisation model for a two-component repairable unit with failure interaction. Comput Indus Eng 2012;63:540–5. [10] Wang W, Banjevic D. Ergodicity of forward times of the renewal process in a block-based inspection model using the delay time concept. Reliab Eng Unit Saf 2012;100:1–7. [11] Taghipour S, Kassaei ML. Periodic inspection optimization of a k-out-of-n load-sharing unit. IEEE Trans Reliab 2015;64:1116–27. [12] Kim DK, Lim JH, Park DH. Optimal maintenance level for second-hand product with periodic inspection schedule. Appl Stochastic Models Bus Indus 2015;31:349–59. [13] Nakagawa T. Advanced Reliability Models and Maintenance Policies. Springer; 2008. [14] Pulcini G. Mechanical reliability and maintenance models. In: Pham H, editor. Handbook of Reliability Engineering. Springer; 2003. p. 317–48. [15] Yuan F, Kumar U. A general imperfect repair model considering time-dependent repair effectiveness. IEEE Trans Reliab 2012;61:95–100. [16] Chien Y, Sheu S. Extended optimal age-replacement policy with minimal repair of a unit subject to shocks. Eur J Oper Res 2006;174:169–81. [17] Huynh K, Castro I, Barros A, Bérenguer C. Modeling age-based maintenance strategies with minimal repairs for systems subject to competing failure modes due to degradation and shocks. Eur J Oper Res 2012;218:140–51. [18] Wang W. Optimum production and inspection modeling with minimal repair and rework considerations. Appl Math Modell 2013;37:1618–26. [19] Park M, Jung K, Park D. Optimal post-warranty maintenance policy with repair time threshold for minimal repair. Reliab Eng Unit Saf 2013;111:147–53. [20] Chang C. Optimum preventive maintenance policies for systems subject to random working times, replacement, and minimal repair. Comput Indus Eng 2014;67:185–94. [21] Zhao X, Al-Khalifa KN, Hamouda AMS, Nakagawa T. First and last triggering event approaches for replacement with minimal repairs. IEEE Trans Reliab 2016;65:197–207. [22] Sheu S, Chang C, Chen Y, Zhang Z. Optimal preventive maintenance and repair policies for multi-state systems. Reliab Eng Unit Saf 2015;140:78–87. [23] Saito Y, Dohi T, Yun W. Kernel-based nonparametric estimation methods for a periodic replacement problem with minimal repair. Proc Inst Mech Eng Part O 2016;230:54–66. [24] Toledo M, Freitas M, Colosimo E, Gilardoni G. Optimal periodic maintenance policy under imperfect repair: a case study of off-road engines. IIE Trans 2016. doi:10. 1080/0740817X.2016.1147663. [25] Barlow RE, Proschan F. Mathematical Theory of Reliability. John Wiley & Sons; 1965. [26] Nakagawa T. Maintenance Theory of Reliability. Springer; 2005. [27] Zhao X, Nakagawa T. Optimization problems of replacement first or last in reliability theory. Eur J Oper Res 2012;223:141–9. [28] Nakagawa T, Zhao X. Maintenance Overtime Policies in Reliability Theory. Switzerland: Springer; 2015. [29] Zhao X, Mizutani S, Nakagawa T. Which is better for replacement policies with continuous or discrete scheduled times? Eur J Oper Res 2015;242:477–86.
[𝐻((𝑗 + 1)𝑇 ) − 𝐻(𝑗 𝑇 )][𝐻(𝑗 𝑇 )]𝐾 [𝐻(𝑗 𝑇 ) − 𝐻(𝑛𝑇 )],
×
=
𝑛 ∑
× [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 ) − 𝐻((𝑗 + 1)𝑇 ) + 𝐻(𝑗𝑇 )][𝐻(𝑛𝑇 ) − 𝐻(𝑗𝑇 )] > 0,
𝑛=0 𝑛 ∑
−
[𝐻 (𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑁−1 ∑
𝑁−1 ∑
∞ ∑
∞ ∑ 𝑛=𝑁
𝑛=0
×
[m5GeSdc;May 15, 2017;12:4]
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑁
10
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Reliability Engineering and System Safety 000 (2017) 1–10
Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Comparisons of replacement policies with periodic times and repair numbers Xufeng Zhao a,b,∗, Cunhua Qian c, Toshio Nakagawa d a
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China Department of Mechanical and Industrial Engineering, Qatar University, Doha, 2713, Qatar c School of Economics and Management, Nanjing Tech University, Nanjing, 211800, China d Department of Business Administration, Aichi Institute of Technology, Toyota, 470-0392, Japan b
a r t i c l e
i n f o
Keywords: Periodic replacement Minimal repair Replacement delay Replacement last Unit failure
a b s t r a c t Periodic replacement policies modeled with the history of minimal repairs have been studied extensively. However, in the viewpoint of cost rate, there is no literature to compare replacement polices which are carried out at some periodic times and at a predetermined number of repairs. In this paper, we compare these two types of replacement policies analytically from the optimizations of the integrated models. It will be shown that there always exists a degradation model when any bivariate replacement policy is optimized and this is just the best choice of the comparisons. Not only that, the approaches of whichever occurs first and last are applied to model the above two types of policies, which are named as replacement first and replacement last, respectively, and their comparisons are also made. In addition, we delay the policy at repair to periodic time for easier replacement, and the modified replacement model, which is named as replacement overtime, is compared with the original ones. Numerical examples are also given and agree with all analytical discussions. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction In order to be strong enough in production capacity, the total productive maintenance (TPM) has been indispensable in Japanese industry since 1970s, which involves an innovative concept of maintaining equipments by everyone in the organization [1,2]. TPM generally uses periodic and predictive maintenances to aim at maximizations of equipment utilization and production stability. Periodic maintenance [3] consists of periodically inspecting, cleaning and servicing equipment and replacing deteriorated parts to prevent serious breakdown and process problem, while predictive maintenance [4,5] is a method to operate the equipment to the limit of its service life, by measuring and analyzing data about deterioration at routine diagnosis and minor repairs. Normally, maintenances are more easily to be performed at periodic times in applications, e.g., a complete maintenance in TPM should be carried out on monthly holidays if the equipment must be stopped and a long time is required for maintenance [2]. Theoretical research works on periodic patterns in maintenance plans were studied extensively. Periodical inspection intervals were optimized to detect soft failures of a complex repairable unit, while hard failures create opportunities for additional inspections of all soft-type components [6,7]. Maintenance policies with periodic inspections were observed for the systems with
∗
several failures [8] and failure interactions [9]. Inspections are carried out periodically on the unit using the delay time concept during two failure state processes [10]. A k-out-of-n load-sharing unit that is periodically inspected to detect failed components was studied [11]. An optimal number of periodic inspections and its maintenance level to minimize the expected total warranty cost for the second-hand product during the warranty period were derived [12]. In order to achieve just-in-time (JIT) principle in TPM [1,2], repairs are included in maintenance schedules in response to all non breakdown deteriorations and resume quickly the operation of equipment after repairable failure. Not only that, the service life of important part can be predicted based on diagnosis at repairs for predictive maintenance decisions in TPM [3]. Repair models have been studied especially for large and complex systems, which consist of many kinds of units [13]. In recent works, models for repairable unit subjected to minimal repair [14], imperfect repair considering time-dependent repair effectiveness [15], age-based replacement with repair for shocks and degradation [16,17], inspection modeling for repairs [18], post-warranty maintenance with repair time threshold [19], random working models with replacement and minimal repair [20,21], etc., have been studied extensively. More recently, preventive maintenance should be planned jointly with the right type of repairs to achieve the best performance for a multi-state
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.ress.2017.05.015 Available online xxx 0951-8320/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: X. Zhao et al., Reliability Engineering and System Safety (2017), http://dx.doi.org/10.1016/j.ress.2017.05.015
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Reliability Engineering and System Safety 000 (2017) 1–10
unit has been studied [22]. A nonparametric estimation method for periodic replacement problem with minimal repairs has been proposed [23]. A case study of periodic maintenance policy under imperfect repair for off-road engines has been surveyed [24]. It has been well-known in reliability theory that the cost rate model for periodic replacement is formulated by using the cumulative hazard function H(t), where H(t) presents the expected number of failures during [0, t] when the failure rate of a repairable unit remains undisturbed by minimal repairs [25,26]. Recent periodic maintenance models based on the history of failures/repairs can be found extensively in above literatures. However, in the viewpoint of cost rate, there is no study until now to compare replacement polices that are carried out at some periodic times and at a predetermined number of repairs, which becomes the main problem to be discussed in this paper. Furthermore, in order to achieve the target of maximizations of equipment utilization and production stability in TPM [3], the approaches of modelings for replacement first [26], replacement last [27], and replacement overtime [28,29] are used to formulate our replacement models, that is, three approaches of whichever triggering event occurs first, whichever triggering event occurs last, and replacing over a planned measure will be taken into considerations in replacement policies, respectively. It has been shown in [21,27–29] that replacement last and replacement overtime could let the equipment operate as long as possible, which will also be discussed and compared for the models in this paper. Our comparisons start from formulating the integrated models with two types of maintenances that are planned at periodic times and at repair numbers. Obviously, when the optimal maintenance policies are obtained in separative models, it will be very easy to compare them with numerical examples in viewpoint of cost rates. However, we compare these two types of maintenances analytically from the optimizations of the integrated models. It will be shown that there always exists a degradation model when any bivariate replacement policy is optimized and it is just the best choice of the comparison, which agrees with the comparisons obtained in [21]. In addition, other comparative results among replacement policies are obtained analytically and numerically.
curs first. Then, the probability that the unit is replaced at time NT is 𝑃 𝐾 (𝑁𝑇 ), and the probability that it is replaced at failure K is PK (NT). Denoting 𝑃 𝑗 (𝑡) ≡ 1 − 𝑃𝑗 (𝑡), and noting that 𝑃𝑗 (𝑇 ) = 𝐾 ∑
𝑇
𝑃𝑗 (𝑇 ) =
𝑗=1 ∞ ∑ 𝑗=𝐾
𝑝𝑗−1 (𝑡)ℎ(𝑡)d𝑡,
∫0
𝑇
∫0
𝑃𝑗 (𝑇 ) =
𝑇
∫0
𝐾 ∑
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡,
𝑗=1
𝑝𝑗−1 (𝑡)ℎ(𝑡)d𝑡,
𝑃 𝑗 (𝑇 ) =
∞ ∑
𝑃𝐾−1 (𝑡)ℎ(𝑡)d𝑡,
∞
∫𝑇
𝑗=𝐾
∞
∫𝑇
𝑃 𝑗 (𝑇 ) =
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡, ∞
∫𝑇
𝑃𝐾−1 (𝑡)ℎ(𝑡)d𝑡,
the mean time to replacement is (𝑁 𝑇 )𝑃 𝐾 (𝑁 𝑇 ) +
𝑁𝑇
∫0
𝑁𝑇
𝑡d𝑃𝐾 (𝑡) =
∫0
𝑃 𝐾 (𝑡)d𝑡,
(1)
and the expected number of minimal repairs until replacement is 𝐾−1 ∑ 𝑗=0
=
𝑗𝑝𝑗 (𝑁𝑇 ) + 𝐾𝑃𝐾 (𝑁𝑇 ) = 𝐾 −
𝐾 ∑ 𝑗=1
𝑃𝑗 (𝑁𝑇 ) =
𝑁𝑇
∫0
𝐾 ∑ (𝐾 − 𝑗)𝑝𝑗 (𝑁𝑇 ) 𝑗=0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡,
(2)
Therefore, the expected repair and replacement cost rate is 𝑁𝑇
𝐶𝐹 (𝑁, 𝐾; 𝑇 ) =
𝑐 𝑅 + 𝑐 𝑚 ∫0
𝑁𝑇 ∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑃 𝐾 (𝑡)d𝑡
,
(3)
where 𝑐𝑅 = replacement cost at time NT or at repair K, and 𝑐𝑚 = cost of each minimal repair. The integrated model in (3) includes the following replacement policies: When the unit is replaced at time T (0 < T < ∞), 𝐶(𝑇 ) ≡ lim 𝐶𝐹 (1, 𝐾; 𝑇 ) = 𝐾→∞
𝑐𝑅 + 𝑐𝑚 𝐻(𝑇 ) , 𝑇
(4)
which agrees with the expected cost rate of the original periodic model with time T [25,26]. When the unit is replaced at time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞),
2. List of assumptions
𝐶(𝑁; 𝑇 ) ≡ lim 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) = 𝐾→∞
In this section, the following assumptions for failure, minimal repair and replacement are given:
𝑐𝑅 + 𝑐𝑚 𝐻(𝑁𝑇 ) 𝑁𝑇
(𝑁 = 1, 2, ⋯).
(5)
When the unit is replaced at repair 𝐾 (𝐾 = 1, 2, ⋯),
i. Failures of an operating unit occur at a nonhomogeneous Poisson 𝑡 process with mean value function 𝐻(𝑡) ≡ ∫0 ℎ(𝑢)d𝑢. Let pj (t) and Pj (t) be the respective probabilities of exact number j of failures and at least number j of failures occur in [0, t], then
𝐶 (𝐾 ) ≡ lim 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) = 𝑁→∞
𝑐𝑅 + 𝑐𝑚 𝐾 ∞
∫0 𝑃 𝐾 (𝑡)d𝑡
(𝐾 = 1, 2 ⋯).
(6)
When the unit is replaced at repair 𝐾 (𝐾 = 1, 2, ⋯) or at time T (0 < T < ∞), whichever occurs first,
∑ [𝐻(𝑡)]𝑗 −𝐻(𝑡) e and 𝑃𝑗 (𝑡) ≡ 𝑝𝑖 (𝑡) (𝑗 = 0, 1, 2, ⋯), 𝑗! 𝑖=𝑗 ∞
𝑝𝑗 (𝑡) =
𝑃 𝑗 (𝑇 ) =
𝑇
𝐶𝐹 (𝐾; 𝑇 ) ≡ 𝐶𝐹 (1, 𝐾; 𝑇 ) =
where 𝑃0 (𝑇 ) = 1 and 𝑃𝑗 (∞) = 1. ii. The unit undergoes minimal repairs at failure events, and begins to run again after repairs. It is also assumed that the failure rate h(t) remains undisturbed by minimal repairs, i.e., the unit after each minimal repair has the same failure rate as before failure. iii. A new unit is replaced at time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞), at repair number 𝐾 (𝐾 = 1, 2, ⋯), and at the next periodic time (𝑛 + 1)𝑇 when a number 𝐾 (𝐾 = 1, 2, ⋯) of repairs have been done during [𝑛𝑇 , (𝑛 + 1)𝑇 ]. iv. The times for minimal repair and replacement are negligible, and let cR be the cost of replacement and cm be the cost of minimal repair at each failure, where cR ≥ cm .
𝑐𝑅 + 𝑐𝑚 ∫0 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 𝑇
∫0 𝑃 𝐾 (𝑡)d𝑡
.
(7)
3.2. Optimum policies When the failure rate h(t) ≡ dH(t)/dt increases strictly with t to ℎ(∞) = ∞, we find optimum replacement policies of T, N and K for the above expected cost rates and comparisons among them are made in terms of cost rates. 3.2.1. Optimum T∗ , N∗ and K∗ We find optimum T∗ , N∗ and K∗ to minimize C(T) in (4), C(N; T) in (5) and C(K) in (6), respectively. (1) Optimum T∗ Optimum T∗ (0 < T∗ < ∞) to minimize C(T) satisfies [25,26] 𝑐 𝑇 ℎ(𝑇 ) − 𝐻(𝑇 ) = 𝑅 , 𝑐𝑚
3. Replacement first 3.1. Expected cost rate
(8)
and the resulting cost rate is
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at a repair number 𝐾 (𝐾 = 1, 2, ⋯), whichever oc-
𝐶(𝑇 ∗ ) = 𝑐𝑚 ℎ(𝑇 ∗ ). 2
(9)
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 1 Optimum T∗ in (8) and K∗ in (11) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 . cR /cm
1.0 2.0 5.0 10.0 20.0
𝑚 = 2.0
𝑚 = 3.0
T∗
C(T∗ )/cm
K∗
C(K∗ )/cm
T∗
C(T∗ )/cm
K∗
C(K∗ )/cm
1.00 1.41 2.24 3.16 4.47
2.00 2.83 4.47 6.32 8.94
2 2 5 11 20
2.26 3.01 4.59 6.40 9.00
0.79 1.00 1.36 1.71 2.15
1.89 3.00 5.53 8.77 13.92
1 2 3 6 10
2.24 3.36 5.76 8.97 14.08
(a) From (9) and (13), we obtain 𝐶𝐹 (𝐾; 𝑇𝐹∗ ) ≥ 𝐶(𝑇 ∗ ). That is, the original periodic replacement policy in (4) is more economical than that in (7) when a number K of minimal repairs is predefined. (b) Optimum policy to minimize CF (K; T) in (7) is (𝐾𝐹∗ = ∞, 𝑇𝐹∗ = 𝑇 ∗ ), in other words, a degradation model with T∗ is always found for the bivariate optimum policy of CF (K; T). That is, C(T∗ ) < C(K∗ ) in (4) and (6), which indicates the original periodic replacement policy in (4) is more economical than that in (6) when a number K of minimal repairs is optimized.
Table 2 Optimum 𝑇𝐹∗ in (12) and its cost rate when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
𝑇𝐹∗
𝐶 (𝐾 ; 𝑇𝐹∗ )∕𝑐𝑚
1.09 1.69 3.39 6.21 11.85
2.18 3.38 6.77 12.41 23.70
1.00 1.42 2.27 3.44 5.73
2.00 2.83 4.54 6.88 11.46
1.00 1.41 2.24 3.19 4.80
2.00 2.83 4.47 6.37 9.61
In addition, forming the inequality 𝐶𝐹 (𝐾 + 1; 𝑇 ) − 𝐶𝐹 (𝐾; 𝑇 ) ≥ 0 for given T,
(2) Optimum N∗ Forming the inequality 𝐶(𝑁 + 1; 𝑇 ) − 𝐶(𝑁; 𝑇 ) ≥ 0 for given T, 𝑐 𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ) ≥ 𝑅 . 𝑐𝑚
𝑇
∫0 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡 𝑇 ∫0
(10)
𝐿(𝑁 + 1) − 𝐿(𝑁) = (𝑁 + 1)
∫𝑁𝑇
[ℎ(𝑡 + 𝑇 ) − ℎ(𝑡)]d𝑡 > 0.
Therefore, there exists a finite and unique minimum N∗ (1 ≤ N∗ < ∞) to minimize C(N; T) which satisfies (10). If 𝐻(2𝑇 ) − 2𝐻(𝑇 ) ≥ 𝑐𝑅 ∕𝑐𝑚 , then 𝑁 ∗ = 1. Noting that
if T ≥ T∗ , where T∗ is given in (8), then 𝑁 ∗ = 1.
∞
∫0 𝑝𝐾 (𝑡)d𝑡
−𝐾 ≥
𝑐𝑅 , 𝑐𝑚
(11)
3.2.2. Optimum 𝑇𝐹∗ and 𝐾𝐹∗ In order to compare the optimum policies of C(T) and C(K) in (8) and (11), the optimizations of C(K; T) in (7) are discussed as follows. Differentiating CF (K; T) with respect to T and setting it equal to zero for given K, optimum 𝑇𝐹∗ satisfies ∫0
𝑃 𝐾 (𝑡)[ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 =
𝑐𝑅 , 𝑐𝑚
(𝑁+1)𝑇
∫𝑁𝑇
𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
(14)
∫0
𝑃 𝐾 (𝑡)d𝑡
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
𝑐𝑅 , 𝑐𝑚
(15)
and forming the inequality 𝐶𝐹 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
(12)
𝑁𝑇
∫(𝑁−1)𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
and the resulting cost rate is 𝐶𝐹 (𝐾; 𝑇𝐹∗ ) = 𝑐𝑚 ℎ(𝑇𝐹∗ ).
𝑐𝑅 , 𝑐𝑚
3.2.4. Optimum 𝑁𝐹∗ and 𝐾𝐹∗ In order to compare the optimum policies of N and K for C(N; T) in (5) and C(K; T) in (7), the optimizations of CF (N, K; T) in (3) are discussed as follows. We find optimum 𝑁𝐹∗ and 𝐾𝐹∗ to minimize CF (N, K; T) for given T. It has been shown above that if T > T∗ , then 𝑁𝐹∗ = 1 and 𝐾𝐹∗ is given in (14), and if 𝑇 = 𝑇 ∗ , then 𝑁𝐹∗ = 1 and 𝐾𝐹∗ = ∞. When T < T∗ , forming the inequality 𝐶𝐹 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
whose left-hand side increases strictly with K to ∞ [26]. Thus, there exists a finite and unique minimum K∗ (1 ≤ K∗ < ∞) to minimize C(K) which satisfies (11).
𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
3.2.3. Numerical example Tables 1 and 2 present optimum T∗ in (8), K∗ in (11), 𝑇𝐹∗ in (12) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 . It shows numerically in Table 1 that both T∗ and K∗ decrease with m and C(T∗ ) < C(K∗ ), and in Table 2 that 𝑇𝐹∗ decreases with K to T∗ and 𝐶(𝑇 ∗ ) ≤ 𝐶(𝐾; 𝑇𝐹∗ ) when 𝑚 = 2.0.
(3) Optimum K∗ Forming the inequality 𝐶(𝐾 + 1) − 𝐶(𝐾) ≥ 0, ∞
∫0
This result will be used to compare optimum policies of N and K for C(N; T) in (5) and C(K; T) in (7) in the following discussion when T ≤ T∗ and in Table 3 when T > T∗ .
𝐻(2𝑇 ) − 2𝐻(𝑇 ) > 𝑇 ℎ(𝑇 ) − 𝐻(𝑇 ),
∫0 𝑃 𝐾 (𝑡)d𝑡
𝑇
(a) If T > T∗ , then (19) > cR /cm , and there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) to minimize C(K; T) which satisfies (14) for given T. (b) If T ≤ T∗ , then 𝐾𝐹∗ = ∞, i.e., the policy with repair K becomes meaningless when a given T is small enough.
lim 𝐿(𝑁) = ℎ(∞) = ∞,
𝑁→∞ (𝑁+1)𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
whose left-hand side increases strictly with K to that of (8). Thus, we obtain
Denoting L(N) as the left-hand side of (10), 𝐿(1) = 𝐻(2𝑇 ) − 2𝐻(𝑇 ) > 0,
𝑝𝐾 (𝑡)d𝑡
𝑇
𝑁𝑇 ∫(𝑁−1)𝑇
(13)
𝑃 𝐾 (𝑡)d𝑡
𝑁𝑇
∫0
(𝑁+1)𝑇
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
(𝑁+1)𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 <
𝑐𝑅 . 𝑐𝑚
(16)
Noting that ∫𝑁𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑃 𝐾 (𝑡)d𝑡 increases strictly with N to ∞ and increases strictly with K to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 (Appendix A), the left-hand side of (15) increases strictly with K to that of (10). Thus, there exists a finite and unique minimum 𝑁𝐹∗ (1 ≤ 𝑁𝐹∗ < ∞) which satisfies (15), and 𝑁𝐹∗ decreases with K to N∗ given in (10) and 𝑁𝐹∗ ≥ 𝑁 ∗ .
Noting that the left-hand side of (12) increases strictly with T from 0 to ∞ and increases strictly with K to that of (8). Thus, there exists a finite and unique 𝑇𝐹∗ (0 < 𝑇𝐹∗ ≤ ∞) which satisfies (12), and 𝑇𝐹∗ decreases strictly with K to T∗ given in (8) and 𝑇𝐹∗ ≥ 𝑇 ∗ . Therefore, the following two comparative results are obtained: 3
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 3 Optimum N∗ in (10) and K∗ in (14) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. 𝑇 = 1.0
cR /cm
1.0 2.0 5.0 10.0 20.0
𝑇 = 5.0
N∗
C(N∗ ; T)/cm
K∗
CF (K∗ ; T)/cm
N∗
C(N∗ ; T)/cm
K∗
CF (K∗ ; T)/cm
1 1 2 3 4
2.00 3.00 4.50 6.33 9.00
∞ ∞ ∞ ∞ ∞
2.00 3.00 6.00 11.00 21.00
1 1 1 1 1
5.20 5.40 6.00 7.00 9.00
1 2 6 11 23
2.26 3.01 4.59 6.40 8.98
Table 4 Optimum 𝑁𝐹∗ in (15) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐹∗
𝐶𝐹 (𝑁𝐹∗ , 𝐾; 𝑇 )∕𝑐𝑚
1 2 4 ∞ ∞
2.19 3.38 6.77 12.41 23.70
1 2 2 4 6
2.00 3.00 4.56 6.88 11.46
1 2 2 3 5
2.00 3.00 4.50 6.38 9.61
Table 5 Optimum 𝑁𝐿∗ in (23) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=1
𝐾=5
𝐾 = 10
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
𝑁𝐿∗
𝐶𝐿 (𝑁𝐿∗ , 𝐾; 𝑇 )∕𝑐𝑚
1 1 2 3 4
2.08 2.96 4.50 6.33 9.00
1 1 2 3 4
2.75 3.21 4.53 6.33 9.00
1 2 2 3 4
3.52 3.84 4.80 6.36 9.00
Forming the inequality 𝐶𝐹 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0, 𝑁𝑇
∫0
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾 (𝑡)d𝑡
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
𝑐𝑅 , 𝑐𝑚
which agrees with that of (11). Thus, 𝐾𝐹∗ increases strictly with N to K∗ and 𝐾𝐹∗ ≤ 𝐾 ∗ . Comparing (19) with the left-hand side of (10),
(17)
𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ) > 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 )
and forming the inequality 𝐶𝐹 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝐹 (𝑁, 𝐾; 𝑇 ) > 0, 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
𝑁𝑇
∫0
𝑃 𝐾 (𝑡)d𝑡 −
𝑁𝑇
∫0
𝑐 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < 𝑅 . 𝑐𝑚
>𝑁 [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] − 𝐻 (𝑁 𝑇 ). Thus, we obtain
(18)
(a) If N > N∗ , then there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) which satisfies (17). (b) If N ≤ N∗ , then 𝐾𝐹∗ = ∞, i.e., the model in (3) is degraded into the one in (5).
Substituting (15) for (18), 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
(𝑁+1)𝑇
<
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
,
It is easy to understand that when the approach of whichever occurs first is applied, replacement should be done as soon as possible when any policy is first triggered, which goes somewhat against the goal of TPM. This result will be used to compare the optimal policies of replacement last in the following section to decide which policy is more economical and also will be shown in Table 6.
which always holds for any N, as 𝑁𝑇
∫0
𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾−1 (𝑡)d𝑡
(𝑁+1)𝑇
< ℎ(𝑁𝑇 ) <
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇 ∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
.
This follows that 𝐶𝐹 (𝑁, 𝐾 − 1; 𝑇 ) > 𝐶𝐹 (𝑁, 𝐾; 𝑇 ), and optimum policy to minimize CF (N, K; T) is (𝑁𝐹∗ = 𝑁 ∗ , 𝐾𝐹∗ = ∞). That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of CF (N, K; T), which indicates replacement policy in (5) is more economical than that in (7). 𝑁𝑇 𝑁𝑇 Furthermore, noting that ∫0 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫0 𝑝𝐾 (𝑡)d𝑡 increases strictly with K to h(NT) and increases strictly with N (Appendix B), the left-hand side of (17) increases strictly with K to 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ).
3.2.5. Numerical example Table 3 presents optimum N∗ in (10) and K∗ in (14) and their cost rates when 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. It shows that when T ≤ T∗ in Table 1, 1 ≤ N∗ < ∞ and 𝐾 ∗ = ∞ and C(N∗ ; T) < CF (K∗ ; T), and when T > T∗ , 𝑁 ∗ = 1 and 1 ≤ K∗ < ∞ and C(N∗ ; T) > CF (K∗ ; T), which agree with the above analytical results. Table 4 presents optimum 𝑁𝐹∗ in (15) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. It shows that 𝑁𝐹∗ decreases with K to N∗ in (10). The comparison between Table 4 and Table 5 will be shown in the following section.
(19)
Thus, if 𝑁 𝑇 ℎ(𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) > 𝑐𝑅 ∕𝑐𝑚 , then there exists a finite and unique minimum 𝐾𝐹∗ (1 ≤ 𝐾𝐹∗ < ∞) which satisfies (17). Furthermore, noting that the left-hand side of (17) increases strictly with N to
4. Replacement last
∞
∫0 𝑃 𝐾 (𝑡)d𝑡 ∞
∫0 𝑝𝐾 (𝑡)d𝑡
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 0, 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at a repair number 𝐾 (𝐾 = 0, 1, 2, ⋯), whichever
− 𝐾,
4
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Reliability Engineering and System Safety 000 (2017) 1–10
4.2. Optimum 𝐾𝐿∗
Table 6 Optimum 𝐾𝐹∗ in (17) and 𝐾𝐿∗ in (25) when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. 𝑁 =1
cR /cm
1.0 2.0 5.0 10.0 20.0
Forming the inequality 𝐶𝐿 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ 𝑁𝑇 + 𝑃 ( 𝑡 )d 𝑡 ∞ ∫𝑁𝑇 𝐾 ∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 [ ] ∞ 𝑐 − 𝐻 (𝑁 𝑇 ) + 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥ 𝑅 , ∫𝑁𝑇 𝑐𝑚
𝑁 =5
∞
𝐾𝐹∗
𝐾𝐿∗
𝐾𝐹∗
𝐾𝐿∗
∞ ∞ ∞ ∞ ∞
0 0 5 10 20
1 2 5 10 20
0 0 0 0 0
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
and forming the inequality 𝐶𝐿 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∞ ∫𝑁𝑇 𝑝𝐾−1 (𝑡)ℎ(𝑡)d𝑡 𝑁𝑇 + 𝑃 ( 𝑡 )d 𝑡 𝐾 ∞ ∫𝑁𝑇 ∫𝑁𝑇 𝑝𝐾−1 (𝑡)d𝑡 [ ] ∞ 𝑐 − 𝐻 (𝑁 𝑇 ) + 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < 𝑀 . ∫𝑁𝑇 𝑐𝑚
occurs last. From the policy definition, it can be understood for replacement last that the unit could operate as long as until any replacement is last triggered, which is somewhat consistent with the goal of TPM. Then, the probability that the unit is replaced at time NT is PK (NT), and the probability that it is replaced at repair K is 𝑃 𝐾 (𝑁𝑇 ). The mean time to replacement is (𝑁 𝑇 )𝑃𝐾 (𝑁 𝑇 ) +
∞
∫𝑁𝑇
𝑡d𝑃𝐾 (𝑡) = 𝑁𝑇 +
∞
∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡,
𝑗𝑝𝑗 (𝑁𝑇 ) + 𝐾 𝑃 𝐾 (𝑁𝑇 ) = 𝐾 +
𝑗=𝐾
= 𝐻 (𝑁 𝑇 ) +
𝐾 ∑ 𝑗=1
∞ ∑ 𝑗=𝐾
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡
(20)
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡.
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡
∞
𝑁𝑇 + ∫𝑁𝑇 𝑃 𝐾 (𝑡)d𝑡
.
(21)
(22)
Clearly, comparing with the expected cost rates in (3)–(6), we obtain
𝐶𝐿 (𝑁, 0; 𝑇 ) = 𝐶𝐹 (𝑁, ∞; 𝑇 ) = 𝐶(𝑁; 𝑇 ),
𝑁𝐿∗
(𝑁+1)𝑇 ∞
−
∫𝑁𝑇
[ 𝑁𝑇 +
𝑃𝐾 (𝑡)d𝑡
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 ≥
∞
∫𝑁𝑇
] 𝑃 𝐾 (𝑡)d𝑡 − 𝐻 (𝑁 𝑇 )
𝑐𝑅 , 𝑐𝑚
𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡 ∞
−
∫𝑁𝑇
(27)
𝑁𝑇 e−𝐻 (𝑁 𝑇 ) 𝑁 𝑇 e−𝐻 ((𝑁 −1)𝑇 ) −𝐻 (𝑁 𝑇 )>𝑁 𝑇 ℎ(𝑁 𝑇 )−𝐻 (𝑁 𝑇 )> ∞ − 𝐻 (𝑁 𝑇 ). ∞ ∫𝑁𝑇 e−𝐻(𝑡) d𝑡 ∫(𝑁−1)𝑇 e−𝐻(𝑡) d𝑡
(23)
Thus, we obtain
and forming the inequality 𝐶𝐿 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) > 0, ∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡 [
.
then there exists a finite and unique minimum 𝐾𝐿∗ (1 ≤ 𝐾𝐿∗ < ∞) which satisfies (25). Furthermore, noting that the left-hand side of (25) increases strictly with N from that of (11). Thus, 𝐾𝐿∗ decreases strictly with N from K∗ and 𝐾𝐿∗ ≤ 𝐾 ∗ . Comparing (27) with the left-hand side of (10),
𝐾𝐿∗
We find optimum and to minimize CL (N, K; T) in (22) when T < T∗ . Forming the inequality 𝐶𝐿 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0,
∫𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡
𝑐 𝑁𝑇 e−𝐻 (𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) < 𝑅 , ∞ 𝑐𝑚 ∫𝑁𝑇 e−𝐻(𝑡) d𝑡
4.1. Optimum 𝑁𝐿∗
𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
to ∞. Thus, if
𝐶𝐿 (1, 0; 𝑇 ) = 𝐶(𝑇 ).
(𝑁+1)𝑇
,
𝑁𝑇
> ℎ(𝑁𝑇 ) >
𝑁𝑇 e−𝐻 (𝑁 𝑇 ) − 𝐻 (𝑁 𝑇 ) ∞ ∫𝑁𝑇 e−𝐻(𝑡) d𝑡
𝐶𝐿 (0, 𝐾; 𝑇 ) = 𝐶𝐹 (∞, 𝐾; 𝑇 ) = 𝐶 (𝐾 ),
∫𝑁𝑇
𝑁𝑇
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)d𝑡
This follows that 𝐶𝐿 (𝑁, 𝐾 + 1; 𝑇 ) > 𝐶𝐿 (𝑁, 𝐾; 𝑇 ), and optimum policy to minimize CL (N, K; T) is (𝑁𝐿∗ = 𝑁 ∗ , 𝐾𝐿∗ = 0). That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of CL (N, K; T), which also indicates replacement policy in (5) is more economical than that in (7). ∞ ∞ Furthermore, noting that ∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 increases with ∞ − 𝐻 ( 𝑁 𝑇 ) − 𝐻( 𝑡 ) K from e ∕ ∫𝑁𝑇 e d𝑡 to ∞, and increases strictly with N from ∞ 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 (Appendix D), the left-hand side of (25) increases strictly with K from
∞
𝑐𝑅 + 𝑐𝑚 [𝐻(𝑁𝑇 ) + ∫𝑁𝑇 𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡]
∫(𝑁−1)𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
Therefore, the expected repair and replacement cost rate is 𝐶𝐿 (𝑁, 𝐾; 𝑇 ) =
>
which always holds for any N, as
∞
𝑃 𝑗 (𝑁𝑇 ) = 𝐻 (𝑁 𝑇 ) +
𝑁𝑇
∞
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑗 − 𝐾)𝑝𝑗 (𝑁𝑇 ) ∞
(26)
Substituting (24) for (25),
and the expected number of repairs until replacement is ∞ ∑
(25)
𝑁𝑇 +
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡 < (𝑁+1)𝑇
∞
∫𝑁𝑇
(a) If N < N∗ , then there exists a finite and unique minimum 𝐾𝐿∗ (1 ≤ 𝐾𝐿∗ < ∞) which satisfies (25). (b) If N ≥ N∗ , then 𝐾𝐿∗ = 0, i.e., the model in (22) is degraded into the one in (5).
] 𝑃 𝐾 (𝑡)d𝑡 − 𝐻 (𝑁 𝑇 )
𝑐𝑅 . 𝑐𝑚
(24)
Using the comparative method in [21], we conclude for replacement first and last that:
(𝑁+1)𝑇
Noting that ∫𝑁𝑇 𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡∕ ∫𝑁𝑇 𝑃𝐾 (𝑡)d𝑡 increases strictly with N to ∞ and increases strictly with K from [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 (Appendix C), the left-hand side of (23) increases strictly with K from that of (10). Thus, from the assumption of T < T∗ , there exists a finite and unique minimum 𝑁𝐿∗ (0 ≤ 𝑁𝐿∗ < ∞) which satisfies (23), and 𝑁𝐿∗ increases with K from N∗ given in (10) and 𝑁𝐿∗ ≥ 𝑁 ∗ . Table 5 presents optimum 𝑁𝐿∗ in (23) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 4, it can be found that replacement last is more economical than replacement first when K is smaller or cM /cm is larger.
(a) If N < N∗ , then 1 ≤ 𝐾𝐿∗ < ∞ and 𝑁𝐹∗ = ∞, i.e., we should plan the policy of replacement last rather than replacement first. (b) If N > N∗ , then 1 ≤ 𝐾𝐹∗ < ∞ and 𝑁𝐿∗ = 0, i.e., we should plan the policy of replacement first rather than replacement last. (c) If 𝑁 = 𝑁 ∗ , then 𝐾𝐹∗ = ∞ and 𝐾𝐿∗ = 0, i.e., replacement should be planned only at a predefined time NT in (5). Table 6 presents optimum 𝐾𝐹∗ in (17) and 𝐾𝐿∗ in (25) when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Clearly, this numerical result agrees with the above analytical discussions. 5
ARTICLE IN PRESS
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X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10 Table 7 Optimum 𝐾𝑂∗ in (32) and its cost rate when 𝑇 = 1.0 and 𝐻(𝑡) = 𝑡𝑚 .
5. Replacement overtime It has been indicated above that replacement policies are more easily to be performed at periodic times, e.g., monthly holidays. We next delay the replacement at repair K during [𝑛𝑇 , (𝑛 + 1)𝑇 ] to the next periodic time (𝑛 + 1)𝑇 for the above replacement first and last, which is also a possible plan during the busy production state for the JIT principle in TPM.
cR /cm
1.0 2.0 5.0 10.0 20.0
5.1. Replacement first Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 1, 2, ⋯ ; 0 < 𝑇 < ∞) or at the next periodic time when a number 𝐾 (𝐾 = 1, 2, ⋯) of minimal repairs have occurred, whichever occurs first. Then, the probability that the unit is replaced at time NT is 𝑃 𝐾 (𝑁𝑇 ), and the probability that it is replaced over repair K is PK (NT). Letting 𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )] be the probability that i repairs occur in [𝑛𝑇 , (𝑛 + 1)𝑇 ], then the probability that the unit is replaced at time (𝑛 + 1)𝑇 (𝑛 = 0, 1, 2, ⋯) when more than K repairs have occurred in [𝑛𝑇 , (𝑛 + 1)𝑇 ] is 𝐾−1 ∑ 𝑗=0
=
𝑝𝑗 [𝐻 (𝑛𝑇 )]
∞ ∑
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}.
The mean time to replacement is (𝑁 𝑇 )
𝐾−1 ∑ 𝑗=0
=𝑇
𝑁−1 ∑ 𝑛=0
𝑝𝑗 [𝐻 (𝑛𝑇 )] +
𝑁−1 ∑
𝐾−1 ∑
𝑛=0
𝑗=0
[(𝑛 + 1)𝑇 ]
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}
(28)
−
𝑗=0
=
𝑛=0 𝑗=0
𝑝𝑗 [𝐻(𝑛𝑇 )]
∞ ∑ 𝑖=𝐾−𝑗
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ).
−
∞ ∑
(30)
𝑛=0
where
∑𝑁−1
𝑄(𝑁, 𝐾; 𝑇 ) =
𝑛=0
∞ ∑ 𝑐 [𝐻((𝑛 + 1)𝑇 − 𝐻(𝑛𝑇 ))]𝑃 𝐾 (𝑛𝑇 ) ≥ 𝑅 , 𝑐𝑚 𝑛=0
∑∞ 𝑁→∞
𝑁−1 ∑
𝑐𝑅 , 𝑐𝑚
(33)
𝑃 𝐾 (𝑛𝑇 )
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑁−1 ∑ 𝑛=0
𝑄(𝑁, 𝐾 − 1; 𝑇 )
(31)
𝑃 𝐾 (𝑛𝑇 ) −
𝑁−1 ∑ 𝑛=0
𝑐𝑅 . 𝑐𝑚
(34)
𝑁−1 ∑ 𝑛=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑐𝑅 , 𝑐𝑚
(35)
𝑃 𝐾 (𝑛𝑇 ) −
𝑁−1 ∑ 𝑛=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 . 𝑐𝑚 (36)
Substituting (33) for (36), 𝑄(𝑁, 𝐾 − 1; 𝑇 ) < [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )].
(32)
which always holds for any N, as 𝑄(𝑁, 𝐾 − 1; 𝑇 ) ≤ [𝐻(𝑁𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] < [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]. This follows that COF (N, K; T) decreases strictly with K, and the opti∗ = 𝑁 ∗ , 𝐾 ∗ = ∞), which mum policy to minimize COF (N, K; T) is (𝑁𝑂𝐹 𝑂𝐹 shows the same result as above in Section 3.2.4. That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of COF (N, K; T), which indicates replacement policy in (5) is more economical than that in (31).
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) , ∑𝑁−1 𝑛=0 𝑝𝐾 (𝑛𝑇 )
𝑄(𝐾; 𝑇 ) ≡ lim 𝑄(𝑁, 𝐾; 𝑇 ) =
𝑃 𝐾 (𝑛𝑇 )
and forming the inequality 𝐶𝑂𝐹 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
𝑛=0 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) . ∑ 𝑇 ∞ 𝑛=0 𝑃 𝐾 (𝑛𝑇 )
and
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑄(𝑁, 𝐾; 𝑇 )
∑∞
𝑃 𝐾 (𝑛𝑇 ) −
3.35 4.08 6.27 9.15 14.17
The left-hand side of (33) increases strictly with N to ∞ and increases strictly with K to that of (10). Thus, there exists a finite and unique ∗ (1 ≤ 𝑁 ∗ < ∞) which satisfies (33), and 𝑁 ∗ decreases minimum 𝑁𝑂𝐹 𝑂𝐹 𝑂𝐹 ∗ ≥ 𝑁 ∗. ∗ with K to N given in (10) and 𝑁𝑂𝐹 Forming the inequality 𝐶𝑂𝐹 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
We find optimum 𝐾𝑂∗ to minimize CO (K; T) in (31) when T < T∗ . Forming the inequality 𝐶𝑂 (𝐾 + 1; 𝑇 ) − 𝐶𝑂 (𝐾; 𝑇 ) ≥ 0, 𝑄(𝐾; 𝑇 )
𝑁−1 ∑ 𝑛=0
𝐶𝑂 (𝐾; 𝑇 ) = lim 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) 𝑐𝑅 + 𝑐𝑚
1 1 1 3 4
𝑛=0
5.1.1. Optimum 𝐾𝑂∗ When the unit is replaced at the next periodic time over repair K, the expected cost rate is
=
2.31 3.03 4.62 6.43 9.02
[𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )]
(29)
Therefore, the expected repair and replacement cost rate is ∑ 𝑐𝑅 + 𝑐𝑚 𝑁−1 𝑛=0 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) = . ∑ 𝑇 𝑁−1 𝑛=0 𝑃 𝐾 (𝑛𝑇 )
𝑁→∞
1 1 3 8 16
and forming the inequality 𝐶𝑂𝐹 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) > 0,
(𝑖 + 𝑗)𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑁−1 ∑ 𝑛=0
𝑁−1 ∑ 𝑛=0
𝑗𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
𝐶𝑂 (𝐾𝑂∗ ; 𝑇 )∕𝑐𝑚
𝑛=0
𝑃 𝐾 (𝑛𝑇 ),
𝑁−1 ∑ 𝐾−1 ∑
𝐾𝑂∗
[𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]
and the expected number of repairs until replacement is 𝐾−1 ∑
𝐶𝑂 (𝐾𝑂∗ ; 𝑇 )∕𝑐𝑚
∗ and 𝐾 ∗ 5.1.2. Optimum 𝑁𝑂𝐹 𝑂𝐹 In order to compare the optimum policies of N and K for C(N; T) in (5) and CO (K; T) in (31), the optimizations of COF (N, K; T) in ∗ ∗ (30) are discussed as follows. We find optimum 𝑁𝑂𝐹 and 𝐾𝑂𝐹 to ∗ minimize COF (N, K; T) in (30) when T < T . Forming the inequality 𝐶𝑂𝐹 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐹 (𝑁, 𝐾; 𝑇 ) ≥ 0,
𝐾−1 ∑ 𝑗=0
𝑚 = 3.0
𝐾𝑂∗
which increases strictly with K to ∞ (Appendix E). Thus, there exists a finite and unique minimum 𝐾𝑂∗ (1 ≤ 𝐾𝑂∗ < ∞) which satisfies (32). Table 7 presents optimum 𝐾𝑂∗ in (32) and its cost rate when 𝑇 = 1.0 and 𝐻(𝑡) = 𝑡𝑚 . Comparing with Table 3 when 𝑇 = 1.0 and 𝑚 = 2, 0, 𝐾𝑂∗ < 𝐾 ∗ , and 𝐶𝑂 (𝐾𝑂∗ ; 𝑇 ) < 𝐶𝐹 (𝐾 ∗ ; 𝑇 ) when cR /cm becomes larger. It is easy to understand that repair cost would possibly increase due to replacement delay so that optimum replacement time should be earlier than the original one, and also, replacement overtime would be more economical than the original one if repair cost becomes smaller.
𝑝𝑖 [𝐻 ((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑖=𝐾−𝑗
𝑚 = 2.0
𝑛=0 [𝐻((𝑛 +
1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) , ∑∞ 𝑛=0 𝑝𝐾 (𝑛𝑇 ) 6
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Reliability Engineering and System Safety 000 (2017) 1–10 Table 8 ∗ in (33) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Optimum 𝑁𝑂𝐹 𝐾=1
cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=5
𝐾 = 10
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐹
∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 )∕𝑐𝑚
1 2 3 4 8
2.27 3.03 5.19 8.80 16.01
1 2 2 3 5
2.50 3.48 4.62 6.54 10.27
1 1 2 3 5
2.50 3.00 4.67 6.45 9.24
Table 9 ∗ in (40) and its cost rate when 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Optimum 𝑁𝑂𝐿 𝐾=1
cR /cm
1.0 2.0 5.0 10.0 20.0
𝐾=5
𝐾 = 10
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
∗ 𝑁𝑂𝐿
∗ 𝐶𝑂𝐿 (𝑁𝑂𝐿 , 𝐾; 𝑇 )∕𝑐𝑚
1 2 2 3 5
2.31 3.02 4.50 6.33 9.00
2 2 2 3 5
3.18 3.55 4.67 6.35 9.00
2 2 3 3 5
3.99 4.27 5.10 6.47 9.00
Furthermore, noting that Q(N, K; T) increases strictly with K to [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] and increases strictly with N (Appendix E), the left-hand side of (35) increases strictly with K to
Therefore, the expected repair and replacement cost rate is ∑ 𝑐𝑅 + 𝑐𝑚 {𝐻(𝑁𝑇 ) + ∞ 𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 )} 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) = . ∑ 𝑁𝑇 + 𝑇 ∞ 𝑛=𝑁 𝑃 𝐾 (𝑛𝑇 ) (39)
𝑁 [𝐻 (𝑁 𝑇 )−𝐻 ((𝑁 − 1)𝑇 )]−𝐻 (𝑁 𝑇 )<𝑁 [𝐻 ((𝑁 + 1)𝑇 )−𝐻 (𝑁 𝑇 )]−𝐻 (𝑁 𝑇 ),
Clearly, comparing with the expected cost rates in (5), (30) and (31), we obtain
which agrees with (10). Thus, we obtain ∗ (1 ≤ (a) If N > N∗ , then there exists a finite and unique minimum 𝐾𝑂𝐹 ∗ < ∞) which satisfies (35) 𝐾𝑂𝐹 ∗ = ∞, i.e., the model in (30) is degraded into the (b) If N ≤ N∗ , then 𝐾𝑂𝐹 one in (5).
𝐶𝑂𝐿 (0, 𝐾; 𝑇 ) = 𝐶𝑂𝐹 (∞, 𝐾; 𝑇 ) = 𝐶𝑂 (𝐾; 𝑇 ), 𝐶𝑂𝐿 (𝑁, 0; 𝑇 ) = 𝐶𝑂𝐹 (𝑁, ∞; 𝑇 ) = 𝐶(𝑁; 𝑇 ). ∗ ∗ We find optimum 𝑁𝑂𝐿 and 𝐾𝑂𝐿 to minimize COL (N, K; T) in (39) when T < T∗ . Forming the inequality 𝐶𝑂𝐿 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ ∑ [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )] 𝑁 + 𝑃 𝐾 (𝑛𝑇 )
This result will be used to compare the optimal policies of replacement last in the following section to decide which policy is more economical and also will be shown in Table 9. ∗ Table 8 presents optimum 𝑁𝑂𝐹 in (33) and its cost rate when ∗ ≤ 𝑁 ∗ and 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 4, 𝑁𝑂𝐹 𝐹 ∗ ∗ 𝐶𝑂𝐹 (𝑁𝑂𝐹 , 𝐾; 𝑇 ) < 𝐶𝐹 (𝑁𝐹 , 𝐾; 𝑇 ) when cM /cm becomes larger or given K becomes smaller.
𝑛=𝑁
− 𝐻 (𝑁 𝑇 ) −
𝑛=𝑁
𝑗=𝐾
𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
= 𝑁𝑇 + 𝑇
∞ ∑ 𝑛=𝑁
∞ ∑
[(𝑛 + 1)𝑇 ]
− 𝐻 (𝑁 𝑇 ) −
𝑗=0
− 𝐻 (𝑁 𝑇 ) −
𝑗=𝐾
𝑗𝑝𝑗 [𝐻 (𝑁 𝑇 )] +
= 𝐻 (𝑁 𝑇 ) +
∞ ∑ 𝑛=𝑁
∞ 𝐾−1 ∑ ∑ 𝑛=𝑁 𝑗=0
𝑝𝑗 [𝐻(𝑛𝑇 )]
∞ ∑ 𝑖=𝐾−𝑗
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 . 𝑐𝑚
(41)
𝑛=𝑁
(37)
∞ ∑ 𝑛=𝑁
and the expected number of repairs until replacement is ∞ ∑
(40)
∗ (0 ≤ 𝑁 ∗ < ∞) Thus, there exists a finite and unique minimum 𝑁𝑂𝐿 𝑂𝐿 ∗ increases with K from N∗ given in (10) and which satisfies (40), and 𝑁𝑂𝐿 ∗ ≥ 𝑁 ∗. 𝑁𝑂𝐿 Forming the inequality 𝐶𝑂𝐿 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) ≥ 0, [ ] ∞ ∑ ̃(𝑁, 𝐾; 𝑇 ) 𝑁 + 𝑄 𝑃 𝐾 (𝑛𝑇 )
{𝑝𝑗 [𝐻(𝑛𝑇 )] − 𝑝𝑗 [𝐻(𝑛 + 1)𝑇 ]}
𝑃 𝐾 (𝑛𝑇 ),
∞ ∑ 𝑛=𝑁
𝐾−1 ∑
𝑛=𝑁
𝑐𝑅 , 𝑐𝑚
𝑛=𝑁
Suppose that the unit is replaced at a periodic time 𝑁𝑇 (𝑁 = 0, 1, 2, ⋯ ; 𝑇 > 0) or at the next periodic time when a number 𝐾 (𝐾 = 0, 1, 2, ⋯) of repairs have occurred, whichever occurs last. Then, the probability that the unit is replaced at time NT is PK (NT), and the probability that it is replaced over repair K is 𝑃 𝐾 (𝑁𝑇 ). The mean time to replacement is ∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
and forming the inequality 𝐶𝑂𝐿 (𝑁 − 1, 𝐾; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∑ [𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 )] 𝑁 + 𝑃 𝐾 (𝑛𝑇 )
5.2. Replacement last
(𝑁 𝑇 )
∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) ≥
𝑐𝑅 , 𝑐𝑚
(42)
and forming the inequality 𝐶𝑂𝐿 (𝑁, 𝐾 − 1; 𝑇 ) − 𝐶𝑂𝐿 (𝑁, 𝐾; 𝑇 ) > 0, [ ] ∞ ∑ ̃(𝑁, 𝐾 − 1; 𝑇 ) 𝑁 + 𝑄 𝑃 𝐾 (𝑛𝑇 )
(𝑖 + 𝑗)𝑝𝑖 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]
𝑛=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ).
(38)
− 𝐻 (𝑁 𝑇 ) −
∞ ∑ 𝑛=𝑁
7
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑃 𝐾 (𝑛𝑇 ) <
𝑐𝑅 , 𝑐𝑚
(43)
ARTICLE IN PRESS
JID: RESS
[m5GeSdc;May 15, 2017;12:4]
X. Zhao et al.
Reliability Engineering and System Safety 000 (2017) 1–10 Table 10 ∗ ∗ in (33) and 𝐾𝑂𝐿 in (42) when Optimum 𝐾𝑂𝐹 𝑇 = 1.0, 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. cR /cm
1.0 2.0 5.0 10.0 20.0
where ̃(𝑁, 𝐾; 𝑇 ) ≡ 𝑄
𝑁 =1
It has been shown that replacement first includes: (i) The unit is replaced at time T, (ii) the unit is replaced at time NT, (iii) the unit is replaced at repair K, and (iv) the unit is replaced at repair K or at time T. We have obtained the comparative results as follows: Policy (i) is more economical than those (iii) and (iv). Policy (ii) is more economical than that (iv) only when T ≤ T∗ , which has also been obtained from the optimization of replacement last. When we compare the policies of replacement first and replacement last, i.e., (v) the unit is replaced at time NT or at repair K, whichever occurs first, and (vi) the unit is replaced at time NT or at repair K, whichever occurs last, it has been found that whether replacement last is more economical or not depends on the scale of the predefined policy and the cost ratio of replacement and repair. Finally, we delay replacement at repair K to periodic time, i.e., (vii) the unit is replaced at the next periodic time when a number K of repairs have occurred. Comparing with the original policy (iii), its optimum policy should be done early due to increasing repair cost. This modification of repair K has also been applied for replacement first and replacement last, i.e., (viii) the unit is replaced at time NT or at the next periodic time over repair K, whichever occurs first, and (ix) the unit is replaced at time NT or at the next periodic time over repair K, whichever occurs last. Similar comparisons have been obtained.
𝑁 = 10
∗ 𝐾𝑂𝐹
∗ 𝐾𝑂𝐿
∗ 𝐾𝑂𝐹
∗ 𝐾𝑂𝐿
∞ ∞ ∞ ∞ ∞
0 0 3 8 16
1 1 3 8 16
0 0 0 0 0
∑∞
𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) . ∑∞ 𝑛=𝑁 𝑝𝐾 (𝑛𝑇 )
Substituting (40) for (43), ̃(𝑁, 𝐾 − 1; 𝑇 ) < [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )], 𝑄 which does not hold for any N. This follows that COL (N, K; T) increases strictly with K, and the optimum policy to minimize COL (N, K; ∗ = 𝑁 ∗ , 𝐾 ∗ = 0), which shows the same result as above in T) is (𝑁𝑂𝐿 𝑂𝐿 Section 4.2. That is, a degradation model with N∗ for given T is always found for the bivariate optimum policy of COL (N, K; T), which indicates replacement policy in (5) is more economical than that in (39). ̃(𝑁, 𝐾; 𝑇 ) increases strictly with K Furthermore, noting that 𝑄 (Appendix F), the left-hand side of (42) increases strictly with K from ∑ −𝐻(𝑛𝑇 ) 𝑁 ∞ 𝑛=𝑁 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]e − 𝐻 (𝑁 𝑇 ) ∑∞ −𝐻(𝑛𝑇 ) 𝑛=𝑁 e
Acknowledgement This paper was made possible by Qatar National Research Fund (No. PDRA1- 0116-14107) and National Natural Science Foundation of China (No. 71371097).
>𝑁 [𝐻 ((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 )] − 𝐻 (𝑁 𝑇 ), Appendix A
which agrees with that of (10). Thus, we obtain ∗ (1 ≤ (a) If N < N∗ , then there exists a finite and unique minimum 𝐾𝑂𝐿 ∗ 𝐾𝑂𝐿 < ∞) which satisfies (42) ∗ = 0, i.e., the model in (39) is degraded into the (b) If N ≥ N∗ , then 𝐾𝑂𝐿 one in (5).
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 1, 2 ⋯ , (𝑁+1)𝑇
𝐻1 (𝑁, 𝐾; 𝑇 ) ≡
Finally, we conclude for replacement first and last that:
∫𝑁𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇
∫𝑁𝑇
𝑃 𝐾 (𝑡)d𝑡
increases strictly with N from H1 (0, K; T) to h(∞) and increases strictly with K from H1 (N, 1; T) to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 .
∗ < ∞ and 𝑁 ∗ = ∞, i.e., we should plan the (a) If N < N∗ , then 1 ≤ 𝐾𝑂𝐿 𝐹 policy of replacement last rather than replacement first. ∗ < ∞ and 𝑁 ∗ = 0, i.e., we should plan the (b) If N > N∗ , then 1 ≤ 𝐾𝑂𝐹 𝑂𝐿 policy of replacement first rather than replacement last. ∗ = ∞ and 𝐾 ∗ = 0, i.e., replacement should be (c) If 𝑁 = 𝑁 ∗ , then 𝐾𝑂𝐹 𝐿 planned only at a predefined time NT in (5).
Proof. Note that ℎ(𝑁𝑇 ) < 𝐻1 (𝑁, 𝐾; 𝑇 ) < ℎ((𝑁 + 1)𝑇 ),
lim 𝐻1 (𝑁, 𝐾; 𝑇 ) = ℎ(∞),
𝑁→∞
𝐻((𝑁 + 1)𝑇 ) − 𝐻 (𝑁 𝑇 ) . 𝑇
lim 𝐻1 (𝑁, 𝐾; 𝑇 ) =
𝐾→∞
∗ in (40) and its cost rate when 𝑇 = 1.0, Table 9 presents optimum 𝑁𝑂𝐿 𝐻(𝑡) = 𝑡𝑚 and 𝑚 = 2.0. Comparing with Table 8, it can be found that replacement last is more economical than replacement first when K is smaller or cR /cm is larger. ∗ in (33) and 𝐾 ∗ in (42) when 𝑇 = Table 10 presents optimum 𝐾𝑂𝐹 𝑂𝐿 𝑚 1.0, 𝐻(𝑡) = 𝑡 and 𝑚 = 2.0, which also agrees with the above analytical ∗ ≤ 𝐾∗ discussions. Table 10 has the same property with Table 6, and 𝐾𝑂𝐹 𝐹 ∗ ≤ 𝐾∗ . and 𝐾𝑂𝐿 𝐿
Forming 𝐻1 (𝑁 + 1, 𝐾; 𝑇 ) − 𝐻1 (𝑁, 𝐾; 𝑇 ),
6. Conclusions
which follows that H1 (N, K; T) increases strictly with N from H1 (0, K; T) to h(∞). Similarly, forming 𝐻1 (𝑁, 𝐾 + 1; 𝑇 ) − 𝐻1 (𝑁, 𝐾; 𝑇 ),
(𝑁+2)𝑇
∫(𝑁+1)𝑇
𝑃 𝐾 (𝑡)ℎ(𝑡)d𝑡
(𝑁+1)𝑇
−
∫𝑁𝑇 (𝑁+2)𝑇
=
In the viewpoint of cost rate, replacement polices that are carried out at periodic times NT and at repair numbers K have been compared analytically from the optimizations of the integrated models. In addition, other comparative results among replacement policies, such as replacement first and replacement last, replacement overtime and its original model, have also been obtained analytically and numerically. Our comparisons have started from formulating the integrated models with two above replacement polices, which have been widely used in total productive replacement (TPM) management in Japanese industry.
∫(𝑁+1)𝑇
𝐾 ∑ 𝑗=0
𝑗=0
(𝑁+1)𝑇
8
∫𝑁𝑇
𝑝𝐾 (𝑡)
(𝑁+2)𝑇
∫𝑁𝑇
𝐾−1 ∑ 𝑗=0
𝑝𝑗 (𝑡)d𝑡 {𝐾−1 ∑ 𝑗=0
𝑃 𝐾 (𝑢)d𝑢
∫(𝑁+1)𝑇
(𝑁+1)𝑇
𝑝𝑗 (𝑡)ℎ(𝑡)d𝑡
∫𝑁𝑇
(𝑁+1)𝑇
=
𝑃 𝐾 (𝑢)ℎ(𝑢)d𝑢
𝑃 𝐾 (𝑡)
∫𝑁𝑇 𝐾 ∑
∫𝑁𝑇
{
(𝑁+1)𝑇
−
(𝑁+1)𝑇
∫𝑁𝑇
𝑗=0
d𝑡 > 0,
𝑝𝑗 (𝑢)d𝑢
(𝑁+1)𝑇
∫𝑁𝑇
(𝑁+1)𝑇
∫𝑁𝑇
}
𝑃 𝐾 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
(𝑁+1)𝑇
𝐾−1 ∑
𝑃 𝐾 (𝑡)d𝑡
𝑝𝑗 (𝑢)ℎ(𝑢)d𝑢 }
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
d𝑡
ARTICLE IN PRESS
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Reliability Engineering and System Safety 000 (2017) 1–10
(𝑁+1)𝑇
=
∫𝑁𝑇
𝑝𝐾 (𝑡)
(𝑁+1)𝑇
+
∫𝑡
(𝑁+1)𝑇
∫𝑁𝑇
𝑡
})
𝑝𝑗 (𝑡)
Proof. Note that
d𝑡.
̃ 1 (𝑁, 𝐾; 𝑇 ) < ℎ((𝑁 + 1)𝑇 ), ℎ(𝑁𝑇 ) < 𝐻
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
̃ 1 (𝑁, 𝐾; 𝑇 ) = ℎ((𝑁 + 1)𝑇 ). lim 𝐻
Thus, by using the similar method of Appendix A, Appendix C can be proved. □
d𝑡
} 𝑝𝐾 (𝑢)[ℎ(𝑢) − ℎ(𝑡)]d𝑡 d𝑢,
𝑡
∫𝑁𝑇
̃ 1 (𝑁, 𝐾; 𝑇 ) = ℎ(∞), lim 𝐻
𝑁→∞
𝐾→∞
}
(𝑁+1)𝑇
∫𝑡 {
̃ 1 (0, 𝐾; 𝑇 ) to h(∞) and increases strictly increases strictly with N from 𝐻 with K from [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 to ℎ((𝑁 + 1)𝑇 ).
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
∫𝑁𝑇
𝑗=0
{
𝑝𝐾 (𝑡)
(𝑁+1)𝑇
=
(𝐾−1 { ∑
𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢
Noting that ∫𝑁𝑇
[m5GeSdc;May 15, 2017;12:4]
Appendix D
the above equation is (𝐾−1 { }) (𝑁+1)𝑇 𝑡 ∑ 𝑝𝐾 (𝑡) 𝑝 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢 d𝑡 ∫𝑁𝑇 ∫𝑁𝑇 𝑗 𝑗=0 (𝐾−1 { }) (𝑁+1)𝑇 𝑡 ∑ − 𝑝𝑗 (𝑡) 𝑝𝐾 (𝑢)[ℎ(𝑡) − ℎ(𝑢)]d𝑢 d𝑡 ∫𝑁𝑇 ∫𝑁𝑇 𝑗=0 (𝐾−1 (𝑁+1)𝑇 ∑ 𝑡 = 𝑝𝐾 (𝑢)𝑝𝑗 (𝑢)[ℎ(𝑡) − ℎ(𝑢)] ∫𝑁𝑇 ∫ 𝑗=0 𝑁𝑇 {[ ]𝐾 [ ]𝑗 } ) 𝐻(𝑡) 𝐻(𝑡) × − d𝑢 d𝑡 > 0, 𝐻(𝑢) 𝐻(𝑢)
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , ∞
̃ 2 (𝑁, 𝐾; 𝑇 ) = 𝐻
∫𝑁𝑇 𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡 ∞
∫𝑁𝑇 𝑝𝐾 (𝑡)d𝑡 ∞
increases strictly with N from 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 to h(∞) and increases strictly ̃ 2 (𝑁, 0; 𝑇 ) to h(∞). with K from 𝐻 Proof. Note that ̃ 2 (𝑁, 𝐾; 𝑇 ) < ℎ(∞), ℎ(𝑁𝑇 ) < 𝐻 ̃ 2 (𝑁, 𝐾; 𝑇 ) = lim 𝐻 ̃ 2 (𝑁, 𝐾; 𝑇 ) = ℎ(∞). lim 𝐻
𝑁→∞
which follows that H1 (N, K; T) increases strictly with K from H1 (N, 1; T) to [𝐻((𝑁 + 1)𝑇 ) − 𝐻(𝑁𝑇 )]∕𝑇 . □
𝐾→∞
Thus, by using the similar method of Appendix B, Appendix D can be proved. □
Appendix B Appendix E
For 0 < T < ∞, 𝑁 = 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , 𝑁𝑇
𝐻2 (𝑁, 𝐾; 𝑇 ) ≡
∫0
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , Q(N, K; T) increases strictly with N from H(T) to Q(∞, K; T) and increases strictly with K from Q(N, 0; T) to 𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 ).
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑁𝑇 ∫0
𝑝𝐾 (𝑡)d𝑡 ∞
increases strictly with N from H2 (1, K; T) to 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡 and increases strictly with K from H2 (N, 0; T) to h(NT).
Proof. Note that
Proof. Note that
𝐾→∞
𝐻2 (𝑁, 𝐾; 𝑇 ) < ℎ(𝑁𝑇 ),
lim 𝑄(𝑁, 𝐾; 𝑇 ) = 𝐻(𝑁𝑇 ) − 𝐻((𝑁 − 1)𝑇 ).
Forming 𝑄(𝑁 + 1, 𝐾; 𝑇 ) − 𝑄(𝑁, 𝐾; 𝑇 ),
1
lim 𝐻2 (𝑁, 𝐾; 𝑇 ) =
, ∞ ∫0 𝑝𝐾 (𝑡)d𝑡
𝑁→∞
𝑁 𝑁−1 ∑ ∑ [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 ) 𝑝𝐾 (𝑛𝑇 )
lim 𝐻2 (𝑁, 𝐾; 𝑇 ) = ℎ(𝑁𝑇 ).
𝑛=0
𝐾→∞
Differentiating H2 (1, K; T) with respect to T, 𝑝𝐾 (𝑇 )
𝑇
∫0
𝑛=0
−
𝑝𝐾 (𝑡)[ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 > 0,
𝑇
∫0
𝑝𝐾+1 (𝑡)ℎ(𝑡)d𝑡
𝑇
∫0
𝑝𝐾 (𝑡)d𝑡 −
𝑇
∫0
𝑝𝐾 (𝑡)ℎ(𝑡)d𝑡
𝑇
∫0
𝑛=0
𝑛=0
𝑁−1 ∑ 𝑛=0
𝑝𝐾 (𝑛𝑇 )
𝑝𝐾 (𝑛𝑇 )[𝐻 ((𝑁 + 1)𝑇 )−𝐻 (𝑁 𝑇 )−𝐻 ((𝑛 + 1)𝑇 ) + 𝐻(𝑛𝑇 )] > 0,
which follows that Q(N, K; T) increases strictly with N from H(T) to Q(∞, K; T). Similarly, forming 𝑄(𝑁, 𝐾 + 1; 𝑇 ) − 𝑄(𝑁, 𝐾; 𝑇 ),
𝑝𝐾+1 (𝑡)d𝑡,
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾+1 e−𝐻(𝑛𝑇 )
we have 𝐿1 (0; 𝐾) = 0 and
𝑛=0
𝑇 d𝐿1 (𝑇 ; 𝐾) = 𝑝𝐾 (𝑇 ) 𝑝 (𝑡)[𝐻(𝑇 ) − 𝐻(𝑡)][ℎ(𝑇 ) − ℎ(𝑡)]d𝑡 > 0, ∫0 𝐾 d𝑇
−
which follows that H2 (N, K; T) increases strictly with K from H2 (N, 0; T) to h(NT). □
=
𝑗=0
𝑛=0
𝑗=0
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
{
(𝑁+1)𝑇
∫𝑁𝑇
(𝑁+1)𝑇
∫𝑁𝑇
𝑛 ∑ [𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )] 𝑗=0
𝑃𝐾 (𝑡)ℎ(𝑡)d𝑡
+
𝑃𝐾 (𝑡)d𝑡
𝑁−1 ∑
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑛
9
[𝐻(𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑁−1 ∑
×
For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ ,
𝑁−1 ∑
𝑁−1 ∑
𝑛=0
Appendix C
̃ 1 (𝑁, 𝐾; 𝑇 ) ≡ 𝐻
𝑁 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
= 𝑝𝐾 (𝑁𝑇 )
which follows that H2 (N, K; T) increases strictly with N from H2 (1, K; ∞ T) to 1∕ ∫0 𝑝𝐾 (𝑡)d𝑡. Similarly, denoting 𝐿1 (𝑇 ; 𝐾) ≡
𝑁−1 ∑
} .
[𝐻(𝑗𝑇 )]𝐾+1 e−𝐻(𝑗𝑇 )
ARTICLE IN PRESS
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Reliability Engineering and System Safety 000 (2017) 1–10
Noting that
=
𝑁−1 ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
=
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑁
̃(𝑁, 𝐾; 𝑇 ) increases strictly with K to h(∞). which follows that 𝑄
𝑗=𝑛
References
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
[𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑗=0
the above equation is 𝑁−1 ∑
𝑛 ∑ [𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑛=0
𝑗=0
[𝐻 (𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
× [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 ) − 𝐻((𝑗 + 1)𝑇 ) + 𝐻(𝑗𝑇 )][𝐻(𝑛𝑇 ) − 𝐻(𝑗𝑇 )] > 0, which follows that Q(N, K; T) increases strictly with K from Q(N, 0; T) to 𝐻 (𝑁 𝑇 ) − 𝐻((𝑁 − 1)𝑇 ). □
Appendix F ̃(𝑁, 𝐾; 𝑇 ) increases For 0 < T < ∞, 𝑁 = 0, 1, 2, ⋯ and 𝐾 = 0, 1, 2, ⋯ , 𝑄 ̃(0, 𝐾; 𝑇 ) to h(∞) and increases strictly with K from strictly with N from 𝑄 ̃(𝑁, 0; 𝑇 ) to h(∞). 𝑄 Proof. Note that ̃(𝑁, 𝐾; 𝑇 ) = ℎ(∞), lim 𝑄
𝑁→∞
̃(𝑁, 𝐾; 𝑇 ) = ℎ(∞). lim 𝑄
𝐾→∞
̃(𝑁 + 1, 𝐾; 𝑇 ) − 𝑄 ̃(𝑁, 𝐾; 𝑇 ), Forming 𝑄 ∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
𝑛=𝑁+1 ∞ ∑
−
𝑛=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )]𝑝𝐾 (𝑛𝑇 )
= 𝑝𝐾 (𝑁𝑇 )
∞ ∑ 𝑛=𝑁
∞ ∑
𝑝𝐾 (𝑛𝑇 )
𝑛=𝑁 ∞ ∑
𝑛=𝑁+1
𝑝𝐾 (𝑛𝑇 )
𝑝𝐾 (𝑛𝑇 )[𝐻((𝑛 + 1)𝑇 )−𝐻(𝑛𝑇 )−𝐻((𝑁 + 1)𝑇 )+𝐻 (𝑁 𝑇 )] > 0,
̃(𝑁, 𝐾; 𝑇 ) increases strictly with N from 𝑄 ̃(0, 𝐾; 𝑇 ) which follows that 𝑄 to h(∞). ̃(𝑁, 𝐾 + 1; 𝑇 ) − 𝑄 ̃(𝑁, 𝐾; 𝑇 ), Similarly, forming 𝑄 [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾+1 e−𝐻(𝑛𝑇 )
𝑛=𝑁
∞ ∑
[𝐻(𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 )
𝑗=𝑁
∞ ∑
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑛=𝑁 ∞ ∑
∞ ∑
[𝐻(𝑗𝑇 )]𝐾+1 e−𝐻(𝑗𝑇 )
𝑗=𝑁
[𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 )][𝐻(𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑛=𝑁 ∞ ∑
×
□
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[𝐻((𝑗 + 1)𝑇 ) − 𝐻(𝑗 𝑇 )][𝐻(𝑗 𝑇 )]𝐾 [𝐻(𝑗 𝑇 ) − 𝐻(𝑛𝑇 )],
×
=
𝑛 ∑
× [𝐻((𝑛 + 1)𝑇 ) − 𝐻(𝑛𝑇 ) − 𝐻((𝑗 + 1)𝑇 ) + 𝐻(𝑗𝑇 )][𝐻(𝑛𝑇 ) − 𝐻(𝑗𝑇 )] > 0,
𝑛=0 𝑛 ∑
−
[𝐻 (𝑛𝑇 )]𝐾 e−𝐻(𝑛𝑇 )
𝑁−1 ∑
𝑁−1 ∑
∞ ∑
∞ ∑ 𝑛=𝑁
𝑛=0
×
[m5GeSdc;May 15, 2017;12:4]
[𝐻 (𝑗𝑇 )]𝐾 e−𝐻(𝑗𝑇 ) [𝐻 (𝑛𝑇 ) − 𝐻(𝑗𝑇 )]
𝑗=𝑁
10