Age Based and Partial Preventive Replacement Policies subject to Shocks

Online Access: www.absronline.org/journals

*Corresponding author: Behrouz Khorshidvand, Industrial Management Department, Kar Higher education Institute, Qazvin, Iran. E-Mail: [email protected]

813

Management and Administrative Sciences Review

Volume 4, Issue 5

Pages: 813-820

September 2015

e-ISSN: 2308-1368

p-ISSN: 2310-872X

Age Based and Partial Preventive Replacement Policies subject to

Shocks

Ashkan Ayough1*, Behrouz Khorshidvand2, and Akbar Alem Tabriz3

1. Productivity Management Department, Iran center for management studies (ICMS), Tehran, Iran 2. Industrial Management Department, Kar Higher education Institute, Qazvin, Iran 3. Industrial Management Department, Faculty of management and accounting, Shahid Beheshti University, Tehran, Iran

In this article, two systems subject to shocks occurring based on a non-homogeneous Poisson

process (NHPP) and failure rate are analyzed. The first system is consisted of a single unit

that partial preventive replacements (PPR) of lower indenture level components can be

done. The second system is consisted of single unit with age based replacement policy

(ABR). In PPR system, occurrence of a defeat causes system stopping and consequently

system will be received minimal repairs and following the occurrence of a shock the system

has received inspection. Also this system is replaced after time T preventively. In ABR

system a defeat leads to unit stopping and accordingly the unit has received minimal repairs

and after a shock has received inspection. This system is replaced at time T or at times less

than T preventively, due to failure of unit. The systems will be replaced with new and the

same types, so analysis done based on average cost rate. Two examples are studied and

optimal replacement age which minimizes expected cost is obtained and according to results

and comparison of models, some analyses are provided.

Keywords: Non-homogeneous Poisson process (NHPP), Partial preventive replacement (PPR), Age based replacement (ABR), Minimal repairs, Shock process, Inspection, Log linear process (LLP).

INTRODUCTION

Replacement policy is an important issue that has

been widely applied to the actual production

settings. In general, there are two major policies,

maintenance policy and replacement policy.

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Maintenance policy implies that an optimal

maintenance time is based on mean down time or

average working time. Replacement policy implies

that an optimal replacement time interval exists

after the long-time running of the system. Thus this

means that in the real production, we should find

out the optimal change time to make the loss of the

system minimum (Meng, 2012). Age based policy

is quite common and easy-to-implement in

practice. Under a basic age based replacement an

operating system is replaced at particular age or at

failure, whichever occurs first (Barlow, 1960). In

partial preventive replacement (PPR) models

policy will assume that, in our equipment, PPR of

lower indenture level components can be done, at

certain operating times Ti restoring the entire

equipment to its initial failure rate. However, it is

also common that after a certain number of PPRs,

these will be more expensive than a complete

preventive equipment replacement (PR) (Marquez,

2007). In the literature, developments on

replacement models based on the shock have

provided satisfactory results for the maintenance

operation. But in these researches, shock is not

considered as an independent parameter of failure.

In this article, both shock process and failure rate

are unavoidable and play an important role in

determining the optimum replacement age. The

main contribution of this article is developing

shock process and failure rate with ABR and PPR

models. These models are applied to industrial

equipment, due to the structure of their

components (e.g. cutting tools, hydraulic

structures, compressor blades and electronic

boards). This article constructed as follows:

literature are briefly reviewed in section 2. Model

description such as shock process and maintenance

models is given in the section 3. Assumptions and

notations are explained in the section 4. Numerical

examples for proofing models optimization and

analyses are given in the section 5. Finally

conclusions are drawn in the section 6.

LITERATURE REVIEW

The classical replacement policies proposed by

Barlow and Hunter, first researches following this

work studied total replacement models such as

constant interval replacement and age based

replacement. Regarding a system subject to shocks

which may cause system failures; Esary studied the

survival probability of a system with homogeneous

Poisson process (HPP) shocks. A-Hameed and

Proschan further considered a non-homogeneous

Poisson process (NHPP) case. Maintenance

optimizations have proposed for systems exposed

to risks by using of log linear process (Guo et al.,

2013). An extended replacement policy have

studied for a system subject to non-homogeneous

pure birth shocks by using non-homogeneous pure

birth Poisson process and developed the model to

optimize the number of repairs before replacement,

as well (Sheu et al., 2013). Comprehensive review

have explained total replacement models such as

constant interval replacement and age based

replacement models as well as shock based

replacement, partial replacement, replacement

with imperfect maintenance and inspection models

(Marquez, 2007). Age based maintenance have

studied strategies with minimal repairs for systems

considering degradation and shocks (Huynh et al.,

2011). Optimal bivariate preventive maintenance

policy presented with NHPP shock model and

cumulative damage model (Yen-Luan Chen, 2012).

In these researches, shocks are not intended as an

independent and immediately defeat factors, as

well as simultaneous evaluation of shock and

failure rate not performed and comparison

between PPR and ABR systems not checked that in

this article we will review and provide these issues

in several examples.

MODEL DESCRIPTION

Importance of models


815

The systems that are subject to external shocks by

occurring any of these shocks can be failure and

then the inspection to prevent larger and more

costly downtime should be done. Given that the

shocks have cumulative damage property and

occur randomly, thus the inspection with lower

cost can be done. Also the systems can be defeat

because of their failure rates that this event is

function of systems life cumulative distribution, in

other words it can be called an internal factor.

Shock process

In this article, a cumulative shock model is

considered to explain the shock process. The

probabilities for the shock damages to occur in

different time intervals are assumed to be

independent. The log linear process (LLP) is very

flexible and has been widely used to describe the

occurrence of random events (Guo, 2013).

N (T) is the number of events by time T. A

stochastic process is a non-homogeneous Poisson

process for some small value h if:

(i) N(0) = 0

(ii) Non − overlaping increments are independent

(iii) P(N(T + h) − N(T) = 1) = Λk

(T)h + o(h)

(iv) P(N(T + h) − N(T) > 1) = o(h)

For all T and where in little o notation limh→0

o(h)

h= 0

Thus, we consider stochastic shocks occur in a non-

homogeneous Poisson process (NHPP).

To simulate this process by using of LLP, intensity

function is

Λ(T) = η𝑒𝑐𝑇; 𝐾 ∈ (0, ∞), 𝑐 ∈ (−∞, ∞) (1)

Let N (T) demonstrate the number of shocks until

time T, then the expected number of shocks until

time T, denoted by D (T) is given by

D(T) = E[N(T)] = ∫ η𝑒𝑐𝑇𝑇

0

𝑑𝑠

= {

𝜂

𝑐(𝑒𝑐𝑇 − 1)

𝜂𝑇

∀𝑐 ≠ 0∀𝑐 = 0

(2)

Also, the probability distribution of N (T) is

𝑃(𝑁(𝑇) = 𝑛)(𝐷(𝑇))𝑛𝑒−𝐷(𝑇)

𝑛!

(3)

Age based replacement (ABR) policy

In this case, the preventive replacement is done

after equipment reaches a certain operating time-

age T. In case of equipment failure a corrective

replacement (CR) is done and the next preventive

replacement (PR) is scheduled after T units of time.

We again want to calculate the best T which

minimizes total expected cost per unit time

(Marquez, 2007)

Figure1. State of ABR policy over time

Partial preventive replacement (PPR) policy

In this policy we assume that, the total preventive

replacement of the equipment (PR) is done after

(k–1) partial preventive replacements (PPR). For

equipment subject to (i–1) PPRs with (i<k), the next

PPR will be done after Ti units of operating time

since the last PPR (or PR in case i=1). In case of

failure, a minimal repair will be carried out; this

will take the equipment back to operation but

without restoring its failure rate (Marquez, 2007)


816

Figure2. State of PPR policy over time

MODEL ASSUMPTIONS AND NOTATIONS

Assumptions

1 When T=0, the systems begin to work and

under external shocks constantly

2 If the systems work until time T properly,

replace preventively

3 For ABR system if the unit breaks before time

T, replaces correctively

4 If in time interval [0, T] unit suffers external

shocks, inspection done

5

Preventive and corrective replacement,

minimal repairs and inspection time are

negligible

Function Notations

R (T) The survival or reliability function and

means that probability of operation until

T

F (T) The failures function and mean that

probability of failure until T

f(x) The Probability density function of failure

D (T) The expected number of shocks until time

T

β (T) The expected number of failures until

time T

λ (x) Failure rate function

µ (T) For ABR system, the expected length of

the failure cycle.

V (T) The expected time of system operation

until time T

K (T) The expected cost of system until time T

C (T) Total expected cost per unit time

Cost Notations

CR The preventive replacement cost rate

CF The corrective replacement cost rate

CPPR The partial preventive replacement cost rate

CPR The total preventive replacement cost rate

CM The minimal repairs cost rate

CI The inspection cost rate

For PPR system, long term average cost rate is

C(T) =K(T)

V(T)

(4)

By considering partial preventive policy, the

expected cost of system until time T is

K(T) = CPPR [k − 1] + CPR + CI [D(T)]+ CM[β(T)]

𝐾(𝑇)= 𝐶𝑃𝑃𝑅 [𝑘 − 1] + 𝐶𝑃𝑅

+ [𝐶𝐼 (∑ ∫ 𝜂𝑒𝑐𝑠𝑑𝑠𝑇𝑖

0

𝑘

𝑖=0)]

+ [𝐶𝑀 (∑ ∫ 𝜆(𝑥)𝑑𝑥𝑇𝑖

0

𝑘

𝑖=0)]

(5)

Also, in this case the system works until k time to

carry out the PPR, thus the expected time of system

operation is


817

𝑉(𝑇) = ∑ 𝑇𝑖

𝑘

𝑖=1

(6)

Finally, taking into account equations (4), (5) and

(6) average cost rate is

𝐶(𝑇)

= (𝐶𝑃𝑃𝑅 [𝑘 − 1] + 𝐶𝑃𝑅

+ [𝐶𝐼 (∑ ∫ 𝜂𝑒𝑐𝑠𝑑𝑠𝑇𝑖

0

𝑘

𝑖=0)]

+ [𝐶𝑀 (∑ ∫ 𝜆(𝑥)𝑑𝑥𝑇𝑖

0

𝑘

𝑖=0)]) / ∑ 𝑇𝑖

𝑘

𝑖=1

(7)

And for ABR system, we have

𝐶(𝑇) =𝐾(𝑇)

𝑉(𝑇)

In this system, the expected cost until time T is

𝐾(𝑇) = [𝐶𝑅(𝑅(𝑇))] + [𝐶𝐹(𝐹(𝑇))] + [𝐶𝐼(𝐷(𝑇))]

+ [𝐶𝑀(𝛽(𝑇))]

𝐾(𝑇) = [𝐶𝑅 (∫ 𝑓(𝑥)𝑑𝑥∞

𝑇

)]

+ [𝐶𝐹 (∫ 𝑓(𝑥)𝑑𝑥𝑇

0

)]

+ [𝐶𝐼 (∫ 𝜂𝑒𝑐𝑠𝑑𝑠𝑇

0

)]

+ [𝐶𝑀 (∫ 𝜆(𝑥)𝑑𝑥𝑇

0

)]

(8)

Also, the length of failure cycle can be estimated

calculating the expected value of the failure

distribution now truncated in T as follows

(Marquez, 2007)

µ(T) =∫ xf(x)dx

T

0

F(T)

(9)

Thus, expected time of system operation until time

T is

𝑉(𝑇) = [𝑇(𝑅(𝑇))] + [µ(𝑇)(𝐹(𝑇))]

𝑉(𝑇)

= [𝑇 (∫ 𝑓(𝑥)𝑑𝑥∞

𝑇

)]

+ [∫ 𝑥𝑓(𝑥)𝑑𝑥

𝑇

0

𝐹(𝑇)(∫ 𝑓(𝑥)𝑑𝑥

𝑇

0

)]

(10)

Finally, taking into account equations (4), (8) and

(10) average cost rate is

C(T)

=[CR(∫ f(x)dx

∞

T)] + [CF (∫ f(x)dx

T

0)] + [CI (∫ ηecsds

T

0)] + [CM (∫ λ(x)dx

T

0)]

[T(∫ f(x)dx∞

T)] + [

∫ xf(x)dxT

0

F(T)(∫ f(x)dx

T

0)]

(11)

NUMERICAL EXAMPLES

PPR system

In this system, it is estimated that cumulative

function followed the Weibull distribution. In this

distribution λ is scale parameter and β is shape

parameter. If β>1 there is an aging process, if β=1

there is a constant process and if β<1 there is a

mortality process. We consider λ=0.5 and β=2. Also

we have CPR=80, CPPR=65, CM=35, CI=18 by

using of an approximate calculation of cost records

and shock parameters are η=0.6 and c=0.05. Also

numbers of partial preventive replacements (k-1)

with respect to structure are 1 and 3 respectively.

Weibull probability density function of failure is

𝑓(𝑥, 𝜆, 𝛽) =𝛽

𝜆(

𝑥

𝜆)

𝛽−1

𝑒−(𝑥𝜆

)𝛽

(12)

By using MATLAB we have

Table1. Total expected Cost per unit time

C( k , T)

T

k=2 k=4

0.1 749.827 712.327


818

0.2 401.354 382.604

0.3 294.548 282.048

0.4 248.159 238.784

0.5 225.936 218.436

0.6 215.797 209.547

0.7 212.563 207.206

0.8 213.644 208.956

0.9 217.602 213.436

1.0 223.575 219.825

Figure3. Total expected Cost per unit time

Shows the optimal point in the figures.

In PPR system, when β=2 (aging) and k=2, the

optimal partial preventive replacement time is 0.7

and cost rate is 212.563 and when k=4, the optimal

partial cycle is 0.7 but cost rete at this time is

207.206. With the increasing number of PPRs, the

cost rate has decline trend. In addition, by

increasing the number of PPRs, total cost rate

decreases.

ABR system

And now, we studied without lower indenture

level components, Similar to previous system, a

shock causes system will be received inspection

and a defeat causes system will be stopped and

received minimal repairs. In addition system

replaces with a same type at time T preventively or

at times less than T due to failure.

In this system, it is estimated the cumulative

function followed the Weibull distribution. In this

distribution λ is scale parameter and β is shape

parameter. We consider λ=0.5 and β=2 and we

have CR=80, CF=88, CM=35, CI=18 similarly to

previous system and shock parameters are η=0.6

and c=0.05.

By using MATLAB we have

Table2. Total expected Cost per unit

T C (T) T C (T)

0.1 839.018 0.6 354.710

0.2 468.605 0.7 386.677

0.3 367.308 0.8 429.456

0.4 336.172 0.9 481.494

0.5 336.157 1.0 541.737

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8 9 10

C(K

,T)

C(2,T)

C(4,T)


819

Figure4. Total expected Cost per unit time

In ABR system, optimal replacement cycle is 0.5

and at this time system cost rate is 336.157.

Comparison of models

Now, we compare PPR system with k=2 and ABR

system. In PPR system replacement time is more

than ABR with lower cost. As a result, if the system

is replaceable for lower indenture level

components have longer life than ABR system.

Also the cost rate in PPR system is lower than ABR

system.

Figure 5. Comparison of PPR (K=2) and ABR systems

CONCLUSIONS

In this paper, two optimal replacement policies

(PPR and ABR) by considering shock process are

presented and studied. We have shown optimal

replacement time through numerical examples.

Also, analyses are provided by considering

replacement policy type. The models in this article

will assist in replacement policies decision making

by considering internal defeats and external shocks

and will help managers to achieve the lowest cost

by considering reliability rate. In fact, by using

data mining documentation for maintenance of a

system and fitted to suit the cumulative function

and its parameters and estimate of maintenance

costs, through the use of proposed models the best

time to get a replacement system. Besides, these

models can use for warranty period time

determining when a system is new or after an

overhaul maintenance. Future research can

formulate models with the addition parameters

such as degradation or corrosion and also expand

models application by considering other

cumulative distributions and structure of

components.

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Age Based and Partial Preventive Replacement Policies subject to Shocks

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