POLICY FOR A SYSTEM WITH IMPERFECT MAINTENANCE

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International Journal of Engineering Advanced Research eISSN: 2710-7167 | Vol. 5 No. 1 [March 2023]

Journal website: http://myjms.mohe.gov.my/index.php/ijear

Chin-Chih Chang¹ and Yen-Luan Chen^2*

1 Department of Distribution Management, Takming University of Science and Technology, Taipei, TAIWAN

2 Department of Marketing Management, Takming University of Science and Technology, Taipei, TAIWAN

*Corresponding author: [email protected]

Article Information:

Article history:

Received date : 9 February 2023 Revised date : 17 March 2023 Accepted date : 23 March 2023 Published date : 30 March 2023

To cite this document:

Chang, C., & Chen, Y. (2023).

POLICY FOR A SYSTEM WITH IMPERFECT MAINTENANCE.

International Journal of Engineering Advanced Research, 5(1), 19-23.

Abstract: A very strong assumption in most maintenance policies in literature is that there is an unlimited supply of spare units, or in other words, a spare unit for preventive replacement is always available if needed. However, as is often the case, the procedure lead time of a spare unit is not negligible, and therefore, a spare unit ordering policy that determines when to place an order and what amount of an order should be incorporated into the maintenance strategies.

In this paper, we investigate an ordering-replacement policy for a deteriorating system with imperfect maintenance. As a failure occurs, the system suffers one of two types of failure and implements imperfect maintenance: type-I (repairable) failure is rectified by a minimal repair, and type-II (non-repairable) failure is removed by a corrective replacement. A spare unit is regularly ordered associated with system operation time T for preventive replacement, or emergently ordered at the occurrence of any non-repairable failure for corrective replacement, whichever takes place first. The main objective is to determine an optimal continuous spare unit ordering schedule T* for preventive replacement that minimizes the mean cost rate function of the system in a finite time horizon. The existence and uniqueness of optimal policy are derived analytically and computed numerically.

Keywords: Spare unit ordering, Lead time, Replacement, Minimal repair, Optimization.

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1. Introduction

The significance of maintenance has been increasingly realized as systems are becoming more large-scale and complex, where catastrophic failures may cause heavier damages and a social sense of instability (Nakagawa, 2006). Maintenance actions, especially replacement behaviors are widely adopted to avoid disastrous failures and to decrease economic losses in various industrial systems. Generally, replacement operations arranged before system failure and after system failure are called preventive replacement (PR) and corrective replacement (CR), respectively. Many PR policies have been proposed and discussed in the past few decades, see for example, Zhao et al. (2017), Chang et al. (2019), and Mizutani et al. (2020).

Most replacement policies in literature assume that a spare unit is immediately available at any time whenever it is needed. However, this prerequisite is often unrealistic or impossible. The procedure lead time of a spare unit should not be negligible. Once we take account of the lead times, we should consider an ordering policy that determine when to order a spare and when to replace the operating unit after it has begun operation (Sheu and Liou, 1993; Chien et al., 2010).

As a result, a tradeoff of the ordering time of the spare unit for replacement should be optimized. Many spare ordering policies have been proposed and discussed in the past few decades, see for example, Nakagawa and Osaki (1974), Sheu and Griffith (2001), Sheu and Chien (2004), and Chang (2018). However, the ordering time of the replacement unit in the above literatures is almost from zero. Therefore, a spare unit ordering policy from different perspectives is worthwhile to propose and discussed. More spare ordering policies are proposed and optimized from a variety of perspectives, see for example, Sheu and Liou (1993), Chien (2009), Chien and Chen (2010), Cheng and Li (2012), Sheu et al. (2013), and Zhang et al.

(2021).

In this paper, we take up a time ordering-replacement policy for a system with two failure types involving repairable one and unrepairable one. A repairable failure is rectified by a minimal repair, and an unrepairable failure is removed by a corrective replacement (CR). The spare unit is regularly ordered at time T for preventive replacement (PR), otherwise the spare unit order is emergently placed at the first unrepairable failure for corrective replacement (CR). The original system is replaced by its spare unit for PR or CR if available or it will wait until the arrival of the spare unit. The continuous T policy is scheduled strategically and the optimal ordering-replacement time T^* can be determined explicitly by minimizing its mean cost rate.

2. Model Description

A totally new system with a lifetime X has a general distribution F t( )P X( t), survival function F t( ) −1 F t( ), and probability density function f t( ) F t( ) t. Then the failure rate (or the hazard rate) r t( ) f t( ) F t( ) is assumed to increase from r(0)=0 to

( ) lim ( ) r t r t

  → .

When the deterioration system has failed, two types of failures and imperfect maintenance are considered. A type-I failure (repairable failure) occurs with probability q and is followed immediately by a minimal repair. A type-II failure (non-repairable failure) occurs with probability p ( 1= −q) and requires a corrective replacement (CR). Note that the system hazard rate ( )r t remains undisturbed by any minimal repair (Barlow and Hunter, 1960).

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A spare unit with lead time L for preventive replacement (PR) is regularly ordered at time T. If the type-II failure occurs before the regular order, then the system is shut down and an expedited order for corrective replacement (CR) is made immediately at the failure time instant with lead time L.

Related costs for operating this ordering-replacement procedure are defined as follows. The cost for a corrective replacement is C_CR, and the cost for a preventive replacement is C_PR (

CR PR 0

C C  ). The cost for each minimal repair is c_m. The cost for an expedited order is c_e, and the cost for a regular order is c_r (c_ec_r 0). The shortage cost per unit of time for an arrival spare unit is c_s.

3. Formulation and Optimization

Let Z be the waiting time until the first unrepairable failure, then from Brown and Proschan (1983), the survival function of Z is directly obtained

( ) ( ) exp( ( ))

F tz P Z =t − p t , (1) and distribution function F t_Z( ) 1 −F t_Z( ), probability density function f t_z( ) F t_z( ) t. Let U and V denote the length and the total cost of a replacement cycle, respectively. The mean cycle length ^{E U}

( )

per replacement cycle can be obtained as

( )

₀^T z^{( )d}

E U = L+



F t t. (2) The mean total cost ^{E V}

( )

per replacement cycle is

( ) ( ) ( )

( ) (

⁽ ⁾⁺⁰

)

^{( )}

( )d + ( ) ( )d .

PR CR PR z r e r z

T L T L

s T z m z

E V C C C F T L c c c F T

c L ⁺ F t t c ⁺ F t qr t t

= + − + + −

+ −

 

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Thus, the mean cost rate becomes

( ) ( ) ( )

( ) ( )

( ) (

⁰

)

0

( )+ ( )

( )d + ( ) ( )d

( )d

PR CR PR z r e r z

T L T L

s z m z

T

T z

C C C F T L c c c F T

c L F t t c F t qr t t CR T E V

E U L F t t

+ +

+ − + + −

+ −

 =

+

 



. (4)

We derive an optimal ordering time policy T^* that minimizes ^{CR T}

( )

. Differentiating ^{CR T}

( )

with respect to T and setting it equal to zero, we see that ^^{CR T}

( )

^{ =}^T ⁰ if and only if

( )

_PR

Q T =C , (5) where

( ) ( ) ( ) ⁽ ⁾ ⁽ ⁾

(

⁰

) (

⁰

)

( )d ( ) ( )

( )d ( ) ( )d ,

T

z CR PR z r e r z

T L T L

s z m z

T

Q T T L F t t C C F T L c c c F T

c L F t t c F t qr t t



+ +

 + − − + − − −

− − −



 

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( ) ( ) ( )

,

( )

( ) ( )

( ) | ( )

( ) ( )

z z

CR PR e r z s z L m

z z

f T L F T L

T C C c c r T c F L T c qr T L

F T F T

  − ^ ⁺ ^+ − + + + ^ ⁺ ^

   ,(7)

( ) ( ) ( ) ( )

r T  f T F T = pr T , ^F

(

^{L T}^|

)

^ ^{F T}^z^{( )}⁻^{F T}^z⁽ ⁺^L⁾. (8)

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Both functions F_{z L},

(

L T|

)

and r t( ) have the same monotone properties (see Barlow and Proschan, 1965, p. 23). Suppose the failure rate r t( ) is increasing with t, it is clear that ^

( )

^T

is increasing with T, and it causes that ^{Q T}

( )

is also increasing with T. Furthermore, let

(

^C^CR ^C^PR

)

^{F L}^z^{( )} ^c^r ^{c L}^s

(

⁰^L^{F t t}^z^{( )d}

) (

^c^m ⁰^L^{F t qr t}^z^{( )} ^{( )d}^t

)

  ⁻ ^{+ +} ⁻L



⁺



, (9)

( ) (

⁰

)

0

( ) ( )d ( )d

CR PR e s m z

z

C C c c L c F t qr t t L F t t





− + + +

 +



. (10) If ^

( )

⁰ ^^^and^

( )

^{ }^^{, then}Q

( )

⁰ CPR and Q

( )

 CPR, there exists a finite and unique

T* which minimizes ^{CR T}

( )

and the optimal mean cost rate is ^{CR T}

( ) ( )

^* ⁼^ ^T^* ^.

4. Numerical Example

Consider a system whose lifetime X follows a Weibull distribution F t( )P X( t) with scale parameter =0.1 and shape parameter  =3, i.e., F t( )= −1 exp[ (0.1 ) ]− t ³ . For the purpose of illustration, parameters are chosen to be C_PR =500, C_CR =1000, c_e=50, c_r =40,

s 20

c = , c_m =100, and L=2. The spare unit is preventively ordered at time T, the optimal ordering time point is shown in Table 1.

Table 1: Optimal Ordering Policy and Its Mean Cost Rate Function Mean Cost Rate

q T^* CR T( ^*)

0.9 10.7718 65.11774

0.8 9.9678 70.63175

0.7 9.3626 75.26069

0.6 8.8839 79.28053

0.5 8.4923 82.85166

0.4 8.1637 86.07611

0.3 7.8826 89.02343

0.2 7.6384 91.74329

0.1 7.4237 94.27251

Based on the numerical results, we have the following observations:

• The finite and unique optimal solutions exist for our policy as predicted by our results.

• The optimal ordering time T^*increase as the minimal repair probability q increases, stating that a shorter ordering time is arranged for a higher unrepairable failure probability.

• As might be mean, the optimal mean cost rate decreases as the minimal repair probability q increases, meaning that a higher minimal repair probability is better for a minimum mean cost rate.

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Chang, C.C. (2018). Optimal age replacement scheduling for a random work system with random lead time. International Journal of Production Research, 56(16), 5511-5521.

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