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Economic Quality Control Vol 21 (2006), No. 1, 127 – 141

An Optimization Methodology for Condition Based Minimal and Major Preventive Maintenance P. Naga Srinivasa Rao and V.N. Achutha Naikan

Abstract: This paper proposes a condition based preventive maintenance (CBPM) policy for Markov deteriorating systems. The proposed model considers deterioration and random failures with minimal and major preventive maintenance (PM). Minimal repairs are carried out after every random failure, and the device is getting replaced after the occurrence of the deterioration failure. The system undergoes random inspections to assess the condition; mean time between inspections is exponentially distributed. Based upon the observed condition of the device, do nothing, minimal PM, and major PM may take place. Minimal PM makes the system one deterioration stage younger, while Major PM makes the system τ deterioration stages (τ > 1) younger. The proposed models consider increasing intensity for the random failures. An exact recursive algorithm computes the steady-state probabilities of the system. Optimal solutions of the model are derived based on two criteria, viz., (a) availability maximization, and (b) total cost minimization. Keywords: Condition based preventive maintenance, minimal preventive maintenance, major preventive maintenance, deterioration failure, random failure, minimal repair, stair-step failure rare.

1

Introduction

Preventive maintenance is defined as the activity undertaken regularly at pre-selected intervals, while the device is satisfactorily operating, to reduce or eliminate the accumulated deterioration [14], while repair is the activity to bring the device to a non-failed state after it has experienced a failure. Preventive maintenance on deteriorating devices extends the useful life in two ways. One by reducing the accumulated deterioration level, i.e. reducing the previously occurred deterioration. The other by reducing the deterioration rate of the device after performing the PM, i.e., reducing the future deterioration. Overhauls, replacements of the components etc, are the PM activities in the former case. Later case includes tightening and adjustments, balancing, lubrication, etc. Generally, there exist two types of preventive maintenance schemes, i.e. condition based and time based preventive maintenance [9]. For condition based preventive maintenance, the action taken after each inspection is dependent on the state of the system. It could be no action, or minimal maintenance to recover the system to the previous stage of degradation, or major maintenance to bring the system to as good as new state. For

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P. Naga Srinivasa Rao and V.N. Achutha Naikan

time based preventive maintenance, the preventive maintenance is carried out at predetermined time intervals to bring the system to as good as new state [18]. In this paper we focus on condition based preventive maintenance. When the value of the extended life is larger than the cost of preventive maintenance (this cost could be cost of downtime, repair expenses, revenue lost, etc.), it is worthwhile to carry out preventive maintenance. Optimal preventive maintenance policies are determined for repairable devices by minimizing total cost, maximizing availability, or optimizing some other objective. Dekker [3, 4], Pham and Wang [13], Hongzhou Wang [7], are the recent research reports on maintenance. Sim and Endrenyi [14, 15], Suprasad and Leland [17], Dongyan and Trivedi [5, 6] modeled the PM policies that reduce the accumulated deterioration level. Canfield [2], Park et al [12] considered the PM policies that reduce the future deterioration. This paper deals the PM activities that reduce the current deterioration level of the device. Suprasad and Leland [17] modeled the deterioration process as k up or operating states and one down or failed state. They assume that mean time between inspections of the device is exponentially distributed, and replacement or complete overhaul of the device takes place, if the identified deterioration stage of the device at an inspection exceeds a threshold n (n is a preventive replacement threshold, 1 < n < k) or in the deterioration failed state. The model maximizes the availability of the device by simultaneously optimizing n and inspection frequency. Sim and Endrenyi [14] modeled the deterioration process with k operating states and one deterioration failed state. They have also considered random failures with constant intensity independent of the deterioration stage of the device. Periodic preventive maintenances are performed which makes the device one deterioration stage younger. Moustafa [11] found a closed form solution for availability considers the same problem with multiple failure modes in each deterioration stage. Sim and Endrenyi [15], Hosseini et al [8], and Dongyan and Trivedi [5, 6] consider multiple maintenance actions viz., minimal and major maintenance. In [15] a statistic-based policy is proposed by putting a limit on number of minimal maintenance activities performed before a major maintenance or overhaul. In [8, 5, 6] condition based maintenance policies are analyzed. No maintenance, minimal maintenance, and major maintenance are the maintenance alternatives; any one may take place based upon the observed condition of the device. Minimal maintenance restores the device to the previous deterioration state, whereas major maintenance restores to as good as new state. Hosseini et al [8], and Dongyan and Trivedi [6] consider exponentially distributed mean time between inspections and repairs, whereas in [5] the authors consider the mean time between inspection and repair as generally distributed. Borgonovo et al [1] consider stepwise increasing failure rate with PM that makes the device one-deterioration stage younger using Monte Carlo simulation methods. All the above articles consider the effectiveness of major preventive maintenance as replacement/complete overhaul that brings the system to the as good as new condition. However, in actual practice, maintenance (minimal/major) is imperfect, and it may not bring the device to the as good as new state. Therefore, modeling the major preventive

Condition Based Minimal and Major Preventive Maintenance

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maintenance as imperfect that makes the device to τ deterioration states younger (τ > 1) is a more general approach. The above mentioned minimal and major PM modes are special case of our proposed model when τ ≥ k2 − 1. Inspired by the above observations, a condition-based minimal and major preventive maintenance policy is proposed in this paper for a deteriorating device considering: • non-zero inspection times, • increased intensity for random failures, and • general effectiveness for major PM activity that leads to τ deterioration stages recovery (τ > 1). For modeling the increasing intensity (e.g. Weibull hazard rate) of random failures we use a stair-step approximation algorithm [16]. The accuracy of the algorithm depends on the number of steps (in our case number of stages in deterioration process). The accuracy of the stair-step approximation algorithm increases with the number of states. But increasing states of the deterioration process makes the effect of the PM insignificant, if the recovery is only one stage. Therefore, by allowing more than one stage recovery and using stair-step approximation algorithm our model gives better results for required accuracy (by increasing the number of states in the deterioration process). The rest of the paper is organized as follows: Section 2 presents nomenclature and notation used in the proposed models. Assumptions of the proposed model and the model are given respectively in Section 3 and Section 4. State transition equations and recursive solutions for that are presented in Section 5 and Section 6 respectively. Section 7, discusses the optimization procedure. Example and discussion are given in Section 8, and conclusions are presented in Section 9.

2

Nomenclature and Notation

2.1

Nomenclature

The categories in each type are mutually exclusive and exhaustive. • Failure type 1. Deterioration: A process, where the important parameters of a device gradually worsen. If left unattended, the process leads to deterioration failure. 2. Random failure: A failure for which the time of occurrence is represented by a random variable. • Restoration type

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1. Preventive maintenance or simply Maintenance: A restoration wherein an unfailed device, at pre-selected (but not necessarily identical) intervals, has its accumulated deterioration reduced or eliminated. 2. Repair: A restoration, by which a failed device has returned to a working condition. • Maintenance and repair types 1. Minimal maintenance: Maintenance of limited effort and effect. Deterioration is modeled by discrete stages, minimal maintenance restores the device to the previous deterioration stage or to deterioration stage 1, whichever is worse. 2. Major maintenance: It restores the device to the previous τ deterioration stages or to deterioration stage 1, whichever is worse. (τ > 1). 3. Minimal repair: Repair, by which the device is returned to the operable state it was in just before failure. 4. Overhaul/replacement: Repair wherein the device is returned to “as good as new”. 2.2

Notation

k k1 k2 i Di RFi P mi P Mi Ii DF Pdf

: : : : : : : : : : :

Ppm

:

PpM

:

Prf

:

1 λd 1 λi 1 λI 1 µdf 1 µpm

: : : : :

number of stages of deterioration before deterioration failure threshold deterioration level for performing minimal PM activity threshold deterioration level for performing major PM activity state of device (1 ≤ i ≤ k) index for deterioration states index for random failure states index for minimal preventive maintenance states index for major preventive maintenance states index for inspection states index for deterioration failure state steady-state probability that the device is being overhauled following a deterioration failure steady-state probability that the device is out of service due to minimal preventive maintenance steady-state probability that the device is out of service due to major preventive maintenance steady-state probability that the device is out of service due to minimal repair after a random failure mean sojourn time of the device in a deterioration stage mean time to random failure in ith deterioration state mean time between inspections mean duration of overhauling the device following a deterioration failure mean duration of minimal preventive maintenance


1 µpM 1 µi 1 µI

A TC Cdf Crf Cpm CpM CI Cdt CT T Cdf T Cf r T Crf T Cpm T CpM T CI T Cdt

3

: : : : : : : : : : : : : : : : : : :

131

mean duration of major preventive maintenance mean duration of repair after random failure in ith state mean duration of inspection steady-state availability of the device total cost per unit operating time cost of deterioration failure cost of random failure cost of performing a minimal PM activity cost of performing a minimal PM activity cost of performing an inspection cost of down time per unit time time between successive replacements or overhauls total cost of deterioration failures in a CT total cost of failure replacements in a CT total cost of random failures in a CT total cost for minimal P M s in a CT total cost for major P M s in a CT total cost for inspections in a CT total cost of down time in a CT

Assumptions and Model

3.1

Device

1. The device is continuously operating except when removed for maintenance or repair. 2. Failure types: deterioration and random, and it is assumes that the failure states and stage of the device can be discovered without any inspection. 3. The device has a deterioration failure immediately following the completion of k stages of deterioration. 4. Random failures show an increasing rate or intensity with increasing accumulation of deterioration. 5. The device is repairable. The failure type is self-announcing. The mean time to repair depends on the failure type. 3.2

System

The system consists of the device and maintenance activities. 1. The duration of each deterioration stage has an exponential distribution with rate λd .

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2. Following deterioration failure, the device is overhauled or replaced, i.e., restored to “as good as new”. The overhaul duration is exponentially distributed. 3. Random failure occurs at an increasing rate or intensity, depending on the deterioration stage of the device. Minimal repairs are performed at random failures that brings the device to just before failure condition. Duration of mean time to minimal repairs is exponentially distributed. 4. Inspections, having exponentially distributed duration time, are performed on the system to know the condition of the device. 5. The device undergoes a minimal PM action, when an inspection reveals a deterioration stage i with k1 ≤ i < k2 , and undergoes a major PM action if the inspection reveals a deterioration stage i with i ≥ k2 . 6. Preventive maintenance durations are exponentially distributed. 7. There are no transitions between the failure and maintenance states.

4

The Model

A continuous time Markov chain (CTMC) model for the proposed minimal and major preventive maintenance scheme is shown in Fig. 1. States D1 to Dk are the k operating states in increasing order of deterioration. D1 is the best operating state and Dk the worst. Di represents the state in which the device is in ith deterioration stage and is in working condition. Ii represents the state in which the device is in ith deterioration stage and under inspection. Pmi is the state where the device is in ith deterioration stage and is under minimal PM activity. P Mi is the state where the device is in ith deterioration stage and is under major PM activity. RFi is the state where the device is in ith deterioration stage and is under repair after a random failure. The device is inspected after a random period that is exponentially distributed with mean λ1I . By this model, the device experiences no maintenance, when the deterioration stage i is determined by an inspection to be i < k1, experiences a minimal preventive maintenance if k1 ≤ i ≤ k2, by which the device is restored to i − 1 deterioration stage with a mean 1 and experiences a major preventive maintenance if k2 ≤ i ≤ k, by which duration of µpm 1 . When the the device is restored to i − τ deterioration stage with a mean duration of µpM device is in random failure state minimal repair takes place that brings the device to just before failure condition in operating mode, with a mean duration µ1 . When the device is in deterioration failure state DF , replacement is carried out to bring the device to as good as new state, with a mean duration µ1df .


5

133

State Transition Equations

Using a Markov approach for the state transitions, the Chapman-Kolmogorov equations [10] are obtained, which are converting into a set of linear equations (also called system balance equations) representing the steady-state conditions of the process of interest. Their solution yields the long-term state probabilities. The Chapman-Kolmogorov equations for the considered process depend upon the real values of the limit for starting minimal PM (k1), limit for starting for major PM (k2), and recovered deterioration states by performing major PM (τ ). Based upon the realization of the above parameters, the system can fall into any one of the following five cases: a) b) c) d) e)

(k2 − τ ) > (k1 − 1) (k2 − τ ) > (k1 − 1) (k2 − τ ) ≤ (k1 − 1) (k2 − τ ) ≤ (k1 − 1) (k2 − τ ) ≤ (k1 − 1)

and and and and and

(k − τ ) ≤ (k2 − 2) (k − τ ) > (k2 − 2) (k − τ ) ≤ (k2 − 2) (k − τ ) > (k2 − 2) (k − τ ) ≤ (k1 − 1)

Each of the above cases yields a different set of linear first order partial differential equations and a different solution. The remaining part of this section presents the linear first order differential equations for the first of the above given five cases, i.e., for Case a. Steady state solutions for Case a are presented in next section. The remaining four cases have been treated analogously and the results can be obtained from the authors. Case a: (k2 − τ ) > (k1 − 1) and (k − τ ) ≤ (k2 − 2) ∂Pd1 (t) = (λ1 + λd + λi )Pd1 − (µrf Prf + µdf Pdf + µI PI1 + µPf 1 ) ∂t ∂Pdn (t) = (λn + λd + λI )Pdn − (λd Pdn−1 + µI PIn + µPf n ) ∂t for n = 2, . . . , k1 − 2 ∂Pdk1−1 (t) = (λk1−1 +λd +λI )Pf k1−1 −(λd Pdk1−2 +µI PIk1−1 +µPIk1−1 +µm Pmk1 ) ∂t ∂Pdn (t) = (λn + λd + λI )Pdn − (λd Pdn−1 + µPf n + µm Pmn+1 ∂t for n = k1, . . . , k2 − τ − 1) ∂Pdn (t) = (λn + λd + λI )Pdn − (λd Pdn−1 + µPf n + µm Pmn+1 + µM PM n+τ ) ∂t for n = k2 − τ, . . . , k − τ ∂Pdn (t) = (λn + λd + λI )Pdn − (λd Pdn−1 + µPf n + µm Pmn+1 ) ∂t for n = k − τ + 1, . . . , k2 − 2

(1) (2)

(3) (4)

(5)

(6)

134


∂Pdn (t) = (λn + λd + λI )Pdn − (λd Pdn−1 + µPf n ) ∂t for n = k2 − 1, . . . , k ∂Pdf (t) = λd Pdk − µdf Pdf ∂t ∂PIn (t) = µI PIn − λI Pdn ∂t for n = 1, . . . , k ∂Pmn (t) = µI PIn − µm Pmn ∂t for n = k1, . . . , k2 − 1 ∂PM n (t) = µI PIn − µM PM n ∂t for n = k2, . . . , k ∂Pf n (t) = µPf n − λn Pdn ∂t for n = 1, . . . , k

(7)

(8) (9)

(10)

(11)

(12)

Along with the above equation the following compatibility equation is used for obtaining the steady-state probabilities. k Pdi (t) + PIi (t) + Pmi (t) + PM i (t) + Pf i (t) + Pdf (t) = 1 (13) i=1

For the steady state representation, the left hand sides of all the partial differential equations for the five cases are equated to zero. This gives a set of linear equations. Recursive solutions for the equations in Case a are presented in next section. The solutions of the remaining four cases have also been determined and can be obtained from the authors.

6

Recursive Solution for the Steady-State Probabilities

The solutions for the balancing equations are obtained by means of a recursive algorithm. The solutions for the steady state balancing equations in Case a presented in Section 4 are obtained are as follows: µdf Pdf Pdk = (14) λd λd + λI Pdn for n = k1, . . . , k2 − 1 (15) Pdn−1 = λd λd + λI λI Pdn − Pdn+1 for n = k2 − 2, . . . , k − τ + 1 (16) Pdn−1 = λd λd


λd + λI λI Pdn − (Pdn+1 + Pdn+τ ) for n = k − τ, . . . , k2 − τ Pdn−1 = λd λd λd + λI λI Pdn − Pdn+1 for n = k2 − τ − 1, . . . , k1 Pdn−1 = λd λd λI Pdk1−2 = Pdk1−1 − Pdk1 λd Pdn−1 = Pdn for n = k1 − 2, . . . , 2

135

7

(17) (18) (19) (20)

Optimization

The optimization of the proposed maintenance model can be done by either maximizing the availability or by minimizing the total costs. By doing so the optimal solution specifies the following parameter values: • The mean time between inspections

1 . λpi

• The limit for starting minimal PM activities k1. • The limit for starting major PM activities k2. For example, the optimal value λ∗pi may be obtained by maximizing A=

k

Pdi

(21)

i=1

In most practical cases, the minimization of the total costs rather than maximization of the availability is of interest. The cost components include maintenance, repair, and outof-service costs. A simple model for the cost analysis is proposed by assigning costs (CI , Cpm , CpM , Crf , Cdf , Cdt ) for the unit times of the various outages (inspections, minimal PM, major PM, repairs after random and deterioration failures, and down time), and calculate the corresponding mean cost for a deterioration replacement cycle (CT ), where CT is the expected time between successive deterioration replacements. From these costs, the long-term cost per operating hour or cost per unit operating time is calculated for decision-making purpose. If CT is the expected time between successive deterioration replacements, then during one CT there is exactly one deterioration replacement. Therefore, 1 1 ⇒ CT = (22) CT · Pdf = µdf µdf · Pdf

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7.1


Cost Model

Expected costs per CT for deterioration failures, preventive maintenance, random failures, and down time are calculated as follows: • Expected cost of deterioration failure during CT For a deterioration failure cycle CT , there exists exactly one replacement due to deterioration failure and, therefore, E[T Cdf ] = Cdf

(23)

• Expected cost for performing inspections during CT : E[T CI ] = µI CI CT

k

PIi

(24)

i=1

• Expected cost for performing minimal PM activities during CT E[T Cpm ] = µpm Cpm CT

k

Pmi

(25)

i=1

• Expected cost for performing major PM activities during CT E[T CpM ] = µpM CpM CT

k

PM i

(26)

i=1

• Expected cost for random failures during CT E[T Crf ] = Crf CT

k

µPf i

(27)

i=1

• Expected cost of down time during CT E[T Cdt ] = Cdt CT

Pdf +

k

(PIi + Pmi + PM i + Pf i )

(28)

i=1

The long-term cost per unit time is obtained as: 1 E[T C] = (E[T Cdf ] + E[T CI ] + E[T Cpm ] + E[T CpM ] + E[T Crf ] + E[T Cdt ]) CT

(29)

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7.2

Optimization Procedure

1. Depending upon the accuracy required and/or deterioration replacement rate choose an appropriate unit for (a) number of deterioration states (k), and (b) mean time between inspections

1 . λI

2. Set k1 = 1. 3. Set k2 = k1 + 1. 4. Set

1 λI

equal to one.

5. Calculate different deterioration state probabilities using the solutions of the balancing equations (see Section 4). 6. Calculate the availability A (λI ) for time between inspections

1 . λI

7. Using equations from (22) to (29) find the expected total costs for various cost components, and cost per operating unit time. 8. Increase

1 λI

by one, and return to Step 3 until

1 λdf

is maximum.

9. Select the PM interval time λ1I that maximizes the availability or minimizes the cost per operating hour whichever be the criteria, note corresponding k1, and k2 values. 10. Increase k2 by one and repeat the steps from Step 4 to Step 9. 11. Increase k1 by one and repeat steps from Step 3 to Step 10. 12. Based on optimization criteria select a policy from all the selected policies in Step 9.

8

Examples and discussion

Assume the following set of technical input parameters where the times are given in weeks. 1 k λ1d µ1pI µpm 20 5 0.01 0.1

1 µpM

1 µdf

1 µi

2

10

5

Let the random failures of the device have a Weibull intensity with scale parameter η = 100, shape parameter β = 3. The cost data are given in thousands as: Cpi Cpm CpM Crf Cdf Cdt 1 2 15 10 100 1

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Table 1 shows sample calculation of maximum availability and minimum total costs for k1 = 10 and k2 = 15. It can be shown that the optimal mean time between inspection (OMTBI) for the above data are 2 weeks, and 8 weeks respectively, for availability maximization and total cost minimization objectives. After calculating maximum availability and minimum total costs at different values of k1 and k2 with 1 ≤ k1 < k and 1 < k2 ≤ k, select a policy based upon one of the objective, namely maximum availability or minimum total costs. Table 1: Maximum availability and minimum total costs for the example k1 = 10 and k2 = 15. Object k1 k2 TBI T.Cost (week) Availability Availability maximization 10 15 2 961 0.94592 Total cost minimization 10 15 8 622 0.92554 Optimal maintenance policies for the given example are displayed in Table 2. Comparison of condition based minimal PM, major PM, and both minimal and major PM policies are also given in Table 2. The proposed optimal maintenance policies are studied by changing various input parameters of the system. Table 2: Optimal maintenance policies for the given example problem. Aspect Performing both minimal and major maintenance Performing minimal PM only

Performing Major PM only

Object Availability maximization Total cost minimization Availability maximization Total cost minimization Availability maximization Total cost minimization

k1 k2 OTBI T.Cost (week) Availability 11 18 2 601 0.9777 11 14

6

384

0.963

2

–

2

905

0.975

2

–

4

690

0.967

–

10

4

616

0.945

–

10

11

497

0.939

The impact of the four cost parameters Cdf , Cpm and CpM , CI and Crf . The following observations were made: • Increasing inspection cost and repair cost at random failure lead to reduced k1, and k2 values, while increasing maintenance cost (minimal and major) lead to larger k1, and k2 values. • Increasing replacement cost at deterioration failure has hardly any effect on both k1 and k2 values.


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• Increasing maintenance cost (minimal and major), and inspection cost considerably increases the optimal mean time between inspection (OMTBI). • Increasing maintenance (minimal, and major), inspection, and repair costs results in larger optimal total cost and smaller optimal availability. • Increasing cost of deterioration failure leads to better availability along with smaller optimal total costs per unit time. From this we conclude that by increasing the cost of deterioration replacement, the solutions for both objective functions (cost minimization and availability maximization) come closer. Next, for the four time parameters, namely the mean inspection time λ1I , the mean mainte1 1 nance (minimal and major) time µpm and µpM , the mean replacement time at deterioration 1 failure µdf , and the mean repair time at random failure µ1i in the ith deterioration state, the following results were obtained: • If all the four time parameters are increased, optimal total cost increases, and availability decreases. • Increasing maintenance time (minimal, and major), and mean repair time at random failure decreases the OMTBI, and increasing mean inspection time increases the OMTBI . • Increasing all the four time parameters have negligible affect on the k1 value. • Increasing mean inspection time and mean repair time (both at random, and deterioration failure) reduces the k2 value. Increasing mean maintenance time (minimal, and major) increases the k2 value.

9

Conclusions

The proposed model assumes Markovian deterioration and considers the three PM actions to do nothing, minimal maintenance, and major maintenance. The model is capable of handling time varying failure rates, and more than one state recovery by performing a PM activity. Based up on the observed condition at an inspection, the optimization procedures specifies: • The deterioration states i in which no preventive maintenance is required (i < k1). • The deterioration states i that requires minimal maintenance (k1 ≤ i < k2). • The deterioration states i that require major maintenance (k2 ≤ i ≤ k). The proposed approach results either in an optimal maintenance policy with respect to the total costs, or with respect to the availability of the device.

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P. Naga Srinivasa Rao and V.N. Achutha Naikan Reliability Engineering Centre Indian Institute of Technology Kharagpur-721 302 West Bengal, India.