Optimum scheduling of a new maintenance program under stochastic degradation

Mieroeleetron. Reliab., Vol. 29, No. 1, pp. 57-71, 1989. Printed in Great Britain.

0026-2714/8953.00+ .00 © 1989 Pergamon Pre-.spie

OPTIMUM SCHEDULING OF A NEW MAINTENANCE PROGRAM UNDER STOCHASTIC DEGRADATION B.D. SIVAZLIAN Department

of Industrial and Systems Engineering, University of Florida, Gainesville, Florida 32611, U.S.A.

(Received for publication

31 December

1987)

ABSTRACT In this paper we consider an intermittently

operating system which

is under periodic review and which is subject to random aging or degradation.

The optimal schedule when initiating a new maintenance

program is analyzed so as to minimize The scheduled maintenance rather than time. incorporate costs,

cost.

is based on the level of system degradation

The functional equation approach enables one to

such factors as the fixed cost of maintenance,

the discount

a given period.

I.

the long term discounted

factor and the probability

The analysis

the recurring

distribution

of aging in

is illustrated with an example.

INTRODUCTION We consider an intermittently

periodic

operating system which is under

review and which is subject to random degradation.

of degradation

The amount

during each review period is a random variable indepen-

dently and indentically

distributed

from period to period.

A typical

example would be a truck where the number of miles registered per month would measure

the level of degradation,

The system degradation known as recurring costs. system,

a major maintenance

necessary, program, assumed

induces increases

in operational

In order to compensate (or overhaul)

whose time is negligible

for the usage of the

relative to the review period,

the amount of degradation

initiated.

associated with each maintenance.

is

At the end of

since last maintenance

and if this amount exceeds a specified level, is immediately

if

The maintenance

to bring the system to a level as good as new.

maintenance

costs,

program is initiated,

at the beginning of a review period.

each review period, measured,

the review being month.

say R, a

There is a fixed cost K

We assume that a new maintenance 57

is

58

B, D. SIVAZL1AN

program is being contemplated and our objective is to determine at which level of R, say R 1 should the new maintenance program be initiated,. Also what should be the optimum level of R, say R 2, at which the new maintenance program should proceed in the future. Problems of maintenance can sometimes be categorized as replacement type problems.

The literature work has dealt with either continuous

review system or periodic review system operating under a variety of policies using undiscounted or discounted costs.

Among the more recent

publications one should note the work of Bergman and Klefsgo Okumoto and Elsayed (1986) and Nakagawa

(1983), Nakagawa (1986).

1 - following maintenance, new;

(1984) Nakagawa

(1983),

(1984), Tapiero

Most of these papers assume that: the system is brought to a level as good as

2 - the optimum maintenance policy consists in the determination of a single critical number; 5 - cost is undiscounted; 4 - the same maintenance policy is used over and over again. A dyadic policy was considered by Sivaz!ian and Iyer (1981) where two critical numbers had to be determined optimally. where following maintenance,

Another variant

the system is hysteretic,

that is, is

brought to a level which is not necessarily as good as new was studied by Brown, Mahoney and Sivazlian

(1982).

In general,

these variants are

considerably more complex than the standard problems and will not be considered here.

The problem under investigation assumes a discounted

cost and studies the optimal scheduling of a proposed new maintenance program over an existing one: the scheduled maintenance is based on the level of system degradation rather than time.

II.

PROBLEM STATEMENT We assume that time is divided into equal intervals known as

periods.

During each period the system deteriorates or degrades or

ages.

Following the last maintenance or overhaul,

the system age is

zero.

The aging could be measured on any selected scale, e.g., it could

be the number of hours of operation of the system, or it could be the number of miles registered. Assume that the amount of aging over successive time periods is a non negative continuous random variable,

identically and independently

O p t i m u m scheduling

59

distributed

from period to period with distribution

probability

density function @(~),0<~<-,

f

#(~)=

function 0(~),

and finite moments.

Thus

(i)

#(u)du

~0 The structure of the maintenance economic

factors affecting

policy usually will depend on the

the operation of the system.

Two such

factors must be identified.

a.

The cost associated

in

overhauling

the system which we shall denote

by the fixed cost K (although K could be a function of the age at which the system is overhauled).

b,

The cost associated in operating period.

If x denotes

the age of the system at the beginning of the

period following a decision, the end of the period. operating

the system for an additional

then the system will be age x + ~ at

Let C(x)dx be the recurring cost of

the system between ages x and x + dx.

cost of operating

the system for one period,

age x the beginning of the period,

h(x) =

In general,

iZfl

[C(u)du]

Then the expected

given that it is of

is

0 < x < ®

~(~)dE

L(x) will be a non-decreasing

(2)

function of x, assumed

differentiable. The maintenance period,

policy used herein is:

"When at the end of a

the system age exceeds a level R, overhaul

We shall denote by u, 0 < u < i. the discount rate measured

in

$/

is no new maintenance expected

($)(period). policy.

the system". factor or interest

We consider first the case when there

A functional equation

for the total

cost over an infinite horizon period is formulated and solved.

The optimal value R* of R is obtained. to solve the case when a new maintenance Finally an example is presented.

Next. we use the results derived program is to be instituted.

60

B.D. SIVAZLIAN

Ill.

i,

THE DISCOUNTED STOCHASTIC NEW MAINTENANCE PROGRAM

MODEL OVER INFINITE HORIZON

- CASE OF NO

The Functional Equation

age

T

/i

.l ! l

i

overhaul FIGURE I.

time

A Periodic Review Maintenance System

Let f(x) = total expected discounted cost over an infinite horizon period given that the initial system age is x, 0 & x < R; this is a conditional expected cost given x, which is also a function of R. Then f(x) satisfies the following functional equation (see figure I) R f(x) = L(x) + ~ J f(y) ~(y-x) dy J vx

+ a[Z + f(O)] { ~(y-x)dy,

(3)

JR To explain this functional equation, we note that the first term on the right-hand side represents the expected cost incurred during the first period.

The second term is the expected cost incurred when starting the

second period with the system age y, x < y < R, when there is no overhaul, discounted to time origin.

The third term is the expected

cost incurred when starting the second period with an overhauled system, that is age zero, discounted to time origin. Let

g(x) = L(x) + a [K + f(O)] f® @(y - x)dy

(4)

JR Then R

f(x) = g(x) ÷ ~ f

f(y) ~(y - x)dy x

(5)

O p t i m u m scheduling

61

The solution of this integral equation, which is the Volterra type, is

f(X)

=' g(x) + n=1~~n J~x g(u) o(n)(u -

0(n)(.)

where

refers

to

the

x)du

(6)

convolution of 0(.) with itself.

n th

Let O ( u ) d u = @(x)

(7)

Relation (4) can be writte9 g(x)

= L ( x ) + (~K 0(R - x ) + a f ( O )

Substituting (8) in (6) f(x)

:

L(x)

we

obtain,

+ ~K # ( R - x )

÷ af(O)

@(R - x )

(8)

#(R - x)

R

+ ~ n I [L(u) + aK O(R - u) + af(O) O(R - u)]o(n)(u - x)du n=l Jx

(9)

At x = 0 f(O)

L(O)

=

+

aK O(R)

af(O) )(R)

+

+n=l ~ ~n fl [L(u) +

aK 0(R

- u)]

o(n)(u)du

+ ~ n |~r(o) ~(R - u) ~(n)(u)du n=l

(io)

#~0

Hence

f(O)

R

m

r

11 - a ~ ( R ) - u Z a n L n=l

;(R- o, .'n'(u,du] R

= L(O)

+ aM O(R)

an

+ QK

Ii ;(R - u)o(n)(u)du

n=l

+

n=l~ ~n

(n)

~0 L(U) o(n)(u)du

Let T(U) MR 29: I-E

=

Z .no(n)(u) n=l

O~;u<~

(12)

B. D. SIVAZL~N

62

Then, the Laplace transform of x(u), say T(s), expressed in terms of the Laplace transform of 0(u), say $(s) is

~(S) =

~ an[$(S)] n n=l

uS(s)

(13)

. since l~$(s)l < i

i - a%(s)

Consider the expression X

M(x) = ~(x) ÷

~ n

I

n=l

~(x - U) ,(n)(u)du

0 ~

x <-

(14)

0

The Laplace transform of this expression is

$(S)][$(s)]n s

$(s) + ~ anFi

i M(S) = ~ -

S

n=l

--

-s

m

n

n=0

i - =%(s) =!÷_i.

=$(s)

s

s i - =$(s)

i s

i - ~ =

i

$(s)

s i - =$(s)

1 ~(s) s I - =%(s)

(i5)

Formal inversion yields

M(x)

= i

--1

-

(x

(X

i

x

.Z n

0(n)(u)du

0n=l

x

1

- -

(16)

[ x(u)du 00

Now compute the quantity R

1 - aS(R) = 1 - a [I - -1 -l- aa

T(u)du] 0

--(I- a)[i + ~ T(u)duI

Optimumscheduling

63

Using ( l h ) and (17), expression (11) can be written as R

(18)

f(O) [1 - aM(R)] = L(O) ÷ I L(u) T(u)du + aKM(R) J0

Solving for f(0),

L(O) +

|n(u) T(u)du

+ ~KN(R)

#o

f(o) =

1

-

aN(R)

R

L(O) ÷ =

-

K

+

oL(U) T(u)du + K

(19)

1 - aM(R)

Using (17) in (19), we get FR K + L(0) + L L ( u ) vu K + f(0) = R

T(u)du

(I - a) (i + I T(u)du) J0

PR K + L(O) + I L(u) T(u)du J0 (i - a) [z ÷ f(0)] = R

or

(20)

i + / T(u)du J0 Note that f(0) is a function of R and we are seeking for the value of R that minimizes f(0).

2.

The Optimal Value of R The optimal value of R, R*, can be obtained by differentiating

with respect to R and setting the result equal to zero. R

I *

(20)

Thus

R

T(uld

L(R) ~(R)

+ L(0) +

L(u) ~(u) du

T(R) = 0

(21)

#0

or R

R

L(R) - L(O) + L(R) JOIT(u) du - J/oL(U) T(U) du

=K

(22)

64

B . D . SIVAZLIAN

Proof: From equation (9)

f(x)

= L(x)

+ a[K + f ( O ) ]

Z an n=l

#(R - x )

L(u) ¢(n)(u - x)du

[

R

+ =[K + f ( O ) ]

r. a n=l

n

+(a

-

x)

÷

u) ®(n)(u - x)du

vX

Let v = u - x, then

f(x)

= L(x)

÷ a[K ÷ f(O)]

{

~(B

-

an

(R - x - v) ¢(n)(v)dv

n=l

R-X

+

I v0

L(x ÷ v) ~ a n 0(n)(v)dv n=l

U s i n g (12) and ( 1 4 ) ,

we g e t R-X

f(x)

= L(x)

* a [K ÷ f ( 0 ) ]

M(R - x ) ÷ /

L ( x * v) T ( v ) d v

!

vO

From (16) R-X

f(x) = L ( x ) +

a [i - 1 [- ~a

T(u)du] [K + f(0)] v0

~R-x J + I L(x + v) T(v)dv

(27)

J0 Differentiate

f(x) with respect to R, set the result equal to zero, and

obtain R-X ~f(x)

= =[K

+ L(R)

+ f(o)]

{ -

T(R - x )

= 0

{(1

[K ÷ f ( O ) ]

+,u,du

Thus

T(R -

x)

- a)

- L(R)}

R-x

(28) 0

Optimum scheduling

65

Factoring out and interchanging the order of integration yields B

u

L(R) - L(O) + fL'(u) f ,(v)dvdu = K qO

(23)

aO

Thus

L'(u)

1 +

dv

(24)

du = K

Let

L'(x) = ~(x)

and X

T(x) = loT(V) dv

then B

I ~(U) [1 + T(u)]du = K

(25)

0 Note that if L(x) is a strictly increasing function of x, then (25) has a unique solution in R. The formal process of differentiating (20) and setting the result equal to zero yields at optimality (denoted by *) em

K + L(0) + (i = a ) [ K

+ f*(O)]

L(U) r(u)du #0

=

I +

Ii

*~(u)du

= L(R')

3-

(26)

Equivalence Between the Minimization of the functions f(O) and f(x)

Proposition: If L(x) is a strictly increasing function of x, the unique root R=R" that minimizes f(O) with respect to R, minimizes also f(x) with respect to R.

Thus, the initial age x of the system does not affect the

determination of the optimal policy in this situation.

66

B.D. SIVAZL~N

From (26), it is clear that R=R* satisfies (28) since anal(0) ,~R=R* = 0 .

IV.

A VARIATION OF THE DISCOUNTED STOCHASTIC MODEL OVER INFINITE HORIZON - INSTITUTION OF A NEW MAINTENANCE PROGRAM

Assume that the old maintenance program is to be replaced by a new maintenance program having a different fixed cost and which may induce different degradation process on the system than the old one.

The

policy to be instituted is as follows:

If the initial age x of the system is less than RI, let the system operate until at the end of a review period its age exceeds for the first time R I.

Immediately, initiate the new maintenance program

on the system.

Let then the system operate until at the end of a

review period its age exceeds for the first time R 2 at which epoch the system is overhauled, and so forth.

If the initial age x of

the system is greater than RI, then initiate immediately the new maintenance program. Use index 1 to refer to the old maintenance program and index 2 to refer to the new maintenance program.

The following system of

functional equations are obtained:

fl(x) = Ll(X) + ~

I

Rlfl(y

@l(y

x)dy

x

0 < x < R1

(29)

0 w
(30)

1

f2(w) = L2(w) + u

I

R2f2(t

e2(t

w)dt

w

o K2 f2oll °2t w0t R2 It

is

evident

that

the general equation

(30) d e f i n e d

for

all

0 ~ w < R2

may be used in particular to obtain an expression for K 2 + f2(O) which is the expected cost associated to the second alternative (corresponding to initiating immediately a new maintenance policy). proposition, M'n22 f2(w) is equiavalent• to M'n22 f2(0).

Also, from our The objective is to

Optimumscheduling determine the optimal values of R 1 and R 2.

67

Despite the apparent

difficulty of this problem, it can be solved easily on the basis of the analysis performed in the previous section. From equation (30), using equation (25), the optimal value of R 2 is given as the solution to the equation

R2* ,

I

L2 (u) [1 + ~(u)]

du =

K2

(31)

00 where x

T2(X) :

f

T2(u)du

(32)

0

and

T2(u) =

(33)

E ~n ~2 (n)(u) n=l

Then, from (26):

(34)

(i - ~) [K 2 + f2*(O)] = L2(R2*) From (27) the formal solution to (29) is .RI-X [K2 + f2(O)]

r.RI-X

+ jI 0

LI(X + ~) Tl(~)d~

(35)

where

Tl(U) =

~ ~n$1(n)(u) n=l

(36)

Differentiating (35) formally with respect to R 2, we obtain (note fl(x) is a function of R 1 and R2)

~fl(x)

~-R2 - 0 -

~f2(0)

~-R2

and this yields for the optimal value of R 2, R2", defined by (31). Differentiating (35) formally with respect to R 1 and setting the result to zero yields

68

B . D . SIVAZLIAN

afl -- -

(1

-

~) [K2

+

f2(O)]

TI(R

1

-

x)

nl(a 1) TI(R 1

+

-

x)

= 0

At optimality

LI(RI*) = (i - a) [X 2 + f2*(0)] = L2(R2* ) Thus. having determined the optimal value of

(37)

R2*, the

optimal value of

RI* can be determined from (37) (see figure 2).

q,

hi(x) LI(RI*)

; L2(R2*) • . . . . . . . . . . . . . . . . . . .

I .........

I

) RI* FIGURE 2.

R2*

X

D e t e r m i n i n g t h e O p t i m a l V a l u e o f R1 f r o m t h e O p t i m a l V a l u e o f R2

It should be noted that the determination of R2* depends on the value of K 2 and L2(.). whereby the determination of RI* depends on LI(.) but is independent of K 1.

This is to be expected,

since the cost K 1

associated with the last maintenance program is a "sunk" cost, and does not affect any future decisions. We next provide an exsmple to illustrate the calculations.

V.

EXAMPLE:

AGING HAS AN EXPONENTIAL DISTRIBUTION

Assume first no new maintenance p r o g r a m .

0(~) = ke -x~.

k > 0, 0 < ~ < ®

C(u) -- au.

a > O. u > 0

Let

AND C(u) IS LINEAR

Optimum scheduling

69

Then

L(x)

and

=

(au d u ) ke -X~ dE

L' (x) =

= ~ x + a

V

a

~k ~k

k + s

z(s)

Also

ak

=

~k

k + s - ak

X(I-~)

+

s

k+s

Hence

r(u)

ake -k(l'a)u

=

Thus X

T(x) = [ a k e -X(l-c~)u du ~0

a

=

[i

- e -k(l-a)x]

Equation (25) becomes R 4~

Xa

J

J0

[i

+ ~ ~- a -

1 -

~

e -k(l-~)u] du = K

or

R~

Ii [I - ~e-k(l-a)U]du = Kk(l-a) a

Thus R* is the unique solution to the equation R* - ~

[i - e -k(l-a)R*] - Kk(1-a)a

or

e -k(1-~)R*

= 1 +

Let Z = k(l-a)R*.

Y=I NR 2 9 : 1 - F

+

2

k(1-~)R*

Then the intersection of the two curves

Kk2(1-e) 2 a~

Kk2(1-a)

Z G

70

B.D. S|VAZLIAN

and Y = e

-Z

will yield the value of Z.

This is graphically illustrated in figure 3-

Consider now the introduction of a new maintenance program, and assume that

@i(£) = kle-kl £

,

Cl(U) = alu

k I > O, 0 < £ <

a I > O, u > 0

@2(£) = k2e-k2£

,

k 2 > O, 0 < £ < ®

C2(u) = a2u

,

a 2 > O, u > 0

aI a2 Clearly. ml(x ) = ~ii × and Lm(x) = ~22 x

1 ÷

KX2(l-~)2 me

0 )

Z k(l-a)R*

FIGURE 3.

Solving the Equation

First we determine R2*.

e

-Z

= 1 +

Kk2(l-a) 2

Z

a~

(x

Let Z 2 = k2(l - ~)R2*, then Z 2 is the

unique solution to the equation

K2k~(l - 6) 2

Z2

e-Z2 = 1 + a2~

Once Z 2, hence R2*, is determined, El* is obtained from the equation

LI(RI*)

= L2(R2* )

or

aI a2 ~ii RI" = ~22 R2*

Optimumscheduling VI.

71

CONCLUSION

In this paper we considered an intermittently operating system which is under periodic review and which is subject to random degradation.

The optimal policy to schedule a new maintenance program

was analyzed so as to minimize the long term discounted cost.

The

scheduled maintenance was based on the level of system degradation rather than time.

The functional equation approach enabled us to

incorporate such factors as the fixed cost of maintenance,

the recurring

costs, the discount factor and the probability distribution of aging in a given period.

Although we have not addressed in this paper the

problem of a comparative study between alternative maintenance program, nevertheless the developed methodology should form the basis to perform future research in this area.

REFERENCES

[i]

Bergman, B. And B. Klefsgo, "TIT Transforms and Age Replacements with Discounted Costs", Naval Research Logistics Quarterly, 30, 631-639 (1983).

[2]

Brown, J. F., J. F. Mahoney and B. D. Sivazlian, "Hysteresis Repair in Discounted Replacement Problem", liE Transactions 15, 156-165 (1983).

[3]

Nakagawa, T., "Periodic Inspection Policy with Preventive Maintenance", Naval Research Logistics Quarterly, 31, 33-34 (1984).

[4]

Nakagawa, T., "Optimal Policy of Continuous and Discrete Replacement with Minimal Repair at Failure", Naval Research Logistics Quarterly, 31, 543-550 (1984).

[5]

Nakagawa, T., "Modified Discrete Preventive Maintenance Policies", Naval Research Logistics Quarterly, 35, 707-715 (1986).

[6]

Okumoto, K. and E. A. Elsayed, "An Optimum Group Maintenance Policy", Naval Research Logistics Quarterly, 30, 667-674 (1983).

[7]

Tapiero, C. S., "Continuous Quality Production and Machine Maintenance", Naval Research Logistics Quarterly, 33, 489-499 (1986).

[8]

Sivazlian, B. D. and S. N. Iyer, "A Dyadic Age-Replacement Policy for a Periodically Inspected Equipment Item Subject to Random Deterioration", European Journal of Operational Research, 6,

315-320 (1981).