A discounted cost relationship

JOURNAL

OF MULTIVARIATE

27, 105-l 15 (1988)

ANALYSIS

A Discounted C. S. &EN*

Cost Relationship

AND THOMAS

University Communicated

H. SAVITS+

of Pittsburgh by the Editors

In Savits (1988. J. Appl. Probab. 4, in press) a very general cost mechanism for a maintained system was considered. There he established a relationship between the expected long run cost per unit time for the age and block maintenance policies. In the present paper a similar relationship is obtained for the expected total a-discounted cost. 0 1988 Academic press, h.

1. INTRODUCTION Recently Savits [3] considered a very general cost mechanism for a maintained system. There he established a relatinship between the expected long run cost per unit time for an age replacement policy and that for a block replacement policy. In this paper we now consider the expected total discounted cost for the same model. Again we show that there is simple cost relationship between the age and block replacement policies. The basic model is first reviewed in Section 2. In Section 3 we prove our main result. Lastly, some further cost relationships are detailed in Section 4.

2.

REVIEW OF THE BASIC MODEL

The model considered in Savits [3] can be described biefly as follows. The basic ingredient consists of a stochastic process {R(t); 0 < t < l}. Here we interpret R(t) as the operational cost of a unit on line during a time interval [0, t). The random variable c designates the time of a major Received March 8, 1988. AMS 1970 subject classifications: Primary 62NOS; Secondary 6OKlO. Key words and phrases: age replacement policy, block replacement policy, renewal reward process, discounting. * Partially supported by AFOSR Grant AFOSR-84-0113. + Supported by ONR Contract NOO14-84-K-0084 and AFOSR Grant AFOSR-84-0113.

105 0047-259X/88

$3.00

Copyright Q 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

106

CHEN

AND

SAVITS

unrepairable breakdown. At this time, we replace the failed item with a new identical unit. Thus we call [ an unscheduled or unplanned replacement. The cost for such an unplanned replacement is cr. The two maintenance policies we consider here are referred to as age replacement and block replacement. In the former case, a scheduled or planned replacement occurs whenever an operating unit reaches age T; in the latter case, a planned replacement occurs at the absolute times T, 2T, . .. . In either case, the cost of a planned replacement is c2. We assume that items put on line are independent and identical units and that both planned and unplanned replacements take negligible time. Throughout this paper, we assume (as minimal requirements) that the stochastic process R has right-hand limits on [0, i) and that R(t+ ) = lim S,t R(s) represents the unit operational cost on [0, t]. We shall sometimes find it convenient to extend R by setting R(f) = R(i) for t > 5. In addition, we assume that R(O+ ) = R(0) = 0 and P{ [ > 0 j = 1. In order to write down to total operational cost for the maintained system, it is convenient to introduce some further notation. First we consider the age replacement maintenance policy. Let { Ri(t); 0 < t 6 ci>, i = 1, 2, ...) be independent copies of (R(t); 0 < t < [}. Define Vi=min(ii,

T), ..+uk

if if

3

k=O ka 1,

(2.1)

and R,+(t) = Ri(t+ ) Ri(?i) + cl z{[,-c T) + QZ{[,> T)

if if

O
for i= 1, 2, ... . Here I, denotes the indicator function of the set A. Then the total operational cost over [0, t] for the age replacement policy, which we denote by KA(f), is given by KA(~)=

f, i=l

(2.2)

R,*(rti)+R,*+,(t-&I

if &
0 (I+

.‘. +(-k

if if

k=O k = 1, 2, ...

DISCOUNTED

and

Q(t)=

R,(t) I

Next, let {Qi(t); and set

107

COST RELATIONSHIP

k iCIR,(ii)fkc,+Rk+,(I-uk)

if

O
if

ak
06 t}, i= 1, 2, .... be independent

Q:(t) = ;i;:fi. I

c2

if if

(2.3)

copies of {Q(t); 0 < t }

OT.

(2.4)

Then the total operational cost over [0, t] for the block replacement policy, denoted by K,(t), is given by

Ks(t)= i Q?(T)+Qk*+,(t-kT)

(2.5)

i= I

ifkT
C,(t) = C,(c 0 = ECK,(t)l and

(2.4)

C,(t) = C,(c T) = ECK,(t)l, respectively. Consequently, the expected long run cost per unit time is given by the ratio of the average cost per cycle to the average length of a cycle, i.e.,

J (T)= lirn -= CA(t’ T) ‘CR*(?)1 A t-m t EC?1 and

(2.7) JB(T)=

Cd& lim r=

t-m

T) ECQ*(T)l T

.

The above results follow from the theory of renewal reward process (cf., Ross [2]). We are, of course, making the implicit assumption that E[(R*(q)l] and E[lQ*(T)I] are finite. If we denote the corresponding numerators by A(T) = E[R*(q)] and B(T) = E[Q*( T)], respectively, then it was shown in Savits [3] that W)=I

A( T- x) &J(x) PA7-j

(2.8)

108

CHEN

AND

SAVITS

where U(x) =CrzO P(cr,
3. DISCOUNTED

COST RELATIONSHIP

In this section we will establish a similar relationship between the discounted costs for the age and block maintenance policies. In order to define the notion of discounting, however, we need to assume that, with probability one, the cost functions KA(f) and KB(f) generate a signed measure on [0, co). This is indeed the case when the cost parameters c, and c2 are nonnegative and R(t) is a nondecreasing process. In order to avoid some technical considerations, we shall henceforth only consider the situation described immediately above. So let a > 0. We then define the a-discounted cost over [0, t] by

K?)(t) = b. r, eeau dK,,(u)

and

(3.1) Kg)(t)=!

e-‘“dKB(u), (0.11

where KA and K, are given by (2.2) and (2.5), respectively. The total a-discounted cost is obtained by replacing (0, t] with (0, co). First we consider the age replacement case. Then Jr)(T)

= lim E[K2)(t)] ,-a,

=aE

=E f

e --a” Z&(v) dv

(0,~)

eecrUdK,(u)

1

1

In the last step we used the expression (2.2). We now consider each sum separately.

DISCOUNTED

COST

109

RELATIONSHIP

For the second sum, we write a f

E[~i”‘e-““R:+,(~-~~)dv] 5t

k=O

=a

f

E[ll”e~““e~‘ilR:+,(R)dw] 0

k=O

= a f

E[epzSk]

E [ jqk” e-““R,*+

=aE

[I

1(w) dw]

0

k=O

’ eebWR*(w)

dw](

0

f

{E[c”~]*~)

k=O

= (1 -E[e-*“I)-‘&

f e-““R*(w)dw].

The second and third equalities above follow from independence and the identically distributed assumptions. Next, we write the first sum as

e

-‘”

i

R:(qJdu]

i=l

= f

E[R,*(qi)epa5’]

= f

i=l

E[e-“qiR.

,* (q.)]I E[e-“‘i-l]

i=l

=(l-E[e-*“])-‘E[e-““R*(rl)l.

Consequently,

= E[~(o,,I

epawdR*(w)l 1 -E[e-‘“1

We shall denote the numerator

(3.2)

’

by A(“)(T), i.e., ecawdR*(w)

1 .

(3.3)

110

CHEN

AND

SAVITS

It is the expected cc-discounted cost over one cycle. For the denominator, we can also write

where G(x) = P{ i > x} is the survival function of [. Since

we note that JA( T) = ii: ctP’( T). Recall that J,(T) Eq. (2.7).

(3.4)

is the expected long run cost per unit time given in

(3.5) Remark. One can also derive the result (3.2) from a renewal equation approach. More specifically, if C@)(t) = E [K?‘(t)], one can show that Cy)(f) satisfies the renewal equation -l”EIR*(q

+s

(O.tl

e

-=C~)(t-

A u)] dv+ep”‘E[R*(q

A t)]

x) dG*(x),

where G*(x) = P{q
lim E[Z$)(t)]=E t+ix,

By the same technique as illustrated

and

epa”dKB(u)

1 .

above, it is easy to derive

(3.6)

DISCOUNTED

111

COST RELATIONSHIP

In this case we denote the numerator

by B@‘(T), i.e.,

1

e-="dQ*(w)

.

Our main goal in this section is to relate A(‘)(T) proceed as in Savits [3]. Since f fO.rl

e-=w'dQ*(w)=a~~e-a-

(3.7) and B(“)(T).

We

Q*(v)du+e-"'Q*(t)-Q*(O),

we can rewrite jco, T, e-“*’ de*(w)

for ck < T< ak + 1 as

pL2wde*(w) = $ [ep”“-‘R,!“)(cj) s(0. Tl e j=l +

eCzukR~~

+ e-“3cl]

,(T-

ak)

cbTc2

+

using Eqs. (2.3) and (2.4). Here we set RI*)(f)

= a It e-av Ri(u+) 0

dv+e-“‘R,(t).

It can be thought of as the a- discounted operational line for a time interval [0, 1). Consequently, B’*‘(T) = E j e-““de*(w) [ (0,Tl

==0211 E

1

e --OLw de*(w); ak < T< ak+,

(0. Tl

k=O

cost of the ith unit on

=E[R’,“‘(T)+eeaT

I

i

{e-a~-lR~a)(~j)+e-““c,}

j=l

1

+e-““kR~~,(T-ak)+e-“Tc,;ak
-t f

ak+l

<

T]

E[e-““kR~~,(T-a,)+e-aTc,;ak
k=O

We now consider th terms in the first sum in more ak+l=ak+ik+lT we have

detail.

Since

112

CHEN

E[epaukRpJ

I([k+

() + eCauk+‘cL;

AND

SAVITS

ak+

1<

T]

=E[e~““kR~~l(~k+,)+e~““k+lc,;~k
Hence, the first sum is given by f

E[eCaukRpJ

l([k+ 1) +e-Zuk+lc,;

ck+ 1< T]

k=O -““E[R’“‘(()

+

e-ai

C,;[<

=5

e-““E[R’“)(S)+e~“rc,;r
T-X]

P(a,EdX)

&I(X)

where, as before, U(x) = cpzo P(ak < x) is the renewal function generated by iI, i-2, ... . Similarly, we can write the terms in the second sum as T
E[e-““kR~~,(T-ak)+e-rTC,;(Tk<

=E{e-““‘E[R’“‘(T-x)+e-“‘T-X)cZ;~~T-~]~,=.t;ak
and so -=Cz; ok < T< fJk+ 1]

k!. E Ceeaok&$(T-fT,)+e

=s

CO.=)

e -2-~E[R(CL)(T-~)+e-a(r-l)c,;

4’> T-x]

&I(x).

But, A’“‘(T)

= E [

e-“” W.vl

dR*(w)

1

R(u+)du+e~“T(R(T)+cZ);r~T

I

DISCOUNTED

Consequently,

113

COST RELATIONSHIP

we obtain the result P(T)

= [

e-““A(“)( T- x) dU(x).

(3.8)

CO.T)

We summarize the results of this section in the following theorem. (3.9) THEOREM. Under the model of Section 2 with cost parameters cl and c2 nonnegative and R(t) a nondecreasing process, the expected total a-discounted cost for the age and block replacement policies are given by

and

respectively, where --bw de*(w)]. EC! (0. Tl e

A’“‘(T) = E[J(,,,, Furthermore,

B@)(T)=j

e -““A”‘(

eeaw dR*(w)]

and

I#“‘( T) =

T- x) dU(x).

CO.T)

(3.10) Remarks: (i) It is clear from the proof that the cost parameters ci and c2 need not be constants. Everything remains as above if cl and c2 are random variables. Moreover, we may allow ci and c2 to be different for the two polices of age and block replacement. In this case, the form of (3.8) changes slightly. See Savits [3] for further details. (ii) One can readily show that if we define a subdistribution function dG(u), and let W be the associated H on CO, 0~)) by f4x)=~~o,xle-aU renewal function generated by H, then dW(x) = e-ax dU(x). Thus we many write (3.8) as

B’*‘(T) = j Consequently,

CO.7)

A’*‘( T- x) dW(x).

we can also write A’“‘(T)=E’“‘(T)-j-

B”‘(T-x)dH(x). CO.7-j

114

CHEN AND SAVITS

4. OTHER COST RELATIONSHIPS Thus far we have established relationships between A(T) and B(T) and also between A’“‘(T) and BaO( T). We complete the cycle by considering the relationship between A(T) and A’*‘(T) and also between B(T) and B”‘(T). Clearly A(T) = A”‘(T) and B(T) = B’O’( 7’). It thus remains to express A@‘(T) and B’“‘(T) in terms of A(T) and B(T), respectively. As in Section 3, we shall assume that R(t) is a nondecreasing process and that c1 and c2 are nonnegative. In addition, we shall assume that the functions A(T) and B(T) are right-continuous and of bounded variation on compact intervals. (4.1) THEOREM. (i) (ii) ProoJ

Under the above conditions, we have

A(a)(T)=j~O,T1 e-axd.4(x)+E[cZepor(i A “1. B’“)(T) = jco,r, ePax dB(x) + epmTE[c,]. We will only prove (i) since (ii) is similar. Consider

s

e pzXdA(x)=a[o’e co,77 =E

-‘“A(v)

dv + e-“‘A(T)

a ‘e-““{R(v)+c,}dv;@T [J 0

[I [,

+E

a

-I-E

T

e-““{R([)+c,}dv;~
i T

ep”“{R(v)+c,}dv;~2T

a

0

+ epzTEIR(I)

[,

1 1 1

+ c,; i < T]

+e-“TE[R(T)+c2;C> =E

- A(0)

T] -E[c2]

a ~e~“R(v+)dv+e~a’{R(~)+cI};~
+Ea

D

e -““R(v+)dv+e-“r{R(T)+c2};~>T

0

+E[c,(l-e-‘i);[
- E[c,

Thus we have the desired conclusion.

e-=(c A T)].

1 1

DISCOUNTED COST RELATIONSHIP

115

In the above derivation we replaced R(o) with R(u+ ) in two integrations. This is permissible since an increasing function can have only countably many discontinuities.

REFERENCES FELLER, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York. [2] ROSS, S. M. (1970). Applied Probability Models with Optimization Applications. Holden-Day, San Francisco. [3] SAVITS, T. H. (1988). A cost relationship between age and block replacement policies. J. Appl. Probab. 4, in press. [l]