Optimal ordering policies and optimal number of minimal repairs before replacement

Microelectron. Reliab., Vol. 32, No. 7. pp. 995--1002,1992.

0026-2714/9255.00+ .00 © 1992PergamonPress Ltd

Printed in Great Britain.

OPTIMAL ORDERING POLICIES AND OPTIMAL NUMBER OF MINIMAL REPAIRS BEFORE REPLACEMENT

SHEY-HUEI SHEU Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, Republic of China CHING-TIEN LIOU Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, Republic of China and BEE-CHING TSENG Department of Industrial Management, National Taiwan Institute of Technology, Taipei, Taiwan, Republic of China (Received for publication 20 February 1991)

Many maintenance policies in the literature have assumed that whenever a unit is to be replaced, a new unit is immediately available. However, if the procurement lead time is not negligible an odering policy should determine when to order a spare and when to replace the operating unit. This paper presents a model for determining the optimal ordering point and the optimal number of minimal repairs before replacement which include the optimal number of minimal repairs before replacemrnt of Park as a special case. We derive the expected cost per unit time in the steady-state as a criterion of optimality and seek the optimum policy by minimizing that cost. Finally, we present the numerical examples for illustration. 1. INTRODUCTION Preventive maintenance policies are of great importance in modern technology. Many authors have studied replacement policies, inspection policies, and their modifications. For example, Barlow and Proschan discussed age replacement, block replacement and inspection policies[3, pp. 84-118]. But many of them have assumed that whenever a unit is to be replaced, a new unit is immediately available. However if, as is often the case, the procurement lead time is not negligible, we should consider an ordering policy that determines when to order a spare and when to replace the operating unit after it has begun operating. Barlow and Hunter[2] considered the case of periodic replacement or overhall at times T,2T,3T,... (for some T>0) and minimal repair if the system failed otherwise. They considered cost c 2 of replacement and c I for each minimal repair. Makabe and Morimura[5,6,7] and Park[10] proposed the new replacement model where a system is 995

996

SI~Y-HuEI SI,~Uet

al.

replaced at n - t h failure, and discussed the optimum policy. These policies are commonly used with complex systems such as computers, airplanes, and large motors. Several authors[8,9,12,15,16] have treated ordering policies for a non-repairable unit, wherein an order for a spare is placed at failure or at a planned order time, whichever occurs first; the original unit is replaced as soon as the ordered spare is delivered, or the delivered spare is put into inventory until the original unit fails. In this article we proposed a model for determining the optimal odering point n and the optimal number k of minimal repairs before replacement which include the 'optimal number of minimal repairs before replacement' of Park as a special case. The model is described explicitly at the beginning of the next section. We derive the expected cost per unit time in the steady--state from this model. The optimal ordering point n and the optimal number k of minimal repairs before replacement which would minimize the expected cost per unit time are discussed. In section 2 the model is described, then the expected cost per unit time in the steady--state is found. In section 3 we present the numerical examples for illustration.

2. MODEL AND ANALYSIS In this paper we consider a one-unit system where each failed unit is repairable and each spare is provided, after a lead time, by an order. We assume that the original unit begins operating at time 0. An order for a spare is made at time Sn where Sn is the time of the n - t h failure. The cost of each order is c2. After the lead time L, the spare is delivered. If the k - t h failure occurs before Sn+L , then the unit is shutdown and replaced at time Sn+L with cost c s per unit time suffered for shortage. If the k - t h failure occurs after Sn+L , then the unit is replaced at time Sk by the spare in inventory with inventory cost c h per unit time. An original unit is replaced at the k-th failure, or at Sn+L , whichever comes last. The minimal repair is performed for failures in the interval [0,Sk), where Sk is the time of the k - t h failure. The cost of each minimal repair is c 1. After a replacement the procedure is repeated. We assume all failures are instantly detected and repaired. Assume that the system has a failure time distribution F(x) with finite mean # and has a density f(x). Then, the failure rate(or the hazard rate) function is r(x)--f(x)/F" (x) and the cumulative hazard function is R(x)=

(y)dy, which has a relation

F(x)--expf-R(x)} , where F'(x)=l-F(x). It is further assumed that the failure rate function r(x) is continuous, monotone increasing, and remains undisturbed by minimal repair. Let Yi denote the length of the i-th successive replacement cycle for i=1,2,.... Let t

R i denote the operational cost over the renewal interval Yi' Thus/(Yi, Ri) t constitutes a renewal reward process and the pairs (Yi' Ri)' i=1,2,.., are independent and identically

Ordering policies and minimal repairs

997

distributed. If D(t) denotes the expected cost of operating the unit over the time interval [0,t], then it is well-known that lira D-~tt E[R1] t-~® =-~-~'

(2.1)

(see, e.g.,Ross[13,p.52].) We shall denote the right-hand side of (2.1) by B(n,k). We assume that all failures are instantly detected and repaired. Let N(t) denote the number of minimal repairs performed on the system of age t. If no planned replacement are considered (i.e., k=®), then IN(t), t_>0}is a non-homogeneous Poisson process with intensity r(t) (see Barlow and Proschan[3], pp. 06-97). We now give a derivation of the expression for E[R1] and ELY1]. First, we need the following result. Le.t IN(t), t_>0t be a non-homogeneous Poisson process with intensity r(t), t_>0 r t and R(t)=EN(t)=J0r(u)du. Denote the successive arrival times by $1,$2,.....,Sn,... Then LEMMA

:

(a) the probabolity density function of the random variable Sn is given by exp{-R(x)I(R(x)) n-1 fs (x)= n (n-l)! (b) the joint probability density function of Sn and Sk where n
(9..2)

(Z3)

Proof : (a)

fSn(Xl=--~x P(Snn) ® exp{_R(x)}(R(x))J j=n exp{-R(x) }(R(x))n-lr(x) (n-I)!

(b)

fSn,Sk(X,y)=fSk i Sn(YIx)fSn(X) =---~-yP(Sk_k-n) = . . . ~ ~®

exp{-(R(y)_R(x))I(R(y)_R(x))J) j!

j =k-n exp{-(R(y)-R(x)) }(R(y)-R(x))k-n-lr(y) (k-n-1) ! exp{-(R(y)-R(x) } (R(y)-R(x))k-n-lr(y) ( k-n-1 ) ! exp{-R(x) }(R(x))n-lr(x) (n-l) ! exp {-R(y) }(R(x)) n-1 (R(y)-R(x))k-n-lr(x)r(y) (n-l) !(k-n-1)! We are now ready to derive the expression for E[R1] and E[Y1]. First note that

Thus fSn,Sk(x'y)=

SHEY-HuEISHEUet

998

f

Sn+L

YI=~ Sk

al.

if Sn
Sk_>Sn+L

and so E[Y1]=P(Sn<_SkSn+L)E[Sk ]. Using the LEMMA we can write rm¢x+L

Z[Vl]=(JoJx

oo

so

so

fSn,Sk(X,y)dydx)(IoXfSn(X)dx+L)+(IOIx+LfSn,Sk(X,y)dydx)

Oo

•(IoXfSk(X)dx)

(2.4)

n--1 ® ® ~ exp{_R(x)}(R(x))Jcl x But E[Sn]=I0XfSn(X)dX=IoP(Sn>x)dx=I0 ~ j! • j=0 n--1 I:I:+Lfsn,Sk(X,y)dydx) (I: dx+L) Thus EiYI]= ( ~ exp{-R(x)}(R(x))J j! j=0 k--1 I®I® I® exp{-R(x)}(R(x"J" j! ax)." +( 0 x+LfSn'Sk (x'y)dydx)( 0j= 0

(2.5)

We also note that RI={ (k-1)Cl+C2+Cs(Sn+L-Sk) if SnSn+L

'

and so E[R1]=P(SnSn+L)((k-1)Cl +c2+ChE[Sk-Sn-L]) =(k-1)cl+e2+P(Sn<_SkSn+L)(ChE[Sk-Sn-L]). Using the LEMMA we can write # rx+L

E[RI]--(k-1)cl+c'+(j0jx

e®rx+L

fSn,Sk(X,y)dydx)(CsJ0Jx (x+L-Y)fsn,sk(x,y)dy dx)

+(I®I~ 0 + LfS~,Sk(X,y)dydx)(chI:I~a_l(y-x-L)fSn,Sk(X,y)dydx). . . . . .

(,.6)

For the infinite-horizon case we want to find n and k which minimize B(n,k), the expected cost per unit time in the steady-state. Recall that ¢ r®cx+L r®cx+L

B(n,k)=t(k-1)Cl+C,+(J0Jx fSn,Sk(X,y)dyd~)(CsJ0Jx(x+L-Y)fSn,Sk(X,y)dy~) ,®,x+L

,®n--1

(JoJ fSn:Sk(X'y)dyd~)(J0j~ o exp{-R(x)}(R(x))J j,

:ix+L)

k--1 +( 0 x+LfSn'Sk ( x ' y ) d y ) ( Oj= 0

J!

The optimal solution n and k which minimize B(n,k) must satisfy the following inequalities : 1. B(n

,k )
3. B(n ,k )<_B(n -l,k )

2. B(n

,k )
4. B(n ,k )_
Ordering policies and minimal repairs

999

It seems to be very clumsy to show the existenceof the optimal n and k . However, we $

*

can find the optimal solution n and k by numerical method. W e present the numerical examples for illustrationin section 3. It is finallynoted that, when L=0, Cs=0, Ch=0, the expected cost per unit time in the steady-state becomes (k-l)Cl+C2

(2.8)

B(k,k)=k_ I ® exp{_R(x)}(R(x))J 0 J{ dx

I

j=O

and this is the same expression as found by Park[10].

3. N U M E R I C A L

EXAMPLE

Consider a system which is assumed to follow a WeibuU TTF distributionin computational amenability. Its cumulative hazard is R(t)=(At) ft.In this case. the expected cost per unit time in the steady-state becomes B(n,k)={(k-l)Cl+C 2 f~ rx+L(x+L y)eXp{--(Ay)fl}Aflkfl2xfln-lyfl--l(yfl-xfl) k-n-1 (n-l), (k-n-l), dyd~x

+o.CsJoJx

(v x L~exp(-(AY)fllAflk:xfln-lyfl-l(Yfl-xfl)k-n-ld,,dxll "{'(1-O)'ChIoJx+LW' ®® (n-1)!(k-n-l)! ~ l'

At(n)

I L)+(I-O)(

Ar(k)

[®[x+L exp{_(Ay)fl~A/3k~2x~-ly~-l(y~_x/3)k-n-1

where

~"m"~OOJx

(n--1) { ( k - n - 1 )

dydx.

}

If the value of n is given, the problem becomes the one-dimensional minimization of (3.1). Similarly, if the value of k is given, the problem becomes the one--dimensional minimization of (3.1). The combination of n and k yielding the lowest overall cost rate is the optimal solution. $

$

Table 3.1 and 3.2 give the results for the optimal solution n and k for the following two different case : Case a : A=0.05, fl=2, Cl--60 , c2=300 , cs=40 , Ch=4 , L=10. Case b : A=0.05,/3--2, Cl--60 , c2--300 , Cs=40 , Ch=4 , L=0. *

$

Table 3.3 gives the results for the optimal solution n and k as the dependence of the ordering cost c 2 where all the other parameters are fixed.

1000

SHEY-HUEISHEUet al.

TABLE 3.1

X..0.005, /)=2, ct..60, cz=300, cs--40, Ch'4, Iml0. k

B(n,k)

(n,k*) 8(n,k*) 1

2

4

5

6

7

8

9

0

23.65

16.41

15.46

15.35

15.46

15.67

15.93

16.21

16.51

(0,4) 15.35

1

25.25

16.73

14.37

13.94

14.00

14.25

14.58

14.94

15.31

(1,4) 13.94

20.77

15.75

13.84

13.54

13.69

13.98

14.34

14.72

(2,5) 13.54

18.97

15.44

13.78

13.48

13.66

13.96

14.32

(3,6) 13.48

18.04

15.36

13.91

13.58

13.75

14.05

(4,7) 13,58

17.53

15.40

14.12

13.77

13.91

(5,8) 13.77

17.25

15.51

14.36

14.00

(6,9) 14.00

17.10

15.65

14.63

(7,10) 14.26

17.05

15.81

(8,11) 14.53

17.06

(9,12) 14.82 B(n*,k*) =B(3,6) =13.48

2 3 n

3

4 5 6 7 8 9

(n*,k)

(0,i)

(0,2)

(1,3)

(2,4)

(2,5)

(3,6)

(4,7)

(4,8)

(5.9}

B(n*,k)

23.65

16.41

14.37

13.84

13.54

13.48

13.58

13.75

13.91

8

9

TABLE 3.2

X~0.005, ~ 2 , ct=60, Ci=300, Cs"40, Ch=4, L=0. k

B(n,k)

(n,k*) 8(n,k*) 1

2

4

5

6

7

0

20.93

17.54

16.64

16.38

16.38

16.51

16.70

16.93

17.18

(0,5) 16.38

1

16.93

14.87

14.51

14.55

14.76

15.03

15.33

15.66

15.98

(1,3) 14.51

13.54

13.44

13.64

13.94

14.29

14.65

15.02

15.38

(2,3) 13.44

12.64

12.95

13.33

13.73

14.14

14.54

14.94

(3,3) 12.64

12.38

12.82

13.27

13.71

14.14

14.56

(4,4) 12.38

12.38

12.87

13.34

13.80

14.23

(5,5) 12.38

12.51

13.01

13.48

13.94

(6,6) 12.51

12.70

13.20

13.67

(7,7) 12.70

12.93

13.42

(8,8) 12.93

13.18

(9,9) 13.18 8(n*,k*)

2 3 n

3

4 5 6 7 8 9

(n*,k)

(I,I)

(2,2)

(3,3)

(4,4)

(5,5)

(6,6)

(7,7)

(8,8)

(9.9)

B(n*,k)

16.93

13.54

12.64

12.38

12.38

12.51

12.70

12.93

13.18

-s(4,4)-s(5,5) -12.38

Ordering policies and minimal repairs TABLE 3.3

x-0.005, /)-2, el-60, c,-40, Ch'4, L-10.

(n*,k*)

B(n*,k*)

120

(1,3)

8.6595

180

(1,3)

10.5626

240

(2,5)

12.1171

300

(3,6)

13.4828

360

(4,7)

14.7632

420

(4,7)

15.9446

480

(5,9)

17.0175

540

(6,10)

18.0329

600

(7,11)

19.0005

CS

REFERENCES 1. H. Asher and H. Feingold, Repairable system reliability: Modeling, Inference, Misconceptions and their cause, Marcel Dekker, New York(1984). 2. R. E. Barlow and L. C. Hunter, Optimum preventive maintenance policies, Operations Research 8, 90-100(1960). 3. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. John Wiley, New York(1965). 4. R. E. Barlow and F. Proschan, Statistical Theory of Reliability. John Wiley, New York(1975). 5. H. Makabe and H. Morimura, A new policy for preventive maintenance, J. Operations Res. Soc. Japan 5, No.3, 110-124(1963). 6. H. Makabe and H. Morimura, On some preventive maintenance policies, J. Operations Res. Soc. Japan 6, NO.1,17--47(1963). 7. H. Makabe and H. Morimura, Some consideration on preventive maintenance, J. Operations Res. Soc. Japan 7,No.4, 154-171(1965). 8. S. Osaki, An ordering policy with lead time, Int. J. Systems Sci. 8, 1091-1095(1977). 9. S. Osaki, N. Kaio and S. Yamada, A summary of optimal ordering policies, IEEE Trans. Reliability R-30, 272-277(1981). 10. K. S. Park, Optimal number of minimal repairs before replacement, IEEE Trans. Reliability 1%-28, 137-140(1979). 11. K. S. Park, Optimal number of minor failures before replacement, Int. J. Systems Sci. 18, 333-337(1987). 12. Y. T. Park and K. S. Park, Generalized spare ordering policies with random lead time, Europ. J. Operations Res. 23, 320--330(1986).

1001

1002

SHEY-HUEISHEUet

aL

13. S. M. Ross, Applied Probability Models with Optimization Application, Holden-Day San Francisco(1970). 14. S. H. Sheu, A generalized model for determing optimal number of minimal repairs before replacement, Europ. J. Operations Res.(in revision). 15. L. C. Thomas, S. Osaki, An optimal odering policy for a spare unit with lead time, Europ. J. Operations Res. 2,409-419(1978). 16. A. D. Wiggins, A minimum cost model of spare parts inventory control, Technometrics 9, 661-665(1967).