R machine repair problem with spares operating under variable service rates

Microelectro,. Reliab., Vol. 32, No. 8, pp. ll71-1183, 1992.

0026-2714/9255.00 + .00 © 1992 Pergamon Press Ltd

Printed in Great Britain.

COST ANALYSIS OF THE M/M/R MACHINE REPAIR PROBLEM WITH SPARES OPERATING UNDER VARIABLE SERVICE RATES K.-H. WANG" and B. D. SlVAELIAN Department

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

(Received for publication

19 March 1991)

ABSTRACT This paper studies the M/M/R machine repair problem consisting of M operating machines with S spares, and R repairmen in the repair facility under steady-state conditions. either cold-standby,

or warm-standby,

Spares are considered to be

or hot-standby.

servicing either at a fast rate or at a slow rate.

The repairmen are

A cost model is

developed in order to determine the optimal number of repairmen and spares simultaneously.

Under the optimal operating conditions,

system characteristics

are evaluated for three types of standby.

i.

INTRODUCTIONAND

several

LITERATURE REVIEW

In this paper, spares are considered to be either cold-standby, warm-standby,

or hot-standby.

or

A standby component is called a "hot

standby" if its failure rate is the s a m e as an operating unit.

The

standby machine i~ referred to as "warm standby" when the failure rate is nonzero and is less than the failure rate of an operating unit, and the standby machine is referred to as "cold standby" when the failure rate is zero.

Each repairman serves at the rate ~l until there are n

(n ~ R) failed machines in the system, at which epoch he switches to the faster rate ~ (#i ~ ~)" in the amount of service.

A decrease in service rate may reflect addition An increase in service rate often arises in

real llfe problem to reduce the level of backlog i t e m s (failed machines) in the queue. Analytic solutions of the Markovlan model for the machine repair problem (MRP) wlth no spares, cold standbys, and warm standbys were first obtained by Feller [I], Toft and Boothroyd [2], and Sivazllan and

*Presently at: Department of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan, People's Republic of China. ll71

1172

K.-H. WANO a n d B. D. SIVAZLIAN Wang [ 3 ] ,

respectively.

Cost (Profit)

p r o b l e m h a v e b e e n s t u d i e d by s e v e r a l

models in the machine repair authors

includin 8 Ashcroft

[4],

Morse [5], Elsayed [6], Gross and Harris [7], White, Schmldt, and Benett [8], Albrlght [9], Hilllard [10], Gross, et el. [11, 12], and Sivazllan and Wang [3], [15].

The machine repair problems with cold standbys were

first considered by Taylor and Jackson [13] and Toft and Boothroyd [2], while the incorporation of a cost (profit) model for cold standbys was studied by Hilliard [I0], Gross, et al. [ll, 12], and for warm standbys system, by Albrlght [9], and Sivazlian and Wan E [3]. The problem considered in this paper is more general than the work of Sivazllan and Wang [3]. General solutions for the MRP with spares (either cold standbys, or warm standbys, or hot standbys) are obtained for two mean repair rates.

This paper should be distlngulshad from

previous works in that: i) it studies the machine repair problem with spares for two types of repair rate; il) it generalizes the MRP with either no spares, or cold standbys, or warm standbys, or hot standbys; ill) the decision variables which are considered in a cost model are the number of repairman and the number of spares.

Previous work considered

either the rate of failure, or the rate of repair, or the number of repairmen, or the number of spares as decision variables.

Many past

results are obtained as special cases of~the present work. We first discuss the practical justification of the model.

Next,

we develop the M/M/R machine repair model with spares for two types of repair rate and show that it generalizes either the no-spare, or the cold-standby, or the warm-standby, or the hot-standby models.

Next,

using a cost model, we determine the optimal values of the number of repairmen and the number of spares simultaneously to minimize the steady-state expected cost per unit time, while maintaining a minimum specified level of system availability.

Finally, several system

characteristics are evaluated under optimal operating conditions for all three types of standby.

2.

PRACTICAL JUSTIFICATION OF THE MODEL

A number of practical in which the repairmen

problems arise

are servicing

either

w h i c h may b e f o r m u l a t e d at e fast

rate

a s one

or at a slow

rate. One p a r t i c u l a r

problem where this

m o d e l was a p p l i e d

was i n t h e

Cost analysis of a repair problem s t u d y o f unmanned a i r v e h i c l e s number o f p i l o t l e s s

aircrafts

In this

system, a given

(UAV) p e r f o r m s i m u l t a n e o u s l y s u r v e i l l a n c e

and/or observation missions. repairs,

(UAV) s y s t e m .

1173

The UAV's may b e s u b j e c t t o f a i l u r e s

and once r e c o v e r e d a f t e r

completing their missions are

catapulted

back i n the a i r by l a u n c h i n g pads which a c t as r e p a i r

facilities.

S p a r e UAVs a r e u s e d t o improve s y s t e m a v a i l a b i l i t y .

total

service

plus its

t i m e o f a UAV i s i t s

launch time.

otherwise.

recovery time plus its

Whenever a l l

The

repair

However, t h e r e c o v e r y t i m e i s e f f e c t i v e

some l a u n c h i n g p a d s a r e empty.

and

time

only if

I t w i l l n o t be a c o n t r i b u t i n g

factor

l a u n c h i n g pads are loaded (busy), the s e r v i c e

time s h o u l d be t a k e n as the r e p a i r / l a u n c h

time (fast

rate).

~Thenever

some l a u n c h i n g p a d s a r e empty, t h e s e r v i c e t i m e s h o u l d be t h e r e c o v e r y time plus the repalr/launch Likewise,

time (slow rate).

in the machine repair problem,

failed machines may first

be inspected in order to identify and locate what needs to be repaired and convey this information busy,

to the repairman.

~Jhen all repairmen are

the service time should be taken as the repair time (fast rate).

Whenever there is at least one idle repairman,

the service time should

be the inspection time plus the repair time (slow rate). Finally,

an increase in service rate may be intentionally

induced

to reduce the level of baoklo 8 items in the queue. It is not unusual to consider spares in real-llfe situations. order to simplify the mathematical as cold standby spares.

In

dlfficultles many authors treat them

One may try to incorporate more realism in the

model by assumln 8 that the spares are subject to casual failures encountered by extraneous uncontrolled

factors.

Then, it is reasonable

to assume that the failure rate of a spare machine is a nonzero constant (i.e., the failure rate of a spare machine is less than or equal to the failure rate of an operatln 8 machine).

In such a case, the spares act

as warm or hot standbys. In general, situations:

there are two reasons why spares are used in real-llfe

i) the system may be required t o operate continuously

long durations,

that is the operational

need to be increased,

efficiency of the system may

and li) the availability

or reliability of the

system needs to be improved by providing sufficient operational

and economic requirements

over

spares.

In general,

will dictate the optimum number of

spares as discussed later on in this paper.

1174

K.-H. WANGand B. D. SIVAZLIAN 3.

PROBLEM

STATEMENT

We consider a model with N - M + S identical machines and R repairmen in the repair facility.

There are M machines

state, and their failure rate is equal to A. are spares with failure rate a (0 ~ = ~ A). swltchover repair,

in an operating

The rest of the S machines We assume that the

time from standby state to operating state, from failure to

or from repair to standby state (or operatin 8 state if the

system is short)

is instantaneous.

We also assume that each of the

operating machines fails independently each of the available spare machines all the others.

of the state of the others, and

fails independently

Whenever one of these machines fails,

replaced by a spare if any is available. spare moves into an operating state, that of an operating machine.

of the state of

it is immediately

We now assume that when a

its failure characteristics

will be

Whenever an operating machine or a spare

fails it is immediately sent to a repair facility where failed machines are repaired in the order of their breakdowns.

Further,

the succession

of repair times and the succession of breakdown times are independently distributed

random varlables.

first-served

Service is provided on a first-come

(FCFS) discipline.

failed machine at a time.

Each repairman can repair only one

If an operating machine fails

(or spare

fails) and one spare is available at an instant when the repairman available,

the failed machine at once 8oes for repair,

put into operation.

If all repairmen are busy,

must wait until a repairman is available. being used and a breakdown occurs,

I

Operating Machines

A <--

and the spare is

then a failed machine

Moreover,

if all spares are

then we say that the system becomes

[

SpareSs

M

[r

J

(i)

(1): The system is short.

Repairmen R

(2): The system is not short.

A: The f a i l u r e rate of an operating machine. a: The f a i l u r e rate of a spare machine. Figure

1.1.

is

The M/M/R Machine Repair Model With Spares.

Cost analysis of a repair problem

1175

short in which case there are less than I4operating machines. Once a machine is repaired, it is as good as new and goes into standby or operating state. When the repair of a failed machine is completed, it is then treated as good as new unless the system is short in which case the repaired machine is sent back immediately to an operating state. The above machine repair model is shown in Figure (1).

4.

STUDY-STATE SOLUTIONS BOB THR U/M/R MACHINE REPAIR MODEL

Let n represent the number of failed machines in the system. For this M/M/R model, the mean arrival rate A, is given by

I44 +

‘n

-

(S

- n)o

(I-l + S - n)A

i

10

n

- 0, 1, 2, .... s

n - s+l, S+2, .... M+S-1

(1)

otherwise.

The mean repair rate pn is given by

pn (

W

n - 1, 2, .... R-l

&

n - R, R+l, .... M+S-N

0

otherwise.

(2)

In steady-state, let PO - probability that no machines are broken down, and 'n - probability that there are n failed machines in the system, where n - 1, ....

M+S-N.

The

steady-state equations for Pn are given by

(i)

For R 5 S

(Mx+Sa)Po - 1rlP1, [MX+(S-n)cr+w]P,- [MA+(S-n+l)a]Pn_l+ (n+l)rlP,+l, 15 n < R-l [MA+(S-n)a+w]P, - [MA+(S-n+l)a]P,_l+ QP,l,

n - R-l

(3) [MA+(S-n)a+Rp]P,- [MX+(S-n+l)a]Pn_l+ QPn+l,

[W-n)A+R~lP,

- [Wn+l)XIPn_l

"N-1 - tipN. and (ii) For R 7 S

+ FQJP,,+~,

R
1176

K.-H. WANG and B. D. SIVAZLIAN (MA+Sa)P 0 - ~IPI (~+(S-n)~,~]P

n -

[MA+(S-n+I)a]Pn. 1 + (n+l)PlPn+ 1,

[ ( N - n ) ~ + ~ ] p n - [(N-n+I)A]Pn. 1 + (n+l)~iPn+ I, n -

[(S-n)A+n~]P

l~n~S S
(4) n - R-I,

[ (N-n+I)A]Pn. 1 + l~Pn+ 1,

[(N-n)A+R4s]p n - [(N-n+I)A]Pn-1

R ~ n < M+S-N

+ P~Pn+I'

APN. 1 - R~P~.

Using the general birth and death results

~j-1 " - J-i ~j

given by

n

Pn "

we obtain the following

(5)

P0'

steady-state

solutions,

respectively,

for R S S

and R > S.

(1) For R ~ S I Pn - - -

n-i "

l~n
[MOA + (S - J)ea]P O,

J-O

n!

(~I/~) n-R+l

Pn

R!

Rn. R

(M-I)I

n-i [MOA + (S - J)O,,]P O,

"

j-O

R
(6)

0~ "s-I (~i/~) n'R+l

S

[M0~ + (s - J)ea]e o,

Pn "

(M+S-n)!

RI R n'R

J-0 S ~ n ~ M+S

and (ii) For R > S i Pn - - -

n-I "

nl

l~n~S

[MeA + (S - j)e=]P O,

J-O n-S-I

(M-I) ! 8 A

S

[Me A + (S - j ) O a ] P O,

Pn " (M+S-n) I n! (M-l)!

S
j-0

(7)

O~ -s-I (~i/~) n-R+l

S [MOA + (S - J)Oa]P O,

Pn "

(M+S-n)!

RI R n'R

j-0 R ~ n ~ M+S

where

0A -

The results

and

ea .

for the hot-standby

obtained by setting a - A in equation

(8)

model with two types of repair rate (6) (or equation

(7)) are given by

Cost analysis of a repair problem (M + S)!

1177

n QA P0'

Pn "

1 ~ n < R

(M + S - n)l n!

(9) (M + S)! (~I/~) n'R+l

Pn --

(M + S - n) l RI R n'R

n 0A PO'

R ~ n ~ M+S.

It should be noted that i) when #i " p' and 0 < a < A, equations (6) and (7) reduce to the existing results for the warm-standby the literature

(see Sivazllan and Wang [3]); il) when P1 " ~' and

= - O, equations

cold-standby iii)

case

w h e n ~1 " ~ '

results

for

case in

(6) and (7) reduce t o the existing results for t h e

in the literature

(see Gross and Harris

S - 0, a n d a - 0, e q u a t i o n

the no-spare

case in the literature

(7) reduces

[7]);

and

to the existing

(see Gross and Harris

[7]). U s i n g equations normalizing

(6), (7), and (9), P0 can be solved from the

equation given by

M+S Z Pn - I. n-O

5. MACHINE AVAILABILITY AND OPERATIVE UTILIZATION

The machine avaflabilit~f represents the fraction of the total time that the machines are running. machine availability

MA-

Followlng Benson and Cox [14], the

(MA) for the M/M/R model is defined as

(io)

1 - {E[N] / N},

where N (- M + S) is the total number of machines,

M+S Z nP n is the expected number of failed machines n-0

E[N] -

system.

and

~nis should not be confused with system availability

in the

defined

later on. The operative utilization represents the fraction of busy repairmen.

(OU) f o r

Following Benson and Cox [14], the operative utilization

t h e M/M/R m o d e l i s d e f i n e d

OU-IR-

The q u a n t i t y repairmen.

as

R-1 ~- (R - n ) P n} / R. n-O

in the numerator

represents

(11)

the expected

number of busy

1178

K.-H. W A N G

6.

COST ANALYSIS

and B. D. SIVAZLIAN

OF T H E M / M / R M A C H I N E R E P A I R M O D E L

We develop an expected cost function par unit time and impose a constraint on the availability in which R and S are decision variables. Our objective is to determine the optimal number of (R, S), say (R*, S*) so as to minimize this function and maintain the availability at a certain level. In state n, let M n m the number of operating machines, Sn - the number of spare (either cold standby, or warm standby, or hot standby) machines, R n - the number of repairmen. Then the values of M n, R~, and Sn are given as follows: I~

0NnNS

-

(12)

+SIi

n

- n

S
Sn -

(13) S
0-
-

(14)

Rn

R
Let E[O] m the expected number of operating machines in the system, E[S] m the expected number of spare machines in the system functioning as standbys, Eli] - the expected number of idle repairmen, E[B] - the expected number of busy repairmen, F(R, S) m the expected cost per unit time in steady-state, A m the minimum fraction of time all M machines are in operation to function properly, A v m the steady-state probability that at least M machines are

in operation to function properly (system availability). Using equations (12) through (14), we obtain the following expressions for E[O], E[S], E[I], and E[B]

E[O] - M -

M+S Z (n - S)P n, n-S+l

(15)

S

E[S] -

Z (S - n)P n, n-O

(16)

R-1

z[I] -

z

n-O

(z-

n)r n,

(17)

Cost analysis of a repair problem ~.[B]

-

r

- ~.[I].

We a s s u m e t h a t

(ls)

the production

machines in operation.

Therefore,

system requires

production

(i.e.,

there

a minimum o f M

the cost per unit

m a c h i n e d o w n t i m e i s c o n s i d e r e d when a l l breakdown occurs

I179

are less

time of each

s p a r e s a r e b e i n g u s e d and a t h a n /4 o p e r a t i n g

machines in the

system).

Let CE m c o s t

per unit

time of e failed

machines are exhausted.

This cost

a production

is not fully

machines

system that

machine after

is the loss

all

in profit

operational

since

(the minimum required) will be operating.

standby

associated less

with

t h a n /4

Thus C g is the

profit loss per unit time per machine. C S m cost per unit time when one machine is acting as a standby. This cost will in general have two components. investment cost per unit time per standby.

The first one is the

The second one is the cost

of operating the standby per unit time, if such standby is warm or hot, which may include energy cost, labor cost, and maintenance C I m cost per unit time when one server is Idle.

cost.

This cost is

measured in terms of lost wages and benefits provided to an idle or free worker who is not productive.

This cost may or may not be the same as

CB • C B ,,, cost per unit time when one server is servicing.

This is

the actual wage and benefit cost provided to a busy worker while performing productive work. In general,

some difficulty arises in practice

in estimating the

expected cost per unit time for a specific cost model due to the determination However,

of accurate estimates

for the various cost parameters.

such accurate estimates are seldom required since at

optimality,

the cost function to the economic model is not too sensitive

to variations

in these cost parameters

as shown later in Table I.

The expected cost per unit time, F(R, S), is given by

Y(g, S) - CE

where

M+S Z ( n - S)P n + CS E[S] + CI E [ I ] + CB E [ B ] , n-S+l

/4+S Z (n - S)P n represents n-S+1

operating

in the system.

(19)

the expected number of machines not

The optimal values of (R, S), say (R*, S*),

can be s o l v e d by the f o l l o w i n g problem which can be s t a t e d

1180

K.-H. W A N G and B. D. SIvAZL~N mathematically as:

Minimize Z - F(R, S) R,

(20)

S

subject to S

Av - n~0 Pn ~ A.

(21)

The decision variables R and S are required t o be integer values and to be determined by an optimization algorithm.

The optimization

problem is an integer nonlinear programming problem with a highly nonlinear and complex objective function of two decision variables R and S subject to a set of constraints.

It is extremely difficult to derive

usable analytic results for the optlmumvalue

(R*, S*).

As a result, a

heuristic approach (computational approximation) is utilized. Computatlonally efficient procedure may be developed to obtain the optimum value (R*, S*).

We use direct substitution of successive values

of R and S into the cost function until the m l n i m u m v a l u e obtained and constraint (21) i s satisfied.

of F(R, S) is

A numerical illustration is

provided by considering the following parameters: M - 10, #i " 1.0, # - 3.0, A - 0.9, C E - $75/day, C s - $50/day, C I - $60/day, C B - $100/day. The upper bound on the decision variables R and S is 2M, with a - 0, or 0.15, or A.

The expected cost F(R, S) is shown in Table (1) for

different values of R and S.

We note that a minimum expected cost per

day of $446.152 and a system availability (Av) of 0.906 is achieved a t R* - 3 and S* - 7. Table i.

The Expected Cost F(R, S) and the System Availability for 0 A - 0.6 and 6a - 0.15.

R•

1

2

3

4

5

480.786 0.021 381.592 0.096 450.308 0.080 552.728 0.061 652.509 0.050 739.634 0.046 813.508 0.044 878.552 0.044

478.495 0.028 349.748 0.273 402.427 0.242 502.089 0.179 605.241 0.143 698.473 0.126 778.352 0.119 847.295 0.117

477.594 0.032 336.934 0.389 370.826 0.537 461.176 0.393 563.335 0.303 660.356 0.258 745.729 0.238 819.282 0.230

477.259 0.033 334.647 0.469 370.954 0.703 441.649 0.707 534.784 0.532 630.854 0.439 719.536 0.393 797.415 0.373

477.143 0.034 338.402 0.526 387.836 0.803 457.398 0.853 529.587 0.814 616.845 0.654 704.428 0.571 784.824 0.532

1

2 3 4

5 6 7

8

Cost analysis of a repair problem

1181

Table 1 -- Continued

1 2 3 4 5 6

7 8

6

7

8

9

477.108 0.034 345.587 0.568 414.201 0.865 489.098 0.924 556.872 0.923 625.217 0.883 705.096 0.755 784.426 0,691

477.100 0.035 354.571 0.600 446.152 0.906 528.423 0.959 596.642 0.968 660.378 0.959 725.136 0.928 798.447 0.836

477.100 0.035 364.286 0.623 481.447 0.932 571.508 0.978 641.419 0.986 704.435 0.985 765.542 0.978 827.832 0.957

477.100 0.035 374.019 0.641 518.727 0.950 616.474 0.988 688.251 0.994 751.547 0.995 811.899 0.993 871.643 0.988

i0

477.100 0.035 383.292 0.655 557.131 0.963 662.373 0.993 735.934 0.997 799.729 0.998 860.046 0.998 919.228 0.997

The numerical results exhibited in Table (I) show that the cost function is unimodal thus resulting in a unique optimal global solution. Although no formal analytic proof is provided, we conjecture at this stage that the function F(R, S) is unlmodal, although not necessarily convex.

From Table (i), we observe that i) for a given number of repairmen, the system availability increases as the number of spares (S) increases; li) the optlmal solution (R*, S*) is unique and the contralnt is satisfied. The minimum expected cost F(R, S), the system availability Av, the values of E[O], E[S], E[I], E[B], E[N], the machine availability (MA), the operative utillzation (OU), at the optlmal values (R*, S*) for A 0.6, 0.9, 1.2, 1.5, 1.8 are shown in Table 2.

Table 2.

System C h a r a c t e r i s t i c s of the M/M/R Machine Repair Model under Optimal Operating Conditions. (I. Cold-Standby: ~ - 0)

A

(R*& S*~ F(R-, S-) [A~ E ] E[S] E[I] E[B] E[N] MA OU

0.6

0.9

1.2

1.5

1.8

(3, 7) 453.219

(4, 9) 578.742

(5, 12) 740.649

(6, 14) 865.713

(8, 13) 980.246

0.921 9.845 3.036 0.256 2.744 4.119 0.758 0.915

0.900 9.787 3.412 0.196 3.804 5.801 0.695 0.951

0.915 9.809 4.653 0.159 4.841 7.538 0.657 0.968

0.910 9.790 5.108 0.136 5.864 9.102 0.621 0.977

0.902 9.790 3.480 0.237 7.763 9.730 0.577 0.970

1182

K.-H. WANO and B. D. SIVAZLIAN

(II. Warm-Standby: a - 0.15)

A' (R*~ S* 1 F(R-, S-) Av E[O] E[S] E[I] E[B] E[N] MA OU

0.6

0.9

(3, 7) 446.152 0.906 9.815 2.804 0.198 2.802 4.381

(4, i0) 603.207 0.909 9.806 3.886 0.141 3.859 6.308

0.742

0.685

0.934

0.965

1.2 (6, i0) 7.51.334 0.915 9.834 3.002 0.281 5.719 7.164 0.642 0.953

(III. Hot-Standby: u -

A

(R*~ S*~ F(R , S ) [~0 E ] E[S] E[I] E[B] E[N] MA OU

1.5

1.8

(6, 15) 880.119 0.907 9.783 5.350 0.092 5.908 9.867 0.605 0.985

(7, 17) 1002.312 0.903 9.768 5.762 0.080 6.920 11.469

0.575 0.989

A)

0.6

0.9

1.2

1.5

1.8

(3, 9) 474.329 0.900 9.804 3.238 0.057 2.943 5.958 0.686 0.981

(5, 9) 637.893 0.911 9.838 2.640 0.156 4.844 6.521 0.657 0.969

(6, 12) 774.791 0.900 9.805 3.255 0.065 5.935 8.941 0.594 0.989

(8, 13) 961.408 0.915 9.841 3.082 0.115 7.885 10.077 0.562 0.986

(9, 15) 1074.971 0.900 9.805 3.261 0.068 8.932 11.934 0.523 0.992

One sees from Table (2) that i) F(R*, S*) increases as A increasesl ll) the optimal value of (R, S), (R*, S*) increases in A.

7. SUMMARY I) General solutions to the machine repair problem with spares and two types of repair rate are obtained. 2) This model generalizes either the no-spare, or the cold-standby, or the warm-standby, or the hot-standby model. 3) The optimal values of the nLu,ber of repairmen and spares are determined so as to minimize the expected cost function associated with a constraint on the system availability. 4) Under the optimal operating conditions, several system characteristics are evaluated for three types of standby.

REFERENCES

i.

W. F e l l e r , An I n t r o d u c t i o n to P r o b a b i l t t 7 Theory and I t s Application, 3rd Edition, Vol. I, John Wiley and Sons, New York (1967).

2.

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3.

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4.

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8.

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15.

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