8
A. KUMAR / R. LAL: Profit Optimization in a Redundant System 725 ZAMRI 60, 72s -73‘2 (1YSO) A. KUMAR / H. LAL Profit optimization in A Two-Unit Maintained Standby Bedundant System Die Arbeit beschaftigt sich mit einem doublierten Nystem bei kalter Reserve mit exponentiell verteilter Auafallzeit und be- liebigen Reparaturzeitverteilungen. Profitgleichungen wurden unter allgemeinsten Kostenstrukturen abgeleitet, wobei die Profitrate des Systems in einem Zustand von drei Parametern abhangt - dem gegenwartigen Zustand, dem n ~ h s t e n Inspektionszustand und der Zeit, zu der die Profitrate gemessen wird. Die Laplace- Transformierten des Profits, den das System in einer gegebenen Zeit abwirft, wenn die Diskontrate exponentiell ist, werden erhalten. Das asymptotische Verhalten des Profits wird ebenfalls diskutiert. Weiterhin wird der Profit ale MaJ fur die Ejfektivitat des Systems vorge- schlagen. Fur den Fall, daJ in jedem Zustand zwischen verschiedenen Moglichkeiten der Steuerung zu entscheiden ist, wird ein Optimierungsproblem zur Bestimmung der optimalen Steuerungsstrategie formuliert, welches den Langzeitprofit maxi- whiert. Ho w a r d s Iterationsmethode wird benutzt, um einen Algorithmus zur Bestimrnung der optimalen StratPgie fur tlas System zu mtwiekeln. The poper deals with a %unit cold etandby redundant system with exponential failure-time and general repair-time dis- tributions. Profit eqwations for the system have been developed under the most generalized cost structure in which the earning rate of the system in a state depends upon three parameters -present state, future state of visit and time at which the earning rate is measured. L a p l a c e transforms of the profit that the system will earn in a giventime when the discounting rate is ex- ponential, are obtained. Limiting behaviour of the profit has also been discussed. Further, profit has been suggested as the measure of system’s effectiveness. When different alternatives are available to a decision ntaker in each state to operate the system, an optimization problem to determine an optimal operating policy for the system which maximizes long term profit has been formulated. Howard’s policy iteration method is used to develop an algorithm to determine an optimal policy for the system. Introduction It is the complexity and multiplicity of influential factors which make the operational environment of standby redundant systems quite sophisticated and involved. Some of the factors are critical as they have direct and signi- ficant impact upon system’s performance whereas there are factors that have marginal contribution in generating environment under which systems have to operate. In order to have better understanding of a standby system, its operational conditions and the interactions between the two, economic analysis is of paramount importance, More precisely cost feasibility and cost sensitivity are two major measures in any evaluation study. Unfortunately, econorriics of standby redundant systems has not received sufficient attention of workers in this area. Of course, some of the papers did concentrate on economic aspects of standby systems (MINE and KAWAI 191, KUMAR [GI), but they are not significant when compared with the number of articles appeared to describe the stochastic behaviour of such systems by obtaining mean-time-to-system failure, steady-state availability, mean recurrence time to a state etc. (BRABSON and SHAH [l], NAKAGAWA and OSAKI [Ill, KUMAR [7], GOPALAN and D’SOUZA [3], KISTNER and SUURAMANIAN [5], CHOW [2]). For a 2-unit parallel redundant system with good, degraded and failed states a main- tenance policy was discussed that maximizes the net expected profit rate from the system over an infinite tin], span (MINE and KAWAI [9]). Later, they considered inspection and replacement policy for one unit systeni (MINE and KAWAI [lo]). As expected profit is one of the most important paranieters in economic evaluation of standby redundant systems, a few papers have appeared to obtain analytic expressions for expected profit in a standby system ope- rating under different operational environment (KUMAR [6], KUMAR [7], KUMAR and LAL [8]). In all the papers, a siniple cost slructure is considered viz., a) fixed earning (loosing) rate of the system in each state; b) fixed rewards (costs) at the time of transitions; c) no discounting of payments received in future. In a recent paper (KUMAR and LAL 181) the authors have given a hint to relax c) only. However, the other two assumptions a ) and b) also need be relaxed as situations do arise when the earning rate of a system depends not only on the present state but also depends upon the future state of visit and it is not fixed throughout the duration of stay in a state. Further, in the above paper two optimization problenis were posed for future work. The purpose of the present paper is two-fold: i) to present the most generalized cost model and discuss profit i.e. to relax a), b) and c) simultaneously; ii) to present a solution procedure to optimization problem 2 given in KUMAR and LAL [8]. This problenl id to deterrninc optimal maintenance policy (maximized profit) in each state for a standby system.

Profit Optimization in A Two-Unit Maintained Standby Redundant System

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Page 1: Profit Optimization in A Two-Unit Maintained Standby Redundant System

A. KUMAR / R. LAL: Profit Optimization in a Redundant System 725

ZAMRI 60, 72s -73‘2 (1YSO)

A. KUMAR / H. LAL

Profit optimization in A Two-Unit Maintained Standby Bedundant System

Die Arbeit beschaftigt sich mit einem doublierten Nystem bei kalter Reserve mit exponentiell verteilter Auafallzeit und be- liebigen Reparaturzeitverteilungen. Profitgleichungen wurden unter allgemeinsten Kostenstrukturen abgeleitet, wobei die Profitrate des Systems in einem Zustand von drei Parametern abhangt - dem gegenwartigen Zustand, dem n ~ h s t e n Inspektionszustand und der Zeit, zu der die Profitrate gemessen wird. Die L a p l a c e - Transformierten des Profits, den das System in einer gegebenen Zeit abwirft, wenn die Diskontrate exponentiell ist, werden erhalten. Das asymptotische Verhalten des Profits wird ebenfalls diskutiert. Weiterhin wird der Profit ale MaJ fur die Ejfektivitat des Systems vorge- schlagen. Fur den Fall, daJ in jedem Zustand zwischen verschiedenen Moglichkeiten der Steuerung zu entscheiden ist, wird ein Optimierungsproblem zur Bestimmung der optimalen Steuerungsstrategie formuliert, welches den Langzeitprofit maxi- whiert. H o w a r d s Iterationsmethode wird benutzt, um einen Algorithmus zur Bestimrnung der optimalen StratPgie fur tlas System zu mtwiekeln.

The poper deals with a %unit cold etandby redundant system with exponential failure-time and general repair-time dis- tributions. Profit eqwations for the system have been developed under the most generalized cost structure in which the earning rate of the system in a state depends upon three parameters -present state, future state of visit and time at which the earning rate is measured. L a p l a c e transforms of the profit that the system will earn in a giventime when the discounting rate i s ex- ponential, are obtained. Limiting behaviour of the profit has also been discussed. Further, profit has been suggested as the measure of system’s effectiveness. When different alternatives are available to a decision ntaker in each state to operate the system, a n optimization problem to determine a n optimal operating policy for the system which maximizes long term profit has been formulated. H o w a r d ’ s policy iteration method is used to develop a n algorithm to determine a n optimal policy for the system.

Introduction

It is the complexity and multiplicity of influential factors which make the operational environment of standby redundant systems quite sophisticated and involved. Some of the factors are critical as they have direct and signi- ficant impact upon system’s performance whereas there are factors that have marginal contribution in generating environment under which systems have to operate. In order to have better understanding of a standby system, its operational conditions and the interactions between the two, economic analysis is of paramount importance, More precisely cost feasibility and cost sensitivity are two major measures in any evaluation study. Unfortunately, econorriics of standby redundant systems has not received sufficient attention of workers in this area. Of course, some of the papers did concentrate on economic aspects of standby systems (MINE and KAWAI 191, KUMAR [GI), but they are not significant when compared with the number of articles appeared to describe the stochastic behaviour of such systems by obtaining mean-time-to-system failure, steady-state availability, mean recurrence time to a state etc. (BRABSON and SHAH [l], NAKAGAWA and OSAKI [Ill, KUMAR [7], GOPALAN and D’SOUZA [3], KISTNER and SUURAMANIAN [5], CHOW [2]). For a 2-unit parallel redundant system with good, degraded and failed states a main- tenance policy was discussed that maximizes the net expected profit rate from the system over an infinite tin], span (MINE and KAWAI [9]). Later, they considered inspection and replacement policy for one unit systeni (MINE and KAWAI [lo]).

As expected profit is one of the most important paranieters in economic evaluation of standby redundant systems, a few papers have appeared to obtain analytic expressions for expected profit in a standby system ope- rating under different operational environment (KUMAR [6], KUMAR [7], KUMAR and LAL [8]). I n all the papers, a siniple cost slructure is considered viz.,

a ) fixed earning (loosing) rate of the system in each state; b) fixed rewards (costs) a t the time of transitions; c) no discounting of payments received in future.

In a recent paper (KUMAR and LAL 181) the authors have given a hint to relax c) only. However, the other two assumptions a ) and b) also need be relaxed as situations do arise when the earning rate of a system depends not only on the present state but also depends upon the future state of visit and it is not fixed throughout the duration of stay in a state. Further, in the above paper two optimization problenis were posed for future work.

The p u r p o s e of the present paper is two- fo ld : i) to present the most generalized cost model and discuss profit i.e. to relax a), b) and c) simultaneously; ii) to present a solution procedure to optimization problem 2 given in KUMAR and LAL [8]. This problenl

id to deterrninc optimal maintenance policy (maximized profit) in each state for a standby system.

Page 2: Profit Optimization in A Two-Unit Maintained Standby Redundant System

720 A. KUMAR / R. LAL: Profit Oplimizatiori in a Redundant Systeiti

As oiir objective is not in the dircctiori of coiriylicating system configuration, but i t is to sllow the feasibility of profit evaluation of a standby systern iinder the most generalized cost structure, we take a usual 2-unit cold &andby system for the purpose of analysis. We divide the paper in the following two parts:

P a r t I: Profit of the system. P a r t 11: Optimal maintenance policy for the system.

IJI part 1 we superinipose the most generalized reward structure (HOWASU [4J) on the standby systcni and develop profit equations under the generalized set-up. The equations are solved for a particular case when the earning ratc in a state is a function of the present state and the future state of visit and is independent of time and transition rewards are also constant. Part I1 deals with determining an optimal maintenance policy for the system in each state that maximizes the expected profit rate of the system. We again make use of HOWABD'S Policy Iteration 141

]'art I: Profit of tho syciteni

T h e s t a n d b y s y s t e m i) 'Iherc is a 2-unit cold standby reduiidant system ; uriits are identical. ii) The failure-time distribution of the operative unit is exponential with rate A and the rppair-tillle is genpri11,

iii) The system stutes and trai~sitions between them art: say Il(%

So: One unit is operative, the other is as standby; (can go to Sl) Sl: One unit is under repair, the other is operative; (can go to So or SJ. S2: Oiie unit is under repair, the other waits for repair; (can go to XI).

(UI~ANYON i d 811~11 [l], ~ ~ U M A ~ L 16, 71). 'l'lie systeiti is up in So -- and it iS down in &. The proccss generiLted by the system model is semi-MARKov

T h e gene ra l i zed r c w a r d s t r u c t u r e

i) While the system occupies 8, having chosen a successor state S,, it earns reward a t a rate yti(o) a t a time G

ii) When the transition from St to A') is actually made a t somc t h o Z, thc proccss carns a bor~ris b,,(z), a fixed

iii) The discounting rate is exponcntial with ratea, i.a. a unit sum of money at time t in tlie future has a worth

after entering S,, This is the yield rate of 8, a t time o when the successor state is X,.

sum.

or present value e at today, a >= 0.

N o t a t i o n s

vi(t, a) : the expected present value of the reward the process will generate in a time interval of length t if it is placed in S$ a t the beginning of this interval. v , (O) : the additional fixed payment if the system occupies S, a t the end of the interval taken in v,(t, m ) . p t , : the one-step transition probability from S, to S,. hfl( t ) : the holding-time probability distribution function of the system in S, before making a transition to 8,.

f * ( y ) denotes tlic LAPLAW transform of it function evaluated aL s, e.g. f*(.s)

J ... implies J ..., unless stated otherwise.

lit,(t) : the capital letters in general stand for the continuous distribution function of the corresponding lower cuse

a, A&) = & Pzikf(t).

J J' t: - a [ f(t)dt.

w 0

0 - denotes the complement, e.g. f( t) = 1 - f@)-

t

e.6'. llL,(t) =- s h L j ( l ) dl. 0

S y s t e m p r o f i t e q u a t i o n s It has been shown in HOWARD [a] that for any state St,

1

v t ( t , a) = yz( t , a ) + e -at v,(O) H J t ) -t rg ( t , a ) + 2' p t , s h&) c-ar q ( ( t - z), @) cla , 1) j 0

Taking the LAPLACE transform of (1) -(3) and writing in matrix notation one obtains

where double bar (=) below a letter stands for )matrix and singlc bar (-) denotes vector and a denotes box opera- tion, e.g.

if A = ((q)) , B = ( ( b t f ) ) , c' = A u B + cii = uijbi, if C = ((c~,)) .

Page 3: Profit Optimization in A Two-Unit Maintained Standby Redundant System

727 A. KUMAR / R. LAL: Profit optimization in a Redundant System

In order to write profit equations for the model under consideration, let us write the required paralneters for the model. The following parameters can be obtained directly from BRANSON and SHAH [l] or KUMAR [B, 71 as this model is a special case of these models.

pol = 1 ,

hO1(t) = 1 e-At, h,,(t) = ___ AlZ(t) =

h,(t) = I h o l ( t ) , h,(t) = e-nt g(t) + L e-At G(t) ,

p , = f e-Ai g( t ) dt = g*(1) , p12 = J L e-At G'(t) dt = g * ( L ) , p21 = 1 ,

( 5 ) e-Af g(t) L e-"L(t) R eav 00 hzl(y) = =-- f e-d' g ( t ) dt , 9*(4 ' s*(4 ) 9 * ( 4 21

n,(t) = l b z 1 ( t ) where y is the remaining repair time of the unit which was under repair a t the instant when S, is entered.

Substituting the required values into (2)-(3) from (5 ) , one gets

(6)

(7)

Yo(4 a) = YOl(t, a) cat >

y1(t, a) = yl0(t,a) J e-"g(z) d t + ylz(t,a) 1 J e-d7 GCz) dz , 00 00

and

where

Now a

#

ro(t, a) = f eca7 y,,(a) du + e--ar bol(z)] dt , 0

(9)

(10)

t rl(t7 a) = ;ePnt g( t ) e-au ylo(a) da + e- b,,(z) dz + 1 J e--Lt Rt) [ SeaU ylz(a)du + e-ar b,,(z)

0 1 0 0

bd d 1 -ab

1 1 - ab - cd [ L - g - F * ( s + .)]-I = p - ~ - ~

a = L / ( A + a) , b = g*(L + a) , c = L3*(L + &)/(A + c() , d = A(g* (a) - g*(A)) / (A - &) j*(A) .

the quantities required in (1) or (4) are available. They can be substitute- And ijhe discounted profit can be obtained when the system starts in any of the states So, S, and S,. We below consider a particular case and discuss in detail profit evaluation.

A p a r t i c u l a r r e w a r d s t r u c t u r e

To illustrate the computation procedure let us consider the case of constant yield rates and bonuses as below

~ t j ( 0 ) = if , bt,(z) = btr *

Then

0 J if a = O .

Page 4: Profit Optimization in A Two-Unit Maintained Standby Redundant System

728 A. KUMAR / R. LAL: Profit Optimization in a Redundant System

1 L*(s) = y (q"( i l ) - g*(za) -1 !/*(S) -- y"(s ~- a)) ,

I+ * a) - g"(s) y"A) -- g*(a + s) s - A +a

- _____. r l ( s ,a) = - "ag*(a) lYZ1 i" ( s - a

( Y * ( 4 - Y * W + 4) . Ab,, + (s - 1 +a) g * ( n )

Substituting of (15) -(20) into (4), one gets the elements of the expected profit vector - v*(s,a) as below

vo"(s,a) = - [ (1 - cd) {Y*(8, a) + z: HGP \cx + 8) V , ( O ) +ro"(s, a) J 1 - a6 - cd

(30)

L imi t ing behav iour of expec ted p r o f i t

Froin the final value theorem onc lrriows

vz(a ) = lini vg( t , a) = lim s V;(Y, a) for all i = 0, 1, 2 , t-tw s+o

Then it easily follows that

or

It implies the set of simultaneous equations

- v *(a) = [ I - g g*(a)I-l T*b)

v *\a) = ~ * ( a ) + v:(a) = T : ( a ) +

0 a*(a) E*(a) .

m W ) Gb) 3

where

(24)

(25)

(z6)

(37)

(28)

For the particular case when yield rate and bonuses are constant, we get

T:(a) = c Pa,,Ytf iy + c P t l b t l m ) * (29) 3 1.

Therefore, if one is interested in long term profit viz., the expected present values v , (a ) of the entering state Si in LL systeni that will continue indefinitely, the following steps arc straightforward :

i) Befine the system states. Identify the system as up or down in each state. State transitions between dif- ferent states.

ii) Write transition probabilities, holding time probability distribution functions, yield rates along with transition rewards and discounting rate. Compute LAPLACE transforms of holding time probability distribution fuilctions evaluated a t the point s = a, the discounting rate.

Page 5: Profit Optimization in A Two-Unit Maintained Standby Redundant System

iii) Coinpute rt (a) using (28) or (29) for each state S, and substitute in (27) to get a set of sirnllltaneous equa-

For the particular case under discussion we have tions describing profit. The number of equations is equal to the number of states.

(30)

(31), (32)

$*(A +a) ?-;(a) = ‘- + b,, g*(A + (x) + - + b,, ;1-----, (; ) (“bz ) Afa

Part 11: Optimal maintenance! policy for the Bystcni In the ~traintenance of equiprncnts, one is often faced with the probleni of choosing a11 optiiiral alternat ivca frorll a given set of alternatives. More elaborately, to maintain a system, there may be several ~iraintenance sclientes available to a decision maker e.g. ordinary nzait&nunce (OM), costly muintemncc (CM) and highly expensive ttluitz- frt~ccjrce (HEM). Last policy viz. HEM may involve large expenditure but ensures a higher value for operating time of the systeni or mean-time-to-system failure, as compared to other policies. One may be interested in selecting the policy (one alternative from each state) that inaxiniiees the expected profit in long run.

111 this part we give a general formulation for a seini-MARKov decision process applicable to any standby redundant system. To determine the optinial operating (or maintenance) policy for the system we apply HOWARD’S Policy Iteration [4]. To get an insight into the formulation aspect of niaintenance policies let us concentrate on the following disc uss ion :

When a standby redundant system is operating, there may be several options available; OM, CM and HEM v tc . While a unit of the system is under repair, there niay again be different alternatives viz., ordinary ropuir (OK), costly repair (CR), highly expensive repair (HER) etc. In general in each state of the standby system, there may be several operating alternatives available to a decision maker and the problem is to choose one alternative in each state keeping in view soiiie effectiveness criterion viz., choose the alternative that niaxiniizes long term profit rate of the systeiii or minimizes long term cost rate of the system.

Before presenting formal description, we state the following assumpt ionb: a) The earning rate of the system in each state is constant, that is, it neither depends on the future state of

b) whenever, the system changes its state, fixed transition rewards are involved; ( 0 ) there is no discounting.

visit nor on time; and

S e 111 i - MABKoV dec is ion I, r o c e ss hupyose when the system is in Rt , there are vaiious alternatives for its operation. Associated with rach alteriitltivc hay I;, In ,)la, there are process paruwietters : transition probabilities (pfj), holding-hie probability distribution func- tions ( / i : ) ( t ) , earning rate (or loosing rate) (yt) and transition rewards (or costs) (rf,). We assuniea finite but differcnt number of alternatives in each state. The problem is to choose one alternative in each state which may be called a dtcisioii and the set of decisions (one in each state) may be called a policy. W e have to f id the policy that maxi- w i x s t / i c uciruye profit of the system in steady-state.

N o t a t i o n s

i i ~ j l ( t ) : total profit wte expect the system to earn in timc t , if the systterii starts in 8, at tinic t = 0 ;

r i ,

: hubscripts to denote system states: 1, 2, ... , u ;

: earning rate of the system in S, when alternative I; is selected; : fixed transition reward when the sjstern goes from S , to S, and in S,, alternative k was selected; : expected profit of the systein per unit time in steady-state. Then following HOWAND [4] we know that for large t ,

vt( t ) = v, + gt

V, + gpt = qaplf L’ pzfvj 3

(32%)

(33) a here v L is the transient purt of the profit and g is the steady-state purl, and

i = 1, 2, * * * 9 11 1

3

\+here y L p L - pafra7 + y L p L . J

The Steps involved in the pol icy i t e r a t i o n procedi i re are: S t e p I : Define the standby redrindant system i.e. its state space and transitions between them. Identify rip

a i d down states of thc. systeni. E:niiiiic~raie altcrnativr operating ratles in each state. Specify earning rate, repair ~ o s t rate, fixed transition rewards etc. I \ / \ \ I ? I , Jkl. Id) , I I 1 2

Page 6: Profit Optimization in A Two-Unit Maintained Standby Redundant System

.... ____ ~ - . A. KWMAR / R. LAL: Profit Optimization in a Redundant System ._ ~. - - ~ _ _ _ _.______-

730

S t e p 2 : Compute the transition probabilities and the holding time probability distribution functions. S t e p 3: Choose one decision in each state i s . the present policy for which

is maximum. S t e p 4: For the policy in step 3 solve the set of T L equations

vt + gpi = qtpt + for vl , v2, ... , v , ~ - I and g by setting v, = 0.

p i p f for all i = 1, 2, ... , n i

S t e p 5: Using thevalues of v,, v,, ... , v , - ~ and g obtained a t step 4 compute the test quantity

(34)

for each alternative in each state. Choose the alternative in each state for which the test quantity is maxiniurn.

mal policy is reached. S t e p 6 : Examine if the new policy is different from the initial policy. If yes, go to step 4; otherwise the opti-

App l i ca t ion t o a s t a n d b y s y s t e m w i t h e x p o n e n t i a l f a i l u r e a n d e x p on en t i a 1 r e p a i r t i m e d is t r i b u t i o n s

Denote by: L the failure rate of the system, 7 the repair rate of thefailedunit, 'om' theordinaryniaintenance, (or' the ordinary repair; 'cm' the costly maintenance, 'cr' the costly repair,

The following table gives the detailed description of various system parameters : Decisions i n d i f f e r e n t s t a t e s :

In So - 1 : om, 2 : cm. In AS', - 1 : (om, or), 2 : (om, cr), 3 : (om, or), 4 : (cni, cr). In S, - 1 : or; 2 : cr.

State Alter- Transition natives Probabilities

St k P& P f l PX

1 - 1 - 2 - 1 - 1 .67 - .33

.80 - .20 2 3 .67 - .33 4 .95 - .05 1 - 1 -

- 1 - 2

So

4

8 2

Rewards - _ _ _ 6J 6 1

- -5.0 - -2.5 -1.0 -

.8 - -1.0 -

.5 - - -1.0

- .5

-

- -

- -2.5 -2.5 -2.5 -2.5 -

A, 7,

.05, -

.01, -

.05, . I ,

.05, .2,

.01, .I,

.01, 2, , . I , , .2,

~-

- -

So, the initial policy is 1 and policy equations are (:I uo + SPO = 4oPo + V l 2

01 + SPl = 41P1 + PlOVO + PlZV'2 Y

v2 + gP2 = 4zPz + u1 , or, if v, = 0

v1 - v0 - 20g = -1995.00, v1 - .fi7vO + G.87g = 498.70, V , - log = 251,

which gives the solution as V, = 1496.00, v1 = 1001.0, V~ = 0 , g = 75.00.

F i r s t po l icy i t e r a t i o n

Using above values of vo, v l , v2 and g we compute the value of test quantity (TQ)

Earning Rate

~-~ Y! 9%

100 99.75 50 49.975 75 74.776 50 49.775 25 24.836 10 00.874

-25 -25.1 -50 -50.1

( 3 0 )

(37)

Page 7: Profit Optimization in A Two-Unit Maintained Standby Redundant System

731 A. KUMAB / It. LAL: Profit Optimization in a Redundant System

in each state. --

State Alternative Value of T.Q.

8 0 1 75

J',

2 45.025 1 74.976 2 98.725 3 24.986 4 98.116

S, 1 75 2 150.1

SO, the iniprovcd policy is 2 and policy equations are (11 ~1 - VO - 209 = -1995.00,

111 - ,800 + 4g = 199.1 , VI - 5y = 250.5,

and the solution is vo = 999, v1 = M G , v2 = 0, g = 83.10.

state. We observe that value of g has been improved from 75 to 83.10. Again we conipute T.Q. valucs in each

State Alternative Value of T.Q. 1 83.10 2 48.845 1 75.276 2 83.075 3 25.202 4 66.805

S, 1 41.5 2 83.1

8 0

8,

So, the improved policy is 2 and policy equations are

(39)

i:) v1 - oo - aoy = -1996.00, v1 - .8uo + 49 = 199.1, u1 -- 5y = 250.5

This policy is the same as obtained in the previous iteration, hence the optinbal policy is reached and is (i) arid the solution is vo = 999, v1 = 666, v2 = 0 and g = 83.10.

Coirdadi~ig rririarks

IVe have developed profit equations for a 2-uriit shndby redundant system under a generdized cost structure. Thc procedure can be applied to compute profit in any standby redundant system. However, as the size of the state space increases, the equations are quite complex and tedious. Under such circu instances it will be desirable to develope coniputer programnies for the purpose. Further, t,he optimal maintenance policy has been discussed for a siniple 2-unit standby system for illustration but the procedure could easily be applied to other standby systems with exponentail failure-time m d general repair-time distributions. To determine optimal maintenance policy, cotiiputer algorithms will serve a useful purpose for system designers and maintenance engineers.

References 1 BRANSON, M. H. ; SHAH, B., Reliability analysis of systems comprised of units with a r b i h r y repair-tinic distributions, lEEE

2 CHOW, U. K., Availability of some repairable computer systems, lEEE Trans. ltel. R-24, No. 1, 64-G6 (1975). 3 GOPALAN, M. N.; D'SOUZA, C. A,, Probabilistic analysis of a system wit,h two dissin~ilar units subject to preventivz niaintenance

4 HOWARD, R. A., Dynamic Probabilistic Systems. Vol. 11, New York, Wiley 1971. 5 KISTNER. K. P.: SUBRAMANIAN, R., The reliabilitv of a system with redundant repairable components subject to failure, Z.

Trans. Rel. Kt-30, No. 4. 217-223 (1971).

and a single service facility, Operations Research 23, No. 3, 534-548 (1975).

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732 A. KUMAI~ / It. LAL: Profit Optimization in it Nedundant Systeiii

8 K U ~ A R , ASIIOK; LAL, ROSHAN, Stochastic behaviour of a two-unit standby system with contact failure and intermittmtfy

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Z. Operations. Res. 22, 171-187 (1976).

Eingercicht am 10. 1. 1980

Bnschriften: Dr. ASHOK KUMAB, Training Group, Defence Science Laboratory, Jletcalfe House, Uellii-110084, India; ROSHAN L a , Research Fellow, Department of Statistics, Institute of Social Sciences, Agra, India.