Chapter 8. Replacement Theory. Babita Goyal. Replacement theory, time value of money, sudden failure, group replacement,

Transcription

1 Chapter 8 Replacemet Theory Babita Goyal Key words: Replacemet theory, time value of moey, sudde failure, group replacemet, mortality ad prevetive replacemet. Suggested readigs:. Gupta P.K. ad Moha M. (987), Operatios Research ad Statistical Aalysis, Sulta Chad ad Sos, Delhi.. Johso R.D. ad Berard R.S. (977), Quatitative Techiques for Busiess Decisios, Pretice hall of Idia Private Limited 3. Swarup K., Gupta P.K. ad Moha M. (), Operatios Research, Sulta Chad ad Sos, Delhi. 49

2 8. Itroductio The study of replacemet is cocered with the situatios that arise whe some items such as equipmet eed replacemet due to chages i their performace. This chage may either be gradual or all of a sudde. Broadly speaig, the requiremet of a replacemet may be i ay of the followig situatios: (i) (ii) (iii) (iv) A item fails ad does ot wor at all or the item is expected to fail shortly. A item deteriorates ad eed expesive maiteace. A better desig of the equipmet is available. It is ecoomical to replace equipmet i aticipatio of costly failure. I this chapter, we are iterested i the first two situatios. Third situatio has bee dealt whe we studied the pay-off criteria. Whe studyig the problem of replacemet, we may or may ot cosider the time value of moey. 8. Replacemet of equipmet that deteriorates gradually Geerally, the cost of maiteace ad repairig of certai equipmets icreases with time ad ultimately the cost may become so high that it is more ecoomical to replace theses equipmets with ew oes. If the productivity of equipmet decreases with time, this may also be cosidered as a failure. At this poit a replacemet is justified. The costs associated with agig icrease at a icreasig rate whereas the resale value of the equipmet decreases at icreasig rate. The decreasig resale value results i icreasig depreciatio, which is the differece betwee the purchase price ad the resale value. The depreciatio of the item icreases at a decreasig rate. The optimal replacemet policy for such items is to replace the equipmet at a poit where the total cost curve itersects the total depreciatio curve 5

3 Costs Total depreciatio Total operatig cost Time to replace Time Fig. 8.(i) Average total cost Costs Average operatig cost Average depreciatio Time Fig. 8.(ii) (a) Time value of moey does ot chage If the value of moey does ot chage with time, the the user of the equipmet does ot eed to pay iterest o his ivestmets. We wish to determie the optimal time to replace the equipmet. We mae use of the followig otatios: C: Capital cost of the equipmet S: Scrap value of the equipmet 5

4 : Number of years that the equipmet would be i use. f( t): Maiteace cost fuctio. A ( ): Average total aual cost. Two possibilities are there (i) Time t is a cotiuous radom variable I this case the deterioratio of the equipmet is beig moitored cotiuously. The total cost of the equipmet durig years of use is give by TC Capital cost - Scrap value + Maiteace cost C - S + f( t) dt C - S A ( ) TC + f( tdt ) d For miimum cost, A ( ) d C - S - f( t) dt f( ) + C - S f( ) + f( t) dt A( ) d A( ) ad at f( ) A( ) d i.e., whe the maiteace cost becomes equal to the average aual cost, the decisio should be to replace the equipmet. (ii) Time t is a discrete radom variable I this case C - S A( ) TC + f( t ) A () is miimum whe 5

5 A ( + ) A ( ) ad A ( ) A ( ) or, A ( + ) A ( ) ad A ( ) A ( ) A( + ) A( ) C S f( t) f( ) A( ) t + A ( ) + f ( + ) A ( ) + + f( + ) A( ) Similarly A ( ) A ( ) f( ) A( ) Thus the optimal policy is Replace the equipmet at the ed of years if the maiteace cost i the (+) th year is more tha the average total cost i the th year ad the th year s maiteace cost is less tha previous year s average total cost. Example : A firm is cosiderig replacemet of a machie, whose cost price is Rs,,,, ad the scrap value is Rs.,. The ruig (maiteace ad operatig) costs of the machie are as follows: Table 8. Year Ruig cost (Rs.), 5, 8,, 8, 5, 3, 4, Whe should the machie be replaced? Sol: C S Rs.,, Rs., th f( ) Ruig cost i the year. 53

6 TC C - S + f ( t),, + f( t) We prepare the followig table Table 8. Year of service Ruig cost (Rs.) Cumulative ruig cost (Rs.) TC C - S + f ( t) A( ) T C () f() f ( ) (Rs.) (Rs.),,,,,, 5, 7,,7, 53,5 3 8, 5,,5, 38,334 4, 7,,7, 3,75 5 8, 45,,45, 9, 6 5, 7,,7, 8, ,,,,, 8, ,,4,,4, 3,5 Thus A (6) Rs. 8,334 is miimum. Hece replacemet should be doe after every sixth year. Example : A pritig machie costs Rs. 4, whe ew. The followig table gives the expected operatig costs, expected productio (per pages) ad the resale value of the machie Table 8.3 Age Operatig costs (Rs.) 5, 7, 8,,,5 5, Productio,, 9, 8, 5, 3,5 Resale value (Rs.) 3, 5, 9, 5,,, 54

7 The productio cost per uit of productio is Rs. 5. Fid the optimal replacemet policy. Sol: Table 8.4 (Rs.) Age Total O.C. Total productio cost Depreciatio Total cost Average cost 5,, 5, 5, 7, 5,,, 3 8, 5 5,, 44, , 3,, 6, 55 5,5 75 5,,, , 975 6,5,39, 367 The machie should be replaced after every years. Example 3: A factory ower fids from his past records that the costs per year of ruig a machie whose purchase price is Rs. 6, are as follows Table 8.5 Year Operatig costs (Rs.),, 4, 8, 3, 8, 34, Resale value (Rs.) 3, 5, 7,5 3,75,,, The ower has three machies, two of which are two-years old ad the third is oe year old. He is ow cosiderig a ew machie with 5% more capacity tha oe of the old oes. The price of this machie is Rs. 8,. The cost estimates for the ew machie are as follows Table 8.6 Year Operatig costs (Rs.), 5, 8, 4, 3, 4, 5, Resale value (Rs.) 4,,, 5, 3, 3, 3, 55

8 Assume that the loss of flexibility due to fewer machies is isigificat ad he has sufficiet wor i ay of the decisios, what should be the optimal policy? Sol: For the first type of trucs, the average aual costs ca be computed as follows Table 8.7 Age Total O.C. (Rs) Depreciatio (Rs.) Total cost (Rs) Average cost (Rs), 3, 4, 4,, 45, 67, 33,5 3 36, 5,5 88,5 9,5 4 54, 56,5,, , 58,,35, 7, 6,5, 58,,63, 767 7,39, 58,,97, 843 For the ew machie, the average aual cost has bee calculated i the followig schedule Table 8.8 Age Total O.C. (Rs) Total capital cost (Rs) (Total depreciatio) Total cost (Rs) Average cost (Rs), 4, 5, 5, 7, 6, 87, 43,5 3 45, 7,,5, 38, , 75,,44, 36, 5,, 77,,77, 35,4 6,4, 77,,7, 3,67 7,9, 77,,67, 38,43 (i) If the first type of machies is opted, the replacemet should be after every five wees. (ii) If the secod type of machies is opted, the replacemet should be after every five wees. 56

9 (iii) Average aual cost of old machie Rs. 7, Average aual cost of ew machie Rs. 35,4 Equivalet cost of old machie 35,4 Rs. 3,6 3 Thus old machies should be replaced by ew oes. (iv) Three old machies are to be replaced by two ew machies whe the joit ruig cost of the old machies is more the the average yearly cost of two ew machies (i.e., 35, 4 Rs. 7,8 ). The total cost of old machies is calculated below Year Table 8.9 Total aual cost of oe old machie (Rs.) Total aual cost of three give old machie (Rs.) 4, 67,-4, 7, 3,5,5 + 7, 7, 4,75 65, 5 4,75 7,5 The old machies must be replaced by ew machies two years from ow. (b) Time value of moey chages I this case, the ivestor is payig iterest o the moey that has bee ivested. We assume that (i) (ii) The maiteace costs are icurred at the begiig of time periods; ad The maiteace costs icrease with time. Let the moey carry a iterest rate of r% per year, i.e., oe rupee i years time is equivalet to Rs. (+r) - today. 57

10 (+r) - is called the preset worth factor of rupee oe spet years after ow. (+r) is the paymet compoud amout factor of rupee oe spet i years time. Let C: Iitial cost of the equipmet R : Ruig cost of the equipmet i year. ν : Rate of iterest ( + r) - V : Future discouted costs associated with the policy of replacig the equipmet at the ed of each years. The the preset value of V is {( ) ν ν... ν } V C+ R + R + R + + R + + {( C R) ν ν R+ ν R+... ν R } C+ ν R ( ν ) C+ ν R ν Now, V is miimum whe V V ad V + V Now, C+ ν R V+ V V + ν ν ( R ( ν) V ) ν + Similarly, V ν ( R ( ν) V ) V 58 ν

11 Now, ν <, ν beig depreciatio factor, ν > ν > + ν ( ν ) ( ) ( ν ) V V > R > V + ( ν ) ad V V > R < V R < V < R C+ R + νr+ ν R ν R R < W < R + ν + ν ν The expressio, which lies betwee R ad R is called the weighted average cost of all the previous years with weights beig, ν, ν,...ad ν respectively. Thus the optimal replacemet policy is: (a) Do ot replace the equipmet if the ext period s operatig cost is less tha the weighted average of previous costs. (b) Replace the equipmet if the ext period s operatig cost is greater tha the weighted average of previous costs. For ν, R < V < R Example 4: A maufacturer is to mae a choice betwee two machies, say, A ad B, which are priced at Rs. 5, ad Rs. 5, respectively. The aual ruig costs for machie A are Rs. 8, for first five years after which the costs icrease per year by Rs.,. Machie B, which has the same capacity as the machie A, will have a ruig cost of Rs., for first six years, ad after that would icrease by Rs., per year. If the moey is worth % per year, which machie should be purchased? Assume that the scrap value of the two machies is il. 59

12 Sol: r. ν We prepare the followig tables: Table 8.: For machie A Year () R ν ν R C + ν R ν W Table 8.: For machie B Year () R ν ν R C + ν R ν W

13 For machie A, the ruig cost i the ith year of operatios is the least so it should be replaced after every te years. For machie B, the ruig cost i the eighth year of operatios is the least so it should be replaced after every eight years. Further the weighted average cost i te years of machie A is Rs , whereas the weighted average cost i eight years of machie B is Rs So machie B should be purchased. 8.3 Replacemet of equipmets that fail suddely Sice the exact failure time is difficult to predict for those equipmets, which fail suddely, i such cases we try to obtai the probability distributio of failures. It is assumed that the failures occur at the ed of the period, say, t. The the objective is to determie the value of t that miimizes the total cost ivolved i the replacemet. I this case, two types of replacemet are ivolved. I first replacemet, equipmet is replaced as ad whe it fails. I secod type of replacemet, equipmet may be replaced eve before it fails. This id of replacemet is udertae i those situatios whe failure of equipmet results i huge moetary losses (e.g. electricity trasformers) or is fatal i ature (e.g. a pace maer). We defie the followig replacemet policies Idividual replacemet policy: A item is replaced immediately after its failure. This policy is called as the idividual replacemet policy Group replacemet policy: Uder this policy, decisios are tae as to whe the items must be replaced irrespective of the fact that the items have failed or have ot failed, with a provisio that if ay item fails before the optimal time, it may be idividually replaced. For example, electricity bulbs o a 6

14 road are replaced after some time eve if they have ot failed. But a idividual failed bulb is replaced durig the iterval betwee two successive group replacemets also. We defie the followig otatios: M ( t) the umber of survivors at ay time t. N iitial umber of items. pt ( ) P( of failure durig the time-period t) Mt ( ) Mt ( ) Mt ( ) p ( t) P( of survival till time-poit t) s Mt () N We have the followig results: Theorem 8.: (Mortality) A large populatio is subject to a give mortality law for a very log period of time. All deaths are immediately replaced by births ad there are o immigratios or emigratios. The the age distributio ultimately becomes stable ad the umber of deaths per uit time becomes costat, which is give by the size of the total populatio divided by the mea age of deaths. Proof: Let the maximum period of survival is Defie f ( t) umber of births at time t; px ( ) P(of daeth at time x). The Ad x px ( ) E(deaths till time t) f( t x) p( x) f( t+ ) x Now, f( t+ ) f( t x) p( x) (8.) x 6

15 t This is a differece equatio i t. To fid a solutio to this equatio, put f() t Aα, where A is a arbitrary costat ad α is a umber betwee ad. t+ t x Aα Aα p( x) x t t t ( α () α ()... α ( )) A p + p + + p t+ t t t α α p() + α p() α p( ) + Let t. The α α p() α p()... p( ) (8.) This is a liear homogeous equatio of degree + so it must have + roots. For α, L.H.S. of (8.) R.H.S. so is oe of the roots of the equatio. Let the other roots be α, α,..., α. The a geeral solutio to (8.) is give by f( t) A + Aα Aα t t t where A i 's are arbitrary costats to be determied. Also α < α i < i. Lettig t, we have lim f ( t) A t Now, our job is to determie A. For this, defie qx ( ) P(survival upto xor more years of age) P(death before age x) x pt ( ). t Obviously, q () Sice births are immediately replaced by deaths, i.e., A is the log ru umber of births as well as the umber of deaths, so we have 63

16 E(survivals upto age x) A q( x) N A q( x) A x x N qx ( ) Fially, we have to determie the deomiator of this expressio. For this, cosider x ( x+ ) x; ad The we have b f( x) hx ( ) f( b+ ) hb ( + ) f( aha ) ( ) hx ( + ) f( x) x a x a b qx ( ) qx ( ) x x x ( + ) q( + ) q() ( x+ ) q( x) x ( + ) q( + ) ( x+ ) q( x) x But, q ( + ) pi ( ) i Ad qx () qx ( + ) qx () px () qx ( ) ( + ) q ( + ) ( x+ )( px ( )) x x ( x+ ) p( x) x Mea age at death N Hece, we have A Mea age at death Hece the result. 64

17 Example 5: A compay maufactures automobile batteries at a cost of Rs. each. Battery life is subject to the followig mortality schedule Table 8.3 Age (moths) Probability of failure i Age (moths) Probability of failure (i) ext moth (p i ) (i) i ext moth (p i ) The compay has a guaratee policy uder which if a battery fails durig the first moth after purchase, either a refud of the full price or a ew battery is made; if it fails durig the secod moth, a refud of 9/3 of the full price is made; durig third moth, this refud is 8/3 of the full price ad so o util 3 th moth whe the refud is /3 of the full price. At what uit price should the batteries be sold so that o average will the compay brea-eve? Sol: Let the brea-eve price is Rs. P per battery ad 65

18 st p P(a ew battery fails durig the i+ moth) i The average refud o the failed batteries is give by 9 8 p+ p + p p3 P.98P The brea-eve price P is the sum of the expected refud ad the factory cost of the battery. Therefore P +.98P.99 P P.99 Rs. Theorem 8.: (Group replacemet) Let all the items of a system be replaced after a time iterval t with the provisio that a idividual replacemets ca be made if ad whe ay item fails durig this time period. The the optimal policy of replacemet is (i) Group replacemet must be made at the ed of the t th time period if the cost of idividual replacemet for the period is greater tha the average cost per uit time period through the ed of t periods. (ii) Group replacemet must ot be made at the ed of period t if the cost of idividual replacemet at the ed of the period t - is less tha the average cost per uit time period through the ed of t periods. Proof: Let N Total umber of items i the system. C the cost of replacig a item i group. C the cost of replacig a item idividually. Ct ( ) Total cost fuctio of group replacemet after time period t. f ( t) Total umber of failuers durig time period t. 66

19 The, t Ct () NC+ C f( x) x Average cost of group replacemet per uit time durig a iterval of t uits is give by At ( ) Ct () t i NC + C f ( x) t t We wat to miimize Ct () t. I that case wheever Ct ( ) Ct ( ) ( ) ( ) ad Ct + > > Ct t t t+ t We replace all items after time t. Now, C( t + ) C( t) > t + t C f ( t) > C() t t C( t ) C( t) > t t C f ( t ) < C() t t tc f ( t ) < C ( t) < tc f ( t) t- t- NC or, tf ( t ) f ( x) < < tf ( t) f ( x) C x x Hece the optimal replacemet policy is (i) Group replacemet must be made at the ed of the t th time period if the cost of idividual replacemet for the period is greater tha the average cost per uit time period through the ed of t periods. (ii) Group replacemet must ot be made at the ed of period t if the cost of idividual replacemet at the ed of the period t - is less tha the average cost per uit time period through the ed of t periods. 67

20 Example 6: The followig failure rates have bee observed for a certai type of electroic equipmets i a digital system Table 8.4 Ed of the wee Probability of failure till date The cost of replacemet of a idividual failed compoet is Rs. 5. The decisio is made to replace all these compoets simultaeously at fixed itervals, ad to replace the idividual compoets as they fail i service. If the cost of group replacemet is Rs. 3 per uit, what is the best iterval betwee group replacemets? Sol: Let there be, compoets i use, i.e. N,. th p P( a compoets fails durig i wee) i ad N umber of replacemets at the ed of the wee, i,,...8. i We calculate N i as follows: Table 8.5 t Probability of failure till time t pi Ci C i N i i t N p N t i t , Thus N i oscillates till the system acquires a steady state. The expected life of the system is give by 68

21 Expected life 8 i ip i 4.6 wees. Expected umber of failures per wee cost of idividual replacemet 6 5 Rs. 7, For group replacemet Table 8.6 t Idividual replacemet N (t) Total cost of replacemet Average cost (Rs.) (idividual + group) (Rs.) N ( t ) The optimum iterval for group replacemet is 3 wees. The group replacemet cost for this iterval is less tha the idividual replacemet cost, so it is better to adopt group replacemet. Example 7: At t, all the items i a system are ew, each of which has a costat probability p of failure before the ed of the first moth of life ad a probability q ( -p) of failure before the ed of the secod moth. If all the items are replaced as they fail, show that the expected umber of failures f (x), at the ed of the moth x is give by N x+ f( x) ( ( q) ); x,, + q where N is the umber of items i the system. If the cost per uit of the idividual replacemet is C, ad the cost of group replacemet per item is C, fid the coditios uder which 69

22 (a) (b) A group replacemet policy at the ed of each moth is the most profitable; No group replacemet policy is better tha the idividual replacemet policy. Sol: Let N the umber of items expected to fail at the ed of the first moth N p N( q) N the umber of items expected to fail at the ed of the secod moth Nq+ Np Nq+ N( q) N( q+ q ) I geeral, N N q q q q 3 ( ( ) ) N N q+ N p N( q q q... ( q) ) q N( q q q... ( q) )( q) N( q q q... ( q) ) f x N q+ q q + + q 3 x ( ) (... ( ) ) lim f( x) x ( q) N + q x+ N + q Total umber of items i the system mea age is the steady-state expectatio. () For group replacemet policy at the ed of each moth, cost of replacemet is NC. () For group replacemet policy at the ed of every secod moth, cost of replacemet is NC + NpC average cost per moth NC + NpC 7

23 (3) Average life of a item p+ q +q NC cost of idividual replacemet + q (a) A group replacemet at the ed of the first moth will be better tha the idividual replacemet, if the total cost of the group replacemet is less tha the average mothly cost of the idividual replacemet, i.e. N( q) + NC < NC + q C < Cq + q For a group replacemet at the ed of the every secod moth, the total cost of total replacemet will be ( N + N ) C + NC N(- q+ q ) C + NC Average mothly cost of group replacemet N q+ q C + NC (- ) i.e., a group replacemet at the ed of the every secod moth will be better tha the idividual replacemet, if the total cost of the group replacemet is less tha the average mothly cost of the idividual replacemet, i.e. (- ) N q+ q C + NC < NC ( + q) or, C < q ( q) C ( + q) (b) For idividual replacemet policy to be better tha ay of the group replacemet policies, we must have C qc qc( q) ( + q) ( + q) > ad C > or, C C ( + q) + q < ad C < q q ( q) 7

24 T r + q + q But q < < q q ( q) C < + q C q 8.4 Prevetive replacemet If the cost of the failure of the equipmet is much more tha the cost of its replacemet, e.g. a pacemaer or a electroic chip i a aviatio system, the it is advisable to replace that equipmet before it fails. We ow derive the optimal replacemet policy for prevetive replacemet. C C R F Cost of replacemet. Cost of failure (icludig replacemet). T the radom variable deotig the life of the equipmet. F ( t) P( T t) P(the equipmet will ot fail till time poit t). T f( t) P(of failure i time period t T t). ( ) The F () t F ( t ) - f() t T T Suppose that the replacemet is made at the begiig of a iterval. Let the replacemet policy be to replace the equipmet after every period if the item has ot failed earlier, i.e., T r max t The the expected life of the equipmet is T t ( ) E( T ) tf ( t ) f ( t) + T F ( t ) I a iterval of legth, say, ω, the umber of equipmet required is ω. ET ( ) Thus the expected cost of replacemet is 7

25 ω ω ( FT ( t ) ) C F + FT ( t ) C R E( T ) E( T ) AC( T ) ( ) F ( t ) C + F ( t ) C T F T R E( T ) Where AC (T) is the average cost curve. The the T, which miimizes AC (T), is the optimal time to replace the equipmet. Example 8: A small compoet i a machie costs Rs. 5 ad it taes, o average, miutes to replace it. However, i miutes, the machie ca produce goods of worth Rs. 3. the probability of the failure of the compoet icreases with the usage so that after some time of usage, it is advisable to replace the compoet. The cost of the replacemet of the compoet is Rs. 5. The probability distributio of the failures is give i the followig table: Table 8.7 Wees f(t) Fid the optimal time to replace the compoet. Sol: C R Cost of replacemet Rs. 5. C F Cost of failure Rs.3 + Rs.5 Rs. 35. Table 8.8 Wees f(t) F T (t) F T (t-) (-f (t)) ( ) E (T) F ( t ) C + F ( t ) C R T F T AC( T ) The optimal time to replace the chip is at the begiig of the secod wee. 73

26 Problems. Followig data has bee collected from the records regardig the ruig cost of a machie priced at Rs. 6,,. Table 8.9 Year Operatig costs (Rs.),, 4, 5, 35, 5, 8, Resale value (Rs.) 4,,,65,,75,,5, 9, 6, 45, Determie the optimal time of replacemet.. The cost of a ew machie is Rs.,,. Compute the optimal time to replace whe the costs associated with the machie are give below: Table 8. Age (Years) 3 Operatig costs (Rs.) 5, 7, 9, Resale value (Rs.) 8, 65, 5, Assume that the repairs are made at the ed of a period oly if the machie is to be retaied ad ot ecessary if the machie is to be sold. Assume that the cost of capital is %. 3. A populatio of N idividuals is subject to a give mortality law per uit of time. The deaths are immediately replaced by births at the ed of the iterval. No idividual ca survive more tha r periods. Show that the distributio of the umber of deaths ultimately stabilizes to N Average life 4. Cosider the followig replacemet schedule of a compoet i a electroic gadget 74

27 Table 8. Hours i use Probability of failure The cost of the replacemet of the part is Rs. 5 whereas the failure would cost Rs. 3,. What should be the optimal replacemet policy? 5. A truc fleet ower ows 5 trucs. He has a policy of replacig a tire whe it is wor completely. This costs him Rs. 6, per tire. He has bee advised that the replacemet cost ca be reduced to Rs. 4,5 if he replaces tires periodically. The past data has revealed the followig replacemet schedule Table 8. Moths after replacemet Proportio of the tires wor-out What should be the optimal replacemet policy? 6. A pipelie is due for repairs. It will cost Rs.,, for repairs, which will last for 3 years. A alterative is to lay a ew pipelie, which ca wor for years. If the cost of moey is % ad there is o salvage value, what should be the decisio? 7. Let p (t) be the probability that a machie i a group of 3 machies would breadow i period t. The cost of repairig a broe machie is Rs.. Prevetive maiteace esures servicig of all the machies at the ed of T uits of time at a cost of Rs. 5 per machie. Fid the optimum T which would miimize the expected total cost per period of servicig give that.3 for t pt ( ) pt ( ) +. for t,3,....3 for t > 75

28 8. A research team is required to attai a stable level of 5 members. The service schedule of the members is give below Table 8.3 Year Percet resigatios at the ed of the year What should be the umber of recruitmets per year to maitai the required stregth? If the promotio requires at least 8 years of service, what is the average legth of the service after which a ew recruit ca expect promotio? 9. The maagemet of a large hotel is cosiderig the periodic replacemet of the light bulbs fitted i its rooms. There are 5 rooms i the hotel ad each roo has 6 bulbs. The curret policy is to replace the bulbs as ad whe they fail. Per uit cost of replacig failed bulbs is Rs... The ew policy ca cut the costs to up to Rs. 6.. The past data reveals the followig iformatio: Table 8.4 Moths of use Percet of bulbs failig by the ed of the moth What should be the optimal replacemet policy?. If the cost of capital is % per aum, which of the followig ivestmet plas should be opted Table 8.5 Details Pla A Pla B Iitial cost (Rs.),,,5, Salvage value after 5 years (Rs.),5,,8, Aual profit (Rs.) 5, 3, 76