Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS

1 Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS

2 Chapter 1 INTRODUCTION TO MAINTENANCE AND REPLACEMENT MODELS 1.0 MAINTENANCE Maiteace is a routie ad recurrig activity of keepig a particular machie or facility at its ormal operatig coditio so that it ca deliver its expected performace or service without causig ay loose of time o accout of accidetal damage or breakdow. Oce equipmet is desiged, fabricated ad istalled, the operatioal availability of the same is looked after by the maiteace requiremet. The idea of maiteace is very old ad was itroduced alog with iceptio of the machie. I the early days, a machie was used as log as it worked. Whe it stopped workig, it was either repaired/serviced or discarded. The high cost sophisticated machies eed to be properly maitaied/serviced durig their etire life cycle for maximizig their availability. The developmet of mechaizatio ad automatio of productio systems ad associated equipmet, with the accompayig developmet of acillary services ad safety requiremets, has made it madatory for egieers to thik about proper maiteace of equipmet. Maiteace is a fuctio to keep the equipmet/machie coditio by replacig or repairig some of the compoets of the machie. The maiteace cocept is a outlie pla of how the

3 maiteace fuctio will be performed. Based o the feedback obtaied from the users ad the history of the equipmet, detailed procedures are draw to cocretize the maiteace cocept. The procedures developed thus are collectively called the maiteace pla. The developmet of such a maiteace pla is oe of the most importat requiremets of the maiteace program that requires iteractio betwee the user ad the maufacturer. With this iformatio, the maufacturer will be i positio to rearrage the desig as per user s maiteace requiremets. Maiteace fuctio also ivolves lookig after the safety aspects of certai equipmet where the failure of compoet may cause a major accidet. For example, a poorly maitaied pressure vessel such as steam boiler may cause a serious accidet. 1.1 CHALLENGES IN MAINTENANCE: The maiteace fuctio of a moder idustry faces a umber of challeges attributable to: Rapid growth of techology resultig i curret techology becomig obsolete. Such a challege is a frequet oe i Iformatio ad Commuicatios Techology (ICT) idustry where computers ad computers based system (hardware ad Software) is the mai compoet. Advet of ew advaced diagostic tools, rapid repair systems, etc. Advace store maagemet techiques to icorporate modular techologies.

4 Requiremets of keepig both outdated ad moder machies i service. For example, may idustrial orgaizatios have a combiatio of the old machies workig o obsolete techology ad ew systems utilizig the latest techology ad equipmet. The effective maagemet of maiteace aspects uder such challegig circumstaces is ofte a difficult job. Besides the rectificatio of the faults i the equipmet, the activities of the maiteace departmet iclude: Up gradatio of the existig plats ad equipmets ad traiig maiteace persoel to atted the required techical skills. Effective maiteace of the old equipmet for higher availability Cost optimizatio of all maiteace fuctios Improvemet of maiteace activities i the areas of tribology ad terrotechology. Recoditioig of used /userviceable spare parts. Developmet of idigeous sources for parts for import substitutio. Settig up of a effective maiteace iformatio maagemet systems (MIMS). Effective utilizatio of the maiteace workforce Settig up of i house R&D activities for effectig improvemets i maiteace practices.

5 1.2 OBJECTIVIES OF MAINTENANCE: The objectives of maiteace should be formulated withi the framework of the overall orgaizatioal setup so that fially the goals of the orgaizatio are accomplished. For this, the maiteace divisio eeds to esure that: (a) The machiery ad/or facilities are always i a optimum workig coditio at the lowest possible cost (b) The time schedule of deliverig to the customers is ot affected because of o-availability of machiery/service i workig coditio (c) The performace of the machiery /facility is depedable ad reliable. (d) The performace of the machiery /facility is kept to miimum to the evet of the breakdow. (e) The maiteace cost is properly moitored to cotrol overhead costs. (f) The life of equipmet is prologed while maitaiig the acceptable level of performace to avoid uecessary replacemets. Maiteace is also related with profitability through equipmet output ad its ruig cost. Maiteace work ehaces the equipmet performace level ad its availability i optimum workig coditio but adds to its ruig cost. The objective of maiteace work should be to strike a balace betwee the availability ad the overall ruig costs. The resposibility of the maiteace fuctio should, therefore, be

6 esure that productio equipmet /facilities are available for use for maximum time at miimum cost over a stipulated time period such that the miimum stadard of performace ad safety of persoal ad machies are ot sacrificed. These days therefore, separate departmets are formed i idustrial orgaizatios to look after the maiteace requiremets of equipmets ad machies. 1.3 TYPES OF MAINTENANCE SYSTEMS Basically, maiteace ca be divided ito two groups: (a) Breakdow maiteace (b) Plaed maiteace 1.3.1 Breakdow Maiteace ad Its Limitatios The basic cocept of breakdow maiteace is ot to do aythig as log as everythig is goig o well. Hece, o maiteace or repair work is doe util a compoet or equipmet fails or it caot perform its ormal performace. I other words, the maiteace work is called upo whe the machie is out of order, ad repairs are required to brig back the equipmet to its origial workig coditio. If the system is aloe followed, it will lead to poor operatioal availability of the equipmet, as spare parts may ot be readily available. Though it appears to be ecoomical propositio, work would greatly suffer if the machie is ot restored to operatioal coditio at the earliest. I this type of maiteace, durig the repair time, o proper care is take to kow the real cause of the breakdow, which i tur may lead to frequet failures of the same kid.

7 This type of practice is ecoomical for that machiery whose breakdow time ad repair costs are less. But i case of high cost productio systems, there are several limitatios with breakdow maiteace. 1.3.2 Plaed Maiteace: The plaed maiteace is said to be a orgaized type of maiteace. I this type of maiteace, the maiteace activities are plaed well i advace to avoid radom failure. It will be pre determied ot oly the whe ad what kid of the maiteace work, but also by whom it would be udertake. The prerequisites for plaed maiteace iclude the coductio of work study that decides the periodicity of maiteace work. Also the coductio of Time Study helps i suggestig ways ad meas of devisig optimal maiteace schedules for the give system. I plaed maiteace, istructios will be i greater detail ad specific for each type of equipmet. Where safety is of paramout importace, the equipmet coditio should be checked everyday. Hece, the type of maiteace activity to be carried out will deped upo the ature of equipmet ad its workig coditios. The plaed maiteace ca be further classified ito: Scheduled Maiteace (SM) Prevetive Maiteace (PM) Corrective Maiteace (CM) Reliability Cetered Maiteace (RCM)

8 Scheduled Maiteace: This is a stitch-i-time procedure to avoid break-dows. The actual maiteace program is scheduled i cosultatio with the productio departmet, so that the relevat equipmet is made available for maiteace work. The frequecy of such maiteace work is decided well i advace from experiece so as to utilize the idle time of the equipmet effectively. This also helps the maiteace departmet to use their mapower effectively. If the schedule of maiteace is kow i advace, the specialists for the same ca also be made available durig the maiteace period. Though scheduled maiteace is costly compared to breakdow maiteace, the availability of equipmet is ehaced. This practice is used for overhaulig of machies etc. Prevetive Maiteace: It is said to be prevetive maiteace whe plaed ad coordiated ispectios, repairs, adjustmets, ad replacemets are carried out to miimize the problems of breakdow maiteace. This is based o the premise that prevetio is better tha cure. This practice ivolves plaig ad schedulig the maiteace work without iterruptio i productio schedule ad thus improves the availability of equipmet. Uder prevetive maiteace, a systematic ispectio of each item of equipmet or at least the critical parts will be carried out at predetermied times to ufold the coditios that lead to productio stoppage ad harmful depreciatio. There is o

9 readymade prevetive maiteace pla that suits for ay idustry. It should be customized to make it suitable to the requiremets of the particular idustry. Plaig ad implemetatio of a prevetive maiteace practice is a costly affair because it ivolves the replacemet of all deteriorated parts/compoets durig ispectio. However, the higher cost of maiteace usually gets compesated by the prologed operatioal life of the equipmet. To avoid serious breakdows, the prevetive mode of maiteace is usually implemeted i complex plats. Corrective Maiteace: The practice of prevetive maiteace brigs out the ature of repetitive failures of a certai part of the equipmet. Whe such repetitive type of failures are observed, corrective maiteace ca be applied so that reoccurrece of such failures ca be avoided. These types of failures ca be reported to the maufacturer to suggest modificatios to the equipmet. Corrective maiteace ca be defied as the practice carried out to restore the full performace of the equipmet that has stopped workig to acceptable stadards. For example, a IC egie may be i workig coditio, but does ot make its full load because of wor out pisto rigs. If the pisto rigs are replaced, it will brig back the performace of the egie to specified level.

10 Reliability Cetered Maiteace (RCM): It is used to idetify the maiteace requiremets of equipmet. The RCM establishes the fuctioal requiremets ad the desired performaces stadards of equipmets ad these are the related to desig ad iheret reliability parameters of the machie. For each fuctio, the associated fuctioal failure is defied, ad the failure modes ad the cosequeces of the fuctioal failures are aalyzed. The cosequeces of each failure are established, which fall i oe of the four categories: hidde, safety or evirometal, operatioal, ad o operatioal. Followig the RCM logic, preemptive maiteace tasks which will prevet these cosequeces are selected, provided the applicability ad effectiveess criteria for prevetive maiteace are satisfied. The applicability requiremets refer to the techical characteristics ad effectiveess criteria for prevetive maiteace tasks ad the frequecy at which these should be carried out. Effectiveess criteria deped o the cosequeces of the failure; probabilities of the multiple failures for hidde failure cosequeces, acceptable low risk of failure for safety cosequeces, ad ooperatioal cosequeces. Whe the requiremets for plaed maiteace (PM) are ot fulfilled, default tasks iclude failure fidig (for hidde failure, possible redesig of equipmet, procedures ad traiig processes) ad o-schedule maiteace.

11 1.4 BENEFITS OF MAINTENANCE: The high ivolvemet of capital cost i ay productio system expects proportioal returs from the equipmet. These expectatios will be met oly whe the equipmet keeps workig at its ormal performace. It is ofte experieced that the maiteace schedules provided by the maufacturer do ot deliver the required results i terms of the productio out put ad the life of the equipmet. I such cases, therefore, it becomes ecessary to properly maitai the equipmet with extra care i order to obtai the desired levels of productio or service. The followig beefits ca be derived from a well orgaized maiteace system: (a) The miimizatio of breakdow time (b) Improvemet i total availability of the system with their optimum capacity (c) Exteded useful life of the equipmet (d) Safety of the persoel. The cosequeces of dowtime ca be very serious whe the machie is workig i a productio lie, as its failure will shut dow the total system. Followig a proper maiteace schedule the ormal wear ad tear of equipmet ca be reduced. I certai cases, the safety of the persoel is of prime importace ad this also ca be assured by proper plaed prevetive maiteace. For example, all

12 aircraft systems eed to be ispected before ad after a flight as safety of the passegers is of prime importace. 1.5 EFFECTS OF MAINTENANCE: Maiteace, beig a importat fuctio i ay productio system, has far reachig effects o the system. If the right practice of maiteace is ot established for a particular eviromet, it may lead to serious problem of either over maiteace or uder maiteace. The selectio of a particular maiteace policy is also govered by the past history of the equipmet. Cost effective maiteace will help i ehacig productivity. It is therefore, is importat for the team associated with maiteace work, to kow how much to maitai. The ature of the maiteace fuctio affects the life of equipmet. It is kow from experiece that optimum maiteace will prolog the life of the equipmet, ad o the other had, carelessess i maiteace would lead to reduced life of the equipmet ad i some cases a early failure as well. Further, proper maiteace will help to achieve the productio targets. If the availability of the equipmet i good workig coditio is high, the reliability of the productio will also be high. Aother importat effect of the maiteace fuctio is the workig eviromet. If the equipmet is i good workig coditio, the operator feels comfortable to use it other wise there is a tedecy to let the equipmet deteriorate further. To get the desired results i

13 maiteace operatios, there should be selective developmet of skilled, semi skilled, ad uskilled labour. Ad also proper job descriptio is required for the jobs i order to make full use of skilled workforce available. 1.6 EQUIPMENT MAINTENANCE/ REPLACEMENT/ REENGINEERING: To decide the effective mode of maiteace it is essetial to carry out reliability aalysis of critical parts of the equipmet i all moder automated ad semi-automated plats. These critical parts may be idividual pieces of equipmet or a combiatio of parts that from systems. Before cosiderig the purchase of ay capital equipmet, the evaluatio of its reliability is essetial, which directly depeds upo the probability of failures. It is desirable to obtai a reliability idex (umerical value) for each machie which is based o such factors as visual ispectio tests ad measuremets, age, eviromet duty cycle of the equipmet. These umbers, so calculated, represet the reliability of particular equipmet. It is also possible to combie these idices ad express a aggregate reliability idex umber for the complete system. From the evaluatio of the above idex umbers, schedules ca be set for equipmet maiteace. Wherever eeded, the maiteace efforts ca be expaded. From the reliability reports it is possible to determie the actios that are required to maitai the operatioal availability at the desired level. Cost estimates for such

14 maiteace for much maiteace fuctios ca also be prepared based o the reliability iformatio. Similarly, the decisio to replace existig equipmet will require the cosideratio of the followig questios, ecoomic factors ad reliability idex umbers calculated for the existig equipmet. (a) Will the maiteace cost come dow with the replacemet of the old equipmet? (b) Will the cost per uit of productio/service come dow due to automated test features of the mew equipmet? (c) Is the existig equipmet ot sufficiet to meet the future productio/service targets? (d) Will the ew equipmet be eviromet friedly ad provide better safety to operators? (e) Is there ay possibility of addig additioal accessories to existig equipmet i order to make it more versatile for future use, or is the rebuildig of existig equipmet possible through mior modificatios? Optimal replacemet policy of the equipmet ca be determied if reliable estimates of reveue (r etur from equipmet), up keep (maiteace cost) cost ad replacemet costs are available. The equipmet i use i idustries ca be maily divided ito (1) equipmet with dimiishig efficiecy ad (2) equipmet with costat efficiecy. The first category deteriorates with time resultig i icrease i operatig cost icludig maiteace cost, ad secod

15 category operates at costat efficiecy for a certai time period ad the deteriorates suddely. Several models have bee developed usig repair vs. time ad cost, i order to solve the replacemet problem of equipmet with dimiishig efficiecy. Replacemet is cosidered to be the regeeratio poit of whole life where the operatig cost fuctio iitially starts. I practice such methods really work well ad the life of the equipmet/system is ehaced. O the other had the cocept of reegieerig i lieu of replacemet is oe viable model as the operatig cost icreases with time. This model maximizes the gai betwee the operatig costs before ad after the overhauls. Reegieerig ca be perceived as the adjustmet, alteratio, or partial replacemet of a process or product i order to make it to meet a ew eed. Successful implemetatio of reegieerig will improve the equipmet or process performace ad this reduces the maiteace ad operatig costs. 1.7 CONVENTIONAL REPLACEMENT PROBLEM: The replacemet problems are cocered with the issues that arises whe the performace of a item decreases, failure or breakdow occurs. The decrease i performace or breakdow may be gradual or sometimes sudde. The eed for replacemet of items is felt whe, 1. The existig item or system has become iefficiet or require more maiteace.

16 2. The existig equipmet has failed due to accidet or otherwise ad does ot work at all. 3. The existig equipmet is expected to fail shortly. 4. The existig equipmet has become obsolete due to the availability of equipmet with latest techology ad better desig. The solutio to replacemet problem is othig but arrivig at the best policy that determies the time at which the replacemet is most ecoomical istead of cotiuig at a icreased maiteace cost. The Mai objective of replacemet policy is to direct the orgaizatio i may situatios so that it ca take right decisio. For Example, few situatios are: (i) Waitig for complete failure of item or to replace earlier at the expese of higher cost of the item. (ii) Whether to replace the uder performig equipmet with the similar kid of item or by differet kid (latest model) of item. The problem of replacemet occurs i the case of both me ad machies. Usig probability it is possible to estimate the chace of death (failure) at various ages. 1.8 TYPES OF FAILURES As the term failure ecompasses wider cocept, failures ca be discussed uder the followig two categories. (a) Gradual Failure: I this, the failure mechaism is progressive. As the age of a item icreases, its performace deteriorates. This results i:

17 Icreased operatig cost Decreased productivity of the item Decrease i resale value of item (Ex: Mechaical items like pistos, bearig rigs, tyres, etc.,) (b) Sudde Failure: This type of failure ca be observed i the items that do ot deteriorate gradually with age but which fail suddely after some period of service. The time period betwee istallatio ad failure will ot be costat for ay particular equipmet. However the failure patter will follow certai frequecy distributio that may be progressive, retrogressive or radom i ature. Progressive failure: It is said to be progressive failure, whe probability of failure icreases with the age of a item. Ex: light bulbs, tyres etc. Retrogressive failure: Certai items will have more probability of failure i the iitial years of their life ad with the icrease i the life of a item the chaces of failure become less. That is, the ability of the item to survive i the iitial years of life icreases its expected life. Aircraft egies exemplify idustrial equipmets with this type of distributio of life spa. Radom failure: It is said to be radom failure, whe costat probability of failure is associated with equipmet that fails because radom causes such as physical shocks that are idepedet of age. I the case of radom failure, virtually all items fail before agig has ay effect. For example, vacuum

18 tubes, items made of glass or mirror, fruits, vegetables etc may fail idepedet of their age. The replacemet situatios geerally are divided ito the followig four types: (1) Replacemet of capital equipmet whose performace decreases with time, e.g., machie tools, vehicles i a trasport orgaizatio, airplaes, etc. (2) Group replacemet items that fail completely, e.g., electrical bulbs, etc. (3) Problem of mortality ad staffig. (4) Miscellaeous problems. 1.9 REPLACEMENT OF ITEMS THAT DETERIORATE Determiig the optimal replacemet period for a item ca be explaied by cosiderig a example of a vehicle ower whose aim is to fid the best age at which the old vehicle is to be replaced by a ew oe. The vehicle ower iteds to ship cargo as cheaply as possible. The associated costs are: (i) The ruig costs ad (ii) the capital cost of the vehicle These costs ca be summarized as average cost per moth. It ca be observed that the average mothly cost will go o decreasig, with icrease i time. However, there will be a age at which the rate of icrease i ruig cost is cosiderably higher tha the savigs i average capital costs. Thus, at this age it is justifiable to replace the vehicle.

19 1.9.1 Case I: Replacemet Policy For Items Whose Maiteace Cost Icreases With Time, Ad Moey Value Does Not Chage With Time i.e. Costat: Theorem: The maiteace cost of a machie is give as fuctio icreasig with time ad machie s scrap value is costat. (a) Whe time is a cotiuous variable, the replacig the machie whe the maiteace cost is equal to the average aual cost will miimize the average aual cost. (b) Whe time is a discrete variable, the replacig the machie whe the maiteace cost i the (+1) th year becomes greater tha the average aual cost i the th year will miimize the average Proof: aual cost. (a) Whe time t is a cotiuous variable. Let R t =Maiteace cost at time t C= the capital cost of the item S= the scrap value of the item The the aual cost of the item at ay time t = R t + C - S The maiteace cost icurred durig years becomes = R t 0 dt The total cost icurred o the item = P () = R t 0 dt + C-S Hece average total cost is give by F()= P() = 1 R t dt + C S 0 ---(1.1)

20 Now, we have to fid such time for which F() is miimum. Therefore, differetiatig F() with respect to, df() d = 1 R + 2 1 C S R t dt - 2 =0, for miimum of F (), ---(1.2) 0 which gives R = 1 R t dt + C S 0 = P(), by virtue of equatio (3.1), ---(1.3) Hece, the maiteace cost at time = average cost i time (b) Whe time t is a discrete variable Sice the time is measured i discrete uits, the cost equatio (1.1) ca be writte as F()= P() = Rt t1 + C S ---(1.4) By usig fiite differeces, F() will be miimum if the followig relatioship is satisfied : ΔF(-1) < 0 < ΔF() ---(1.5) Now, differecig (3.4) uder the summatio sig by defiitio of first differece, ΔF() = F(+1) -F() Rt C S = t1 1 1 - t1 Rt C S R Rt 1 t1 1-1 = t1 Rt 1 1 + (C-S) 1

21 R 1 = 1 - t1 Rt 1 1 1 + (C-S) 1 1 1 R 1 = R t 1 - t1 ( - C S 1) ( 1) Sice ΔF()>0 for miimum of F(), so R 1 R t 1 > ( 1) t1 C S + ( 1) or R 1 > t1 R t + C S Or R 1 > P()/, by virtue of equatio (1.4) Similarly, it ca be show that R < P()/, by virtue of ΔF(-1) <0 Hece R 1 > ( P()/ ) > R This completes the proof. 1.9.2 Case 2: Replacemet Policy For Items Whose Maiteace Cost Icreases With Time, Ad Moey Value Chages With Costat Rate Moey Value: Whe it is said that Moey is of worth 10% per aum. it ca be iterpreted i two ways: (i) Firstly, ivestig a amout of Rs.100 today is equivalet to ivestig Rs.110 after oe year i.e. if we pla to ivest Rs.110 after oe year it is equivalet of ivestig Rs.100 today. (ii) Secodly, if Rs.100 is borrowed at the rate of iterest 10% per aum ad sped this amout today, the it is required to pay Rs. 110 i a year time.

22 So it ca be iferred that Rs.100 i had today will be equivalet to Rs.110 i had after oe year from ow. I other words, Re.1 i had after oe year is of worth Rs. 1 1.1 i had today. Preset Worth Factor (PWF): Thus ivestig Re.1 after oe year is equivalet to ivestig 1 1.1 rupee today with rate of iterest 10% per aum. Re.1 to be ivested after two years from ow is equivalet to ivestig (1.1) 2 Rupees today. Therefore Re.1 to be spet after years is equivalet to ivestig (1.1) rupees today. The quatity (1.1) is cosidered as preset worth factor (pwf) or preset value of Re.1 to be spet years from ow. I geeral, let moey carry a iterest rate of r, the - 1+r is called the preset worth factor (pwf) or preset value of Re.1 to be spet after years from ow. Discout rate (Depreciatio value): Preset worth of Re.1 to be spet after years from ow is give by - ν= 1+r, where r is the iterest rate. The, ν is called discout rate or depreciatio value.

23 Theorem.: whe the maiteace cost icreases with time ad moey value decreases at costat rate i.e. depreciatio value is give, the the replacemet policy will be: Replace if the ext period s maiteace cost is grater tha the weighted average of previous costs. Proof: Let C= Capital cost of the item to be replaced R i = Maiteace cost icurred at the startig of the i th year r = iterest rate ν= 1+r -1 is the preset worth of Re.1 to be spet a year hece. The proof ca be divided ito two major steps: Step1: To determie the preset worth of total cost Let the item is replaced at the ed of every th year. The year wise preset worth of costs o the item i the successive cycles of years ca be computed as show i Table 1.1. Year 1 2 +1 +2 2 2+1 Preset Worth 1 C+ R 1 R2 R ν (C+ R 1 ) ν R 1 2 Table 1.1: Preset worth(year wise) of future costs C+R ν 2-1 2 R ν 2 1 Assumig that the item has o salvage value at the time of replacemet, the preset value of all future discouted costs associated with the policy of replacig the item at the ed of every year will be give by

24-1 P()= C+R 1 + R 2ν + + R ν + +1 2-1 C+R1 ν +R 2ν + +R ν + 2 2+1 3-1 C+R ν +R ν + +R ν 1 2 Summig up the right had side, we get ad so o P() = 2 C+R1 1+ν +ν + + R 2 2 1+ν +ν + + -1 R ν 1+ν +ν 2 + = -1 C+R 2 1+R ν+ 2 +R ν 1+ν +ν + -1 = C+R +R ν+ +R ν 1 ---(1.6) 1-ν 1 2 [Sice 1 ν<1,the sum of ifiite G.P.is ] 1-ν F() P()= 1-ν, P(+1)= F(+1) +1 1-ν ---(1.7) where, for simplicity, F() = C+ R 1 + + R ν -1 Step2: To determie replacemet policy so that P() is miimum. As is measured is discrete uits, we shall use the method of fiite differece method is used i order to miimize the preset worth cost P (). Therefore, if P(+1) >P()> P(-1), i.e. ΔP() > 0 > ΔP( -1), the P() will be miimum. So by the defiitio of first differece, F(+1) ΔP() = P(+1) P() = +1 1-ν F() - 1-ν from equatio (1.7) = F(+1)(1-ν)-F()(1-ν) +1 (1-ν)(1-ν) +1 N D r r form

25 For coveiece, we first simplify the r N of P() oly. That is r N = F(+1) (1- ν 1 ) F()(1- ν ) =F(+1) - F() + 1 ν F() - ν F(+1) = R ν + ν F() - ν [F()+ν R ] [ F(+1) = F() + R 1 +1 +1 +1 ν ] = +1 v(1- ν)r -ν(1-ν)f() ΔP() = v(1- ν)r -ν(1-ν)f() +1 +1 (1-ν)(1-ν) = ν(1-ν) (1-ν)(1-ν) +1 (1-ν) (1-ν) R +1 - F() ---(1.8) Simply settig (-1) for i equatio (1.8) ΔP(-1) = 1 v (1) v (1)(1) v v 1-1 1-ν R -F(-1) (1-ν) After little simplificatios of RHS ΔP(-1) = 1 v (1) v (1)(1) v v 1 (1-ν) 1-ν R -F() ---(1.9) v (1) v The quatity 1 (1)(1) v v i eq. (1.8) is always positive, sice ν <1. Thus, ΔP() has the same sig as the quatity uder [.] i (1.8), with similar explaatio for ΔP(-1) i (1.9) also. Hece the coditio, ΔP(-1) < 0 < ΔP(), for miimum preset worth expediture becomes. 1-ν 1-ν +1 R -F() < 0 < R -F() 1-ν 1-ν ---(1.10)

26 1-ν 1-ν R < F() < R +1 1-ν 1-ν C+R +R ν+ +R ν R < <R 1+ν+ν + +ν -1 1 2 2-1 +1 F() R < ν < R -1 +1 ---(1.11) ---(1.12) ---(1.13) The expressio betwee R ad R +1 i equatios (1.12),ad (1.13) is called the weighted average cost of previous years with weights 2-1 1, ν,ν,, ν respectively. The value of satisfyig the relatioship (1.10), or (1.12) will be the best replacemet age of the item. This proves the theorem. Aual paymet = Weighted average cost for years. Selectio of best machie: While makig a decisio o solutio of the best machie, various costs are to be take ito cosideratio. The costs that are costat with time for each give machie are to be take ito cosideratio. Some times these costs may differ for each machie. However those costs that are same for the machies uder compariso may be excluded. Cosiderig two machies - M1 ad M2 - selectio of a ecoomically best machie ca be doe by adoptig the followig outlied procedure.

27 Step 1: Fid the best replacemet age for both machies M1 ad M2 by makig use of F() R < ν < R -1 +1 Let the optimum replacemet age for machies M1 ad M2 comes out be 1 ad 2, respectively. Step 2: calculate the fixed aual cost (or weighted average cost) for each machie by usig the formula : x -1 C+R 1+R ν+ 2 +R ν F() 2-1 -1 1+ν+ν + +ν ν Ad i this formula substitute = 1 for machie M1 ad = 2 for machie M 2. Let it be x 1 ad x 2 for machies M1 ad M2 respectively. Step 3: (i) If x 1 < x 2, the select machie M1 (ii) If x 1 > x 2, the select machie M2 (i) If x 1 = x 2, the both machies are equally good. 1.10 REPLACEMENT OF ITEMS THAT FAIL COMPLETELY Cosider a system usually made up of a large umber of low cost items that are proe to failure with age e.g. failure of a resistor i televisio, radio, computer etc. I some cases the failure of a compoet may cause the complete failure of the system. I such cases, the cost of overall failure will be quite higher tha the cost of

28 compoet itself. E.g. the cost of a codeser or tube i a aircraft is little, but its failure may result i total collapse of the airplae. Whe dealig with such situatios, two types of replacemet policies shall be cosidered. (i) Idividual replacemet: I this policy, a item is replaced immediately after its failure. (ii) Group replacemet: I this policy, decisio is about the age whe all the items should be replaced, irrespective of whether the items have failed or ot. I this policy the items that fail before the optimal time, will be replaced idividually. 1.10.1 Case 1: Idividual Replacemet Policy Uder this policy a item is immediately replaced after its failure. To determie the probability of failure (or life spa of ay item), mortality tables are used. To discuss such type of replacemet policy, we cosider the problem of huma populatio. Assumptios: (i) All deaths are immediately replaced by births, ad (ii) There are o other etries or exits. However i reality it is impossible to have these coditios. But, the reaso for assumig the above two is that the aalysis will be easier by keepig the virtual huma populatio i mid. Such models ca be applied to idustrial items, where death of a perso is

29 equivalet to the failure of a item or part ad birth of a perso is equivalet to replacemet. Thus, orgaizatios also face a fairly commo situatio. The followig Mortality Theorem will make the coceptios clear. Mortality Theorem: 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 other etries or exits. The the age distributio ultimately becomes stable ad that the umber of deaths per uit time becomes costat ( which is equal to the size of the total populatio divided by the mea age at death). Proof: For coveiece, let each death occurs just before some time t=w, where w is a iteger ad o member of the populatio ca survive upto ad beyod w+1 time uits, i.e. life spa of ay member lies betwee t=0 ad t=w. Let f(t) = umber if births at time t, p(x) = probability of member will die (fail) just before age x+1, i.e. at age x. Now f(t-x) = the umber of births at time (t-x). The age of such ewly bor members who remai alive at time t will obviously be x. This ca be uderstood from the followig Fig. 1.1. Age Time 0 x x+1 t-x t t+1 Fig. 1.1: Relatio betwee age ad time period

30 Hece, the expected umber of deaths of such alive members at time t is p(x)f(t-x). Therefore, the total umber of deaths at time t will be w x0 f ()() t x p x, t=w, w+1, w+2, Also, total umber of births at time (t+1) = f(t+1). Sice all deaths at time t are replaced immediately by births at time (t+1), therefore w f ( t 1)()() f t x p x, t=w,w+1, ---(1.17) x0 The differece equatio (1.17) i t, may be solved by substitutig f(t) t = A where A is some costat ad < 1 The differece equatio (1.17) becomes w t1 tx A A p() x x0 O dividig by t w A, we get w w w1 w-x w1 w-x p() x or() 0 p x x0 x0 w1 w w1 w2 Or p(0)(1)(2)() p 0 p p w --- (1.18) Sice the sum of all probabilities is uity, so w x0 p() x 1 or w 1() 0p x x0 or 1-[p(0)+p(1)+p(2)+ +p(w)]=0 ---(1.19) Now comparig equatios (1.18) ad (1.19), it is foud that oe solutio of (1.18) is 0 1. But the polyomial equatio (1.18) must have (w+1) total umber of roots. Let the remaiig roots be deoted

31,,,, w, cosequetly, the solutio of differece equatio by 1 2 3 (1.17) will be of the form: t t t f () t A0 A1 1 A2 2 A w w ---(1.20) Where A0, A1,, Aw are costats whose value ca be determied from the age distributio at some give poit i time. Further, it ca be observed that the absolute value of all the remaiig roots is less tha uity i.e. < 1 for i= 1,2,3,,w. i Hece, t 1, t 2,, t w teds to zero as t. Cosequetly, equatio (1.20) becomes f(t)= A 0, which shows that the umber of deaths per uit time ( as well as the umber of births) is costat ad equal to A 0. To show that the age distributio ultimately becomes stable: Let P(x) =the probability of members remai alive loger tha x time uits. The, P(x)= 1- P(survivor will die before attaiig the age x) = 1-[p(0)+p(1)+ +p(x-1)] ad P(0)=1 ---(1.21) Sice the umber of births ad deaths have become costat, each equal to A 0, the expected umber of survivors of age x is also stable at A 0 P(x).

32 As the deaths are replaced immediately (i.e. the umber of births are always equal to the umber of deaths), the size N of total populatio remais costat, i.e. w N A P() x or ---(1.22) 0 x0 A 0 w x0 N P() x ---(1.23) Now the umber of survivors aged 0,1,2,3, ca be calculated from the equatio (1.22) as A 0, A 0 P(1),(2), A 0 P ad so o Fially, if the deomiator i (1.23) i.e. w x0 P() x,is equivalet to mea age at death, the the age distributio will ultimately become stable. To Prove this, differeces] w P() x = x0 w P() x Δ(x) [Δ(x)=(x+1) x =1 by fiite x0 =P(). x x 0 w1 - w x0 ( x 1) Δp(x) w =[P(w+1).(w+1) -0] - ( x 1) Δp(x) ---(1.24) But, P(w+1) = 1-p(0) p(1) p(2)- -p(w) [from equ.(1.21)] = 0 [by virtue of eq (1.19)] x0 Ad ΔP(x) =P(X+1) P(x) = [ 1-p(0) p(1)- p(2)- - p(x)]- [ 1-p(0) p(1)- p(2)- - p(x-1)] = -p(x)

33 Therefore, substitutig the simplified values of P(w+1) ad ΔP(x) i equatio (1.24) to obtai. w P() x = 0+ x0 w ( x 1) p(x) = x0 w1 ()( y p 1) y [settig x+1 = y] y1 = w1 y x prob.[that age at death is y] y1 = mea (expected) age at death. The theorem is thus proved. 1.10.2 Case 2: Group Replacemet Of Items That Fail Completely There are certai items viz. Light bulbs that either work or fail completely. I some cases a system made up of a big umber of similar low cost items that are icreasigly proe to failure with age. While replacig such failed items, always a set-up cost will be there for replacemet. The said set-up cost is idepedet of the umber of items to be replaced ad hece it may be advatageous to replace etire group of items at fixed itervals. Such a policy is referred as group replacemet policy ad foud attractive whe the value of ay idividual item is so less ad the cost of keepig records for age of idividual items is ot justifiable. Group Replacemet Policy: Group replacemet policy is defied i the followig theorem ad later it is explaied by umerical example.

34 Theorem: (a) Oe should group replace at the ed of idividual replacemets for the average cost per period through the ed of (b) Oe should ot group replace at the ed of Proof: idividual replacemet at the ed of average cost per period through the ed of It is proposed to, th t Period if the cost of th t Period is greater tha the th t period. th t Period if the cost of th t period is less tha the th t period. (i) replace all items i group simultaeously at fixed iterval t, whether they have failed or ot, ad (ii) cotiue replacig failed items immediately as ad whe they fail. Let N t = umber of uits failig durig time t N=Total umber of uits i the system C(t)=Cost of group replacemet after time period t C 1 = Cost of replacig a Idividual item C 2 = Cost of replacig a item i group The C(t) = C 1 [ N 1 + N 2 +..+ Nt 1] + C 2 N Therefore, average cost per uit period will be C() t C1[ N1 N2 Nt 1] C2N F() t t t ---(1.25) Now i order to determie the replacemet age t, the average cost per uit period [C(t)/t = F(t), say] should be miimum. The coditio for miimum of F(t) is ΔF(t-1) < 0 < ΔF(t) ---(1.26)

35 C( t 1)()() C t C() t C1N Now, ()( 1)() t C t F t F t F t t 1 t t 1 t tc1n t C()() t / C1N t C t t t( t 1)( 1) t ---(1.27) which must be greater tha zero for miimum F(t), that is C 1Nt >C(t)/t ---(1.28) Similarly, from ΔF(t-1) <0, C1N 1 C() t / t ---(1.29) t Thus from equatios (1.28) ad (1.29), the group replacemet policy is completely established. ILLUSTRATIVE EXAMPLE: for a certai type of light bulbs( 1000 Nos.), followig mortality rates have bee observed: Week : 1 2 3 4 5 Percet failig by the ed of week : 10 25 50 80 100 Each bulb costs Rs.10 to replace a idividual bulb o failure. If all bulbs were replaced at the same time i group it would cost Rs. 4 per bulb. It is uder proposal to replace all bulbs at fixed itervals of time, whether or ot the bulbs have burt out. Ad also it is to cotiue replacig immediately burt out bulbs. Determie the time iterval at which all the bulbs should be replaced? SOLUTION: Let p i = the probability that a ew light bulb fails durig the i th week of its life. Thus p 1 = the probability of failure i 1 st week = 10/100 = 0.10 p 2 = the probability of failure i 2 d week = (25-10)/100 =0.15

36 p 3 = the probability of failure i 3 rd Week = (50-25)/100 = 0.25 p 4 = the probability of failure i 4 th week = (80-50)/100 = 0.3 p 5 = the probability of failure i 5 th week = (100-80)/100 =0.2 Sice the sum of all the above probabilities is uity, the further probabilities p6, p7, p8 ad so o, will be zero. Thus, all light bulbs are sure to burout by the 5 th week. Furthermore, it is assumed that bulbs that fail durig a week are replaced just before the ed of that week. Let N i = the umber of replacemets made at the ed of the th i week. Ad let all 1000 bulbs are ew iitially. Thus, N N =1000 0 0 N1 N0 p1 1000 0.10 =100 N2 N0 p2 N1 p1 1000 0.15 100 0.10 =160 N3 N0 p3 N1 p2 N2 p1 1000 0.25 100 0.15 160 0.10 =281 N4 N0 p4 N1 p3 N2 p2 N3 p1 =377 N5 N0 p5 N1 p4 N2 p3 N3 p2 N4 p1 =350 N 0 N p N p N p N p N p =230 6 1 5 2 4 3 3 4 2 5 1 N 0 0 N p N p N p N p N p =286 7 2 5 3 4 4 3 5 2 6 1 It has bee foud that expected umber of bulbs failig i each week icreases util 4 th week ad the decreases util 6 th week ad agai starts icreasig. Thus, the umber of failures or replacemets will cotiue to oscillate till the system settles dow to a steady state. I

37 steady state the proportio of bulbs burt out i each week is reciprocal of their average life. Idividual replacemet: The mea age of bulbs 1 p1 2 p2 3 p3 4 p4 5 p5 = 1 x 0.1 + 2 x 0.15 +3 x 0.25 +5 x 0.30 +5 x 0.20 = 3.35 Weeks, The umber of failures i each week i steady state =1000/3.35 = 299 Ad the cost of replacig bulbs idividually o failure =10x299 ( at the rate of Rs. 10 per bulb) = Rs. 2990 Group replacemet: The replacemet of all 1000 bulbs at the same time i bulk costs Rs. 4 per bulb ad replacemet of a idividual bulb o failure costs Rs. 10. Costs of replacemet of all bulbs simultaeously are calculated i the Table 1.2. Ed of week Cost of idividual replacemet Total cost of group replacemet (Rs.) Average cost per week (Rs.) 1 100 x 10=1000 1000 x 4 + 100 x 10=5000 5000.00 2 160 x 10=1600 5000 + 160 x 10=5000 3300.00 3 281 x 10=2810 6600 + 281 x 10=9410 3136.67 4 377 x 10=3770 9410 + 377 x 10=13180 3295.00 Table 1.2: Cost of replacemet of bulbs for Bulbs example The cost of idividual replacemet i the 4 th week is greater tha the average cost for 3 weeks. Therefore the optimal replacemet decisio is to replace all the bulbs at the ed of every 3 weeks. Otherwise the average cost from 4 th week owards will start icreasig.

38 1.11 EQUIPMENT RENEWAL The word reewal meas either to rope i ew equipmet i place of old equipmet or repair the old equipmet so that the probability desity fuctio of its future lifetime will be equal to that of ew equipmet. The probability that a reewal takes place durig the small time iterval (t, t+ t) is called the reewal rate at time t. Here time t is measured from the time whe the first machie was started. The reewal rate of equipmet is asymptotically reciprocal of the mea life of the equipmet i.e. h() t 1 = reciprocal of mea life Equipmet reewal comes uder major prevetive maiteace activity which may iclude replacemet of few parts/subsystems or coditioig the equipmet. With respect to this detailed mathematical models are ot discussed here as the mai focus area is o blocks ad block replacemets. Summary: The first chapter discusses the cocepts ad importace of maiteace i productio eviromet. The two types of maiteace Breakdow ad Plaed- are explaied. The objective of maiteace work should be to strike a balace betwee the availability ad the overall ruig costs. Also, the possibilities viz. reegieerig the equipmet, replacemet of the equipmet etc. to esure the equipmet delivers its ormal performace, are discussed. Also this chapter discusses two categories of replacemet techiques for determiig the best replacemet strategies for the items that deteriorate with time ad those do ot deteriorate but fail suddely. These models are discussed with respect to the parameters like maiteace cost, time ad value of moey.