Designing Maintenance Systems

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DESIGNING MAINTENANCE CONTROL

SYSTEMS

John van Rijn

INDEVELOPMENT


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INDEVELOPMENT: Designing Maintenance Control Systems

DESIGNING MAINTENANCE CONTROL

SYSTEMS

Any part of this publication may be fully reproduced or translated provided that the source and author are fully

acknowledged.

Edition 2004.

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Acknowledgement

This document presents a methodology for the design of Maintenance Control

Systems. These maintenance control systems were originally designed for productionplants; however experience shows that they can also be used for maintenance planning

of infrastructure systems. The model here presented is a slightly modified version, of

the model developed by W.M. J. Gereards and C.W. Gits of the University ofTechnology Eindhoven.

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Table of Contents:

1 Introduction .............................................................................................................................. 5

1.1 Relevance........................................................................................................................... 51.2 Maintenance....................................................................................................................... 5

2 Maintenance Control ................................................................................................................ 8

2.1 Maintenance Demand ........................................................................................................ 82.1.1 Condition-based maintenance..............................................................................................................10

2.2 Objectives ........................................................................................................................ 123 Design Framework ................................................................................................................. 14

4 Some Special Issues ............................................................................................................... 20

5 Upgrading............................................................................................................................... 22

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1 INTRODUCTIONConditions for maintenance Over the past decade sustainability has become a core issue in

development and development projects. Who havent heard thecomplaints, they should maintain that road. This time it is the road,which is deteriorated, the next time the discussion is about a watersupply system or any other piece of infrastructure. They could be areference to a local infrastructure agency, utility but often the commentis actually refraining to those who finance infrastructure works.

Morss (1980) indicated a number of reasons why certain developmentprocesses are not sustained. The same applies for infrastructure:

Low rates of financial returnsInsufficient revenuesExcessive costsInadequate institution building and individual capacity building

Insufficient timeIndividual time and interestsInstitutional incentivesEconomic policiesPolitics

There are three basic conditions to ensure successful maintenance:It is in the organisations interest to maintain the (infrastructure)productsThere is sufficient technical, managerial and financial capacity atorganisation level to execute the maintenanceThere are effective and efficient financing mechanism in place.

The organisation, responsible for the maintenance should be able tomobilise resources. It is however not necessary that that theorganisation is totally supported by local resources (self-reliance versusself-sufficiency), but that there is an institutional capacity andcommitment to raise the necessary funds to maintain the products/assets (MDF, Netherlands, 1999).

Mr. Olav Ellevset, former director of Tanroads, the agency responsiblefor the maintenance of the national road network in Tanzania, stated onseveral occasions that the income of Tanroads for the financial year2001-2002 only covers 30% the required budget to carry out this task.

The other networks are clearly worse off, because not only is its totalroad length and surface considerable bigger but also they only received30% of the funds made available for the national road network.The situation in Tanzania is quite representative for many low andmiddle-income countries. Donor organisations were in the past veryhesitant to finance any maintenance activities with exception ofreconstruction. Recently some donors changed their main strategytoward the so-called basket funding and sector wide approach. Thisstrategy would allow them to supplement budget for operation and

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maintenance.

However sufficient funding is only one condition for sustainableinfrastructure provision. Technical and managerial capacity are twoimportant other conditions. This document will discuss these issues inmore detail.

1.1 RELEVANCE

Delivery of functions Maintenance in an infrastructure-supplying organisation is essentiallyconcerned with the delivery of the functions of the infrastructure thatmay be for production purposes of for example sewer, water, electricityetc. Unavailability of means of production will result in serious deliveryproblems for these organisations. Organisations that provideinfrastructure for road transport may have more flexibility, but thoseorganisations responsible for economic transport in highly congestedareas would certainly argue otherwise. Most customers wish reductionof the periods of unavailable services and goods. Providers canimprove their production and delivery system eliminating failures andapplying preventive maintenance. However even then unavailability islikely to occur, but organisations can accommodate their customers byinforming them in advance that certain services and goods aretemporarily unavailable. This requires transforming unplanned intoplanned unavailability. It is clear that the need for the provision of theinfrastructure influence the need for these measures.

This paper guide engineers and planners with the design ofmaintenance control systems, that match the demands of productionand delivery of goods and services through infrastructure.

1.2 MAINTENANCEProduction process Maintenance in any (water supply) organisation aims at supporting the

production and delivery process. The infrastructure assets help toproduce and deliver the desired goods and services. For example roadauthorities provide road infrastructure in order to make transportpossible. Vehicle operating kilometres could be seen as a product,produced (enabled) by road network.

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Infrastructure systems Infrastructure is a collection of physical elements with a specificproduction and/or delivery function. In most situations a specificauthority is responsible for the exploitation of the infrastructure assets.The physical elements within a specific infrastructure network or assetcould be composed of different items and have different levels ofaggregation. For example in a road network, the most abstract level is

of course the road links. Whenever a road link no longer exists tofunction this would hamper transport considerably. Secondly on the listare the water crossings or crossings over or under other things. Butmany more levels are possible varying from drainage (always crucial)and guardrail to elements like street lightning and traffic controlsystems.

Condition The condition of the infrastructure system or asset is the physical abilityconsidered relevant for fulfilment of the functions of the infrastructure.

As the condition deteriorates due to aging, use and other factors, thesystem or asset may loose its capability to fulfil it function and a failureoccurs. Maintenance assists these organisations in controlling,

preventing and reducing periods of unavailability.

Maintenance There are two types of maintenance activities: Preventive Corrective maintenance.

Preventive maintenance aims at retaining the infrastructure in condition.Corrective maintenance restores the infrastructure in conditionsrequired for its fulfilment of functions. The latter includes activities likereconstruction and rehabilitation.

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2 MAINTENANCE CONTROLAcceptable level Maintenance will be considered acceptable when it is effective in terms

of retaining or restoring infrastructure, as the functions of theinfrastructure requires and its total costs are within an acceptableproportion and are minimised to the maximum extend possible(efficiency).

2.1 MAINTENANCE DEMAND

Demand Maintenance demand specifies what and when. In other words itindicates which activities need to take place and when to retain andrestore infrastructure production and delivery systems in functionconditions. It should be clear that infrastructure systems involve all theassets necessary to produce and deliver the goods and services.Certain systems may be composed of single assets, often buildings, but

others may involve production plants and transportation links (watersupply).

Required availability The required availability depends on the production and delivery plan ofthe organisation. It is always an exogenous input in the maintenancecontrol system. Most professional organisations will make assessmentsabout the consequences of unavailability levels. On the one handunavailability may result in unsatisfied customers and even a loss ofmarket share. It certainly results in lower quality standards. On theother hand high availability standards may involve high costs andsometimes exceeding available budgets.

The maintenance concept is the set of rules or procedures that activateand specify maintenance activities. Maintenance activities can baactivated through 3 rules:

1. Failure-based maintenance;2. Use-based maintenance;3. Condition-based maintenance.

Failure-based maintenance Failure-based maintenance activates a maintenance activity in theevent of failure. This type of rule is always effective. It is efficient if theconsequences of failure are small. Failure-based maintenance onlyresults in corrective maintenance.

Use-based maintenance Use-based maintenance initiates a specified repair after a certainperiod of use. Use-based maintenance rules are in particular effectivewhen the failure rate tends to increase. Its efficiency depends on thevariance of the failure rate. The more narrow the variance, the moreefficient the use-based maintenance rule.It can only be applied when there is a correlation between the conditionand use. The failure-time relation graph presented below presents fourdifferent failure relationships. Time presented on the horizontal axis (g)and Fw (g) presents the number of failing objects in relation to time.

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Only for two objects it is acceptable to initiate maintenance on basis ofits use. Which ones?

Condition-based maintenance initiates a specified maintenance activity

on basis of a value of value of a characteristic property. If the value ofthe characteristic property is lower (or higher) than a fatal limit, than theinfrastructure (element) is no longer able to carry out its functions. Toactivate preventive maintenance activities (repairs and detailedinspections) most organisations work with warning levels. The value ofthe warning level is the value of the fatal limit minus the value of thebuffer. The value of the buffer often correlates with the processing timeto implement repairs. In addition organisations may work withintervention limits. Fatal limits and intervention limits do not necessarilycorrelate. Preventive maintenance interventions should take placebefore deterioration reaches the fatal limit. With the help of use-basemodels it is possible to calculate the most efficient intervention level,

the most cost-effective scenario of providing maintenance. It mayactually be more cost-effective to maintain high condition standards andfrequently maintain the asset with small and cheap interventions thanrehabilitate the asset when it has deteriorated and has reached its fatallimits.

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This type of rule can only be used when it is possible to identify asuitable characteristic property that can be measured over time. Thesemeasurements or inspections cost money. The inspection interval isdetermined by the period between the onset of noticeable deteriorationand the occurrence of actual failure.

Corrective maintenance All maintenance rules may result in corrective maintenance, as not allfailures will be covered by use-based and condition-basedmaintenance. Corrective maintenance cannot be planned and ischaracterised by its ad hoc nature. It is virtually impossible to prepareaccurate budgets for corrective maintenance and organisations areadvised to maintain flexible budgets to deal with corrective

maintenance.

2.1.1 Condition-based maintenance

Characteristic property Condition-based maintenance requires observations of one or morecharacteristics of a technical system or physical element of that system.That characteristic may be used when

It actually reflects the condition of the physical element It is possible to quantify the minimum value of that characteristic It can be measured.

If the characteristic meets these criteria it is considered a characteristicproperty. The development of characteristic property can have fourforms.

a) Linearb) Progressivec) Digressived) Step drop

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If the development is digressive, the value of characteristic property willnever meet the value of the minimum condition (fatal limit) andtherefore it is not suitable to condition-based maintenance. When thecharacteristic property develops suddenly (step drop), it is also not

suitable for condition-based maintenance.Linear development is very suitable to this type of maintenanceactivation. As it should prevent failures a limit is chosen that is slightlyhigher than the fatal limit (warning level or control limit). This limit has topurpose to activate repairs. The difference between the two dependson the lead-time to mobilise the repair.Progressive development can be used to initiate condition-basedmaintenance to measure the actual difference between twomeasurements. When during a time interval x the reduction in value ofcharacteristic property was bigger than y repairs should be initiated.

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2.2 OBJECTIVES

Objectives of maintenance The objectives of maintenance control have to be derived from theprimary functions of the infrastructure like production, delivery andproviding access to goods and services. In addition the goals of theorganisation itself is important. A private company want to maximise itsprofit, where a public organisation usually has multiple objectives.Realising acceptable production costs is a sine qua non for survival ofany organisation in the long run. Many public organisations in low andmiddle-income countries survive through soliciting funds fromelsewhere.Each production process and therefore each infrastructure system haveparticular requirements when it should be operational, and when it maybe partly or completely closed. It may be wishful to guarantee access toa health centre. Many water supply organisations can only supply waterevery other day in the dry season. Electricity companies may do thesame if there resources are scarce for a certain moment. Many links inroad networks in low-income countries are inaccessible for a period inthe year, which may be considered legitimate. The demandedcontinuity of the availability of the infrastructure influences the timing ofmaintenance operations and preference for preventive maintenance.When failures are unacceptable, because this will close the productionprocess, the willingness to pay for preventive maintenance will be veryhigh. At the same time the timing and design of the maintenanceoperations will be so organised that it minimizes discontinuity of theavailability of the infrastructure or its production process.

Demand for preventive

maintenanceNot only the risk of non-functioning can be a reason to developpreventive maintenance rules. When organisations are held responsible(or take the responsibility) for accidents caused due to a lack ofmaintenance, they tend to have a higher preference for preventive

maintenance. Even if it would result in a cost increase. Some roadauthorities in high-income countries are held financially accountable fortraffic accidents or environmentally accidents due to a lack ofmaintenance. Lost court cases changed their perception towardsmaintenance.

Laws and other bindingdocuments

The government may have issued laws with regard to maintenance. Itgoes without saying that the organisation has to address these laws.The organisation may also have contracts with insurance companies oreven product delivery organisations. Insurance companies often onlyaccept contracts with clauses related to minimum required maintenancelevels and compensation. Product delivery organisations will only issue

guarantee papers with a clause related to proper handling, operationand maintenance of the product.

Regular intervals Control of maintenance is easier when maintenance operations areactivated with regular intervals. This results in a general preference forrepetition of preventive maintenance. Distinguish the following modesof preventive maintenance: Variable maintenance

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Recurrent maintenance Cyclical maintenance

Variable maintenance Variable maintenance does not result in any restriction when and whatmaintenance activities should be activated. Thus the maintenanceplanner/engineer has full freedom.

Recurrent maintenance Recurrent maintenance, as here defined, initiates maintenance

operations on equidistant moments. However the load and content ofoperations may vary.

Cyclical maintenance Cyclical maintenance initiates a specific combination of maintenanceoperations on equidistant moments.

Resource constraints Maintenance requires all kinds of resources. All organisations havetheir constraints (procedures) to make resources available towards thistask. Although ideally the procedures should meet the maintenancedemands this may not always be acceptable to the organisations. Thedesign of the maintenance control system should take into accountthese procedures. The warning level should accommodate theprocessing and implementation time for the repair, in order to preventfailures.

Many organisations have budget limitations for immediateexpenditures. If the estimated amount exceeds a certain figure theproposed operation (like a major overhaul) requiresmanagement/planning decisions of higher echelons. To avoid the riskof failure, maintenance planners/engineers could opt for operations,which reduces the impact of the failure but economically patches thetechnical physical element up longer than the ordering lead-time1.Similar actions may be necessary if the risk of failure or theconsequences of failure are considered unacceptable and the requiredlead-time for the overhaul (for pure technical reasons) takes to long.

1 Time required to complete the required operation, including time for decision taking

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3 DESIGN FRAMEWORKDesign framework The design framework of a maintenance control system describes the

steps to be taken to plan maintenance. It should be noted beforehandthat often knowledge about failure development and impact ofmaintenance operations is lacking. The maintenance control systemaims therefore at satisfying the maintenance demands rather thanoptimising the design. The framework should of course guarantee theobjectives of maintenance control. The framework in particulargenerates maintenance rules.

A maintenance rule is a combination of maintenance operations andthe way they will be activated.

Six steps Maintenance planning consist of 6 steps:1. Initiating of maintenance2. Defining Maintenance Operations3. Limiting intervals4. Clustering operations5. Harmonising of intervals6. Grouping of operations

Initiating maintenance Initiating maintenance matches each possible failure to an elementarymaintenance rule. The elementary maintenance rule describes the waymaintenance will be initiated (failure-, use, or condition based).The choice, which elementary maintenance rule will be dominant,depends in the first place of its effectiveness. Secondly, on thedemands of the organisation with regard to the operation/usage of theinfrastructure. And thirdly, the planner will look into the efficiency of thedifferent rules.

Different effects Maintenance activities may have different effects. It may not bepossible to model these effects, however. The maintenance activitymay result in:1. Change of progression of the failure2. Change actual deterioration3. Correct actual deterioration and progression of failure

Moment of intervention

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Time (years)

Failure

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Actual

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Only change in actual deterioration

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years

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Only alters deterioration

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Failures]

Time [years]

Intervention

Corrects progression and actual deterioration

More than one condition Most technical systems develop different failures. Asphalt pavementsdevelop ruts, roughness and ravels. Usually maintenance activities aretargeted towards a particular failure. However it may have additionaleffects on the progress of other failures. The graphs below show ahypothetical situation in which at moment (a) a maintenance activitythat had positive impact on both the roughness and ravelling, but hadonly marginal impact on the rutting. However it delays the progressionof rut development considerably. Only at moment (b) rut fillinginterventions are implemented. Etc.

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Rutting

Minor failures or failures with only minor consequences for theorganisation and their users, usually results in failure-basedmaintenance rules. The preventive maintenance costs will probably notoutnumber the expected reduction of failure consequences. If it ispossible to initiate maintenance of major failures by both use- andcondition based maintenance, usually the final selection of theelementary maintenance rule is based on efficiency. An importantconsideration is the required additional diagnostic activities todetermine the cause of the failure and to design the repair. The otherconsideration is of course the failure probability density function and itsvariance.

Elementary maintenance rules

The next step is to determine for each elementary maintenance rule,which operation has to take place and how will that operation beactivated. This step results in a list of operational maintenance rules.There are no operational maintenance rules for the failure basedelementary maintenance rules, because it is activated ad hoc, and therepair will be designed after the system has failed.

Specification of repairs andinspections

This activity concerns with the specification of the repairs andinspections in case of condition based maintenance rules and onlyspecification of repairs if elementary maintenance rules are use-basedinitiated.

Maximum time interval The next step is to determine the maximum time interval in which theoperational maintenance rule must have been activated. This stepresults in a list of limited maintenance rules. This interval depends onthe technical and economical life of the particular product.

Incremental life This maximum interval depends highly on the demands of theorganisation and its clients towards the availability of the infrastructureand elementary efficiency. Elementary efficiency is the comparisonbetween the reduction of failures consequences and the cost of themaintenance operations.Usually the maintenance planner has different repair options. For

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example several small repairs versus one big overhaul. A comparisoncan only be made on basis of time- or use based models, even whenthese models have a low correlation to predict the actual failure (andtherefore can not be used to initiate maintenance). As the differentrepairs have different life expectancies, it will be recommendable tocompare them on basis of the incremental project. This means that all

projects are repeated till they have the same total live.

The next step in generating maintenance rules is to search for optionsto combine maintenance rules.

Clustering operations First it is necessary to determine which operations can be clustered.This is both a technical issue and depending on the demands towardsthe availability of infrastructure. Secondly a combined efficiencyanalysis is required.

Combined efficiency Combined efficiency determines whether combining of maintenance

operations results in a cost increase or cost reduction. After all,combing of activities will result in a reduction of cost, because somecosts of activities can be shared like site management, road diversionsand temporarily signposting. But also social cost could be taken inconsideration like hinder to traffic on highly congested roads.But combining of interventions also means that certain physicalelements are maintained earlier than really necessary, which results ofloss of residual values.

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At T1 Element 1 requires maintenance if at T1 also element 2 and 3 aremaintained the loss of the residual values of element 2 and 4 arerespectively the V3-V4 and V1-V4. Alternatively at T2, element 2 needsmaintenance. If at T2 element 3 is also maintained its loss of residualvalue is equal to V2-V4.

Unless the total required life of the infrastructure physical element (likea viaduct or bridge) is beforehand limited because after a certain date itno longer needed, the calculation of the combined efficiency should bebased on a period covering the least common multiple maximuminterval. Otherwise you may come to the conclusion to replace a certainitem x during the maintenance of item z, while in fact the replacement

of item x also could have taken place during the next round ofmaintenance of item z (or any other item).

Element 2 requires maintenance at T3, but as its maintenance can

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easily be combined with the maintenance of element 1, resulting in amajor cost reduction, planners will explore options to maintain element2 while maintaining element 1. Planners may be tempted to maintainelement 2 at T1 when the cost reduction of combined maintenance ishigher than the loss of residual value. However maintaining element 2at T2 will prove to be more efficient as the loss of residual value of

element 2 is a lot lower.

Opportune planning Certain maintenance operations, either repairs or inspections are bestplanned opportune. This means that they are carried out wheneverother works are carried out and they easily can be fit in. Often theseoperations are small but important in terms of impact and they areeasily combinable with other activities.

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4 SOME SPECIAL ISSUES

It is necessary to define the production and delivery objectives of theorganisation. The production outputs of a commercial organisation areeasy to identify, the products. The maintenance systems are designedin such a way that it supports the production process. The infrastructureplanning documents describe processes, which help to assess whetherinvestments like improvements or expansions are feasible, and whichof these investments are the most feasible.

Indicators The indicators used in these assessments do not represent theproduction process. Surely the final objective of the production processis the result in profit (the indicator used), however the productionprocess has many more specific requirements. The financial capacity ofthe organisation influences also the production process. Everycompany strives at high profit rates and wish to sustain itself. It maylack the financial resources to invest to such an extent that it actuallyoptimises its annual profit. Thus the companys profit is at lot lowerlevel. The maintenance system is ideally designed to optimise thislower level profit. In other words if the maintenance system fails, thecurrent profit levels may even decline further.

Affordable? Public sector organisations operate in the exact same way. Roadauthorities may wish to provide high-speed access throughout the year,but because of its financial constraints, is only able to provide slowspeed access during the dry season on large parts of its road network.If the maintenance system is not designed properly, some of the assetsmay deteriorate complete and even slow access during the dry seasonmay no longer be an option.

Setting fatal limits These issues should be taken into account while setting the fatal limits.Often the fatal limits are imposed on the organisation. The supremecourt or the elected bodies may set fatal limits.

Delivery capacity It is recommendable that the organisation makes an analysis of itsdelivery capacity. What is its production level? Is the water-supplyingorganisation able to provide household connections to every house, orcan it only provide a stand post to every street? Is road-supplyingorganisation able to provide motorways with a design speed of 80km/hr to every village or can it only provide dry weather access forvehicles? These production levels influences the design of themaintenance control systems. Motorways have higher maintenancedemands than roads that only have to provide access for vehiclesduring the dry season. Thus this analysis influences the fatal limits.

Cycle planning Maintenance planning and budgeting operate in cycles. Initially it isnecessary to prepare a long-term plan. To prepare this plan, theplanner can only draw upon use-based models and experience data.Even use-based models without any significant reliability may beacceptable for this purpose. During the medium long term or the short-term plans, the long-term plan may be fine tuned on basis of moreaccurate information, among others generated with inspections andmonitoring. The long term and medium long term plans may be based

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on actual maintenance demand, the short-term plan however should bebased on the secured financial resources. Otherwise such a plan wouldbecome completely unrealistic.

While developing the maintenance scenarios, the planner should takeinto account the availability and specific requirements of resources.

There may be specific rules with regard to the availability of theresources. A certain part of the resources should for example be spendbefore the end of the year. A typical constraint for governmentmanaged organisations. Or certain resources may be provided for free,but have quality constraints, like in countries still operating labour tax.These countries could use labour forces for maintenance activities but itis unlikely that these labour forces can carry out specialised tasks.

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5 UPGRADING

One way of reducing maintenance costs is through qualityimprovements of the technical systems. A typical example of this is the

upgrading of the pavements of the roads from an inferior to a higherquality surface. Although upgrading may indeed result in lowermaintenance costs, it also requires investment costs. The basicquestion, planners have to find the answer for:What is the optimal replacement age of the existing technical systemwith the more advanced (higher quality) technical system?

In other words, what is the economic life of the technical system/equipment?

The older the technical system becomes the higher its costs foroperation and maintenance and perhaps even the lower its production.

Economic life

The economic life has been reached when the complementary cost ishigher that monetary value of the production in the same period.New modified equipment Economic degradation of technical systems usually accelerates when

identical items with lower costs are introduced on the markets.

Action 1: Financial evaluation If new technical systems are introduced the first step is to carry out ananalysis to find out if the new technical system actually results in higheryields/profit. Thus the new technical system should meet therequirements set, but also result in more profit. Technical systems, thatdo not generate income, and most infrastructure technical systemsbelong to this category, only have to be evaluated on basis of theirexpenditures. Thus the new technical system only has to be cheaper onthe long run. The Net Present Value and the Internal Rate of Return

methods are financial analysis methods, which will give you an answeron this question. If the new/modified technical systems result in higheryields/profit, action 2 can be used to determine the optimal replacementage of the current technical system with the new one.

Action 2: 10 steps The below-described method compares the direct goodwill with theindirect goodwill of the current technical system. Do the following tensteps:1. Determine current market value of existing technical systems2. Determine current production of existing technical systems3. Determine the depreciation, and variable costs2 of the new/modified

technical systems4. Determine variable costs per production unit of existing technical

system5. Step 3-Step 46. Multiply step 5 with production per year7. Step 6 minus maintenance cost of particular fiscal year8. Step 7 plus residual value at end of fiscal year9. Determine indirect goodwill of step 810. If indirect goodwill is higher than direct goodwill continue with the

2 Variable cost: Incremental cost; increase in cost if the production increases with one unit.

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same exercise with the following years, replace in existing technicalsystem in the year with highest indirect goodwill

An example Suppose technical system X has a technical life of 5 years. The newvalue is US$10,000 and the residual value at the end of its technical lifeis US$ 1,000.

Age (years) Maintenance costs in that year (US$)1 1002 2003 3004 4005 500

The interest is 4%. Technical system X delivers 1000 hours production peryear. After two year a new technical system Y is introduced. The currentmarket value of X is US$ 4000. The variable cost per production unit for X isUS$ 6. The depreciation cost plus variable cost per production unit for Y isUS$ 7.50.

Suppose one more production with X than replacement.

Production x additional depreciation 1000 x (7.5-6) 1500Maintenance cost year 3 300 300Total 1500-300 1200

Residual value after 3 years 2975

Total 1200+2975 4175Net present value 4174/(1.04) 4014.42

4014.42 are higher than 4000 thus repeat calculation.Suppose two more years production with X.



Total 1100+2000 3100Net present value 1200/(1.04) + 3100/(1.04)2

4019.93

4019.93 are higher than 4000 thus repeat calculation.Suppose three more years production with X.



Total 1000+1000 2000Net present value 1200/(1.04) + 1100/(1.04)2 +

2000/(1.04)3

3984.86

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3984.86 is lower than 4000, thus search with the year with highest net presentvalue. (year 4, US$ 4019.93)

Documents

Designing Maintenance Systems