minimizing life cycle cost by managing product dependability via validation plan and warranty

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Int. J. Production Economics 112 (2008) 796–807

Minimizing life cycle cost by managing product reliability viavalidation plan and warranty return cost

Andre Kleynera,�, Peter Sandbornb

aDelphi Corporation, Electronics and Safety Division, P.O. Box 9005, M.S. D34, Kokomo, IN 46904, USAbCALCE, Department of Mechanical Engineering, University of Maryland, MD, USA

Received 10 October 2006; accepted 6 July 2007

Available online 24 July 2007

Abstract

This paper presents a quantitative solution that minimizes the life cycle cost of a product by developing an optimal

product validation plan. Dependability constitutes an integral view of a product’s reliability, availability, maintainability,

quality, and safety. The methodology developed in this paper incorporates several dependability-related activities into a

comprehensive probabilistic cost model that enables minimization of the product’s life cycle cost. The model utilizes the

inverse relationship between the cost of product validation activities and the expected cost of repair and warranty returns.

The model emphasizes the test duration and sample size for the environmental qualification tests performed in a product

validation program. The overall stochastic cost model and its minimization are done with Monte Carlo simulation in order

to account for uncertainties in model parameters. The model is demonstrated on an automotive electronics application.

The results of this work provide application-specific optimal product validation plans and evaluate the efficiency of a

product validation program from a life cycle cost point of view with an emphasis on the cost of validation and product

warranties.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Warranty; Reliability; Life cycle cost; Validation; Automotive

1. Introduction

Life cycle cost (LCC) analysis is a tool thatproduces important metrics for choosing the mostcost-effective approach from a series of alternatives.LCC generally refers all the costs associated with aproduct throughout the product’s life. The exactcontent of LCC varies depending on the horizon ofthe interested party; however, generally LCC

includes conceptual/preliminary design costs, de-tailed design and development costs, productionand/or construction costs, and product use/support/phase-out/disposal costs (Fabrycky and Blanchard,1991). In this paper, LCC is used to refer to thecombination of design, validation, manufacturing,and warranty costs. One important contributor toLCC for many types of products is the cost ofproduct failure. The product characteristics asso-ciated with a product’s potential failures or mal-functions are often summarized by the termDependability, which constitutes an integral viewof product’s reliability, availability, maintainability,

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�Corresponding author: Tel.: +1 765 451 3379;

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E-mail address: [email protected] (A. Kleyner).


quality, and safety (Fernandez, 2001). This paperfocuses on two major quantifiable dependabilitycontributions: reliability and quality.

To assure uninterrupted performance during asystem’s mission life, product testing and validationis conducted as an important part of the develop-ment cycle. Product validation activities normallyinclude reliability analysis and testing (both func-tional and environmental), which are intended toprove that the design satisfies specified quality andreliability requirements. From a supplier’s view-point, the cost of product validation activities is asignificant variable in the overall cost model.Product development activities associated withdependability are presented in Fig. 1.

Clearly product validation activities have a directimpact on the expected warranty cost, although inmany industries the issue of product validation costand its impact on the development program are notgiven sufficient attention especially in the earlystages of product planning.

Traditionally, in the initial phase of the businesscycle during product quoting, the costs of productdevelopment and validation are treated as a one-time expense, and usually not treated in conjunctionwith the rest of the product’s non-recurring costs.This often leads to a customer’s insistence on thehighest possible reliability without proper consid-eration for the costs involved in the process. As anexample, in the mid-1990s, one of the majorautomotive manufacturers was on a quest toimprove quality and reduce warranty claims. Theydecided to approach the problem exclusively fromthe product validation process. The product valida-tion organization calculated the number of testsamples and finds that for a required reliability of0.90 with a confidence level of 90%, the number oftest samples required is 22 (see Appendix A for adiscussion of sample size calculations). If thereliability requirement is raised to 0.99, the numberof required test samples becomes 229. As thereliability approaches 1.0 (100%), the number oftest samples required approaches infinity. Ob-

viously, the suppliers want to minimize validationtesting in order to save money, while the customeroften assumes that more testing by the supplier willsolve all warranty problems—in this case thecustomer desired increasing reliability targets with-out an accurate understanding of the economicbenefits (or lack thereof).

Product warranty is a significant contributor tothe post-manufacturing portion of the LCC. Forexample, according to Nasser et al. (2002), onaverage General Motors spends approximately $3.5billion per year (roughly 22.5 million warrantyclaims) paying dealerships to repair failed partsunder warranty. Original Equipment Manufac-turers (OEMs), the brand name of the product,often penalize their suppliers based on their cost ofwarranty by passing to the suppliers all or part oftheir warranty cost (Balachandran and Radhakrish-nan, 2005). Based on these considerations, suppliersmust make decisions at the beginning of a productdevelopment cycle regarding how much should bespent on product validation and estimate the effectof that spending on the expected warranty cost.Project managers often need to focus their activitieson the dependability-related variables of LCCanalysis, since these are the inputs that can beaffected during the development process.

Fig. 2 shows a qualitative diagram of therelationship between the pursued reliability andthe total cost. The higher the pursued reliability ofthe product, the higher the product developmentcost (the ascending curve). At the same time thehigher the achieved dependability of the product,the lower the cost of the associated warrantyand service (the descending curve). Relationshipssimilar to Fig. 2 have been referred to as ‘contrac-tor’s cost vs. reliability’ (Blischke and Murthy,1994) and ‘dependability vs. non-dependability cost’(Fernandez, 2001).

The sum of those two costs in Fig. 2 resembles aU-shaped curve with a minimum at the lowest sumof product validation and warranty cost, thusminimizing the contribution to total LCC. Unlike

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New Product Quoting

Product Design Manufacturing

Product Testand Validation

Warranty ClaimsProcessing

and Analysis

WarrantyRepairs

Sales

ReliabilityReliability

and Quality Reliability and QualityQuality

Fig. 1. A product development process involving dependability-related activities.

A. Kleyner, P. Sandborn / Int. J. Production Economics 112 (2008) 796–807 797


validation cost, the cost of warranty is not knownupfront and involves a process of modeling andforecasting discussed in detail later in this paper.

Validation activities are defined as the formalprocess of confirming through environmental test-ing, analyses, inspections, and other engineeringactivities that product reliability requirements aremet. Validation potentially improves the product’squality (this assumes that problems discoveredduring validation are fed back into the developmentprocess) and validation also improves the reliabilityprediction for the product thereby making thewarranty cost forecast more accurate. However,validation can be a significant expense that must betraded off against the value gained. Accuratemodeling of the validation costs (and their relationto the reliability and its associated confidence level)coupled with warranty cost modeling enables thecustomer to optimize the reliability targets based onLCC minimization.

In this paper, we develop a life cycle model thatincorporates several key elements of dependabilityto enable the study of a set of strategic choicesfacing engineers and project managers as theydevelop the best product development flow in orderto minimize the total expenses associated withproduct failures.

1.1. Modeling nomenclature

ad design cost of the projectae cost of equipping one test sampleam manufacturing cost on a per unit basis

amnt cost of monitoring one test sample duringvalidation

ap cost of producing one test sampleaparts cost of the spare parts per repair—random

function f2(x; g2)

aPM average cost of preventive maintenanceapv total cost of product validationaW warranty cost per unit per repairb Weibull slope of the failures observed

before the time point tSbT historical Weibull slope for primary failure

mode.C confidence levelD depreciation of test chamberDc(N, tT) dependability reliability cost functionF(t) cumulative distribution function for the

first time to failureFForecast(tML) expected percent of cumulative fail-

ures obtained using the warranty forecastmodel extended to the mission life tML

y vector of design parametersK test equipment capacityk number of recorded failuresL number of service lives the product is tested

forM maintenance costMTBFEQ Mean Time Between Failures of the test

equipment (repairable system)—randomfunction f1(t; g1)

N test sample sizen total production volumenf number of parts expected to fail during the

warranty periodNPM number of preventive maintenances per

yearnsold number of units sold, which approximates

the total number of manufactured unitsO0 overhead expenses (management, certain

fixed costs, etc.)QCorr correction factor, a random variable ob-

tained from the ratio of predicted reliabilityand demonstrated reliability approximatedfrom the similar products.

R reliability of the productR(t) reliability functionRDemo(tML) a demonstrated reliability at the end of

a mission life (see Appendix A for details)RForecast(tML) predictive model reliabilityt timetBogey test duration (one mission life)tML product mission life (e.g., 10 years, 100,000

miles, etc.)

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ost

Total cost

Minimized LCC

Reliability

Warranty/Service

Development

Fig. 2. Theoretical product development cost versus reliability

curve.

A. Kleyner, P. Sandborn / Int. J. Production Economics 112 (2008) 796–807798


trepair duration of repair—random function f3(t; g3)tS hazard rate stabilization pointtT test durationTW warranty period (e.g., 2 years, 36 months,

etc.)Y additional equipment expensesW warranty terms (can be one- or two-

dimensional)WC total warranty cost for a renewable processL(t) cumulative failure intensity function (num-

ber of replacements per unit)gi vector of statistical parameters. These

parameters can be obtained from statisticalanalysis of the repair and failure data of aparticular test facility

Z Weibull scale parameter of the failuresobserved before the time point tS

jrepair repair labor ratejT hourly labor rate of performing the testxi vector of statistical distribution parameters

for the warranty prediction modeld e ceiling function, indicating rounding up to

the next integer

2. Life cycle cost analysis and its dependability-

related variables

Eqs. (1) and (2) show the components of thesupplier’s cost for products in general and auto-motive products in particular (Kleyner et al., 2004).1

LCC ¼ Design costþ Validation cost

þManufacturing costþWarranty costþOverhead:

ð1Þ

Writing Eq. (1) more explicitly produces

LCC ¼ adðy;W Þ þ apvðyÞ þ namðyÞ þ nf ðy;W ÞaWðyÞ þO0.

(2)

The cost of product development that is includedin product quotes is usually based on forecastingmethods, such as analogy models, expert judgment,prototype models, top-down calculations, andothers (Bashir and Thompson, 2001). Thus, ad,which is based on historical development cost of

similar product lines, is assumed to be independentof product dependability factors. The basis tosupport this supposition is discussed in the nextparagraph.

In today’s competitive environment a fixedbudget is often allocated for design and develop-ment. Many companies do not allocate budgets forredesign, since redesign will make the company non-competitive, therefore when redesigns happen theyare dealt with on an emergency basis and usuallyfunded from profits. A similar view is taken withproduct recalls; they are also an emergency activityfunded from profits. Small redesign expensesrelative to the expected product revenue areincluded in the model and the large redesignexpenses are not since there is no accurate way toaccount for them. The majority of the designchanges resulting from failures during validationare assumed to be relatively minor. For example, inthe automotive electronics industry, changes mayinclude circuit board re-mounting, componentderating/uprating, enclosure redesign, seal change,connector type change, etc. Also at the productvalidation stage, the design changes are relativelyinexpensive. Therefore we model the reliability(validation and warranty) portion of the LCC as

Reliability cost ¼ apvðyÞ þ nf ðy;W ÞaWðyÞ. (3)

Eq. (3) is consistent with the reliability cost modeldepicted in Fig. 2, where apvðyÞ is the ascending partof the curve and nf ðy;W ÞaWðyÞ represents thedescending portion, which will be described in moredetail in Section 4 of this paper.

3. Model formulation

Fig. 3 provides the general outline of the LCCmodel. The process starts with the product defini-tion and then splits into the parts representing theascending and descending curves of Fig. 2. The twokey blocks corresponding to the ascending curveinclude the cost of ownership (COO) of the testequipment required to conduct particular environ-mental tests and the contribution of the sample sizecost. The descending curve deals solely with theexpenses related to future product failures, such aswarranty and service costs. After all the key inputsare estimated, the LCC value is simulated, which isfollowed by the minimization process. The five keyinput blocks of Fig. 3 will be addressed in thesubsections that follow.

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1Eq. (1) models the buyer’s perspective on LCC necessary for

the purposes of this paper. Eq. (1) is not the only way to express

LCC. Other formulations exist that focus on different contribu-

tions to the LCC and/or are expressed to satisfy other views of

the problem, e.g., Fabrycky and Blanchard (1991).



3.1. Product specifications

Typically, the first step in product developmentincludes some form of product definition, whichrequires a wide variety of information includingproduct specifications, functionality, usage, andothers attributes. The majority of the productsdesigned to be used by consumers in the real worldare validated using a series of environmental testssuch as those discussed in Lewis (2000). Reliabilityrequirements usually cover a wide variety ofenvironmental tests including temperature, humid-ity, vibration, mechanical shock, dust, electricaloverloads, and many others. Other relevant speci-fications often include warranty terms and othercontractual obligations concerning product serviceand repair.

3.2. Test equipment cost of ownership (COO)

Test and validation of the product is an integralpart of the product development cost. In someindustries, including consumer and automotiveelectronics, the cost of product validation can easilyreach millions of dollars depending on the type ofthe product, its geometry, technology, functionalrequirements, reliability specifications, and otherparameters.

The primary test and validation cost contributorsare test equipment COO, labor cost, test samplepopulation attributed costs, floor space, laboratoryoverheads, and other miscellaneous expenses. Thegeneral COO concept relates to the total cost ofacquiring, installing, using, maintaining, changing,upgrading, and disposing of a piece of equipment

over its predicted useful lifespan. The majorconcepts of COO as it is applied to manufacturingare discussed in LaFrance and Westrate (1993) andDance et al. (1996).

3.3. Test duration and sample size cost

Test sample size also has a large effect on the costof product validation. Each test sample carries thefollowing costs associated with the sample popula-tion:

� Cost of producing a test sample.� Cost of equipping each test sample. In the

electronic industry this would include harnesses,cables, test fixtures, connectors, etc.� Cost of monitoring each sample during the test.

For example, in the electronics industry thisincludes the labor cost associated with: (a) design-ing and building the load boards that simulate theinputs to the electronic units, (b) connecting andrunning the load boards, (c) recording the data,and (d) visual and other types of inspection.

Considering that some tests may run for weeks oreven months, these expenses can be significant. Themathematical aspects of calculating test sample sizeare presented in Appendix A. It is also important tonote that an increase in sample size sometimescauses the growth of the equipment-related costs asa step-function due to the discrete nature of theequipment capacity. For example, if the capacity ofa testing chamber is 25 units of a particulargeometric size, then a test sample of 26 units would

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Product Specifications

Test EquipmentCost of Ownership

Test Duration and Sample Size Cost

Field ReturnAnalysis

WarrantyForecasting

Minimized Lifecycle cost

Ascending Curve

Descending Curve

StochasticSimulation and

Optimization

Fig. 3. Block diagram of the LCC methodology flow.



require two chambers if simultaneous testing isdesired.

Calculation of product validation cost includesboth the COO of test equipment and the expensesassociated with each test sample size. Capital anddepreciation cost (D), which includes acquisition,installation, and cost of scraping, spread over theuseful life of the equipment. Maintenance cost (M),includes both scheduled and unscheduled mainte-nance, plus indirect maintenance cost. Indirectmaintenance includes technician training, lost rev-enue due to the equipment idle time, etc. Miscella-neous costs (Y) include energy cost, floor space,upgrades, insurance, etc. Therefore, the total cost ofproduct validation per test can be represented by

apv ¼ tT jT þðM þDþ Y Þ

365� 24

� �N

K

� �

þNðap þ ae þ amntÞ. ð4Þ

In many cases it is economically advantageous torun the environmental tests beyond the bogey lifetBogey, which is in test time terms equivalent to onemission life. This is often done to reduce the testsample size while demonstrating the same targetreliability (see Appendix A). In these cases the lifetest ratio L given by Eq. (5) is widely used in thevalidation cost calculations in lieu of tT.

L ¼tT

tBogey. (5)

Maintenance cost per year, including both correc-tive and preventive maintenance can be calculated asthe total cost of parts and labor multiplied by thenumber of maintenance actions per year (Kececioglu,2002). Therefore the yearly maintenance can berepresented by the following equation:

M ¼365 days

MTBFEQðtrepairjrepair þ apartsÞ þNPMaPM.

(6)

The presence of a random vector gi reflects theuncertainty associated with equipment maintenanceinformation. The cost of corrective maintenance isnot known in advance and can only be estimatedbased on the prior history or expected reliability ofthe equipment. In a commercial environmentaltesting laboratory, the maintenance cost can be amajor source of uncertainty due to incomplete ormissing maintenance records, long storage times ininventory for spare parts and other maintenancematerials. These effects impact the parameter M

(maintenance cost per year) in Eq. (4) and arereflected in the statistical parameters gi. To deter-mine M, a case study was performed based on datafrom an automotive validation test laboratory andits results were utilized in the automotive example inSection 4 of this paper. In the example, maintenancedata were available only for the last 4 years ofoperation and all the missing records were related tothe past 20 years, effectively making it a left-censored with univariate missing data. It is beyondthe scope of this paper to provide details for thosecalculations; however, a summary of the result isdiscussed in the example.

3.4. Field return analysis and warranty forecasting

This section discusses Fig. 3 blocks responsiblefor the descending portion of the curve in Fig. 2.Warranty and its associated costs are anothersignificant contributor to the LCC. At present, itis difficult for a product developer to have a clearindication of predicted warranty cost when productsare at the conceptual design level (Nasser et al.,2002); however, the ability to estimate the warrantycost with a known uncertainty would provide adistinct engineering and business advantage.A forecast of a product warranty often becomesan important input in the decision-making processassociated with awarding automotive componentbusiness. There exists a multitude of warranty costmodels, many of which are reviewed in Murthy andDjamaludin (2002).

Most companies maintain some form of warrantyreporting system, in which they collect and analyzefield and test failures. This type of information isused for the prediction of future field failures andtheir expected warranty costs as well as for guidingdesign improvements of the current products. Rootcause analysis of automotive electronics warrantyproblems at Delphi Corporation shows that therange of warranty claims contains a large mix ofdifferent types of problems including: (a) initialperformance or quality, (b) manufacturing or assem-bly related, (c) design-related failure or unacceptableperformance degradation due to applied stresses(environment, usage, shipping, etc.), (d) servicedamage and misdiagnosis, (e) software-related pro-blems, and (f) others.

The process of warranty forecasting starts withproduct specifications, where the main designcharacteristics of the product should be defined.Based on the knowledge of the geometry, utilized

ARTICLE IN PRESSA. Kleyner, P. Sandborn / Int. J. Production Economics 112 (2008) 796–807 801


technology, applications, and other parameters wecan determine the products that are similar to theproduct under development. The warranty numbersfor the similar products can be analyzed for failurerates, trends, statistical distributions, and otherproperties. These data can be utilized for thewarranty analysis and prediction.

For free replacement non-renewable warrantypolicies, which are very common in the industry, theexpected total warranty cost can be estimated as anumber of units expected to fail multiplied by theaverage cost of repair. Assuming that the number offailed units can be approximated by the number ofunits sold multiplied by the cumulative percentfailed, the total warranty cost WC for a renewableprocess can be represented as

WC ¼ LðTWÞaWnsold, (7)

where L(t) represents the number of replacementsper unit (see Kaminsky and Krivtsov, 2000), which,according to Rigdon and Basu (2000), is governedby the fundamental renewal equation given in (8):

LðtÞ ¼ F ðtÞ þ

Z t

0

Lðt� tÞdF ðtÞ. (8)

However, it is not uncommon in the supplierindustry to model the replacements per unit by thecumulative distribution function F(t), which is easierto model than L(t), see for example Lu (1998) orMajeske (2003). This approximation is suitable fornon-repairable systems and the systems with a lownumber of the repeat failures. An analysis of severalautomotive electronics products at one of theDelphi remanufacturing centers showed the numberof parts with repeat failures is below 5% and forsome product lines well under 1%. Therefore, thisapproach will be utilized in the case study discussedin Section 4.

Most of the time, warranty reporting systems andproduct validation activities deal with different timehorizons. Product validation is normally intendedto simulate the product mission life tML, which canbe 10–15 years in the automotive industry. How-ever, warranty usually deals with shorter timeintervals, therefore the current warranty reportingsystem does not normally provide enough informa-tion to evaluate the failure rates corresponding tothe product mission life.

Therefore, the best way to link this model withreliability at the end of the mission life R(tML) is torelate the projected numbers to the target reliabil-ities demonstrated during product validation. This

can be accomplished by using a correction factorQCorr, linking the predictive model RForecast(tML)with demonstrated reliability:

RForecastðtMLjxiÞ ¼ QCorrRDemoðtMLjxiÞ. (9)

QCorr in this case is a random variable specific to aparticular product or a product family. Thereforethe calculation of QCorr can be based on the historyfor the similar products or early filed return data forthe existing product. In general, terms QCorr reflectshow well the reliability forecasting model correlateswith the demonstrated reliability obtained duringvalidation testing. The forecasted reliability valuecan be obtained using a variety of the availableforecasting techniques (see for example Attardiet al., 2005; Lu, 1998 or Kleyner and Sandborn,2005) and in many cases is dependent on a series ofstatistical distribution parameters xi utilized in theprediction model.

Since reliability demonstration values obtainedduring product development cycle depend on a testsample size and test duration (see the case study inSection 4 and Appendix A), the statistical distribu-tion parameters xi will depend on validationparameters N and tT as well as on QCorr:

xi ¼ xiðN; tT;QCorrÞ. (10)

Eq. (9) shows that the demonstrated reliabilitywould be reflected in product performance in thefield with the forecasted number of warranty claimsbecoming a function of demonstrated reliability andtherefore of the test sample size and test duration.

3.5. Stochastic simulation and optimization

Based on Eqs. (4), (6), (9) and (10), the reliabilitycost given in Eq. (3) will be a function of test samplesize N and test duration tT in addition to otherparameters. Therefore the target of the optimizationis the reliability cost function Dc(N, tT):

DcðN; tTÞ ¼ apvðN ; tTÞ þ nf ðN; tTÞaW. (11)

In the case of extended life testing given inEq. (5), tT will be replaced by L in Eq. (11). Being asuperposition of the ascending and descendingcurves, Fig. 2, the objective function will have aminima point, which is not an extreme. Achievingthat point by managing dependability variables willprovide the desired target value for the LCC.

The formulation thus far has not assumed specificfunctional forms for the various distributions thatappear in the model. Section 4 applies the model to

ARTICLE IN PRESSA. Kleyner, P. Sandborn / Int. J. Production Economics 112 (2008) 796–807802


an automotive electronics case by assuming specificdistributions for the input parameters and perform-ing a stochastic simulation.

4. Case study

The automotive electronics case study presentedin this section illustrates the methodology developedin this paper. This case study contains manysimilarities to the operation of the design, valida-tion, and quality functional areas in the Electronics& Safety Division of Delphi Corporation. Theactual values in this example have been modifiedto protect the proprietary nature of the data. Thiscase study considers an automotive radio with CDplayer designed for a mission life of tML ¼ 10 years,with a total production volume of 500,000 units,sold to the automotive OEM for $150 each.

Validation cost in this case can be estimated byapplying Eqs. (4) and (6) using the input parameterspresented in Table 1, which comprises the ascendingportion of the curve in Fig. 2.

As mentioned in Section 3.4, the analysis of thedescending portion of the cost curve utilizeswarranty prediction activities. Based on the factthat a typical automotive part is designed for amission life of 10–15 years it would not be expectedto see wear-out failures during either the warrantyor even extended warranty period of 3–7 years,which is confirmed by the analysis of the failurerates during the extended warranty, Fig. 4.

The data suggest that in the majority of cases thewarranty failure model is sufficiently represented by

the combination of the infant mortality and usefullife phases of bathtub curve. A detailed study of theexisting warranty of various product lines ofautomotive parts performed at Delphi Electronics& Safety showed a clear trend of diminishing failurerate for the first 8–18 months followed by aflattening of the failure rate curve for the remainderof the time period where warranty and extendedwarranty data were available as shown in Fig. 4.

Taking this into account, the choice of theexpected failure probability function was based onthe model in Eq. (12) presented in Kleyner andSandborn (2005):

FForecastðtjb; Z; tSÞ ¼ 1� e�ð1þðbðt�tSÞ=tSÞÞðtS=ZÞb; tXtS.

(12)

Therefore, Eq. (9) can be presented as

1� FForecastðtMLjb; Z; tSÞ ¼ QCorrRDemoðtMLÞ. (13)

The common technique of calculating the demon-strated product reliability is presented in AppendixA, where RDemo(tML) ¼ R. Therefore substitutingEq. (18) into Eq. (9) would produce

QCorr ¼1� F ðtMLjb; Z; tSÞ

ð1� CÞ1=NLb . (14)

Time tS is a change point, the coordinate wherethe pattern of data changes requires a differentdata-fitting model. Each of the parameters, b, Z, tS isa random variable and could be represented by astatistical distribution obtained from the warrantyhistory of the product.

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Days

0

5

15

20

25

0 200 600 800 1000 1200 1400 2000

10

Cla

ims/1

000 V

eh

icle

s

400 1600 1800

Vehicle Line 1

Vehicle Line 2

Vehicle Line 3

Vehicle Line 4

Fig. 4. Extended warranty charts compiled from Delphi Corporation’s 5-year warranty data (modified for the reason of propriety).



Solving Eqs. (9), (10), and (12) for distributionparameters would provide the functional relationshipbetween the distribution parameters in Eq. (12) as

Z ¼tS

½� ln½QCorrRDemoðtMLÞ�=1þ bðtML � tSÞ=tS�1=b

.

(15)

The choice of the modeling parameters b, Z, and tSis based the product family and is typically obtainedfrom the prior warranty history for the family ofproducts with the similar features. The examplecan be an XM-satellite automotive radio withsix CD changer and cassette player. Eq. (15) linksthe scale parameter in the warranty model inEq. (12), Z with the validation target reliabilityRDemo. It has been noticed from the warranty dataanalysis that Z fluctuates significantly more than theshape parameter b. The shape of the warrantydistribution remains reasonably consistent withinthe same product line, where the scale parameterZ is more volatile due to the fact that it isdirectly linked with the expected life of the failedpart. Product-specific variable QCorr in Eq. (15) wasgenerated as one of the random inputs for MonteCarlo simulation.

The dependability-related portion of the LCCplays the role of the objective function in theoptimization procedure. A direct search for thevariables C, RDemo, and L was used to minimize theobjective function. Note that C, RDemo are directlylinked to the test sample size N, and L is to testduration tT, which makes it consistent with bothEqs. (12) and (14).

The random input variables simulated as prob-ability distributions are marked ‘‘(random)’’ inthe first column of Tables 1 and 2. The randominputs used for this model were obtained fromthe analysis of the existing automotive data. Good-ness of fit of the existing data was used to determinethe distribution that best describes the analyzeddata.

As expected, some of the inputs to this stochasticsimulation model were correlated. The analysisindicates that the cost of the equipment spare partsand the duration of their corrective maintenancehad the correlation factor r ¼+0.4 (see Fig. 5) andwere simulated as such. Similarly, the data analysisshowed some positive correlation between b and tS.Based on the available data the correlation betweenb and tS was modeled with the correlation factorr ¼+0.2.

4.1. Simulation results

Each simulation was conducted with 10,000iterations, which was based on the convergencecharacteristics of the simulation. The processdemonstrated 3% convergence with 1000 samplesets; therefore 10,000 appeared to be sufficient.

ARTICLE IN PRESS

Table 1

Model inputs for the cost of product validation

Input Symbol,

units

Value

Confidence level (search variable) C 80–90%

Target reliability (search variable) RDemo 0.80–1.0

Number of lives tested (search variable) L 1.0–2.0

Depreciation of test chamber D, $/year 25,000

Additional equipment expenses Y, $/year 10,000

Hourly labor rate for equipment

maintenance

jrepair, $/h 35.00

Hourly labor rate for product testing jT, $/h 30.00

Cost: spare parts (random) aparts, $/year/

chamber

836.21

Time of maintenance repair (random) trepair, h 2.30

Maintenance MTBF, w2-distr (random) Days 313.6

Number of preventive maintenances NPM/year/

chamber

2

Cost of each preventive maintenance aPM, $/

year/

chamber

2000

Maintenance cost (random) M, $/year/

chamber

5067

Test duration (one mission life) tBogey, h 800

Chamber capacity, units K, units 25

Cost of producing one test sample ap, $/unit 2000

Cost of equipping one test sample ae, $/unit 450

Cost of monitoring one test sample amnt, $/unit 500

Table 2

Model inputs for the cost of warranty and service

Input Symbol Value

Production volume n, units 500,000

Mission life tML,

years

10

Failure rate change point (random) tS, days 305.4

Correlation factor: warranty to reliability

(random)

QCorr 0.9

Shape parameter (random) b 0.780

Scale parameter Z, days 101,953

Cost of one warranty claim (random) aW, $/

unit

504.47

Warranty period TW,

days

1095 (3

years)



Fig. 6 shows the results for a standard bogeytesting (1�mission life, L ¼ 1) and an extendedbogey testing (2�mission lives, L ¼ 2). The entiresimulation result Fig. 6 and the results of thesimulation along with 2-D slice are provided forC ¼ 90%. The lowest cost data points are circled onthe slice chart. The optimal reliabilities RDemo are inthe range of RDemo [0.95; 0.98]. The minimum value

of LCC was achieved at C ¼ 90%, R ¼ 0.97,L ¼ 2.0 and equal to $423,696.

4.2. Uncertainty analysis results

An uncertainty analysis was performed togenerate the confidence bounds for the wholeLCC optimization curve. Fig. 7 shows the 725%

ARTICLE IN PRESS

$5,000.00

$7,000.00

0.0 1.0 3.0 5.0 7.0

Repair Duration, hrs

Sp

are

Part

Co

st

$10,000.00

$9,000.00

$8,000.00

$6,000.00

$4,000.00

$3,000.00

$2,000.00

$1,000.00

$-

2.0 4.0 6.0

Fig. 5. Correlated inputs: repair duration and spare parts cost.

L=1

-

L=2

Co

st,

$

3,000,000

2,750,000

2,500,000

2,250,000

2,000,000

1,750,000

1,500,000

1,250,000

1,000,000

750,000

500,000

250,000

0.9 0.91 0 .92 0.93 0.94 0.95 0.96

Target Reliability

0.97 0.98 0.99 1

7,500,00

7,000,00

6,500,00

6,000,00

5,500,00

5,000,00

4,500,00

4,000,00

3,500,00

3,000,00

2,500,00

1,000,00

500,000

-

1,500,00

2,000,00

11.2

Test Lives, L

1.41.6

1.82 0.8

0.90.92

0.940.96

0.980.995

0.997

0.999

Reliability

LC

C$

Fig. 6. Case study simulation results.



confidence bounds of the solution effectively re-creating Fig. 2 with the uncertainty intervals. It wasnoticed that the confidence bounds are becomingnarrower along as R increases demonstrating thatthe uncertainty of the solution is diminishing withthe increasing reliability targets.

This case study demonstrates that the projectmanagement activities can be enhanced by theapplication of dependability principals (specificallyreliability and validation) within the product LCCanalysis. The LCC of the product can be minimizedby properly choosing the product validation pro-gram including test sample sizes, test durations, andthe choice of appropriate reliability targets. Speci-fically, this case study showed that the lowest valueof LCC can be obtained by pursuing reliability of97% with 90% confidence, while testing for theduration of two bogey product lives. As can be seenfrom Fig. 7, this LCC is approximately 30% lowerthan the cost of pursuing 99% reliability, frequentlyrequested in the industry.

5. Conclusions

The methodology presented in this work can beused to evaluate the efficiency of a productvalidation program from a life cycle cost point ofview with an emphasis on the cost of validation andproduct warranties. This methodology also providesa basis on which to optimize the environmentaltest flow during the product validation, therefore

affecting the overall life cycle cost of the product.Including product dependability factors into thebusiness model can potentially help to improve theproject management by minimizing the life cyclecost of the product. The case study of anautomotive electronics product demonstrated thatcommon customer requested reliability targets maynot be the most life cycle cost effective requirements.

In this paper, we have not focused on adetermination of the ‘‘best’’ numerical optimizationapproach to solving the problem, no doubt moreefficient numerical methods could be brought tobear on this problem. Rather we were interested indemonstrating that an optimum can be found. Theemphasis here was made on formulating themethodology and compiling the model with com-prehensible inputs and outputs suitable for optimi-zation by most of the available engineering andmathematical methods.

Appendix A. Calculating test sample size and test

duration

Reliability demonstration attribute tests intendedto prove that reliability of a product is at or above acertain level are often conducted by running aspecific number of test samples under conditionssimulating the field environment and for a durationequivalent to the mission life. Those tests with twooutcomes (pass or fail) are sometimes referred as‘‘test to a bogey’’. Most of the time, the number of

ARTICLE IN PRESS

$2,500,00

$2,000,00

$1,500,00

$500,00

$-

$1,000,00

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

Reliability, R

Co

st

t

25%-title

75%-title

+25%

-25%Median

Fig. 7. Results of LCC uncertainty analysis.



the required test samples are determined by therequested reliability and confidence level and isbased on the binomial distribution (Meeker et al.,2004). Therefore, the basic relationship betweenreliability, confidence level, and the number of testsamples can be expressed as

C ¼ 1�Xk

i¼0

N!

i!ðN � iÞ!Rn�ið1� RÞi. (16)

Since reliability demonstration is one of theparameters that can be controlled by a projectmanager during the development process, it isnatural to use it as one of the metrics in quantifyingthe expected reliability of the product. Under thecondition of no failures, often referred as SuccessRun testing with k ¼ 0, Eq. (16) can be solved forthe test sample size N as

N ¼lnð1� CÞ

ln R. (17)

Based on Eq. (17), the demonstration of reliabilityR approaching 1.0 requires the sample size N toapproach infinity. For example 90% reliabilitydemonstrated with 90% confidence would require22 test samples, where 99.9% reliability with the sameconfidence would require the sample size of 2300.

Another factor, which can significantly affect thetest sample size, is test duration. A relationshipbetween the test sample size and test duration, oftenreferred as Parametric Binomial or Lipson equalityLipson and Sheth (1973), allows the substitution oftest samples for an extended test time and visaversa. This relationship requires the knowledgeabout the wear-out mechanism for the particularfailure mode in form of a Weibull slope bT:

C ¼ 1� RNLbT, (18)

where L is given by Eq. (5). For the case study inthis paper, R ¼ Rdemo(tML), reliability demonstratedat the end of the mission life. It is important to notehere that Eq. (18) is derived under assumption ofSuccess Run testing, i.e., no failures are experiencedduring the test.

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