Optimising Burn-In Procedure and Warranty Policy in Lifecycle Costing

Mahmood Shafiee

Department of Industrial Engineering University of Tehran

Tehran, Iran shafiee@iust.ac.ir

Ming Jian Zuo Department of Mechanical Engineering

University of Alberta Edmonton, Canada

Abstract—In this paper, an optimization model is developed to investigate the optimal burn-in time and warranty length of a product from a manufacturer’s perspective. It is assumed that the cost of a minimal repair to the component which fails at age t is a continuous non-decreasing function of t. Out-of-warranty, if the product fails before its useful life limit, it causes customer dissatisfaction and incurs a penalty cost for the manufacturer. The properties of the optimal burn-in time and optimal warranty policy are also given. We provide a numerical example to illustrate our results. A sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal solution.

Keywords-burn-in; bathtub shaped failure rate; warranty length; post-warranty

I. INTRODUCTION

Manufacturers guarantee the quality of their products by offering to repair or replace a failed unit free of charge (or at a reduced price) for a certain length of time, referred to as the warranty period. According to Murthy and Blischke [1], the two main roles of warranty are (i) protection – for the consumers against defective items and for the manufacturers against consumers’ excessive claims, and (ii) promotion – for product differentiation by manufacturers and by consumers.

Offering warranty results in additional costs to the manufacturer. These warranty costs depend on several factors such as product reliability, warranty terms, maintenance actions and warranty servicing strategy [2]. Since the cost of failure during production usually is lower than during the warranty period, the manufacturers attempt to use effective strategies for reducing the warranty servicing costs, before the items are shipped to customers or put into field operation.

Burn-in is a widely used technique for improving the quality of products after they have been produced. The effect of burn-in is accelerated through higher temperature, higher current density, higher voltage, higher frequency or other stresses to approximate a time period of field use. Those items which fail during the burn-in procedure are repaired and only those that survive the burn-in procedure are considered to be of good quality and put into field operation [3].

An insufficient burn-in or excessive burn-in incurs high cost. Thus a major problem is to decide how long the procedure should continue. The best time to stop the burn-in procedure for a given criterion (minimum failure rate, maximal mean residual life or minimal total cost, etc) is called

the optimal burn-in time. The problem of estimating the optimal burn-in time is discussed widely in the literature. For example, restricting our attention only on publications within the last five years (2007-2011), we mention the following studies. Jiang and Jardine [4] determine the optimal burn-in/preventive replacement policy to reduce the field operational cost of a product with mixed failure distribution. Wu and Clements-Croome [5] develop two burn-in policies for products having dormant states and sold under warranty. Cha and Mi [6] determine the optimal burn-in time for a periodically inspected system. Wu et al. [7] develop a cost model to determine the optimal burn-in time and warranty policy for non-repairable products sold under the FRW/PRW policy. Bebbington et al. [8] determine the optimal burn-in time to maximize mean residual life of a product with bathtub shaped failure rate. Cha et al. [9] determine an upper bound for optimal burn-in time of a warranted product with two types of minor and catastrophic failures. Kim and Kuo [10] determine the optimal burn-in time for repairable non-series systems to maximize reliability. Kwon et al. [11] determine the optimal burn-in time and replacement policy for a product with bathtub shaped failure rate. Cha and Finkelstein [12] consider ‘shocks’ (as a method of burn-in) and determine the optimal severity levels of the shocks to minimize the expected costs. Cha and Finkelstein [13] define two types of risks and investigate the optimal burn-in time, which minimizes the mean number of repairs during the field operation. Shafiee and Chukova [14] investigate the optimal burn-in time and corrective maintenance strategy to minimize the total expected warranty servicing cost. Cha and Finkelstein [15] develop a burn-in model for a population which is composed of two stochastically ordered subpopulations. Ulusoy et al. [16] determine the optimal burn-in time to maximize the joint utility function of reliability and cost. Shafiee et al. [17] investigate the optimal burn-in time and periodic imperfect preventive maintenance strategies for a warranted product. Also, as a good source of references for trade-off models between the cost of burn-in procedure and the reduction of the expected warranty servicing cost, readers can refer to Sheu and Chien [18].

In this paper, we develop an optimization model to investigate the optimal burn-in time and warranty length that minimizes the total mean servicing cost of a product with bathtub shaped failure rate. It is assumed that the cost of a minimal repair to the component which fails at age t is a

978-1-4577-1232-6/11/$26.00 ©2011 IEEE

continuous non-decreasing function of t. Also, a penalty cost will be incurred for the manufacturer when the item fails during post-warranty before the end of its useful life limit.

The rest of this paper is organized as follows. In Section II, the model components are described in detail. In section III, we formulate our cost model. The properties of the optimal solution are also given in this Section. In Section IV, we provide a numerical example to illustrate some of our results. In Section V, we conclude with some discussions of topics for future research.

II. THE COMPONENTS OF THE MODEL

Nomenclature

L useful life limit of the product Jt, Vt the change points of the bathtub curve

)]([)( tFtf

the probability [cumulative] density function of the failure time

F(t) the survival function r(t) the failure (hazard) rate function H(t) the cumulative failure rate function

JJ βα , the parameters of Weibull failure distribution during infant mortality

VV βα , the parameters of Weibull failure distribution during wear-out period

b burn-in time [decision variable] bT the random variable representing the failure time of

the product during the burn-in period (0≤bT ≤ b)

)(trb the failure (hazard) rate function after the burn-in )(bc the burn-in cost )(1 tc the cost of a minimal repair during the burn-in period

][ bME the expected minimal repair cost during the burn-in

)]([ bhE the total expected burn-in cost w warranty period [decision variable]

wT the random variable representing the failure time of the product during warranty period (0≤

wT ≤ w)

)(2 tc the cost of a minimal repair over the warranty period

)]([ bcE w the expected warranty servicing cost

U the residual life of a warranty accomplished burned-in product

uT the random variable representing the failure time of the product during post-warranty (0≤

uT ≤ u)

0pc the fixed penalty cost at 0=uT

)( up Tc the amount of penalty at time uT

[ ]),( wbcE p

the expected penalty cost during post-warranty

[ ]),( wbcE

the total mean servicing cost

A. Bathtub Shaped Failure Rate

Let F(t) be the cumulative distribution function of the product lifetime T. If T has density f(t) on [0,∞ ], then its failure rate function r(t) is defined by r(t)= f(t)/ F (t), where F (t) = 1− F(t) is the survival function of T.

It is widely believed that many mechanical and electronic components have a bathtub failure pattern which comprises three parts. The first part is a decreasing failure rate, known as early failures; the second part is a constant failure rate, known

as random failures; the third part is an increasing failure rate, known as wear-out failures.

Assume that the product has a bathtub-shaped failure rate, r(t) with two change points Jt and Vt ( ∞<≤≤ VJ tt0 ) such that

r(t)=

The time interval ],0[ Jt is called the infant mortality period; the interval ],[ VJ tt is called the normal operating period or useful period and the interval ),[ ∞Vt is called the wear-out period.

B. Expected Burn-In Cost

Consider a fixed burn-in time b and begin to burn-in a new item from time point 0. If the item fails at time bt before burn-

in time b (0≤ bt ≤ b), then it is repaired minimally and the burn-in procedure is continued with the repaired component. After the fixed burn-in time b, the burn-in stops and the product is released to the market. We assume that the cost of a minimal repair to the item which fails at time bt during the burn-in period, )(1 btc

is a continuous non-decreasing

function of bt . Since performing a minimal repair becomes more expensive as the component ages, this assumption is more realistic than considering the minimal repair cost as a constant value.

Let )]([ bhE be the total expected burn-in cost for a repairable product with burn-in time b. This cost includes the cost of burn-in procedure and the expected minimal repair cost during the burn-in period ],0[ b , i.e.,

][)()]([ bMEbcbhE += ,

(2)

where c(b) is the burn-in cost and ][ bME is the expected minimal repair cost during the burn-in period. We assume that the burn-in cost is proportional to the burn-in time with proportionality constant 00 >c , i.e.,

bcbc 0)( = .

Let ],0[ bN be the random variable denoting the number of

minimal repairs during the burn-in period. Conditional on

bb nN =],0[ , suppose that ibT (

bni ,...,2,1= ) is the random variable denoting the time of the i th minimal repair. Therefore, the total minimal repair cost is

∑=

=bn

i

ibb TcM

11 )( .

(3)

Since all the repairs are minimal, the failure process of the item during the burn-in period can be modelled by a non-homogeneous Poisson process (NHPP) with cumulative failure rate function ∫=

tdxxrtH

0)()( . Therefore, the total

number of minimal repairs during the burn-in period has the Poisson distribution with mean H(b), i.e.,

)( ],0[ bNE = H(b) . (4)

strictly decreasing for Jtt ≤≤0 constant for VJ ttt ≤≤ . (1)

strictly increasing for ttV ≤

It is well known that )( ibi THU = (

bni ,...,2,1= ) is uniformly distributed on the interval [0, H(b)]. Therefore, the expected minimal repair cost during the burn-in period,

][ bME , is

( )[ ] ( )],0[1

1 )(][ bib NEUHcEME ×= −

( ))(

)(

)()(

0

11

bHbH

dttHcbH

×= ∫−

∫ ×=b

dttrtc0 1 )()( . (5)

Specifically, when 11 )( ctc = , we get

)(][ 1 bHcME b ×= ,

which can be also immediately obtained from

∑=

=],0[

11

bN

ib cM using the Wald’s equation. By substituting Eq. (5)

into (2), we have

dttrtcbcbhEb

∫ ×+=0 10 )()()]([ .

(6)

C. Warranty Policy

Each product survives the burn-in procedure has the age b. After the burn-in procedure, the product is released to the consumers with warranty length w and is put into field operation immediately after purchase. In this paper, we consider a NFRW (non-renewing free repair warranty) policy. Under a NFRW policy, the manufacturer agrees to rectify, free of charge to the customer, any failures of the item only during the original warranty period w.

If the item fails at time wt (∈ [0,w] ) during the warranty period ( wt is a calendar time over the warranty period) then it is repaired instantly by a minimal repair with cost of )(2 wtc . We assume that )()( 12 tctc ≥ for all t≥ 0, so that the cost of a minimal repair in field operation over the warranty period is higher than that of a minimal repair during the burn-in period.

Let )]([ bcE w be the expected warranty servicing cost of the burned-in product with burn-in time b. Then, the expected warranty servicing cost is

wwb

w

ww dttrtcbcE )()()]([0 2 ×= ∫ ,

(7)

where (.)br is the failure (hazard) rate function of the product after the burn-in procedure. Since we assumed that failures during burn-in are corrected by minimal repair, we have

)()( wwb tbrtr += . Then,

dttrbtcbcEwb

bw )()()]([ 2 ×−= ∫+ .

(8)

D. Expected Penalty Cost During the Post-Warranty

Customer satisfaction with a purchased product depends not only on its performance over the warranty period, but also depends on its performance during the remainder of its useful life. The cost of repairing product failures or providing replacements during the post-warranty period is borne completely by the buyer. Having a failure during the post-

warranty period causes high dissatisfaction to the customers. Each dissatisfied customer can impact on future sales and this has serious implications for manufacturers. These include loss of potential new sales due to the negative word of mouth effect and repeat purchase sales disappearing when customers decide to switch to a competitor [19].

A brief review of the existing literature shows that the proposed cost models focus on minimizing total warranty servicing cost and do not look at customer dissatisfaction after the warranty period has expired. In other words, optimal burn-in time and warranty length need to be considered in the context of the life cycle of the product (for both the buyer and manufacturer). In this paper, we define dissatisfaction in terms of a penalty cost for the manufacturer.

Each burned-in product passing through warranty has the age b+w with u being its residual life during the post-warranty. Let L be the useful life limit (life expectancy) of the product. Therefore, the residual life of the warranty accomplished burned-in product is u = L − b − w. We assume that product failures over the post-warranty period are minimally repaired, with the buyer paying all repair costs.

When product is out of warranty and fails at time

ut (∈ [0,u] ) before it has reached its useful life limit, it incurs an additional penalty cost for the manufacturer. The amount of penalty is a function of the time point of failure ( ut ). Since failures just after the warranty expires cause higher dissatisfaction than those that occur later, )( up tc

is a

continuous decreasing function of ut . Let )],([ wbcE p be the

expected penalty cost for the manufacturer during the post-warranty period. Then,

uub

u

upp dttwrtcwbcE )()()],([0

+×= ∫. (9)

By replacing )()( uub twbrtwr ++=+ , we have

dttrwbtcwbcEL

wb pp )()()],([ ×−−= ∫ +.

(10)

III. THE COST MODEL

The total mean servicing cost of the product includes the expected burn-in cost, expected warranty servicing cost and the expected penalty cost during the post-warranty period. Let

)],([ wbcE be the total mean servicing cost per unit of product, subjected to a burn-in procedure with time b and sold under warranty of length w. Therefore,

)],([)]([)]([),([ wbcEbcEbhEwbcE pw ++= ,

(11)

By substituting Eqs. (6), (8) and (10) into (11), we have

∫∫+

×−+×+=wb

b

bdttrbtcdttrtcbcwbcE )()()()()],([ 20 10

dttrwbtc

L

wb p )()( ×−−+ ∫ + . (12)

Although the small values of b has lower burn-in cost, this strategy doesn’t reduce the warranty servicing cost significantly. Also, offering a short warranty period has lower warranty servicing cost but this strategy produces much higher expected penalty costs. The optimization problem is to select the optimal burn-in time ( ∗b ) and optimal warranty length ( ∗w ) to minimize the total mean servicing cost given by equation (12).

Theorem 1. Suppose the failure rate function r(t) is Bathtub shaped with the first change point Jt . Total mean

servicing cost )],([ wbcE is strictly increasing in b> Jt and the optimal burn-in time occurs no later than the first change point of bathtub failure rate, i.e,

Jtb ≤≤ ∗0 .

Proof: Omitted. ◘

Theorem 2. Suppose the failure rate function r(t), )(tc m and )(tc p are differentiable and r(t) is Bathtub shaped.

Then, the optimal burn-in time ∗b and the optimal warranty length ∗w are the solution of the following equation:

[ ] )()(2)()0()( 2210 wbrwcbrcbcc +×=×++

, (13)

[ ] )()()()( 02 LrwbLcwbrcwc pp ×−−=+×+

Proof: By taking the derivatives of )],([ wbcE with respect to b and w, we obtain,

[ ] )()0()()],([ 210 brcbccwbcEb

×++=∂∂

[ ] )()()()( 02 LrwbLcwbrcwc pp ×−−−+×−− , and

[ ] )()()()()],([ 02 LrwbLcwbrcwcwbcEw pp ×−−−+×+=∂∂.

By putting 0)],([ =∂∂ wbcEb

and 0)],([ =∂∂ wbcEw

, the

following conditions will be obtained. ◘

Theorem 3: It is beneficial for a manufacturer to carry out a burn-in procedure if the total mean servicing cost with burn-in is less than the total mean servicing cost without burn-in (b= 0), i.e.,

)],0([),([ ∗∗∗ < wcEwbcE , (14)

where

∫ ∫∗

∗×−+×= ∗∗ w L

w p dttrwtcdttrtcwcE0 2 )()()()()],0([ .

Proof: Omitted. ◘

IV. A NUMERICAL EXAMPLE

Suppose that the failure rate of a component has a bathtub shaped similar to Jiang and Murthy [20] as follows:

⎪⎩

⎪⎨

⎧

≤−+≤≤≤≤−+

=−

−

ttfortttttforttfortt

tr

VV

VJ

JJ

10

0

1110

2

1

)(

0)()(

β

β

αβλλ

βαλ, (15)

with 0,, ≥VJ ααλ ; 1, >VJ ββ and 0>> JV tt . Fig. 1 is an example of this failure rate for 10 =λ ; 008.01 =α ;

01.02 =α ; 321 == ββ ; 101 =t ; 302 =t and 40=L .

Figure 1. Example of bathtub shaped failure rate

We take the minimal repair cost as an exponentially increasing function of t as { })(exp)( taHctc ii = for 2,1=i with 21 cc ≤ and 0≥a . By modifying Chen and Chien [21], we assume that the penalty cost is a linearly decreasing function of failure time during the post-warranty, and it is modelled by

⎪⎩

⎪⎨⎧

−−>

−−≤≤⎟⎠⎞

⎜⎝⎛

−−−=

wbLxif

wbLxifwbL

xcxc pp

0

01)( 0 ,

(16)

Let 5.0=a ; 100 =c ; 11 =c ; 22 =c and 100 =pc .

Fig. 2 shows that the strategy with burn-in time 2.1≈∗b and warranty length 22≈∗w (month) results in minimum total mean servicing cost for the manufacturer.

Figure 2. Total mean servicing cost for a range of b and w.

We now study the effect of the model parameters on the optimal burn-in time and warranty length.

1c and 2c varying

The optimal burn-in time ( ∗b ) and warranty length ( ∗w ) for various choices of 1c and 2c are given in Table I.

TABLE I. 1c AND 2c VARYING 2c

1.0 2.0 3.0

1c ∗b ∗w ∗b ∗w ∗b ∗w0.50 1.2 22.0 1.5 20.6 2.1 19.70.75 1.1 22.3 1.3 21.2 1.8 20.51.00 1.0 22.5 1.2 22.0 1.5 21.71.25 0.8 23.7 1.1 22.9 1.3 21.91.50 0.5 24.8 0.9 23.9 1.2 22.2

From Table 1, we observe two trends: (1) the optimal burn-in time decreases and the optimal warranty length increase as 1c increases, (2) the optimal burn-in time increases

and the optimal warranty length decreases as 2c increases.

L and 0pc varying

The optimal burn-in time ( ∗b ) and warranty length ( ∗w ) for various choices of L and

0pc are given in Table 2.

TABLE II. l AND 0pc VARYING

Table II shows that an increase in useful life limit of the product increases both the optimal burn-in time and warranty length. When a large penalty cost is likeable to be incurred during the post-warranty, the optimal result for the manufacturer is to choose high burn-in time and long warranty length.

V. CONCLUSION AND EXTENSIONS

In the present paper, a cost function is developed to determine the optimal burn-in time and warranty length for the repairable products under the non-renewing free repair warranty policy.

There is a wide scope for future research in the area of burn-in/maintenance for warranted products. Some of the possible extensions are:

(a) Introduce general repair strategies (additional to minimal repair strategy) during the burn-in time;

(b) Optimising burn-in procedure, warranty policy and servicing strategies to improve customer satisfaction;

(c) We have confined our analysis to the failure free policy. The analysis of other types of warranty policies for example, pro-rata, combination is yet to be carried out.

ACKNOWLEDGMENTS This research is partially supported by the Natural Sciences

and Engineering Research Council of Canada (NSERC).

REFERENCES [1] D. N. P. Murthy and W. R. Blischke, Warranty Management and

Product Manufacture. London: Springer, 2006. [2] W. Y. Yun, D. N. P. Murthy, and N. Jack, “Warranty servicing with

imperfect repair,” International Journal of Production Economics, vol. 111, pp. 159-169, 2008.

[3] W. Y. Yun, Y. W. Lee, and L. Ferreira, “Optimal burn-in time under cumulative free replacement warranty,” Reliability Engineering and System Safety, vol. 78, pp. 93-100, 2002.

[4] R. Jiang and A. K. S. Jardine, “An optimal burn-in preventive maintenance model associated with a mixture distribution,” Quality and Reliability Engineering International, vol. 23, pp. 83-93, 2007.

[5] S. Wu and D. Clements-Croome, “Burn-in policies for products having dormant states,” Reliability Engineering and System Safety, vol. 92, pp. 278–285, 2007.

[6] J. H. Cha and J. Mi, “Optimal burn-in procedure for periodically inspected systems,” Naval Research Logistics, vol. 54, pp. 720-731, 2007.

[7] C. C. Wu, C. Y. Chou, and C. Huang, “Optimal burn-in time and warranty length under fully renewing combination free replacement and pro-rata warranty,” Reliability Engineering and System Safety, vol. 92, pp. 914-920, 2007.

[8] M. Bebbington, C. D. Lai, and R. Zitikis, “Optimum burn-in time for a bathtub shaped failure distribution,” Methodology and Computing in Applied Probability, vol. 9, pp. 1-20, 2007.

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[10] K. O. Kim and W. Kuo, “Optimal burn-in for maximizing reliability of repairable non-series systems,” European Journal of Operational Research, vol. 193, pp. 140-151, 2009.

[11] Y. M. Kwon, R. Wilson, and M. H. Na, “Optimal burn-in with random minimal repair cost,” Journal of Korean Statistical Society, vol. 39, pp. 245–249, 2010.

[12] J. H. Cha and M. S. Finkelstein, “Burn-in by environmental shocks for two ordered subpopulations,” European Journal of Operational Research, vol. 206, pp. 111-117, 2010.

[13] Cha, J-H., Finkelstein, M.S. Stochastically ordered subpopulations and optimal burn-in procedure. IEEE Transactions on Reliability 59, 635–643 (2010).

[14] Shafiee, M. and Chukova, S. (2010) A burn-in/maintenance strategy for minimizing expected warranty cost. In: Proceedings of the 4th Asia-Pacific International Symposium on Advanced Reliability and Maintenance Modeling, 2-4 December, Wellington, New Zealand, 608-615.

[15] J. H. Cha and M. S. Finkelstein, “Burn-in and the performance quality measures in heterogeneous populations,” European Journal of Operational Research, vol. 210, pp. 273–280, 2011.

[16] S. K. Ulusoy, T. A. Mazzuchi, and D. Perlstein, “Bayesian calculation of optimal burn-in time using the joint criteria of cost and delivered reliability,” Quality and Reliability Engineering International, DOI: 10.1002/qre.1167, 2011.

[17] M. Shafiee, M. S. Finkelstein, and S. Chukova, “Burn-in and imperfect preventive maintenance strategies for warranted products,” Proceedings of the IMechE, Part O, Journal of Risk and Reliability, vol. 225, DOI: 10.1177/1748006X11398584, 2011.

[18] S. H. Sheu and Y. H. Chien, “Optimal burn-in time to minimize the cost for general repairable products sold under warranty,” European Journal of Operational Research, vol. 163, pp. 445-461, 2005.

[19] D. N. P. Murthy and N. Jack, “Warranty servicing strategies to improve customer satisfaction,” IMA Journal of Management Mathematics, vol. 15, pp. 111-124, 2004.

[20] R. Jiang and D. N. P. Murthy, “Two sectional models involving three Weibull distributions,” Quality and Reliability Engineering International, vol. 13, pp. 83-96, 1997.

[21] J. A. Chen and Y. H. Chien, “Warranty and preventive maintenance for non-repairable products with failure penalty post-warranty,” Quality and Reliability Engineering International, vol. 23, pp. 107-121, 2007.

0pc 8 10 12

L ∗b ∗w ∗b ∗w ∗b ∗w32 0.7 20.6 1.0 20.2 1.1 21.836 0.9 21.2 1.1 21.6 1.2 22.040 1.1 21.7 1.2 22.0 1.3 22.344 1.2 21.9 1.3 22.5 1.5 22.948 1.3 22.1 1.5 23.1 1.8 23.7