Optimal burn-in time under cumulative free replacement warranty

Optimal burn-in time under cumulative free replacement warranty

Won Young Yuna,*, Yang Woo Leea, Luis Ferreirab

aDepartment of Industrial Engineering, Pusan National University, Pusan 609-735, South KoreabSchool of Civil Engineering, Queensland University of Technology, Brisbane, Qld, Australia

Received 25 July 2000; accepted 6 May 2002

Abstract

In this paper, optimal burn-in time to minimize the total mean cost, which is the sum of manufacturing cost with burn-in and cumulative

warranty-related cost, is studied. When the products with cumulative warranty have high failure rate in the early period (infant mortality

period), burn-in procedure is considered to eliminate the early product failures. After burn-in, the posterior product life distribution and the

cumulative warranty-related cost are dependent on burn-in time; long burn-in period decreases the warranty-related cost, but it increases the

manufacturing cost. The paper provides a methodology to obtain the total mean cost under burn-in and cumulative warranty. Properties of

the optimal burn-in time are analyzed here. Numerical examples and sensitivity analysis are used to demonstrate the applicability of the

methodology derived in the paper. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Burn-in; Cumulative warranty; Bathtub failure rate; Mean total cost

1. Introduction

Most durable products are sold today with some type of

warranty to protect consumers from unexpected early

product failures. The manufacturer’s warranty-related

costs are becoming a significant portion of production

cost. High product reliability in the early period of the

product life cycle is crucial to reduce warranty-related costs.

In the product life cycle, there are generally three phases

in terms of failure rate. Three phases can be represented by

the bathtub pattern as shown in Fig. 1.

Definition. Let us assume that the product has a bathtub

failure rate, rðtÞ if there exist 0 # t1 # t2 # 1 such that

rðtÞ is

strictly decreasing; for 0 # t # t1;

constant; for t1 # t # t2;

strictly increasing; for t2 # t;

8>><>>:

where t1 and t2 are called the change points of rðtÞ: The time

interval ½0; t1� is called the infant mortality period; the

interval ½t1; t2� where rðtÞ is flat and attains its minimum

value is called the normal operating period or useful period;

the interval ½t2;1� is called the wear-out period.

To reduce damage from early failures, a burn-in

procedure is carried out by operating the products under

electrical or thermal conditions that approximate the

working conditions in field operation before shipment to

customers (e.g. IC chips for large systems, satellites, and

space ships).

Early failures result in a high warranty cost which is

dependant on burn-in period and warranty type. We

consider a special warranty, cumulative free replacement

warranty (cumulative FRW) which is distinguished from

standard warranty policies. Rather than covering each item

separately for a period T, the cumulative warranty covers the

group of items for a total service time of nT.

This paper is related to economic optimization problem

of how long burn-in procedure should be to minimize the

total mean cost. The latter is the sum of the manufacturing

cost with burn-in and the warranty cost under cumulative

warranties. The studies for burn-in testing began with the

advent of transistors in the early 1950s (Kececioglu and Sun

[5], Kuo and Kuo [6], Leemis and Beneke [7], Block and

Savits [2]). Nguyen and Murthy [11] first proposed a model

to determine the optimal burn-in time for products sold with

warranty. They considered two types of warranty policy

(failure-free and rebate policies) and derived the total cost as

the sum of the manufacturing cost and the warranty cost for

repairable and nonrepairable products. Mi [8] considered

the same situation as Nguyen and Murthy [11] by assuming

0951-8320/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 95 1 -8 32 0 (0 2) 00 0 49 -2

Reliability Engineering and System Safety 78 (2002) 93–100

www.elsevier.com/locate/ress

* Corresponding author. Tel.: þ82-51-510-2421; fax: þ82-51-512-7603.

E-mail address: [email protected] (W.Y. Yun).

http://www.elsevier.com/locate/ress

that burn-in procedure continued until the first products

surviving the burn-in period, although they used a different

cost structure. Mi [8] also proved some important proper-

ties, namely that the optimal burn-in time occurs no later

than the first change point of bathtub failure rate. Mi [9]

compared policies with renewable warranties and burn-in.

Monga and Zio [10] studied the reliability-based design of a

series–parallel system considering burn-in, warranty and

maintenance. They assumed that the system life cycle cost

included costs of burn-in, warranty, installation, preventive

maintenance, and minimal repair. They obtained optimal

values of system design, burn-in time, preventive mainten-

ance intervals, and replacement time. Kar and Nachlas [4]

studied warranty and burn-in strategies together in order to

examine the possible benefits of coordinated strategies for

product performance management.

In this paper, a cumulative FRW is considered as follows.

Cumulative FRW [1]. A lot of n items are warranted for a

total (aggregate) period of nT. The n items in the lot are used

one at a time. If the total lifetimes of the whole batch,

Sn , nT, free replacement items are supplied, one at a time,

until the first instant when the total lifetimes of all failed

items plus the service time of the item then in use is at least

nT.

This type of policy is applicable to components of

industrial and commercial equipment bought in lots as

spares and used one at a time as items fail. Example of

possible applications is mechanical components such as

bearings and drill bits. The policy would also be appropriate

for military or commercial airline equipment such as

mechanical or electronic modules in airborne units.

Burn-in procedure [8]. We consider a nonrepairable

product and for given burn-in period b, all the new product

is tested under environment situation similar with field

condition failed product during burn-in period is replaced

with a new one. Burn-in test is continued until the first

product surviving the burn-in period b is obtained.

In addition, we assume that the products have a bathtub

failure pattern. Given cumulative FRW and bathtub failure

rate, we consider optimal burn-in problem.

2. Total mean cost model

In this section, we derive cost parts under burn-in and

cumulative FRW and property of optimal burn-in period is

proved.

Theorem 1. If products are sold with cumulative FRW

without burn-in, the total mean warranty cost is given by

wCFðT ; nÞ ¼ ðc0 þ c4ÞMCFðnTÞ ð1Þ

where

MCFðnTÞ ¼X1j¼0

FðnþjÞðnTÞ

and FðnÞðxÞ is n-fold convolution of FðxÞ with itself.

Proof. See Ref. [1] (p. 248–9).

Nomenclature

FðtÞ failure time cumulative distribution function

f ðtÞ failure time probability density function�FðtÞ survival function ¼ 1 2 FðtÞ

rðtÞ hazard rate function ¼ f ðtÞ= �FðtÞ

b burn-in time

FbðtÞ distribution function of product survived

burn-in period b ¼ ðFðb þ tÞ2 FðbÞÞ= �FðbÞ

T warranty period on individual item

MðtÞ renewal function with interarrival time distri-

bution FðtÞ

MbðtÞ renewal function with interarrival time distri-

bution FbðtÞ

MCFðtÞ expected number of replacement

MCF;bðtÞ expected number of replacement with burn-in

period b

c0 manufacturing cost per item without burn-in

c1 fixed set-up cost of burn-in per item

c2 cost of burn-in per item per unit time

c3 shop replacement cost per failure during burn-

in period, includes the manufacturing cost and

set-up cost

c4 extra replacement cost per failure during the

warranty period

hðbÞ random manufacturing cost incurred until the

first item survives burn-in period

vðbÞ expected manufacturing cost ¼ E½hðbÞ�

kCFðn; bÞwarranty cost for a lot size n, and burn-in

period b

wCFðn; bÞ mean warranty cost ¼ E½kCFðn; bÞ�

CCFðT ; nÞ total mean cost with warranty period T, lot

size n, without burn-in

CCFðT ; n; bÞ total mean cost with warranty period T,

lot size n, and burn-in period b

Fig. 1. Bathtub failure pattern.

W.Y. Yun et al. / Reliability Engineering and System Safety 78 (2002) 93–10094

2.1. Mean burn-in cost

Under given burn-in, let Z be the total burn-in time until

first product survives the burn-in time b, and X1;…;Xn; be

independently and identically distributed lifetimes of the

products with common c.d.f. F. Let also h 2 1 be the

random variable that is the number of shop replacements

until the first product surviving burn-in time is obtained.

Denoting Xbi as XilXi # b; the manufacturing cost

incurred until the first product survives the burn-in time b

is given by [8]

hðbÞ ¼ c0 þ c1 þ c2Z þ c3ðh2 1Þ; ð2Þ

where

Z ¼Xh21

i¼1

Xbi þ b:

E½h2 1� ¼½1 2 �FðbÞ�

�FðbÞ¼

FðbÞ

�FðbÞ: ð3Þ

Product lifetime Xbi has the following distribution:

Pr Xbi , x

n o¼

1 x . b

FðxÞ

FðbÞ0 # x # b

0 otherwise

8>>><>>>:

for i ¼ 1; 2;… ð4Þ

Thus

E Xbi

h i¼

ðb

01 2 Pr Xb

i , xn o� �

dx

¼1

FðbÞ

�bFðbÞ2

ðb

0FðxÞ dx

� �

for i ¼ 1; 2;…

ð5Þ

since h 2 1 is a stopping time with respect to i.i.d. random

variables {Xbi }

E½Z� ¼FðbÞ

�FðbÞ

�1

FðbÞ

�bFðbÞ2

ðb

0FðxÞ dx

� �þ b

¼

ðb

0½1 2 FðxÞ� dx

�FðbÞ: ð6Þ

Therefore, the mean time burn-in cost, vðbÞ; is

vðbÞ ¼ E½hðbÞ� ¼ c0 þ c1 þ c2

ðb

0

�FðtÞ dt

�FðbÞþ c3

FðbÞ

�FðbÞ:ð7Þ

These results (Eqs. (2), (3) and (7)) are identical with those

of Mi [8].

Additionally, by the result of Mi [8], the lifetime of the

product which has survived burn-in time b is Xh 2 b, and is

distributed according to

Pr{Xh 2 b . s} ¼�Fðb þ sÞ

�FðbÞ; ;s $ 0: ð8Þ

2.2. Mean cumulative warranty cost

For a product survived burn-in time b, its distribution

function is

FbðtÞ ¼Fðb þ tÞ2 FðbÞ

�FðbÞ; t $ 0: ð9Þ

Using FbðtÞ and Theorem 1, we can obtain expected

warranty cost with burn-in. Let MbðtÞ be the renewal

function with interarrival time distribution FbðtÞ; and

MCF;bðtÞ be the expected number of replacement under

Cumulative FRW, then

MbðxÞ ¼X1j¼1

FðjÞb ðxÞ; ð10Þ

and

MCF;bðnTÞ ¼ MbðnTÞ2Xn21

j¼1

FðjÞb ðnTÞ: ð11Þ

Let NðT ; n; bÞ be the number of replacement under

cumulative FRW with burn-in procedure and the warranty

cost is given by

kCF;bðT ; nÞ ¼ ðhðbÞ þ c4ÞNðT ; n; bÞ: ð12Þ

The expected warranty cost is given by

wCF;bðT ; nÞ ¼ E½kCF;bðT ; nÞ� ¼ ðvðbÞ þ c4ÞMCF;bðnTÞ: ð13Þ

2.3. Total mean cost

Under burn-in period, b and cumulative FRW, nT, the

total mean cost is given by

CCFðT ; n; bÞ ¼ n £ vðbÞ þ wCF;bðT ; nÞ

¼ n £ vðbÞ þ ðvðbÞ þ c4ÞMCF;bðnTÞ; ð14Þ

and the per-item total mean cost is given by

CUCFðT ; n; bÞ ¼ vðbÞ þ wCF;bðT ; nÞ=n

¼ vðbÞ þ ðvðbÞ þ c4ÞMCF;bðnTÞ=n: ð15Þ

For special case, if burn-in is not executed, the total mean

cost is given by

CCFðT ; nÞ ¼ n £ c0 þ wCFðT ; nÞ

¼ n £ c0 þ ðc0 þ c4ÞMCFðnTÞ; ð16Þ

For a cumulative FRW policy with burn-in period, optimal

W.Y. Yun et al. / Reliability Engineering and System Safety 78 (2002) 93–100 95

burn-in time b p to minimize the total mean cost function,

Eq. (14) occurs no later than first change point t1 in Fig. 1.

Proof. For cumulative FRW, assume that burn-in procedure

is implemented at certain time b0 . t1. Then, from Eq. (14),

the total mean cost is given by

CCFðT ; n; b0Þ ¼ n £ vðb0Þ þ ðvðb0Þ þ c4ÞMCF;b0ðnTÞ;

and if the burn-in time b ¼ t1; the total mean cost is given by

CCFðT ; n; t1Þ ¼ n £ vðt1Þ þ ðvðt1Þ þ c4ÞMCF;t1ðnTÞ:

The failure rates of Fb0ð·Þ and Ft1

ð·Þ are given by

rb0ðtÞ ¼

f ðb0 þ tÞ= �Fðb0Þ

�Fðb0 þ tÞ= �Fðb0Þ¼

f ðb0 þ tÞ

�Fðb0 þ tÞ¼ rðb0 þ tÞ; and rt1

ðtÞ

¼ rðt1 þ tÞ:

Note that F has the bathtub shaped failure rate rðtÞ:

Therefore

�Fb0ðtÞ ¼ exp 2

ðt

0rb0

ðxÞ dx

�¼ exp 2

ðt

0rðb0 þ xÞ dx

�;

and

�Ft1ðtÞ ¼ exp 2

ðt

0rt1ðxÞ dx

�¼ exp 2

ðt

0rðt1 þ xÞ dx

�:

Obviously, �Fb0ðtÞ # �Ft1

ðtÞ; and Fb0ðtÞ $ Ft1

ðtÞ; since rðb0 þ

xÞ $ rðt1 þ xÞ; for b0 ^ t1; for all t $ 0;and n-fold con-

volution functions FðnÞb0; FðnÞ

t1are the same relation as Fb0

; and

Ft1: That is, FðnÞ

b0ðtÞ $ FðnÞ

t1ðtÞ; for n $ 1.

Thus, by using above relation, for all b0 . t1

MCF;b0ðnTÞ ¼

X1j¼n

FðjÞb0ðnTÞ $ MCF;t1

ðnTÞ ¼X1j¼n

FðjÞt1ðnTÞ;

since for all j, FðjÞb0

$ FðjÞt1:

Also, it is easily seen that vðb0Þ $ vðt1Þ; for all b0 . t1.

Therefore, by using the two results, for MCF;bðnTÞ; vðbÞ;

CCFðT ; n; b0Þ $ CCFðT ; n; t1Þ; ;b0 $ t1:

A

3. Numerical examples

In this section, we consider some distribution functions

and optimal burn-in times for each one.

3.1. Mixture of two exponential distributions

First we consider a hyper-exponential distribution of

which the distribution is

FðtÞ ¼ p½1 2 expð2ðl1tÞÞ� þ ð1 2 pÞ½1 2 expð2ðl2tÞÞ�

with 0 # p # 1: ð17Þ

Fig. 2 shows the optimal burn-in time and the related cost

for warranty period T, lot size n ¼ 1;…; 4 when c0 ¼ 126;

c1 ¼ 2; c2 ¼ 0:1; c3 ¼ 130; and c4 ¼ 80: As the lot size n

increases, the optimal burn-in time b p decreases. In this

Fig. 2. Optimal burn-in time and per-item total menu cost for mixture of two exponential distributions with p ¼ 0:2; l1 ¼ 44; l2 ¼ 0:1 when c0 ¼ 126; c1 ¼ 2;

c2 ¼ 0:1; c3 ¼ 130; and c4 ¼ 80:


case, when the lot size is 4, no burn-in is the best economical

policy.

3.2. Mixed-Weibull distribution

Consider the following mixed-Weibull distribution:

FðtÞ ¼ p 1 2 exp 2ðt=a1Þb1

� �j kþ ð1 2 pÞ

� 1 2 exp 2ðt=a2Þb2

� �h k

with 0 # p # 1:

ð18Þ


for warranty period T, lot size n ¼ 1;…; 4; and cumulative

FRW, when c0 ¼ 100; c1 ¼ 2; c2 ¼ 0:1; c3 ¼ 102; and c4 ¼

40:

3.3. Weibull-exponential distribution

Chou and Tang [3] introduced the following distribution

to model the failure pattern of product in the burn-in studies.

The failure rate function is given by

rðtÞ ¼bð1=aÞbtb21 0 # t # t1

bð1=aÞbtb211 t1 # t;

8<: ð19Þ

where 0 , b , 1 is the shape parameter, a is the scale

parameter, and t1 is the change-point. Because 0 , b ,

1; the failure rate function is strictly decreasing in the

interval ½0; t1� and stays at a constant level bð1=aÞbtb211 ;

for t1 # t: Fig. 4 shows the optimal burn-in time and

the related cost for warranty period T, lot size n ¼

1;…; 4; and cumulative FRW, when c0 ¼ 126; c1 ¼ 2;

c2 ¼ 0:01; c3 ¼ 127; and c4 ¼ 95: As the lot size n

increases, the optimal burn-in time b p decreases; however,

the difference between no burn-in and optimal burn-in

policy is small in each case.

3.4. Weibull-exponential–Weibull distribution

We use the following distribution to model the bathtub

failure pattern. The failure rate function is given by

rðtÞ ¼

b1ð1=a1Þb1 tb121 0 # t # t1

b1ð1=a1Þb1 t

b1211 t1 # t , t2

b1ð1=a1Þb1 t

b1211 þ b2ð1=a2Þ

b2 ðt 2 t2Þb221 t2 # t

8>><>>:

ð20Þ

where 0 , b1 , 1 and b2 . 1 are the shape parameters, a

is the scale parameter, and t1 and t2 are the change-points.


for warranty period T, lot size n ¼ 1;…; 4; and cumulative

FRW, when c0 ¼ 126; c1 ¼ 2; c2 ¼ 0:01; c3 ¼ 127; and

c4 ¼ 95: As the lot size n increases, the optimal burn-in time

b p decreases.

The optimal burn-in time clearly depends on the model

parameter ci, and the underlying distribution function F.

Thus, we consider a Weibull-exponential–Weibull distri-

bution with a1 ¼ a2 ¼ 12; b1 ¼ 0:55; b2 ¼ 1=0:55; t1 ¼ 2;

t2 ¼ 8 and use a numerical study to investigate how the lot

size, n and the individual warranty period T affect the per-

item total mean cost and the optimal burn-in time. The

effects of the model parameter ci on the optimal burn-in time

are also studied. Table 1 shows the effects of the lot size n

Fig. 3. Optimal burn-in time and per-item total mean cost for mixed-Weibull case with p ¼ 0:1; a1 ¼ 1:2; b1 ¼ 8; a2 ¼ 100; b2 ¼ 30 when c0 ¼ 100; c1 ¼ 2;

c2 ¼ 0:1; c3 ¼ 102; and c4 ¼ 40:


and the individual warranty period T on the per-item total

mean cost and the burn-in time, respectively, for cumulative

FRW.

It can be seen from Table 1 that as the lot size n increases,

the optimal burn-in time b p decreases; this means that if we

sell a large size of products with cumulative FRW, the short

burn-in is economical, and when the individual warranty

period T increases, the optimal burn-in time b p increases;

this result is identical with the known existing results. The

change of the lot size n is more sensible to the change of the

optimal burn-in time b p than that of the individual warranty

period T. The effect of the cost parameter ci is shown in

Fig. 4. Optimal burn-in time and per-item total mean cost for Weibull-exponential case with a ¼ 12; b ¼ 0:55; and t1 ¼ 2 when c0 ¼ 126; c1 ¼ 1; c2 ¼ 0:01;

c3 ¼ 127; and c4 ¼ 95:

Fig. 5. Optimal burn-in time and per-item total mean cost for Weibull-exponential Weibull case with a1 ¼ a2 ¼ 12; b1 ¼ 0:55; b2 ¼ 1=0:55; t1 ¼ 2 and

t2 ¼ 8:


Table 2

Optimal burn-in time for Weibull-exponential–Weibull with a1 ¼ a2 ¼ 12; b1 ¼ 0:55; b2 ¼ 1=0:55; t1 ¼ 2; and t2 ¼ 8

T ¼ 8; n ¼ 2;

c0 ¼ 100

c3/c0

c4/c0, 1.00 c4/c0, 1.02

0.5 0.8 1.0 2.0 0.5 0.8 1.0 2.0

c1/c0 0 c2/c0 0 0.0632 0.1415 0.1964 0.4423 0.0562 0.1319 0.1856 0.4297

0.0001 0.0631 0.1413 0.1962 0.4420 0.0562 0.1317 0.1855 0.4294

0.001 0.0624 0.1401 0.1945 0.4393 0.0556 0.1305 0.1839 0.4268

0.01 0.0564 0.1282 0.1793 0.4138 0.0503 0.1195 0.1695 0.4021

0.01 c2/c0 0 0.0656 0.1442 0.1991 0.4444 0.0585 0.1345 0.1883 0.4318

0.0001 0.0655 0.1441 0.1989 0.4441 0.0585 0.1344 0.1881 0.4315

0.001 0.0648 0.1428 0.1973 0.4414 0.0578 0.1332 0.1866 0.4289

0.01 0.0586 0.1307 0.1818 0.4159 0.0523 0.1220 0.1720 0.4042

0.02 c2/c0 0 0.0680 0.1470 0.2018 0.4465 0.0608 0.1372 0.1910 0.4339

0.0001 0.0679 0.1468 0.2016 0.4462 0.0608 0.1370 0.1908 0.4336

0.001 0.0672 0.1455 0.2000 0.4435 0.0601 0.1358 0.1893 0.4310

0.01 0.0608 0.1333 0.1844 0.4179 0.0544 0.1245 0.1746 0.4062

0.05 c2/c0 0 0.0754 0.1552 0.2100 0.4528 0.0679 0.1452 0.1991 0.4402

0.0001 0.0753 0.1550 0.2098 0.4525 0.0678 0.1451 0.1989 0.4399

0.001 0.0745 0.1537 0.2081 0.4498 0.0671 0.1438 0.1972 0.4373

0.01 0.0676 0.1409 0.1920 0.4240 0.0609 0.1319 0.1821 0.4123

c3/c0

1.05, c4/c0 1.10, c4/c0

0.5 0.8 1.0 2.0 0.5 0.8 1.0 2.0

c1/c2 0 c2/c0 0 0.0467 0.1182 0.1702 0.4113 0.0331 0.0975 0.1465 0.3819

0.0001 0.0467 0.1181 0.1701 0.4110 0.0331 0.0974 0.1463 0.3816

0.001 0.0462 0.1170 0.1687 0.4085 0.0328 0.0965 0.1451 0.3793

0.01 0.0418 0.1072 0.1556 0.3850 0.0297 0.0886 0.1340 0.3577

0.01 c2/c0 0 0.0488 0.1208 0.1728 0.4134 0.0349 0.0999 0.1490 0.3840

0.0001 0.0488 0.1207 0.1727 0.4131 0.0349 0.0998 0.1488 0.3837

0.001 0.0483 0.1196 0.1713 0.4106 0.0346 0.0989 0.1476 0.3814

0.01 0.0437 0.1096 0.1580 0.3871 0.0313 0.0908 0.1363 0.3597

0.02 c2/c0 0 0.0510 0.1233 0.1755 0.4155 0.0368 0.1023 0.1514 0.3861

0.0001 0.0509 0.1232 0.1753 0.4152 0.0367 0.1022 0.1513 0.3858

0.001 0.0504 0.1221 0.1739 0.4127 0.0364 0.1013 0.1501 0.3835

0.01 0.0457 0.1120 0.1604 0.3891 0.0330 0.0930 0.1386 0.3617

0.05 c2/c0 0 0.0576 0.1311 0.1833 0.4218 0.0425 0.1095 0.1589 0.3923

0.0001 0.0575 0.1310 0.1832 0.4215 0.0425 0.1094 0.1588 0.3920

0.001 0.0569 0.1298 0.1817 0.4190 0.0421 0.1085 0.1575 0.3897

0.01 0.0516 0.1192 0.1678 0.3952 0.0382 0.0996 0.1456 0.3678

Table 1

Optimal burn-in-time, and per-item total mean cost for Weibull-exponential–Weibull case with a1 ¼ a2 ¼ 12; b1 ¼ 0:55; b2 ¼ 1=0:55; t1 ¼ 2 and t2 ¼ 8

when c0 ¼ 100; c1 ¼ 1; c2 ¼ 0:01; c3 ¼ 102; and c4 ¼ 70

n ¼ 1 n ¼ 2 n ¼ 3 n ¼ 4

b p CUFR b p CUFR b p CUFR b p CUFR

T 1 0.1013 144.7905 0.0000 109.4184 0.0000 103.2515 0.0000 101.7564

2 0.1636 167.1283 0.0000 119.1756 0.0000 107.8769 0.0000 104.0556

4 0.1715 207.7131 0.0124 144.0542 0.0000 124.7433 0.0000 116.0949

6 0.1742 248.2421 0.0650 175.4177 0.0303 152.6116 0.0131 140.8109

8 0.1727 288.8050 0.1080 214.3014 0.0887 189.0428 0.0772 175.7549

10 0.1507 332.7030 0.1276 258.2180 0.1188 231.9296 0.1153 218.3473

12 0.1408 380.1895 0.1330 304.9668 0.1286 278.5800 0.1271 265.1836


Table 2. It can be seen that when the fixed set-up cost c1

increases, the optimal burn-in time b p increases. That is, as

the burn-in set-up cost c1 increases, there is a corresponding

increase in the burn-in cost. Thus the optimal burn-in time

b p is increased until warranty cost reduction is compensated

for. When warranty cost reduction does not cover the burn-

in cost increment, the optimal burn-in time b p decreases.

When the cost of burn-in per item per unit time c2 increases,

the optimal burn-in time b p decreases. When the shop

replacement cost c3 increases, the optimal burn-in time b p

decreases. When the extra replacement cost during the

warranty period c4 increases, the optimal burn-in time b p

increases.

4. Conclusions

In this paper, the cost model, and the determination of

optimal burn-in time to minimize a total mean cost for

product sold under cumulative warranty are studied. When

the products have failure rate with the infant mortality

period, that is, the probability that the product fails in the

early period is greater than that in the useful period, burn-in

procedure must be taken into consideration to reduce

manufacturing cost.

To determine optimal burn-in time, however, we

consider burn-in and warranty costs under cumulative free

replacement warranty. Then for distributions with infant

mortality period, the total mean cost is calculated, and

optimal burn-in time is obtained. Then sensitivity analysis

on cost parameters was analyzed. The main conclusions

from the analysis based on numerical studies with specific

distributions and restricted ranges of cost parameters are as

follows.

For large lot size n, short burn-in time is better and for

very short or very long period, T, short burn-in is better in

respect of total mean cost. For cost parameters, if c1 or c4 are

large, we run burn-in test longerly and if c2 or c3 are large,

we finish burn-in shortly.

Our results will be a good reference to manufacturers for

military products or components when they estimate the

warranty costs and try to find burn-in procedure because

cumulative FRW is usually used in military contracts. For

further research, other warranty policies can be considered

and some cost estimation problems for buyers are also

promising areas.

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Optimal burn-in time under cumulative free replacement warranty