A Bayesian life test sampling plan for products with Weibull lifetime distribution sold under warranty

Reliability Engineering and System Safety 53 (1996) 61-66 ELSEVIER

PII:

S0951-8320[96)00024-5

© 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/96/$15.00

A Bayesian life test sampling plan for products with Weibull lifetime distribution sold under warranty Y. I. K w o n Dept. of Industrial Engineering, Chongju University, Chongju, Chungbuk 360-764, Republic of Korea (Received 5 May 1995; accepted 7 February 1996)

A Bayesian life test sampling plan is considered for products with Weibull lifetime distribution which are sold under a warranty policy. It is assumed that the shape parameter of the distribution is a known constant, but the scale parameter is a random variable varying from lot to lot according to a known prior distribution. A cost model is constructed which involves three cost components; test cost, accept cost, and reject cost. A method of finding optimal sampling plans which minimize the expected average cost per lot is presented and sensitivity analyses for the parameters of the lifetime and prior distributions are performed. © 1996 Elsevier Science Limited.

1 NOMENCLATURE N n r tr w~ w2

Cs c, cr

c,

r-th order statistic and Epstein 2 proposed a test with hybrid censoring which combines both Type I and Type II censoring. A two-stage test with Type II censoring at each stage was p r o p o s e d by Bulgren & H e w e t t e 3 and one with hybrid censoring by Fairbanks. 4 Sequential life testing in the exponential case was considered by Epstein & Sobel 5 and the truncated sequential test was proposed by Aroian. 6 For a Weibull distribution, sampling plans with known shape p a r a m e t e r s are given in the US Defense D e p a r t m e n t Quality Control and Reliability Technical Reports TR-3, 7 TR-4, s and TR-6. 9 H a r t e r & M o o r e l° present a modification of the sampling plans in MIL-STD-781B ~t which can be used for a Weibull distributions. Fertig & Mann ~2 proposed a sampling plan with Type II censoring for Weibull distribution and Schneider ~3 considered one based on Type II censored data which can be used for both the Weibull and the lognormal distributions. Conducting life tests are usually expensive because the tests are time consuming and destructive in that some or all of the products on test fail during the tests. W h e n prior information on the quality of a product is available, sample size a n d / o r test duration may be reduced by utilizing this information in designing life test sampling plans. Therefore, it is worthwhile to consider Bayesian life test sampling plans based on the cost model. Costs related to life test sampling plans are test cost, accept cost, and reject cost. Test cost consists of the costs associated with the length of

lot size sample size a fixed integer the r-th failure time in a sample of size n time limit of total rebate warranty time limit of prorated warranty (w2-> Wl) cost of sampling and putting an item on test cost per unit of test time cost of rejecting an item cost associated with an external failure.

2 INTRODUCTION For consumer durable products, one of the important quality characteristics is the lifetime of the products. Sampling plans to determine acceptability of a product with respect to lifetime are called life test sampling plans. In this article, a life test sampling plan for a product sold under a warranty policy is presented. In life testing, a fixed n u m b e r of items are often tested simultaneously and testing continues for some fixed period of time (Type I censoring) or until some fixed n u m b e r of items on test fail (Type II censoring). Traditionally, life test sampling plans have largely been based upon statistical criteria. For exponential distributions, Epstein & Sobel 1 proposed a test based on Type II censored data and one based only on the 61

62

Y. I. K w o n

test duration and the number of test items. Accept cost is the external failure cost for the products in the accepted lot. Reject cost means scrap or reprocessing costs for products in the rejected lot. A life test sampling plan based on a cost model for the Weibull distribution with known shape parameter is considered by Soland 14 and one for one parameter exponential distribution with Type II censoring by T h y r e g o d ] 5 Nigm & Ismail ~6 extended Thyregod's works to two parameter exponential distribution. Dunsmore & Wright ~v proposed a Bayesian sequential sampling plan based on costs for exponential distribution. Bat & Kwon l~ proposed an economic life test sampling plan for repairable products in which failures of products occur according to the power law process. Nowadays, almost all products sold to the consumer carry a warranty of some kind. Therefore, in designing acceptance sampling plans for a product sold under warranty, it is worthwhile to consider the costs associated with the warranty policies. The commonly used warranty policies are the failure free policy and the prorated rebate policy; see Menke, ~'~ Heschel 2° and Nguyen & Murthy. 2~ The supplier makes full compensation in the failure free policy and prorated compensation in the prorated rebate policy for all failures occurring during the warranty period. Both policies are special cases of the general rebate policy proposed by Thomas 22 in which the failure free policy is used during the period (0, Wl) and the prorated rebate policy is used during (w~, w2) where w~ -< w2 are pre-specified positive constants. In this paper, a Bayesian life test sampling plan for products with Weibull lifetime distribution which are sold under the general rebate warranty policy is considered. Little work has been done on the design of Bayesian life test sampling plans for Weibull distributions based on costs except for the work of Soland] 4 He assumed that costs for the accepted and rejected lots are linear in the mean of the life distribution and did not consider the effect of the lot size and sample size in the construction of the cost model. However, most of the models in the literature on Bayesian sampling plans based on costs consider accept and reject costs as functions of lot size and sample size; for example, see Hald, 23 Guenther, 24 Tagaras & Lee, 25 Thyregod, ~5 Dunsmore & Wright, w and Nigm & Ismail. ~6 We consider Bayesian life test sampling plans for a Weibull distribution in which the cost model is constructed as a function of lot size and sample size.

have a Weibull distribution with probability density function (p.d.f.) F 0-i/3tt3 ie ,~/o, t > 0 , f(tlO'13) = [0, otherwise.

(1)

We consider the situation where the value of the shape parameter /3 is known, since, in Weibull life testing procedures, it is frequently the case that an appropriate value of /3 is known from previous test experiences. Soland ~4 gives a justification for this situation. Letting y = t ~, eqn (1) can be transformed into the exponential p.d.f, with mean lifetime 0. However, the problem of designing Bayesian life test sampling plans for a Weibull distribution with a known shape parameter is different from that for exponential distributions because the distribution and moments of the lifetime directly influence the cost structures and the design of sampling plans. The parameter 0 is itself a random variable, varying from lot to lot according to a known distribution. When the value of the shape parameter /3 is known, the conjugate prior for 0 is an inverted gamma distribution with p.d.f. g(O) =Fb"F(a)l k0,

l0 ("+l)e t,lO 0 > 0 , otherwise.

(2)

where a and b are known constants and F(.) denotes the gamma function. Most often, the parameters a and b are to be estimated from the past history. Note that the choice of an inverted g a m m a prior for 0 is equivalent to choosing a gamma prior for A = 1/0. Waller et al. 2~" proposed a method for selecting the parameters of the gamma prior distribution. The method requires information in the form of two distinct percentile values and can be used to determine the parameters of the inverted gamma prior distribution. To use the method, we need to specify 0t and 02 such that for Ot < 02,Pr(® < Or) = p~ and P r ( ® < 02) =P2. The solution of the above pair of simultaneous equations selects the pair of values for a and b that determine the prior distribution summarizing the available information. It is assumed that after an accept/reject decision, products in the accepted lot are sold under the general rebate warranty policy and those in the rejected lot are scrapped or reprocessed. 3.1 Test procedure

3 THE MODEL Suppose that lots of size N are submitted for inspection and the lifetimes of the individual products

Life testing with Type II censoring is considered: n items drawn at random from a lot are placed on test simultaneously and the test is terminated when r

Bayesian life test sampling plan (r <- n) of them have failed. Letting tl -< t2 -< ... ---~t~ be the r smallest lifetimes,

V, = ~ T~i + (n - r)T~

(3)

63

Therefore, the expected average cost per lot for a given (n,r,c) plan with lot size N is

K(n,r,c) = fo~ K(n,r, clO)g(O)dO

i--1

is sufficient for 0 and it follows that the decision rule may be based on Vr, i.e., accept if V~ > c and reject otherwise where c is a pre-specified positive constant. 3.20C

= csn + GB(n,r) + ( N

n){Cp + H(r,c)},

-

(7)

where

B(n,r) = £~ B(n,rlO)g(O)dO

curve

r(a -/3-1),,, From the results of Epstein & Sobel, ~ it follows that, for a given 0, V~ follows a gamma distribution with parameters r and 0. Therefore, for given n, r, and c, the probability of accepting a lot is given by

F(a)

X (;-1)(n - r + k + 1) -°3+wl3,

tt(r,c) = L(r, c l O ) =

O - T ( r ) - ~ t ~ ~e '/°dt.

~{/3 + 1~%' ( - 1)kn! o tq\fl]kz~-_o'--'--(r- 1)!(n - r)!

(4)

(S)

{A(0,¢3) - cp}L(r, clO)g(O)dO

,-1 = Z [(co - cp)

c.(b + c) a+k

k-o

we -- wl

3.3 Expected test time

× Since the test is terminated at the r-th failure time, the expected test time is (see Lieblein 27)

(b + c + t13)-(~+k~dtl~(c,k),

(9)

and

B(n,rlO ) = E(TrIO)

ckbaF(a + k) ~(c,k) - k!F(a)(b + c) °+k" ~=o (r - i-)v.-~n--- r) !

× (~,-l)(n - r + k + 1) -(13+1)//3.

(5)

Letting Ka and Kr be the expected cost per lot for the special cases of acceptance and rejection without test, respectively, then

3.4 Expected accept cost per product

K~ = coN 1

(b + t~)-~dt W2 -- W1

According to the general rebate warranty policy, the cost of accepting an item with life time T is

[

ca, c*(T) = |c,(w2L0,

T ) / ( w 2 - Wl)

if

T
if if

Wl <- T < w 2 , T>w2.

K r =

[

1{

wle wf/o

~

w2e-W~/o

W2 -- WI

= co[1

- -

1

W2 - - W1

f~: e-'~/°dt].

JWl

cpN.

4 OPTIMAL SAMPLING PLAN

Therefore, for given 0, the expected accept cost per item is given by

A(O) = Ca 1 + - -

and

(6)

3.5 Expected average cost

For given 0, the average cost per lot is

r ( n , r , cJO) = Csn + c,B(n,rlO) + ( N - n)[A(O)L(r, clO ) + Cp{1 - L(r,c]O)}].

The optimal sampling plan can be obtained by minimizing eqn (7) with respect to (n,r,c). A three-dimensional search over (n,r,c) can be used for finding the optimal values n*, r*, and c*. We first determine optimal numbers c* and n* for fixed r and then determine r* which minimizes the expected average cost. The following theorem shows that, for fixed r, K(n,r,c) is unimodal with respect to c and provides the optimal value c*. T h e o r e m 1. Define d(r,c) = cp - c, + w - 2--cow ,

fwi2(

b+c , °+r b+c+t13) dt.

Then, for given r, (i) c* = 0 when d(r,0) -->0 and

(10)

Y. L K w o n

64

(ii) c* is the unique solution to d ( r , c ) = 0 when d(r,0) < 0. Proof. The expected average cost depends on c only through the term H ( r , c ) and we have

l b ' T ( a + r) c y + , d(r,c), r(a)r(r)(b + Cr

OH(r,c)/Oc ;M(r,c)/Oc -

c,(a + r)(b + c) ''+~ -1 W2 -- WI

×

t~/(b+c+t

¢) .... + l d t > O '

and d(r,~c) = Cp > O. Therefore, for given r, it follows that (i) when d(r,O)> O, K ( n , r , c ) is increasing in c and therefore c* = O, and (ii) when d(r,O)- c,,, c* = 0 for r > 1 because d(r,0) is decreasing in r and d ( r = z ¢ , 0 ) = c p - c a . This implies that the optimal decision is to accept the lot without test if cp >- c,,. Next, the optimal sample size n* for given r, and finally r* should be determined. It is difficult to show analytically whether the expected cost is unimodal with respect to n and r. Numerical studies over wide ranges of the values of the related parameters, however, indicate that a) for fixed r, c,* is unimodal with respect to n and b) the expected cost is unimodal with respect to r. The above results may be helpful for finding optimal sampling plans and a procedure for finding the optimal sampling plans when Cp < c , can be described as follows: Step 1. Compute K, and Kr. Set K0 = Min{Ka,K~}, r = 0 , and Km = 2. Step 2. Set r = r + 1 and compute d(r,0). Step 3. If d(r,0) < 0, go to step 4. Otherwise, go to step 2. Step 4. Determine c* by solving the equation d(r,c) = 0. Set n = r and K,,(r) = ~c. Step 5. Set n = n + 1 and compute K(n,r,c*). Step 6. If K ( n , r , c * ) > K,,(r), set K* = K,,(r) and go to step 7. Otherwise, set n,* = n, Kin(r) = K(n,r,c*); go to step 5. Step 7. If K* > K .... go to step 8. Otherwise, set K,, = K*, rl = r, nl = n*, c~ = c*; go to step 2. Step 8. If K ( n ~ , r l , c t ) < K o , the optimal plan is ( n * = n l , r * = r l , c * = c l ) . Otherwise, the optimal decision is to accept the lot without test if Ko = K,, and to reject it otherwise.

5 A

NUMERICAL

EXAMPLE

Control relays used in various electronic devices are manufactured in batches of N = 3000. A general rebate warranty policy with w~ = 180 and w2 = 360 unit times is used for the products. From the life test history, it is known that the lifetimes of the relays have a Weibull distribution in which /3 = 0.8 and parameter 0 varies from lot to lot according to an inverted gamma distribution with parameters a = 3.0 and b = 875. The estimated cost components in dollars are c , = 2 . 8 , c, = 1.2, Cp=2.5, and c,,=9.5. The optimal sampling plan is (n*,r*,c*) = (62, 4, 1095) and the corresponding expected cost per item is 2.131. Optimal sampling plans for various lot sizes are given in Table 1. It shows that optimal decisions for lots of size N-< 300 is to accept the lot without test. Optimal sampling plans in the table seem to be robust to changes in lot size. For selected combinations of the parameters /3, a and b, optimal sampling plans and their expected costs are given in Table 2. We can see that the differences in the optimal expected costs for different values of/3, a and b are relatively small and that optimal sampling plans (n*,r*,c*) are not sensitive to small changes in the parameters/3, a and b. When the values of the parameters /3, a and b are unknown, estimates of the parameters from the life test data of the previous batches may be used. The effect of using incorrect estimates of the parameters is studied in terms of percentage errors (PEs) denoting the relative increases in the expected costs. PEs in using incorrect values of/3, a and b are given in Fig. 1 and those for different values of/3 are given in Fig. 2 in which P D means percentage deviation of /3 from the true value/30, i.e., P D = 100(/3 - /3o)//3o%- In Figs 1 and 2, the values of the parameters are selected so that probabilities of failure during the warranty period for each case are the same since the parameters can be T a b l e 1. O p t i m a l s a m p l i n g plans for j0 = 0.8, a = 3.0, b = 875, w~ = 180, w2 = 360, c= = 2.8, c, = 1.2, cp = 2.5 a n d ca = 9.5

Lot Size N

n*

r*

c*

Expected Cost per Item

300 400 500 600 800 1000 1200 1500 2000 3000 4000 5000

0 68 68 75 75 75 75 62 62 62 62 62

-2 2 3 3 3 3 4 4 4 4 4

0 517.5 517.5 802.5 802.5 802.5 802.5 1095 1095 1095 1095 1095

2.362 2.348 2.309 2.282 2.239 2.214 2.197 2.176 2.154 2.131 2.120 2.114

Bayesian life test sampling plan Table 2. Optimal sampling plans for selected combinations of parameters ~, a and b /3

a

0.6

b

3 5 8 3 5 8 3 5 8 3 5 8 3 5 8 3 5 8

0.8 1.0 1.2 1.6 2.0

n*

r*

269.6 59 460.4 59 746.6 59 874.9 62 1494.0 62 2423.1 62 2839.5 50 4848.6 50 7863.7 50 9215.0 47 15735.4 47 25520.3 78 73902.2 42 127103.5 42 206964.9 78 778356.6 39 1338687.0 39 2179807.0 75

4 4 4 4 4 4 4 4 4 4 4 3 4 4 3 4 4 3

c*

Expected Cost per Item

375.0 367.5 362.5 1095.0 1042.5 975.0 3180.0 2932.5 2557.5 9265.0 8137.5 3757.5 100676.5 98522.5 69713.0 877440.0 801822.0 445441.5

2.173 2.269 2.339 2.132 2.225 2.290 2.097 2.183 2.237 2.055 2.130 2.165 2.202 2.298 2.363 2.129 2.212 2.256

estimated from the failure data observed during the warranty period. PEs are small when the parameters are not far from the true values and therefore the proposed sampling plan seems to be reasonably robust to small changes in the parameters/3, a and b.

6 CONCLUDING REMARKS In this article a Bayesian life test sampling plan for products with Weibull lifetime distribution which are sold under the general rebate warranty policy is proposed. A n inverted gamma prior distribution on

O O _

,/:.,]

ou~E

/ / :',:

t--

/' /

a=5.0

o if?

' ,," / ,'

65

$

O

$

$~

,/"/ / ,

° o

0= 1 " 2 I¢

i

i

-- 2 0 I 0

[

i

I

I

i

i

10 .O

i

i

I

i

i

,-"

i

0 i0

i

I

1010

i

i

i

/

20"0

eD(%) Fig. 2. Percentage errors for different values of/3 when true values are/3o =0.8, 1.2, and 2.0. the scale parameter is considered and a cost model is constructed which consists of test cost, accept cost, and reject cost. The m e t h o d of finding optimal sampling plans which minimize the expected average cost per lot is presented. Numerical studies show that the sampling plans based on the expected cost are reasonably robust to small changes in the parameters of the prior distribution. The optimal sampling plans in the example were obtained numerically on an IBM PC-486 computer. In most cases, they can be obtained in a few minutes.

References 1. Epstein, B. & Sobel, M., Life testing. J. Am. Stat. Ass., 48 (1953) 485-502. 2. Epstein, B., Truncated life tests in the exponential case. Annals of Math. Stat., 25 (1954) 555-564. 3. Bulgren, W. & Hewette, J., Double sample test for hypotheses about the mean of an exponential distribution. Technometrics, 15 (1973) 187-190. 4. Fairbanks, K., A two-stage life test for the exponential parameter. Technometrics, 30 (1988) 175-180. 5. Epstein, B. & Sobel, M., Sequential life test in the exponential case. Annals of Math. Stat., 26 (1955) 82-93. 6. Aroian, L.A., Some comments on truncated sequential tests for the exponential distribution. Industrial Quality Control, 21 (1964) 309-312. 7. TR-3, Sampling Procedures and Tables for Life and

Reliability Testing Based on Weibull Distribution (Mean Life Criterion), US Dept. of Defense, Washington, ~

L

0.6

~

~

I

0.7

~

~

~

"1 "~

0.8

~

~

~

I

0.9

~

D.C., 1961. 8. TR-4, Sampling Procedures and Tables for Life and

~

1.0

Fig. 1. Percentage errors in using incorrect values of /3, a and b when true values are/3 = 0.8, a = 3.0 and b = 875.

Reliability Testing Based (Hazard Rate Criterion),

on

Weibull Distribution

Dept. of Defense, Washington, D.C., 1962. 9. TR-6, Sampling Procedures and Tables for Life and

Reliability

Testing

US

Based on

Weibull Distribution

66

10. 11. 12. 13. 14.

15. 16. 17.

18.

Y. L Kwon (Reliable Life Criterion), US Dept. of Defense, Washington, D.C., 1963. Harter, H.L. & Moore, A.H., An evaluation of exponential and Weibull test plans. IEEE Trans. Reliab., 11-25 (1976) 100-104. MIL-STD-781B, Reliability T e s t s : Exponential Distribution, US Dept. of Defense, Washington, D.C., 1967. Fertig, K.W. & Mann, N.R., Life-test sampling plans for two-parameter Weibull population. Technometrics, 22 (1980) 165-177. Schneider, H., Failure-censored variables-sampling plans for lognormal and Weibull distributions. Technometrics, 31 (1989) 199-206. Soland, R.M., Bayesian analysis of the Weibull process with unknown scale parameter and its application to acceptance sampling. IEEE Trans. Reliab., 11-17 (1968) 84-90. Thyregod, P., Bayesian single sampling plans for life testing with truncation of the number of failures. Scandinavian J. Star., 2 (1975) 61-70. Nigm, A.M. & Ismail, M.A., Bayesian life test sampling plans for the two parameter exponential distribution. Commun. Star. Simula. Computa., 14 (1985) 691-707. Dunsmore, I.R. & Wright, D.E., A decisive predictive approach to the construction of sequential acceptance sampling plans for life times. Appl. Stat., 34 (1985) 1-13. Bai, D.S. & Kwon, Y.I., Economic designs of life test sampling plans for repairable products. Engng

Optimization, 20 (1993) 287-302. 19. Menke, W.W., Determination of warranty reserves. Management Sci., 15 (1969) 542-549. 20. Heschel, M., How much is a guarantee worth?. Industrial Engng, 3 (1971) 14-15. 21. Nguyen, D.G. & Murthy, D.N.P., A general model for estimating warranty costs for repairable products, l i e Trans., 16 (1984) 379-386. 22. Thomas, M.U., Optimum warranty policies for the nonrepairable items. 1EEE Trans. Reliab., !1-32 (1983) 282 -288. 23. Hald, A., The compound hyper geometric distribution and a system of single sampling inspection plans based on prior distributions and costs. Technometrics, 2 (1960) 275-340. 24. Guenther, W.C., Rectifying inspection for nonconforming items and the hald linear cost model. J. Quality Tech., 17 (1985) 81-85. 25. Tagaras, G. & Lee, H.L., Optimal Bayesian singlesampling attributes plans with modified beta prior distribution. Naval Res. Log., 34 (1987) 789-801. 26. Waller, R.A., Johnson, M.M., Waterman, M.S. & Martz, H.F., Gamma prior distribution selection for Bayesian analysis of failure rate and reliability. Nuclear Systems Reliab. Engng and Risk Assessment, X (1977) 584-606. 27. Lieblein, J., On moments of order statistics from the Weibull distribution. Annals of Math. Stat., 26 (1955) 330-333.