8D

- i APPENDIX 8D. LIFETIME DISTRIBUTIO NS

TABLE OF CO NTENTS

8D.1 INTRODUCTION

................................ ................................ ................................ .. 8D

- 1

8D.2 DERIVATION OF WEIBUL L DISTRIBUTION PARAM ETERS

...................... 8D

- 1

8D.3 LIFETIME DISTRIBUTIO NS FOR DISHWASHERS

................................ ......... 8D

- 2

LIST OF TABLES

Table 8D.3.1 Distribution Parameters for Dishwashers

................................ ................... 8D

- 2

LIST OF FIGURE S

Figure 8D.3.1 Likelihood that Dishwasher Will Fail

................................ ........................ 8D

- 28D

- 1 APPENDIX 8D. LIF ETIME DISTRIBU TIONS

8D.1 IN TRODUCTION

The U.S. Department of Energy ( DOE ) characterized the lifetime of both product classes

of dish washers being considered for new energy efficiency standards ( standard

- sized and

comact )

. DOE characterized dish washer l ifetimes using a Weibull probability distribution that

ranged from the minimum to maximum lifetime estimates , as described in chapter 8

, section

8.2.3

. The Weibull distribution is recommended for evaluating lifetime data because it can be

shaped to match low, most likely ( or average ) , and high values . The probability of exceeding the

high value is contained in the long tail of the Weibull distribution

. 1

, 2

8D.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

Weibull distributions utilize available data to assign low, average, and high values to a

random variable that has unknown distribution parameters. DOE appli ed Weibull distribution s to

product lifetime data to de riv e low, average, and high lifetime value s

, along with a percentile

containing a high value. A similar approach is described in a technical note to the softw are

Crystal Ball , which uses a most likely value in place of an average value. 3 The Weibull

distribution can be defined as:

W here :

L

= location

,

α = scale , and

β

= shape

.

T he cumulative distribution therefore is :

Based on available data, Weibull distribution parameters are specified as follow s.

1. The output deviates must be greater than the expert opinion of low value

. 2. The average

, X

avg , must be equal to the average value from the available data

. 3. The high value, xb, must correspond to some particular percentile point ( e.g. , 95

percent or 90 percent )

.8D

- 2 The values for the parameters in the equations we re determined using the approach

outlined in Crystal Ball’s technical note

. 3

Crystal Ball can be used to check a solution by specifying a Weibull distribution that has

the calculated parameters (location, scale, and shape) in an assumption cell , then generat ing a

forecast that equals th at assumption. The f orecast histogram and statistics will confirm whether

the Weibull distribution match es the desired shape.

8D.3 LIFETIME DISTRIBUTIO NS FOR DISH WASHERS

Table 8 D .3.1 shows the average, minimum, and maximum lifetime s plus maximum

percentile values used to determine the Weibull distribut ion parameters α and β for residential

dish washers

. DOE estimated th at the maximum lifetime percentile for both standard

- sized an d

compact was 99 percent

.

Table 8D.3.1 Distribution Parameters for Dishw as hers

Values Weibull Parameters

Minimum Average

years Maximum

years Maximum

years Percentile Alpha

(scale) ( % ) Beta

(shape)

5 15.4 50 99 16.29 2. 31

Figure 8 D .3 .1 show the Weibull distribution for the lifetime of both standard

- sized and

compact dish washers

.

Figure 8 D .3.1 Likelihood that Dishw asher Will Fail 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0%

0 5 10 15 20 25 30 35 40 45 50 

### Weibull Retirement Function for

Dishwashers (mean = 15.43 years)8D

- 3 REFERENCES

1 B arnes, P.R., J.W. Van Dyke, B.W. McConnell, S.M. Cohn, and S. L. Purucker

. The

Feasibility of Replacing or Upgrading Utility Distribution Transformers During Routine

Maintenance . 1995

. Oak Ridge National Laboratory: Oak Ridge, TN.

2 Karr, T. Making the Most of Life Test Data

. Appliance Magazine

. October 2003. (Last

accessed march 23, 2010.)

< www.appliancemagazine.com/ae/editorial.php?article=197&zone=215&first=1 >

3 Crystal Ball. Technote: Derivation of Weibull Distribution Parameters When the Minimum,

Most Likely Value, and a Percentile are Known . 2007. (Las t accessed December 4, 2008.)

Available from: www.crystalball.com/support/simulation/cbl_gen_021A.html

.