8

- C

- i APPENDIX 8

- C. LIFETIME DISTRIBUTIO NS

TABLE OF CO NTENTS

8

- C.1 INTRODUCTION

................................ ................................ .............................. 8

- C

- 1
8

- C.2 DERIVATION OF WEIBUL L DISTRIBUTION PARAM ETERS

..................... 8

- C

- 1
8

- C.3 LIFETIME DISTRIBUTIO NS FOR CLOTHES WASHERS

.............................. 8

- C

- 2

LIST OF TABLES

Table 8

- C.3.1 Distribution Parameters for Clothes W ashers

................................ ........... 8

- C

- 2

LIST OF FIGURE S

Figure 8

- C.3.1 Likelihood that Clothes Washer Will Fail

................................ ................ 8

- C

- 28

- C

- 1 APPENDIX 8

- C. LIF ETIME DISTRIBU TIONS

8

- C.1 IN TRODUCTION

The U.S. Department of Energy ( DOE ) characterized the lifetime of both product classes
of clothes washers being considered for new energy efficiency standards ( front

- loading and top

loading )

. DOE characterized clothes 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

8

- C.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 software 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 )

.8

- C

- 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 matches the desired shape.

8

- C.3 LIFETIME DISTRIBUTIO NS FOR CLOTHES WASHERS

Table 8

- C.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
clothes washers

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

- and top

loading units was 99 percent

.

Table 8

- C.3.1 Distribution Parameters for Clothes Washers

Values Weibull
Parameters

Minimum Average

years Maximum

years Maximum

years Percentile
% Alpha (scale) Beta (shape)

2 14. 2 50 99 10.126 2.655

Figure 8

- C.3 .1 show the Weibull distribution for the lifetime of both front

- and top

loading clothes washers

.

Figure 8

- C.3.1 Likelihood that Clothes Washer 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 Clothes Washer (mean = 14.16 years)8

- C

- 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

a ccessed 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. (Last accessed December 4, 2008.)
Available from: www.crystalball.com/support/simu lation/cbl_gen_021A.html

.