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 N FOR PORTABLE AIR CONDITIONERS

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

- 2

LIST OF TABLE S

Table 8D

- 3.1 Distribution Parameters for Portable ACs

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

- 2

LIST OF FIGURE S

Figure 8D

- 3.1 Percent of Portable ACs Retiring each Year

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

- 28D

- 1 APPENDIX 8D. LIF ETIME DISTRIBU TIONS

8 D

- 1 IN TRODUCTION

In order to perform a life

- cycle cost analysis, t he U.S. Department of Energy ( DOE )

characterized the lifetime of portable air conditioners (ACs) being considered for new energy efficiency standards. DOE characterized portable ACs lifetimes using a Weibull probability

distribution encomp ass ing lifetime estimates from minimum to maximum, as described in chapter 8, section 8.2. 5

. The Weibull distribution is recommended for evaluating lifetime data

, because it can be shaped to match low, most li kely ( or average )

, or high values . The probability of exceeding the high value is contained in the long tail of the Weibull distribution

. 1

, 2

8 D

- 2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

Weibull distributions utilize data to assign low, average, and high values to a random variable that has unknown distribution parameters. DOE applied Weibull distributions to product

lifetime data for portable ACs to derive low, average, and high lifetime values, along with a percentil e containing a h igh value. A sim ilar approach is described in a technical note to the Crystal Ball software , 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 :

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 ( such as 95

percent or 90 percent )

.8D

- 2

The values for the parameters in the equa tions 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 c onfirm whether the Weibull distribution matches the desired shape.

8D

- 3 LIFETIME DISTRIBUTIO N FOR PORTABLE AIR CONDITI ONERS

Table 8D

- 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 portable ACs

. DOE estimated th at the maximum lifetime percentile for portable ACs was 9 9 percent

.

Table 8D

- 3

. 1 Distribution Parameters for Portable ACs

Value Weibull Parameters

Minimum

years Average

years Maximum

years Maximum

Percentile % Alpha

scale Beta

shape

3 10 .47 20 99 10.66 2. 64

Figure 8D

- 3

. 1 show s the Weibull distribution for the lifetime of portable ACs

. The
lifetime distribution is based on the retirement function, which indicates the number of portable

ACs that fail and are re tired in each year of their life. DOE used an average lifetime of 10.47 years in its analyses.

Figure 8D

- 3

. 1 Percent of Portable ACs Retir ing each Year8D

- 3

REFERENCES

1 Barnes, 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 t he Most of Life Test Data. Appliance Magazine . October 2003. (Last

accessed June 20

, 2014 .)

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

.

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

M ost Likely Value, and a Percentile are Known . 2007. (Last accessed December 4, 2008.)

Available from < www.crystalball.com/support/simulation/cbl_gen_021A.html >

.