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

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- 1

LIST OF TABLES

Table 8

- C.1.1 Commercial Clothes Washers

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

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- 2

LIST OF FIGURE S

Figure 8

- C.2.1 Fraction of the Commercial Clothes Washers Failing, Multi

- Family

....... 8

- C

- 3 Figure 8

- C.2.2 Cumulative Lifetime Length of Commercial Clothes Washers Multi

- Family

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

- C

- 3 Figure 8

- C.2.3 Fraction of the Commercial Clothes Washers Failing, Laundrom at

.......... 8

- C

- 4 Figure 8

- C.2.4 Cumulative Lifetime Length of Commercial Clothes Washers Laundromat

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

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- 4

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- iAPPENDIX 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 commercial 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 ave rage ) , 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 DISTRIBU TION 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 )

.

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- 1The 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
t he 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.

This soluti on can be checked using Crystal Ball by specifying a Weibull distribution with the calculated parameters (location, scale, and shape) in an assumption cell and generate a forecast that equals the assumption. Forecast histogram and statistics verify that th e Weibull distribution matches the desired shape.

Table 8C.2.1 shows the average, minimum, maximum lifetime, and maximum percentile values used to determine the Weibull distribution parameters alpha and beta. For commercial clothes washers, DOE developed two lifetime estimates based on product application

— one for multi

- family applications and another for Laundromats. DOE estimated that product lifetimes did not vary ba sed on whether the washer was top

- loading or front

- loading. DOE estimated that the maximum lifetime percentile for both product applications was 99 percent.

Table 8

- C.1.1 Commercial Clothes Washers

Product Application Expert Opinion Values Weibull Parameters

Minimum
years Average
years Maximum
years Maximum percentile % Alpha (scale) Beta (shape)
Multi

- Family 7 11.3 15.5 99 4.782 2.655
Laundromat 5 7.1 9.3 99 2.391 2.655

Figures 8C.2.1 to 8C.2

. 4 show the Weibull distribution as well as the cumulative Weibull distribution for each commercial clothes washer product application.

8

- C

- 2Figure 8

- C.2

. 1 Fraction of the Commercial Clothes Washers Failing, Multi

- Family

Figure 8

- C.2

. 2 Cumulative Lifetime Length of Commercial Clothes Washers, Multi

- Family 0% 5% 10% 15% 20% 25%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Age of Equipment (Years)Probability of Failure(%)

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Age of Equipment (Years)Probability (%)

8

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- 3Figure 8

- C.2

. 3 Fraction of the Commercial Clothes Washers Failing, Laundromat

Figure 8

- C.2

. 4 Cumulative Lifetime Length of Commercial Clothes Washers, Laundromat 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

0 1 2 3 4 5 6 7 8 9 10 11 Age of Equipment (Years)Probability of Failure (%)

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 1 2 3 4 5 6 7 8 9 10 11 Age of Equipment (Years)Probability (%)

8

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- 4REFERENCES

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. Report No. Report No . ORNL

- 6804/R1. < http://www.ornl.gov/~webworks/cpr/v823/rpt/78562.pdf >

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

. Appliance Magazine , 2003.

< http://www.appliancemagazine.com/print.php?article=197&zone=1&first=1 >

3. Crystal Ball, Technote: Derivation of Weibull Distribution Parameters When the Minimum, Most Like ly Value, and a Percentile are Known, 2007. (Last accessed November, 2013.) < http://www.crystalball.com/support/simulation/cbl_gen_021A.html >

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