## APPENDIX 8C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8C.1 INTRODUCTION ...................................................................................................8C-1 8C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS .......................8C-1

LIST OF TABLES

Table 8C.2.1 Commercial Clothes Washers......................................................................8C-2

LIST OF FIGURE

Figure 8C.2.1 Fraction of the Commercial Clothes Washers Failing, Multi-Family .........8C-3Figure 8C.2.2 Cumulative Lifetime Length of Commercial Clothes Washers, MultiFamily ..........................................................................................................8C-3 Figure 8C.2.3 Fraction of the Commercial Clothes Washers Failing, Laundromat ...........8C-4 Figure 8C.2.4 Cumulative Lifetime Length of Commercial Clothes Washers, Laundromat..................................................................................................8C-4

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## APPENDIX 8C. LIFETIME DISTRIUBTIONS

8C.1 INTRODUCTION DOE characterized the product lifetime using a Weibull probability distribution that ranged from the minimum to maximum lifetime estimates described in chapter 8, section 8.2.3. The Weibull distribution is recommended for applying lifetime data because it can be shaped to match low, most likely or average, and high values, yet still allow some probability of exceeding the high value. The Weibull distribution has a long tail, which the analyst can use to specify a distribution with some chance of exceeding the high value. 1,2



## 8C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS DOE utilized an approach for calculating the Weibull distribution using product lifetime data to determine low, average, high value, and percentile of high value. A similar approach is described by Crystal Ball Technical Note, which uses most likely value instead of average. 3 

## Available data is used to assign low, average, and high values to a random variable with unknown distribution parameters.

The Weibull distribution can be defined as:

Eq. 1

where

L = location Alpha = scale Beta = shape

Therefore, the cumulative distribution is

Eq. 2

Based on available data, Weibull distribution parameters are specified in the following

manner:

1. The output deviates must be greater than the expert opinion for 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%, or 90%)

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## The values for the parameters in Eq.1 and Eq. 2 are determined using the approach outlined in Crystal Ball’s Technote. 3



## This solution 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 the 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 based 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 8C.2.1 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.

8C-20% 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(%)



## Figure 8C.2.1 Fraction of the Commercial Clothes Washers Failing, Multi-Family

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 (%)



## Figure 8C.2.2 Cumulative Lifetime Length of Commercial Clothes Washers, Multi-Family

8C-30% 5% 10% 15% 20% 25% 30% 35% 40% 45%

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



## Figure 8C.2.3 Fraction of the Commercial Clothes Washers Failing, Laundromat

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

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



## Figure 8C.2.4 Cumulative Lifetime Length of Commercial Clothes Washers, Laundromat

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## 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 the Most of Life Test Data , in Appliance Magazine . 2003.

3 

## Crystal Ball. Technote: Derivation of Weibull Distribution Parameters When the Minimum, Most Likely Value, and a Percentile are Known . 2007 [cited; Available from: http://www.crystalball.com/support/simulation/cbl_gen_021A.html

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