### APPENDIX 8C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8C.1 INTRODUCTION ...................................................................................................8C-1

8C.2 DERIVATION OF WEIBULL DI STRIBUTION 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 Cl othes Washers Failing, Multi-Family .........8C-3 Figure 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 usi ng a Weibull probability distribution that ranged from the minimum to maximum lifetime es timates described in chapter 8, section 8.2.3.

The Weibull distribution is reco mmended for applying lifetime data because it can be shaped to match low, most likely or average, and high valu es, yet still allow some probability of exceeding the high value. The Weibull distribution has a lo ng 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 DI STRIBUTION 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 Tec hnical Note, which uses most li kely value instead of average. 3


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

The Weibull distributi on 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 greate r 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%)

8C-1

### 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 Crysta l Ball by specifying a Wei bull distribution with the calculated parameters (location, scale, a nd shape) in an assumption cell and generate a forecast that equals the assumption. Forecast hist ogram and statistics verify that the Weibull distribution matches the desired shape.

Table 8C.2.1 shows the average, minimum, ma ximum lifetime, and maximum percentile

values used to determine the Weibull distributi on 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 Laundr omats. 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 di stribution as well as the cumulative Weibull

distribution for each commercial cl othes 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 Utili ty Distribution Transformers During Routine

Maintenance . 1995, Oak Ridge National Labor atory: Oak Ridge, TN.

2 

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

3 

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

.

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