## APPENDIX 8C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8C.1 INTRODUCTION ...................................................................................................8C-1 8C.2 DERIVATION OF WEIBULL DI STRIBUTION PARAMETERS .......................8C-1 8C.3 COOKING PRODUCTS – COOKTOP AND OVEN LIFETIME DISTRIBUTIONS ...................................................................................................8C-2 8C.4 MICROWAVE OVEN LIF ETIME DISTRIBUTIONS..........................................8C-5

LIST OF TABLES

Table 8C.3.1 Cooktops and Ovens ....................................................................................8C-2 Table 8C.4.1 Microwave Ovens ........................................................................................8C-5

LIST OF FIGURE
Figure 8C.3.1 Fraction of the Electric Cooktops and Ovens Failing..................................8C-3 Figure 8C.3.2 Cumulative Lifetime Length of Electric Cooktops and Ovens....................8C-3 Figure 8C.3.3 Fraction of the Gas Cooktops and Ovens Failing ........................................8C-4 Figure 8C.3.4 Cumulative Lifetime Length of Gas Cooktops and Ovens..........................8C-4 Figure 8C.4.1 Fraction of the Microwave Ovens Failing ...................................................8C-5 Figure 8C.4.2 Cumulative Lifetime Le ngth of Microwave Ovens.....................................8C-6

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## APPENDIX 8C. LIFETIME DISTRIUBTIONS

8C.1 INTRODUCTION For each product class, DOE characteri zed the product lifetime using a Weibull probability distribution that ranged from the mi nimum to maximum lifetime estimates described in chapter 8, section 8.2.3. The Weibull distri bution is recommended for applying lifetime data because it can be shaped to match low, most lik ely or average, and high values, yet still allow some probability of exceeding the high value. Th e Weibull distribution ha s a long tail, which the analyst can use to specify a distribution w ith 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 cum ulative 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%)

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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 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.

8C.3 COOKING PRODUCTS – COOKTOP AND OVEN LIFETIME DISTRIBUTIONS Table 8C.3.1 shows the average, minimum, ma ximum lifetime, and maximum percentile values used to determine the Weibull distribut ion parameters alpha and beta. DOE estimated that the maximum lifetime percentile for both el ectric and gas cooking products was 99 percent.
Table 8C.3.1 Cooktops and Ovens
Expert Opinion Values Weibull Parameters

Product Class Minimum years Average years Maximum years Maximum percentile % Alpha (scale) Beta (shape) Electric Cooktops and Ovens 10 19 28 99 10.126 2.655 Gas Cooktops and Ovens 12 19 26 99 7.876 2.655


## Figures 8C.3.1 to 8C.3.2 show the Weibull di stribution as well as the cumulative Weibull distribution for electric cookt ops and ovens. Figures 8C.3.3 to 8C.3.4 show the Weibull distribution as well as the cumulative Wei bull distribution for gas cooktops and ovens.

8C-2Probability of Failure (%) 11%
10%

9%
8%
7%

6%
5%

4%
3%
2%

1%
0% 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 A ge of Equipment (Years) 

## Figure 8C.3.1 Fraction of the Electric Cooktops and Ovens Failing

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

0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 A ge of Equipment (Years)Probability (%)



## Figure 8C.3.2 Cumulative Lifetime Length of Electric Cooktops and Ovens

8C-3Probability of Failure (%) 15%

10%

5%

0% 0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 A ge of Equipment (Years) 

## Figure 8C.3.3 Fraction of the Gas Cook tops and Ovens Failing

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

0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 A ge of Equipment (Years)Probability (%)



## Figure 8C.3.4 Cumulative Lifetime Length of Gas Cooktops and Ovens

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## 8C.4 MICROWAVE OVEN LIFETIME DISTRIBUTIONS Table 8C.4.1 shows the average, minimum, ma ximum lifetime, and maximum percentile values used to determine the Weibull distribut ion parameters alpha and beta. DOE estimated that the maximum lifetime percentile for microwave ovens was 99 percent.

Table 8C.4.1 Microwave Ovens

Product Class Expert Opinion Values Weibull Parameters

Minimum years Average years Maximum years Maximum percentile % Alpha (scale) Beta (shape) Gas-Fired Pool Heaters 7 9 10 99 2.183 4.806Probability of Failure (%) 

## Figures 8C.4.1 to 8C.4.2 show the Weibull di stribution as well as the cumulative Weibull distribution for microwave ovens.

90%

80%

70 %

60 %

50 %

40 %

30 %

20 %

10 %

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

## Figure 8C.4.1 Fraction of the Microwave Ovens Failing

8C-50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

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



## Figure 8C.4.2 Cumulative Lifetime Length of Microwave Ovens

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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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