(DRAFT)

SENSITIVITY OF BALLAST LIFE CYCLE COST AND PAYBACK ANALYSIS

TO ELECTRICITY PRICE AND ANNUAL LIGHTING HOURS

Xiaomin Liu, James E. McMahon, Diane Fisher and Isaac Turiel

Technology and Market Assessment Group

Energy Analysis Program

Ernest Orlando Lawrence Berkeley National Laboratory

July 26, 1996

1. Introduction

In a recent analysis (LBNL 1996), LBNL examined the impacts of energy efficiency levels

for fluorescent lamp ballasts. Several measures of the impacts were calculated, including lifecycle

costs (LCC), LCC change (the change in LCC between the baseline design and the more efficient

design options) and payback periods. These values were calculated using point estimates

representing typical values of the input variables. Some sensitivity analyses were performed,

using point estimates which bracketed the base case values for the input parameters.

This report describes the process of performing thorough sensitivity analyses to account

for the full range of actual values for key input variables. LCC change and payback periods are

calculated using a probability distribution curve of possible values for the input variables weighted

by their likelihood of occurrence and the Monte Carlo simulation technique, instead of relying on

point estimates. We examine a specific product class of fluorescent lamp ballasts, the 2-lamp

F40T12/ES. The baseline design for this ballast is the energy efficient magnetic ballast, or EEM.

The two more efficient design options considered are the cathode cutout (CC) and the electronic

rapid start (ERS) ballasts. We examine two input variables, electricity price and annual lighting

hours, which contribute signficantly to the economic impacts. This paper shows how variability in

these two inputs affects the LCC change and payback periods. The analyses described here were

presented at a workshop in Washington, DC on June 20, 1996. At the workshop, several changes

in input parameters were suggested (e.g. ballast life and lamp life with CC ballasts) which will

impact these analyses when they are revised.

II. Method

The lifecycle cost and payback calculations, described in LBNL 1996, are repeated many

times using a different input value each time for the parameter being varied, while other input

parameters are held constant (See Appendix A for a list of values of input parameters) . Each

parameter is varied separately, so that the interaction between parameters is not considered. The

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FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
empirical probability distribution of each parameter is estimated from sample data to fairly

represent the range of conditions among U.S. consumers. The distributions were weighted so

that economic benefits and costs of energy efficiency standards could be assessed on a national

scale.

The simulation calculations were done on a Macintosh desktop computer. The

uncertainty model was originally constructed using the program, DEMOS , DEcision Modeling

System, and later transferred to its successor , Analytica (Lumina Decision Systems, 1996). The

software took 1000 samples from the distribution for one variable at a time while other variables

were kept as single point values taken from the conventional engineering and economic analysis.

The sampling method used was the Median Latin Hypercube variation of a Monte Carlo

simulation. Other sampling schemes (Simple Monte Carlo and Random Latin Hypercube) were

also tested but no significant difference in the resulting distribution was observed (see Appendix B

for a more detailed description of the sampling methods).

III. Sensitivity of LCC and Payback to Electricity Price

A. Data

To assess the consumer impacts of energy efficiency standards across the country, we

have combined two data sets (EIA 1995a, EIA 1995b) to obtain commercial sector electricity

prices from EIA financial statistics of 146 major U.S. investor-owned electric utilities and 441

publicly-owned electric utilities. Each utility electricity price is weighted by its commercial sector

kWh sales in 1994. The combined data set consists of 587 utilities and accounts for more than

90% of all electricity sales in the commercial sector. A distribution derived from the combined

data set should reasonably represent the variability of electricity price in the commercial sector

nationwide.

Figure 1 illustrates the relative frequency histogram of electricity price and its statistics.

The horizontal axis presents the range of electricity price and the height of the bars represents the

relative probability of the electricity price. The probability was calculated as the ratio of kWh

sales at a specific price to the total kWh sales.

Figure 2 presents the same information in a different form, as the cumulative probability

distribution of the commercial electricity price. It represents the cumulative probability (y-axis)

that electricity sales will happen at less than or equal to a certain price value (x-axis). Table 1

presents the same information (i.e., percent of electricity sales in commercial sector ). From the

cumulative probability distribution one can also easily find the percent (probability) of electricity

sales in a specified interval. For instance, it can be easily estimated that about 90% of electricity

sales occur at prices ranging from $0.05/kWh to $0.12/kWh (95% of sales are at or below

$0.12/kWh, less 5% that are at or below $0 .05/kWh).

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FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
B. Results

Once the variability of electricity price was specified, the life cycle cost and payback

calculations were carried out through a set of mathematical equations used in a conventional

engineering analysis (LBNL 1996). Figure 3A shows the relationship between change in LCC and

the electricity price. The plot illustrates a linear relationship between the input and the output.

The crossover point (from Figure 3A) is the electricity price at which the LCC is the same for the

baseline and more efficient design. At electricity prices lower than the crossover point, LCC will

increase; at electricity prices higher than the crossover point LCC will decrease for the more

efficient design compared to the baseline. Figure 3A shows that Cathode Cutout (CC) and

Electric Rapid Start (ERS) ballasts tend to reduce LCC (negative LCC change) at prices above

$0.038/kWh and $0.064/kWh respectively with respect to the baseline design, Energy Efficient

Magnetic (EEM).

For CC compared to EEM, at the mean electricity price ($0.0784/kWh), LCC decreases

$7, from $180' for EEM to $173 for CC. Lifecycle costs could decrease by up to $21 (at

$0.162/kWh), or increase by up to $3 (at $0.016/kWh), based on the current range of electricity

prices for U.S. commercial customers. Using the crossover point from Figure 3A applied to the

cumulative probability from Figure 3B, for 98% of the sales-weighted electricity prices (those

greater than $0.038/kWh) there will be a net reduction in LCC in switching from the EEM to the

CC design.

For ERS compared to EEM, at the mean electricity price ($0.0784/kWh), LCC is reduced

$4 to $176. For the current range of electricity prices, LCC could decrease by up to $25 (at

$0.162/kWh), or increase by up to $11 (at $0.016/kWh). For 64% of the commercial electricity

sales (at prices greater than $0.064/kWh) there will be a net savings in LCC.

Figure 4 and Table 2 show results of a similar analysis of the sensitivity of payback period

to the electricity price. The scatter plot shows that there is a non-linear relationship between the

electricity price and payback and that the CC design requires less time to obtain a capital return

(shorter payback period) than the ERS design. The payback period for the CC ballast, compared

to the baseline EEM, ranges from 11 years (at $0.016/kWh) down to about 1.5 years (at

$0.162/kWh), with 99% of the ballasts paying back the initial investment in less than 8 years (the

mean ballast service life). The payback period for the ERS ballast compared to the baseline

ranges from just over 14 years (at $0.016/kWh) to just over 2 years (at $0.162/kWh), with 97%

of the ballast paying back in less than 8 years.

1This differs from the value of $174 in LBNL 1996 because the average commercial electricity price there is

assumed to be $0.075/kWh.

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FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
IV. Sensitivity of LCC and Payback to Annual Lighting Hours

A. Data

The annual lighting hours distribution was developed from Xenergy's XENCAP database

(Xenergy, 1995 ). We received an extract from Xenergy 's lighting equipment database,

aggregated to the 3-digit zip code level. This database contains data from audits of over 32,000

buildings across the U.S., and gives information about operating hours and equipment counts for

various lighting equipment types in the buildings audited. We used data from audits performed in

1990 through 1995 to approximate current practice. These data were used to obtain a count of

four-foot fluorescent lamp ballasts (of which the 2-lamp F4OT12/ES ballast is an example) found

in each of 21 operating hour "bins" (0-8 hrs/wk, 8-16 hrs/wk, etc., corresponding to 0-417 hrs/yr,

417-834 hrs/yr, etc.). This distribution was weighted by building type using data in the 1992

Commercial Building Energy Consumption Survey (CBECS) (EIA 1994).

The absence of a bar for the first lighting hours bin in Figure 5 is an artifact of the

sampling process. As described in Appendix B, in a stochastic simulation each distinct value (or

level) of an input distribution is selected based upon the associated probability, which in this case

is approximated by the percent of the ballast counts within a specific lighting hour level. The

relative frequency of the first lighting hour level is 0.002, and our simulation process picked no

values in the first lighting level.

Figure 6 presents the cumulative probability distribution of the annual lighting hours. It

represents the cumulative probability (y-axis) that lighting hours will be less than or equal to a

certain value (x-axis). Table 3 presents the same information. This cumulative probability

distribution can be used to determine the percent of ballasts operating in a specific interval of

annual lighting hours. For example, the percent of ballast operating between 2000 and 7000

hours per year is 89% (96% operating at less than 7000 hours/year, minus 7% operating at less

than 2000 hours/year).

B. Results

A scatter plot (Figure 7A) of the sensitivity analysis of LCC change to lighting hours

shows a non-linear relationship between the input and the output as well as the points at which the

curves cross the zero net saving line (LCC change = 0). Figure 7A shows that when an EEM

ballast is replaced with a CC ballast, a net savings in LCC occurs for annual operating hours

above approximately 1000 hours/year. A net savings in LCC occurs for operating hours above

2700 hours/year when the EEM ballast is replaced with an ERS ballast. Combining the

sensitivity scatter plot (Figure 7A) with the input distribution histogram (Figure 7B ), we can get

an estimate of the percent of the ballasts in the commercial sector having a net life-time saving for

each of the design options as illustrated in Figure 7B and shown in the chart at the bottom of the

same page.

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FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
For the CC ballast compared to the EEM, the LCC change ranges from a net increase of

about $1.60 (at 834 hrs/yr) to a net reduction of about $8 (at 8760 hrs/yr). A net reduction in

LCC will result for 98.5% of the ballasts, that is those operating more than 1000 hrs/yr. For the

ERS ballast compared to the EEM, the LCC change ranges from a net increase of just above $8

(at 834 hrs/yr) to a net decrease of approximately $6 (at 8760 hrs/yr). A net reduction in LCC

results for 76% of the ballasts, namely those operating more than 2700 hrs/yr. .

The exact same procedures were implemented for the sensitivity of the payback period to

annual lighting hours. Figure 8 and Table 4 present the results. For the CC ballast compared to

the EEM, the payback period ranges from just over 14 years (at 834 hrs/yr) to under 2 years (at

8760 hrs/yr). The payback is less than 8 years for 96% of the ballasts, those operating at less than

about 1500 hrs/yr. For the ERS ballast compared to the EEM, the payback period ranges from

approximately 16 years (at 834 hrs/yr) to just over 2 years (at 8760 hrs/yr). The payback period

is less than 8 years for 76% of the ballasts , that is those operating for less than about 2500

hrs/yr..

V. Summary of Results

This sensitivity analysis shows how the calculated LCC change and payback period for

switching from EEM ballasts to more efficient CC or ERS ballasts is affected by variability in two

important input parameters, electricity price and annual lighting hours.

We find that for the range of commercial electricity prices found in the U.S., substituting

CC ballasts for the baseline EEM ballasts would result in a change of LCC ranging from an

increase of $3 to a decrease of $21, with 98% of the sales-weighted commercial electricity prices

resulting in a reduction in LCC. The payback period for the EEM to CC switch ranges from

approximately 11 years to about 1.5 years. Switching from EEM to ERS ballasts results in a LCC

change ranging from an increase of about $12 to a decrease of $25 for the range of electricity

prices, with 64% of sales-weighted electricity prices resulting in a reduced LCC. The payback

period for this switch ranges from just over 14 years to just over 2 years.

For the range of annual operating hours examined, substituting CC ballasts for EEM

ballasts results in an LCC change ranging from an increase of approximately $ 1.60 to a decrease

of $8, with 98.5% of ballast operating hours leading to a reduction in LCC. The payback period

for the EEM to CC switch ranges from about 14.5 years down to approximately 1.5 years.

Switching from EEM to ERS results in an LCC change ranging from an increase of $8 to a

decrease of $6, with 76% of ballast operating hours leading to a reduction in LCC. The payback

period for changing EEM ballasts to ERS ranges from just over 16 years down to just over 2

years.

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FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
REFERENCES :

Burmaster, David E. and Paul D. Anderson , 1994. Principles of Good Practice for the Use of

Monte Carlo Techniques in Human Health and Ecological Risk Assessments. Risk Analysis,

Vol. 14, No. 4

EIA 1994. Commercial Buildings Characteristics 1992: Commercial Buildings Energy

Consumption Survey. Energy Information Administration, United States Department of Energy.

DOE/EIA-0246(92).

EIA, 1995a. Financial Statistics of Major U.S. Investor-Owned Electric Utilities 1994.

DOE/EIA-0437(94)/1, Energy Information Administration U.S. Department of Energy,

Washington DC.

EIA, 1995b. Financial Statistics of Major U.S. Publicly Owned Electric Utilities 1994.

DOE/EIA-0437(94)12, Energy Information Administration, U.S. Department of Energy,

Washington DC.

Iman, R.L. and W.J. Conover, 1980. Small Sample Sensitivity Analysis Techniques for

Computer Models, with an Application to Risk Assessment. Communications in Statistics,

A17:1749.

Lawrence Berkeley National Laboratory, 1996. Draft Report on Potential Impact of Possible

Energy Efficiency Levels for Fluorescent Lamp Ballasts. LBNL, Berkeley, CA, 94720.

Lumina Decision Systems, 1996. Analytica User Guide. Lumina Decision Systems, Los Altos,

CA 94022.

Morgan, M. Granger and Max Henrion, 1990. Uncertainty: A Guide to Dealing with

Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press.

Ripley, Brian D, 1987. Stochastic Simulation. John Wiley & Sons.

Smith, Roy L, 1994. Use ofMonte Carlo Simulation for Human Exposure Assessment at a

Superfund Site. Risk Analysis, Vol. 14, No. 4.

Xenergy, 1995. Extract from XENCAP database, received by LBNL from Xenergy, Cambridge

Massachusetts, January 1995.

6

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure U.S. Commercial Sector Electricity Prices

0.04

0.03

0.011

------------------------- - ---------------------------------

.» ................»..»-4-

i

0

0

I

0.05

Statistics

0.15

Max 0.1616

Mean 0.0784

Median 0.0719

Min 0.0162

Std. Dev. 0.0226

$/kWh

0.2

Electricity Price ($/k h)

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 2 Cumulative Probability of Commercial Electricity Price

..................................... ........

»...}........ . ........ ..... ........ ..

}- ....... ........ .......

. . ............ .......... ........ . ... ........... ......... ^.. ......

.....» .. ....... ..... ......... .}.._.__....p. .

....... ...... ....... ..

b

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Electricity Price ($/kWh)

T t7 ----»--•.».»..... ----------- 8...»...... d----------- - ----------- ;------------ P ---------- ------------

---------- ----- -------- - - }- - ------- -- ........ ........ ..

....................... ............ :.........

. .

}-

Table 1 Cumulative Probability Table

of Electricity Price

Electricity Price

Less an ($/kWh)

0.02

0.'04

0.06

0.08

0.10

0.12

0.14

0.16

Percent of

Sales

1%

2%

27%

64%®

80% - 94%

99%

99%

0.18 1 100% 1

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 3A .'.L LCC C j a Two --_-" 1174OT12ES

Net Net C - a,mct^cit Ice Depend um

...E ® EEM . .....:..............................:.............. ......_......

$0.064/kWh

0.05

ctrl ku

0.15

WkWh)

0.2

Figure For , Net Savings Occur Above - 8` IfkWh

For Net Savings Occur Above w. 6/ h

I $0.064/kWh

0.04T°

U

0

0

-----•------ ------... ..._ ............................Y............................

........................ .......... .............

.............................. !............................... ........... ---------------------...

I

AN w

Percent Having,-: ".`-

CC ®EE

ERS ®EE -

0.03

$0.038/kWh

0.01 y

-.1

Electricity Price ($/kWh)

Net Sa

0.15 0.2

36% WM^ 64- /

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 4 Sensitivity of PAYBACK To Electricity Price

Two Lamp T1 S

1 6T---*--* --- * ------ * ................................... ............_..............s_ .-........--------..-...-

_..._..._... ------------------•....-••-• ------..........----..----•

............................. ............................. ............................ ...........................

............................. .......................... ........................................................

...........e ®........^ ...................... ...........................

, ---------- .......................................

..

.

..............................

Key

0 1

+ 2

0.05

.......... ................ .........

0.1 0.15

Electricity Price (/kWh)

(CC E

(E - E

Table 2

PAYBACK BY PER

ELE

OF U.S. CO

SALES

0.2

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 5 Ballast Annual. Lighting Hours in U.S.- Commercial Stock

Four Foot Lamp

---- ................ ................ ...... -............ .!_.'.............. --..--.. ... .,

Max 8760

Mean 3791

Median 3754

Min 834

std. Dev. 1479

04

0 20Cr6 1^ 0 6 Ci0 8 00

Lighting Hours

hours/ rs

10K

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 6 Cumulative Probability of Annual Lighting Hours

Four Foot Lamp

- -----------

0.7

0.6

0.5

0.4

-------------------------------------------------------- --- ---------------------

---------- -------------------------------- ---- -------------- ----------- ----------- ------------------

------------------------ ------------------- -- --------------------- -----------

------------- - ----------

------------------------

----------------- ---- ----------------

----------- ---- z ---- -------- --------------------------------------------

----------------------- ---------------------------------------------------------------- - -----------------------

2000 4000 6000 8000 10K

Lighting Hours (Hours/year)

Table 3 Cumulative Probability Table

of Annual Lighting Hours

Lighting Hours

Less Than

(Hours/year)

Percent of Ballasts

in Commercial

Stock

1000 1.5%

2000 7%

3000 30 %

4000 57 %

5000 80 %

6000 92%

7000 96%

:

..8000 98%

8760 100%

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Figure 7A TOTAL LCC CHANGE: Two Lamp F'40T12 S

Net Savings or Net Costs Depend on Lighting Hours

0

0

10

6

6

4

2

0

-2

-41

-61

--------------------- ------------------- ---------------------

------------- ------- ---------------------------

----------- ------------ ------------ ---------- --__...... -....... ... ----------------------

»----------•-------- _ ._ _.».._. _ ...__-»----•------ ------ _------- •-------

2700 hrs --------------------- d--------

----•-------------- ----------- ----------- ------------------------

------------------------

a

---------------

_-----»--.--------- -ti ...... ,»^.----, »»...... »..........

.

---------- -- - »».. ..---- .. ---

.

Ens - EE11--

---------------------

---------- - --•a-- --._M-_.-.*. _,^. ¢ -- - -----------

.._- -------------- ...... --- ------------------ ----- ------------- 1000 hrs

.

- ------ - - -----

CC - EEM 13 93

--_--------------------------------- - -------- -»------ --------- -»a---®- .et --a --^ ------------

2000 4000 6000 8000 10K

Lighting Hours (Hours/year)

figure 7B For CC - EE M, Net Savings Occur Above 1000 Hours/Year

For ERS - EEM, Net Savings Occur Above 2700 Hours/Year

0.14T ----»_.._.»;

0.12

0.02 ------------

1000 hrs -

-

Lighting Hours ours/year

Percent Having:

CC - EEM:

ERS - EEM:

Net

24%

avings

76%

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
t

, Figure 8 itivity of PAYBACK To Lighting Hours

Two Lamp F4 12 S

.cn

to

1 -»«__^».«-----«a. ...«_._...«--°•--- a -•--._ ..»«»..«....P»_«__----«-----•- - «-------- ««.

16 »---- _«_«_.._-_°.-._.._._

12 -._._««__.«μ «......«.««.-______.. ._«.« .............

10

.6

2

8«.«.__•---.»«» «@:..._.««..«... ««.... «««...... {_{p. __......... «-.«..._ ..............

-- --------- - --- - -----

2.000 4000

Key Lighting ours

a 1 (CC - EE

+ 2 (ERS - E

6000 8000

(Hours/year)

10K

Table 4 PAYBACK BY PERCENT OF U.S. BALLASTS

COMMERCIAL STOCK

(BASED ON DISTRIBUTION OF ANNUAL LIGHTING HOURS)

PAY

TH

BA

AN

CK LESS

(Years)

CC - EEM ERS - EEM

2 8% 0%

4 64% 17%

6 91% 58%

8 96% 76%.

10 98% 91%

12 99% 96%

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Appendix A

Lifecycle Cost and Payback Period Inputs

The following input parameters were used to calculate the total lifecycle cost for each

option, the change in LCC between the design options and the baseline, and the payback period

for each option.

The values shown below for operating hours and electricity price are the values at which

each of these parameters was held constant while the other was varied . Electricity price was

varied over the range shown in Figure 1. Annual operating hours were varied over the range

shown in Figure 5.

Table A.I. Input Parameters for LCC analysis

Parameter Value

Annual lighting hours 4230

Baseline ballast consumer price ($) $12.44

Design 1 (CC) ballast consumer price ($) $18.86

Design 2 (ERS) ballast consumer price ($) $28.94

Discount rate 0.08

Electricity price ($/kWh) $0.075

Labor rate, electrician's helper $29.35

Labor rate, electrician $42.10

Lamp price (both lamps, $) $3.28

Lifetime of ballast (hours) 33000

Normalized fixture watts, baseline option 72.7

Normalized fixture watts, design option 1 (CC) 65.6

Normalized fixture watts, design option 2 (ERS) 61.9

Relamp life (hours) 19000

Time to change ballast (minutes) 30

Time to change lamp (minutes) 23.5

A-1

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06
Appendix B

Monte Carlo Simulations

Simulation can be defined as the repetitive analysis of a mathematical model. The Monte

Carlo simulation refers to a class of simulation techniques for decision-making in which

probability distributions are used to obtain input parameters when the model is being evaluated

(Ripley, 1987; Smith, 1994; Burmaster and Anderson, 1994). The distributions are used to

generate random outcomes for each parameter that is considered to be important and has a

probabilistic value.

A distribution can either be approximated by a theoretical one such as a normal or

exponential distribution curve, or represented by an empirical probability table in which the

probabilities of occurrence of various distinct values (levels) are estimated by their relative

frequencies in sampled data or weighting factors from the existing data sources. These random

outcomes are then used in calculations of the decision criterion (or measure of merit), which in

this analysis are life cycle cost and payback period. Since other parameters are entered as fixed

values in all calculations, the variability in the outputs (LCC and Payback) calculated by Monte

Carlo simulation is due entirely to the variation in the parameter under study.

The core of the Monte Carlo simulation is that the outcomes of the parameter under study

must be randomly selected from the distribution representing it. Two sampling methods widely

used in computer simulations are the Simple Monte Carlo and the Latin Hypercube (Morgan and

Henrion, 1990). The Simple Monte Carlo sampling first draws a cumulative probability curve

from a specified distribution (theoretical distribution or empirical probability table). Then a

random generator is used to produce a set of independent, uniformly distributed numbers, R,,

(i.e., between 0 and 1).Through the rule of a reserve function, a set of the input values Xn

corresponding to the cumulative probability curve can be obtained for a given sample size n

(Lumina Decision System, 1996).

However, in a parametric sensitivity study, the ultimate goal of the simulation is not the

randomness but the resulting equidistribution (or equiprobable) properties of the sets of values in

the parameter space (Morgan and Hendon, 1990). To improve this equidistribution a technique,

called Latin Hypercube Sampling (LHS), which uses a stratified sampling approach, was

originated (Iman and Conover, 1980). It divides each input distribution into n (sample size)

equiprobable intervals. A single value is sampled at random from within each of these intervals,

based upon the probability distribution. This produces a sample of n values for each input

distribution which are more uniformly spread out than for strict random sampling. In a variation

of LHS, Median Latin Hypercube, the median of each of the n intervals is chosen instead of a

random value so that a more uniform sampling is produced.

FLUORESCENT LAMP BALLASTS WORKSHOP JUNE 20, 1996 COMMENT 06