Risk Analysis Using Simulation

Risk Analysis using Simulation Page 1 of 29

Risk Analysis using simulation

Introduction Simulation is a process of understanding behavior of a system or

evaluating the various strategies for the operation of the system.

Simulation calculates multiple scenarios of a model by repeatedly sampling values from the probability distributions for the uncertain

variables.

Simulation involves the generation of the artificial history of the system and observation of the artificial history to draw inferences concerning the

operating characteristics of the real system that is presented.

Why Simulation? When it is too costly to do physical studies on the system itself (e.g., trying

alternative layout of a factory, building new facility)

The corresponding analytic models are too complicated to study (e.g. a queuing/transportation network)

When uncertainty, non-stationarity dominates the output results.

Steps in Simulation Model development: System defination, objectives to study, decision

variables, output measures, input variables and parameters.

Data collection: Collect data from the real system, obtain probability distributions of the input parameters by statistical analysis

Model translation: Translate the physical model into computer simulation software.

Simulation runs & Output Data Analysis: Run the simulation upto satisfactory level and use statistical analysis of the ouput data to estimate

the performance measures.


Model Verification and Validation: z Verification: Ensuring that the model is free from logical errors. It

does what it is intended to do.

z Validation: Ensuring that the model is a valid representation of the whole system. Model outputs are compared with the real system

outputs.

Optimization/Experimental Design: Analyze alternative strategies on the validated simulation model.

Sensitivity analysis: Analuze the input data sensitivity in performance measure of the system. Cheking model robustmess.

Tools for simulation @Risk Crystal Ball MatLab Minitab Excel Spreadsheet And so on, etc

Simulation using Excel Spreadsheet

Basic Excel Skills To facilitate the understanding of simulation using Excel, it is necessary to be

familiar with the following features of Excel

Copying formula and cell references Function: It is used in performing special calculations in the Spread sheet

cells. Some of the more common functions that are usually used in

simulation models are including:

9 MlN (range) - finds the smallest value in a range of cells. 9 MAX (range) - finds the largest value in a range of cells 9 SUM (range) - finds the sum of values in a range of cells


9 AVERAGE (range) - finds the average of the values in a range of cells

9 STDEV(range) -finds the standard deviation for a sample in a range of cells

9 AND (condition 1, condition 2, ) -a logical function that returns TRUE if all conditions are true, and FALSE if not.

9 OR (condition1, conditiott2 . . .) - a logical function that returns TRUE if any condition is true, and FALSE if not.

9 IF (condition, value if true, value if false) -a logical function that returns one value if the condition is true and another if the condition

is false.

9 VLOOKIUP (value, table range column number) -looks up a value in a table.

Charts and graphs Excel has many other functions for statistical, financial and other

applications

Others useful features Split Screen: Division of displayed spreadsheet screen Paste Special: Controlling of direct pasting into the cells Column and Row width: Customization of Column and Row size. Displaying Formulas in Worksheets: Showing actual formula in

spreadsheet

Displaying Grid Lines and Row, and Column Headers for Printing Filling a Range with a Series of Numbers. Comment Boxes

Building Simulation Models in Excel However a good design is essential to user understanding. Any good simulation

model design in any spreadsheet should be included with the following features.

A descriptive title


A separate input data section area A separate working space A separate output section that provides the model results. Appropriate formatting such as in currency or comma formatting. Complex calculations should be divided into several cells to minimize the

chances of error and enhance understanding.

Comments should be placed next to formula cells or in comment boxes for explanation, if appropriate.

Example: A Simulation Model for Dave's Candies

Daves Candies is a small family owned business that offers gourmet chocolates

and ice cream fountain service. For special occasions such as Valentines day,

the store must place orders for special packaging several weeks in advance from

their supplier. One product, Valentines day chocolate massacre, is bought for

$7,50 a box and sells for $12.00. Any boxes that are not sold by February 14 are

discounted by 50% and can always be sold easily. Historically Daves candies

has sold between 40-90 boxes each year with no apparent trend. Daves

dilemma is deciding how many boxes to order for the Valentines day customers.

If the order quantity, Q is 70, what is the expected profit?

Formulation: Selling price=$12

Cost = $7.50

Discount price=$6

| If *DQ Profit=selling price*Q-cost*Q................................................[2]

*D=Demand & Q=Quantity


Simulation model in Excel

Model description

The input data are given in Column A and Column B. Simulation results are shown in Columns D through F. IF function is used in Column F to calculate the profit based on order

demand and quantity relation using the equations [1] and [2].


Each row in the results table represents one trial of the simulation. Demand in each row was generated from probability distribution. Since selling boxes range 40-90 boxes/year, the values in Column E were

generated by rolling a die and entered into the worksheet to verify the

formulas for profit in column F.

Probability & Statistic in Simulation

Fundamentals Probability - Probability is a measure of how likely a value or event is to

occur. It can be measured from simulation data as frequency by

calculating the number of occurrences of the value or event divided by the

total number of occurrences. This calculation returns a value between 0

and 1 which then can be converted to percentage by multiplying by 100.

Probabilistic Risk Assessment (PRA) - A risk assessment that yields a probability distribution for risk, generally by assigning a probability

distribution to represent variability or uncertainty in one or more inputs to

the risk equation.

Probability Density Function (PDF) - A function representing the probability distribution of a continuous random variable. The density at a

point refers to the probability that the variable will have a value in a narrow

range about that point.

Probability Distribution - A probability distribution is a set of probabilities associated with all possible outcomes of an uncertain event.

Probability Mass Function (PMF) - A function representing the probability distribution for a discrete random variable. The mass at a point

refers to the probability that the variable will have a value at that point.

Random Variable - A variable that may assume any value from a set of values according to chance. Discrete random variables can assume only a

finite or countable infinite number of values (e.g., number of rainfall events

per year). A random value is continuous if its set of possible values is an

entire interval of numbers (e.g., quantity of rain in a year).


Central Tendency Exposure (CTE) - A risk descriptor representing the average or typical individual in a population, usually considered to be the

mean or median of the distribution.

Confidence Interval - Confidence intervals refers an interval that provides a range where the exact or true probability value of parameters may lie.

Confidence Limit - The upper or lower value of a confidence interval. Cumulative Distribution Function (CDF) - Obtained by integrating the

PDF, gives the cumulative probability of occurrence for a random

independent variable. Each value c of the function is the probability that a

random observation will be less than or equal to c.

Frequency Distribution or Histogram - A graphic (plot) summarizing the frequency of the values observed or measured from a population. It

conveys the range of values and the count (or proportion of the sample)

that was observed across that range.

Monte Carlo Analysis (MCA) or Monte Carlo Simulation - A technique for characterizing the uncertainty and variability in risk estimates by

repeatedly sampling the probability distributions of the risk equation inputs

and using these inputs to calculate a range of risk values.

Point Estimate - In statistical theory, a quantity calculated from values in a sample to estimate a fixed but unknown population parameter. Point

estimates typically represent a central tendency or upper bound estimate

of variability. [Source: U.S. EPA]

Types of Distribution: Continuous distribution: A probability distribution is continuous when

any value between the minimum and maximum is possible (has finite

probability). For example, an uncertainty function describing the possible

annual rainfall in Ithaca, New York next year would be a continuous

function since any value between 0 and some upper limit is possible.


Discrete distribution: A discrete probability distribution has only a finite number of possible values between the maximum and minimum. For

example, an uncertainty function describing the outcome of a coin toss is

discrete since only two values are possible: heads or tails.

Descriptive Statistics

Parameter - A value that characterizes the distribution of a random variable.

Parameters commonly characterize the location, scale, shape, or bounds of the

distribution. For example, a truncated normal probability distribution may be

defined by four parameters: arithmetic mean [location], standard deviation

[scale], and min and max [bounds]. It is important to distinguish between a

variable (e.g., ingestion rate) and a parameter (e.g., arithmetic mean ingestion

rate).

Measures of Central Tendency: Mean, Median, & Mode

Mean: The mean of a set of values is the sum of all the values in the set divided by the total number of values in the set.

=

=N

ii

xN 11

Median: The median is the middle data point when the data is ordered. The middle value (50th percentile) in the ordered sequence of measured

values. For highly skewed data sets the median can give a better


representation than the mean of the middle of the data distribution. The

median is not as comprehensive a measure of the data set as the mean.

Mode: The most likely value or mode is the value that occurs most often in a set of values. In a histogram and a result distribution, it is the center

value in the class or bar with the highest probability.

Measures of Variation

Standard deviation: The standard deviation is a measure of how widely dispersed the values are in a distribution. Equals the square root of the

variance.

1

)(1

2

==

N

xN

ii

Variance: The variance is a measure of how widely dispersed the values are in a distribution, and thus is an indication of the "risk" of the

distribution. It is calculated as the average of the squared deviations about

the mean. The variance gives disproportionate weight to "outliers", values

that are far away from the mean. The variance is the square of the

standard deviation.

Range: The range is the absolute difference between the maximum and minimum values in a set of values. The range is the simplest measure of

the dispersion or "risk" of a distribution

Kurtosis: Kurtosis is a measure of the shape of a distribution. Kurtosis indicates how flat or peaked the distribution is. The higher the kurtosis

value, the more peaked the distribution. The coefficient of kurtosis is

computed as:

41

4)(

=

=N

iix

CK


Skewness: Skewness is a measure of the shape of a distribution. Skewness indicates the degree of asymmetry in a distribution. Skewed

distributions have more values to one side of the peak or most likely value

one tail is much longer than the other. A skewness of 0 indicates a

symmetric distribution, while a negative skewness means the distribution

is skewed to the left. Positive skewness indicates a skew to the right. The

coefficient of skewness is computed as:

31

3)(

=

=N

iix

CS

Coefficient of Variation: It is the ratio of standarad deviation and the sample mean.

=CV

Correlation: It is measure the interdependences of two variables. The sample correlation coefficient is computed as:

yx

N

iii

SSN

yyxxr

)1(

))((1

==

Random numbers & Probability distribution in Simulation It is a numerical description of the outcome of some experiment. Example

of outcome of experiment could be profit values, failure rate or chance of

an event occurrence, etc.

It can generated from a probability distribution. In simulation terminology, a random number is one that is uniformly

distributed in between 0 and 1.

Recall from statistic that uniform probability distribution characterizes a random variable for which all outcomes between a minimum value a and a

maximum value b are equally likely.


Some Important Continuous Distribution Exponential Distribution: The exponential distribution is characterized

by the single parameter (.) The exponential random variable T (t > 0) has a probability density function

f(t) = e-t for t > 0

Where,

= mean number of occurrences per unit time t= number of time units until the next occurrence The cumulative distribution function is

00

( ) 1t

tx x tF t e dx e e = = = for t > 0. Mean: = and

1

Variance: 212

=

Lognormal Distribution: Distribution parameter (, )


Normal Distribution: Distribution parameter (, )


Triangular Distribution : Distribution Parameter (min, m.likely, max)


Uniform Distribution : Distribution parameter (min, max)

Risk Analysis using simulation Page 18 of 29

Some Important Discrete Distribution Binomial Distribution: Distribution parameter (n, p)


Poisson distribution: Distribution parameter ()


Generating random variables in Excel Two properties of probability distribution are useful to generate random numbers:

1. the probability of any outcome is always between 0 and 1 and

2. the sum of the probabilities of all outcomes adds to 1

Random numbers from discrete distribution

Divide the range from 0 to 1 into intervals that correspond to the probabilities of the discrete outcomes.

Use VLOOKUP function (look up value, table array, column index number) or Random number generation function in excel.


Example : Random number generation for Dave's Candies


Random numbers from continuous distribution

Inverse transform method is used in case if there is no direct function to generate random numbers.

Inverse transform method uses the following steps: 1. generate a random number from the Uniform distribution:

u=Uniform(0,1),

2. Calculate inverse cumulative distribution function (ICDF). In Excel, the

function name to calculate the ICDF for Normal distribution NORMINV

and for LogNormal distribution is Lognormal (LOGINV). RAND function

in excel can also be used to generate random numbers from the Uniform

distribution, and apply the built-in functions to calculate the ICDF.

Example: For example, the following formula will return the inverse CDF of the Normal

distribution with mean=1 and standard_deviation=2 evaluated at p=0.2:

=NORMINV(0.2; 1; 2)

Replacing 0.2 with RAND will yield the Normal random number generation formula:

=NORMINV(RAND(); 1; 2)

Similarly for LogNormal distribution the function is

LOGINV (probability, mean, Standard_dev)

For triangular distribution, It may need to use If function to generate random numbers

in Excel. The structure of the IF function is:

=IF(expression, what is returned if true, what is returned if false)

If function first determines which side of the distribution corresponds to the random

number and then evaluates the appropriate formula.

=IF (RAND () < (Mode Min)/(Max Min),Left formula, Right formula)

Let assume,

In a triangular distribution, if Min is a, Mode is b and Max. is c, then

bxaacabRANDaFormulaLeft += )(*)(*() cxbacbcRANDcFormulaRight = )(*)(*())1(


Monte-Carlo Simulation

Figure 1: Monte Carlo analysis to a model.

Monte-Carlo simulation uses the following steps for assessing parameters uncertainty:

i. Select a distribution to describe possible values of a parameter.

ii. Generate data from this distribution.

iii. Use the generated data as probable values of the parameters in the model to

produce output.

Monte-Carlo simulation on Excel Monte-Carlo simulation on excel spreadsheet follows the following steps::

1. Develop the spreadsheet model including a separate input and output

region.

2. Generate random numbers from the assigned probability distribution and

use the random data into the appropriate formula in the simulation model.

3. Repeat step 2 until to obtain the satisfactory output to create a distribution of

results.


4. Make a summary for the descriptive statistics and collect output data in a

frequency distribution or histogram for analysis.

Example: Monte-Carlo simulation on excel for Dave's Candies problem

-

Output

-

-

Example: Methyl-mercury has been inadvertently released to a

Monte Carlo simulation. With 90% (subjective) confidence, what

maximally exposed individual? The following table shows the (sub

distributions for this problem.

nearby lake. Use

is the risk to the

jective) probability


Solution: Step 1: Selecting distribution for model parameters 9 Table 1 is used as input for a Monte Carlo simulation

Table 1: Input data for MCS

Input Region Parameter Distribution Min. Max. Mean/Mode STD Fish concentration(CF), mg/kg Normal 2.06E-01 4.22E-02

Intake of Fish (IF),kg/d Uniform 2.00E-02 1.30E-01 6.50E-02

Methyl mercury RfD (RfDMM), mg/kg-d

Triangle 1.50E-04 3.00E-03 3.00E-04

Body mass (BM), kg Triangle 4.50E+01 1.20E+02 7.00E+01

Step 2: Generate data from this distribution.

9 Table 3 shows the generated dated for this example: Table 3: Random number generation

Sample Number CF IF RfD BM

1 0.209 0.032 5.39E-04 59.203 2 0.268 0.047 9.61E-04 66.066 3 0.203 0.052 2.35E-03 109.907 4 0.199 0.026 2.51E-04 95.667 5 0.140 0.123 3.71E-04 94.443 6 0.230 0.083 6.53E-04 60.822 7 0.191 0.025 2.77E-03 78.394 8 0.182 0.102 1.00E-03 80.310 9 0.197 0.095 1.36E-03 57.432

10 0.220 0.041 6.73E-04 84.090 11 0.218 0.099 7.61E-04 59.938 12 0.198 0.024 1.57E-03 88.117 - - - - - - - - - - - - - - -

96 0.265 0.098 2.90E-04 90.694 97 0.213 0.109 4.57E-04 84.283 98 0.233 0.113 1.55E-03 64.637 99 0.265 0.070 4.98E-04 93.816

100 0.161 0.066 6.63E-04 87.898


Step 3: Use the generated data as probable values in the model to calculate HQ.

9 Used model to calculate the Hazard Index (HI):

RfDBMICHI FF

=

Table 3: MCS to calculate HI using 100 iterations

Sample Number CF IF RfD BM HI

1 0.209 0.032 5.39E-04 59.203 2.12E-01 2 0.268 0.047 9.61E-04 66.066 1.97E-01 3 0.203 0.052 2.35E-03 109.907 4.07E-02 4 0.199 0.026 2.51E-04 95.667 2.13E-01 5 0.140 0.123 3.71E-04 94.443 4.92E-01 6 0.230 0.083 6.53E-04 60.822 4.79E-01 7 0.191 0.025 2.77E-03 78.394 2.16E-02 8 0.182 0.102 1.00E-03 80.310 2.32E-01 9 0.197 0.095 1.36E-03 57.432 2.40E-01

10 0.220 0.041 6.73E-04 84.090 1.59E-01 11 0.218 0.099 7.61E-04 59.938 4.71E-01 12 0.198 0.024 1.57E-03 88.117 3.48E-02 13 0.142 0.037 2.05E-03 64.939 3.92E-02 14 0.268 0.099 1.83E-03 87.285 1.66E-01 15 0.217 0.070 2.17E-03 76.220 9.23E-02 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

94 0.216 0.119 2.01E-03 69.327 1.85E-01 95 0.223 0.078 2.68E-03 74.652 8.68E-02 96 0.265 0.098 2.90E-04 90.694 9.83E-01 97 0.213 0.109 4.57E-04 84.283 6.04E-01 98 0.233 0.113 1.55E-03 64.637 2.63E-01 99 0.265 0.070 4.98E-04 93.816 3.95E-01

100 0.161 0.066 6.63E-04 87.898 1.83E-01


Output Summary: The output from MCS is shown in Fig 1. The frequency distribution for the output from MCS (Monte Carlo simulation) is

shown in Fig 2.

From Monte Carlo simulation, the confidence interval for 90% & 95% are obtained [0.193, 0.251] & [0.188, 0.256] respectively.

This implies that after taking into account the uncertainties on the parameters, one is highly confident (at a subjective level of 95%) that the true HI should lie

between 0.188 and 0.256.

Since the 95% upper confidence limit of HI is still below 1, there is high confidence that the maximally exposed individual for this scenario is not

exposed to an unacceptable level of risk, and remediation should not be

warranted.

Figure 1: MCS output


Forcast: Hazard Index (HI)

0

5

10

15

20

25

0.01

0.11

0.21

0.31

0.41

0.51

0.61

0.71

0.81

0.91

1.01

Hazard Index

Freq

uenc

Figure 2: Frequency distribution of HI

References: Hammonds, J. S. , Hoffman, F. O., and Bartell, S .M., An Introductory Guide to

Uncertainty Analysis in Environmental and Health Risk Assessment managed by

MARTIN MARIETTA ENERGY SYSTEMS, INC. for the U.S. DEPARTMENT OF

ENERGY

James R. Evans, David Louis Olson, Introduction to Simulation and Risk Analysis,

Prentice Hall PTR, Upper Saddle River, NJ, 2001

http://www.cas.lancs.ac.uk/glossary_v1.1/prob.html#pdf

http://www.ltcconline.net/greenl/courses/201/probdist/zScore.htm

Risk Analysis using simulation

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