Monte carlo inventory simulation excel

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MODULE 1: INTRODUCTION TO SIMULATION

Module outline:

• What is Simulation?

• Simulation Terminology

• Components of a System

• Models in Simulation

• Typical applications

• References

WHAT IS SIMULATION? simulation may be defined as a technique that imitates the operation of a real world

system or processes as it evolves over time. It involves the generation of an artificial

history of the system and observation of that artificial history to obtain information and

draw inferences about the operating characteristics of the real system. Simulation

educates us on how a system operates and how the system might respond to changes. It

enables us to test alternative courses of action to determine their impact on system

performance. Before an alternative is implemented, it must be tested. Although

performing tests with the “real thing” would be ideal. This is seldom practically feasible.

The cost associated with changing/improving a system may be very high both in the

term of capital required to implement the change and losses due to interruption in

production operations and other losses. In most cases experimentation with the

proposed alternative is practically impossible. In addition, as the cost of proposed

changes (alternative solutions) increase, so does the cost of physically experimenting.

As an example, suppose a heavy-duty conveyor is being considered as an alternative to

the existing material handling method (by trucks) for improving productivity and

speeding up the production operations in a factory (seeFigre3). It is obvious that

installing the proposed conveyor on a test basis would probably not be cost effective.

Therefore, experimentation with alternative configurations would be practically

impossible. In stead, experimentation with a representative model of the system would

probably make more sense.

Simulation is a means of experimenting with a detailed model of a real system to

Determine how the system will respond to changes in its environment, structure, and its

underlying assumption [Harrel (1996)]. Management Scientist uses a wide variety of

analytical tools to model, analyze, and solve complex decision problems. These tool

include linear programming, decision analysis, forecasting, Queuing theory and

Alternative 1: Use lift-truck

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Point A Point B

(Warehouse) (Factory)

Alternative 2: use a conveyor

Point A

(warehouse ) . . . . . . . . Point B

(Factory)

Figure 3. Material handling alternatives

Simulation. Many of these tools often require the user to make some simplifying model

assumptions or they apply only to special types of problems. For instance, linear

programming applies only to well-structured situations that can be modeled with linear

objective function and linear constraints. In addition, we assume that all data are known

with certainty. Most real world problems exhibit significant uncertainty, which

generally is quite difficult to deal with analytically.

For situations in which a problem does not meet the assumptions required by standard

analytical modeling methods, simulation can be a valuable approach to modeling and

solving the problem. A recent survey of management science practitioners show that

simulation and statistical have the highest rate of application over all other analytical

tools. (1). It should be noted that simulation should not be used indiscriminately as a

substitute for sound analytical models. Many situations exist where analytical tools are

the more appropriate. The modelers need to understand the advantages and

disadvantages of different methods and use them appropriately.

SIMULATION TERMINOLOGY

The art and science of simulation uses a unique set of vocabulary of terms which enables

practitioners communicate specific concepts. We must, therefore consider the meaning

of these terms before we can begin studying actual simulation techniques. The following

list contains the key words and concepts that every modeler should know.

System:

A system as defined here is a group of objects that are joined together in some regular

interaction or interdependence for the accomplishment of some purpose. An example is a

production system manufacturing Television units. The machines, components parts, and

workers operate jointly along the assembly line to produce a good quality television set.

Similarly, the physical facilities of a hospital, its nursing staffs, physicians, and

administrative staff would be an example of a health care system. A jet aircraft is an

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excellent example of a complex system consisting of numerous mechanical, electronic,

chemical and human components. A major corporation, together with its customers and

its suppliers, represent another example of a system containing complex interacting

components

A system is often affected by changes occurring outside of the system. Such changes are

said to happen in the system environment (Gordon 1978). In modeling a system, it is

necessary to determine the boundary between the system and its environment.

Systems can be categorized as Discrete or continuous. A discrete system is one in which

the state variables (see below) change only at a discrete set of points in time. A bank is

an example of a discrete system since the number of customers in the bank (a state

variable), change only when a customer arrives or departs. Figure 1-1 shows how the

number of customers changes only at discrete points in time.

Number of customers 3 -

Waiting in line or Figure 1-1

being served 2 -

1 -

Time

In a continuous system the state variables change continuously over time. An example is

the temperature of a point inside or outside of a steel coil cooling after heat treatment.

Figure 1-2 shows how state variable (temperature) changes over time

Temperature (F)

of a point inside Figure 1-2

a steel coil while

cooling

Time

COMPONENTS OF A SYSTEM

Entity. An object of interest in the system (example: products in an inventory system)

Attribute: A property of an entity (i.e., Weight of the product)

State: The state of a system can be thought of the collection of all variables required to

describe the system at any point in time, with respect to the objective of the

study. The state of the system is determined by assigning a particular value to

each of these variables. In the case of jet aircraft (see above). The state of the

system would be determined by such factors (state variables) as the aircraft’s

Speed, altitude, direction of travel, weather condition, number of passengers

amount of fuel remaining, and operating status. Some of these factors will

remain constant whereas others will vary with time. As a result, the state of the

system can (and often does) change with time. Note that some of these factors

are deterministic, whereas others, like weather conditions, are stochastic.

Event: An instantaneous occurrence that may change the state of the system. In

the

case of the jet aircraft a sudden change in altitude constitutes an event

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Activity: Time-consuming elements of a system whose starting and ending

coincide with event occurrence.

Decision Variables: Those variables whose values can be specified by the

decision maker at the beginning of a problem, independent of other variables.

The value assigned to a decision variable will normally affect the state of the

system under consideration. We can call state variables as dependent variables

and decision variables as independent variables. For instance, in simulating

average queue length in a service station, the number of pumps is a decision

variable while number of people waiting in line is a state variable. Table 1

of the text list some examples of the simulation terminology.

Cause–and-effect relationships: all systems are governed by certain

relationships that describe the interaction between state variables, decision

variables, and system parameters. These relationships may represent physical

laws, statistical correlations, economic principles, and etc. Mathematically, if

we represent sets of state variables, decision variables, and system parameters

as S, X, and P respectively, for a given system the cause –and-effect

relationship can be expressed as:

 (S, X, C) = 0

MODELS.

A model is used to provide some type of description of an actual system. Models can

range from exact physical mock-ups of the system to abstract mathematical

representations. Models of systems may be classified as being physical, graphical, or

symbolic. Physical models also called iconic models may be to the same scale as the

system itself. Example of this sort is an aircraft cockpit model used for pilot training.

Physical models may also be of smaller scale than the system they represent. An Example

is mock-up of building structures used by architects. Some scaled-down physical models

of three-dimensional systems may be two-dimensional, such as scaled templates used in

plant layout design.

Graphical models may be two or three-dimensional representation of systems. They

may be static, such as drawing on a paper, or dynamic such as animated films and

computer graphics. Graphic representations generally enhance communication and

understanding of the abstract models.

Symbolic (mathematical) models are abstract representation of systems and as such

they do not look like the system they represent. In many applications these models are a

more effective way to represent a system because of their ease of construction and

manipulation.

Mathematical models are used to describe the behavior of an actual system. A

simulation model is a particular type of mathematical model of a system. Such models

are comprised of s set of equations that represent the underlying cause-and-effect

relationships within the system.

Suppose the following variables** are used to determine yearly profit for a production

system

P = Gross yearly profit X = Sales volume (# of units)

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S = Sales price per unit

F = Total fixed cost per year including taxes

C = Variable cost per unit

Assuming all other factors could be ignored, we can easily develop an expression

------------------------------------------------------------------------------------------------------ ** these are Decision variables or system Parameters

(mathematical model) for the gross profit for the business as follows:

P= (X)[S – C] –F (1)

Equations (1) constitute a mathematical model for the system. The model is used to

evaluate the state of the system as well as the performance of the system. Assigning a

set of values X, S, C, and F, the model will provide us with quantitative measures of

system’s performance and a set of values for the state variables). In addition, by

specifying different set of values for the four variable (assuming the can take random

values) and evaluating the model repeatedly for each case, we can determine how the

system responds to changes in decision variables or system parameters.

For example suppose:

X: May take any value between 200000 and 320000 (random variable)

S: The company may decide to sell the product for any value say $4/unit to $5.5/unit

C: variable cost/unit changes from one period to another from $3.5 to $4.1

T: total fixed cost plus taxes per year is constant or changes between $300K to 380K

Depending on what random o fixed value each variable assumes, the value of P, the

performance measure of the system will change. The profit (P) may take negative or

positive values.

Types of Simulation models

Simulation models may be classified as being static or dynamic. A static simulation

model, sometimes called Monte-Carlo Simulation represents a system at a particular

point in time. Monte-Carlo is basically a sampling experiment whose objective is to

estimate the distribution of an outcome variable. For example, we may be interested in

the distribution of net profit from a business for the coming year when sales volume, and

variable cost per unit are uncertain. Consider the following simple case:

Let: P= Gross yearly profit

X= Sales volume (# of units)

S= Sales price per unit

F= Total fixed cost per year including taxes

C= variable cost per unit

Assuming all other factors could be ignored, we can easily develop an expression

(mathematical model) for the gross profit for the business as follows:

P= (X)[S – C] –F

We could input many different values for these variables, X and C, into the model and

determine the value of gross profit (P) for each combination of inputs. If we do this, we

will have created a distribution of the possible values of the gross profit. The output

values (and the distribution) provide an n indication of the likelihood of what we might

expect. Monte-Carlo simulation is often used to estimate the impact of policy changes

and risk involved in decision-making. See more examples of Monte-Carlo simulation in

Chapter 2

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Dynamic simulation models represent systems as they change over time. System

simulation Explicitly models sequences of events that happen over time. Therefore,

queuing, inventory, manufacturing problems are addressed with system simulation. As

an example, consider the following simple case. The Dynaco Company produces a

product in a two stages manufacturing system as shown below.

Input M Output

Machine #1 Machine #2

Each machine may break down randomly. A review of the historical records on the time

between breakdowns and repair time for each machine, provide the following

information.

Time Between Breakdowns (TBB) Repair Time (RT)

In hours in minutes

Machine #1 Machine #2 Machine #1 Machine #2

TBB Probability TBB Probability RT Probability RT Probability

5 0.08 5 0.04 10 -20 0.27 10 -20 0.16

10 0.18 10 0.15 20 -30 0.35 20 -30 0.30

15 0.24 15 0.37 30 -40 0.29 30 -40 0.41

20 0.39 20 0.43 40-50 0.06 40 -50 0.11

25 0.08 25 0.01 50- 0.03 50- 0.02

30 0.02 30 0.00

35 0.01

The Company is interested in estimating the Average production volume per week, as

well as the average breakdown cost/week (assume they know repair cost per hour). This

is a dynamic situation since the state of the system could change from one hour to the

next. However the state of the system will change only when the normal operation is

interrupted at discrete points in time because of breakdown of one machine or

simultaneous breakdown of both machines. Therefore, this case must be analyzed using

discrete event (system) simulation. In order to answer these questions, it is necessary to

simulate the operation of the system for n units of time and collect data on units

produced, downtimes, and other desired indexes of operations. In chapter 2 we have

provided a number of examples concerning system simulation.

Simulation models that contain no random variables are classified as deterministic

models. These models have a known set of inputs, which will result in a unique set of

outputs. Deterministic arrival would occur at a shipping/receiving dock if all trucks

arrived at the scheduled arrival time (i.e., one truck every 40 minutes, starting 12:00

noon). A stochastic simulation model has one or more random variables as inputs that

will result in random outputs. Since the outputs are random, they can be considered only

an estimate of the true characteristics of a model. For example, the simulation of a two

stages production system (see above) would involve random times between breakdowns

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(random occurrence time) and random repair times. Thus, the output measures_ the

average production rate per week, the average breakdown cost per week- must be treated

as statistical estimates of the true values of those measures.

It should be noted that a discrete simulation model is not always used to model a discrete

system, nor is a continuous simulation model is used to model a continuous system. In

addition, simulation models may be mixed, both discrete and continuous. The choice of

whether to use a discrete or continuous (or both) simulation model is a function of the

characteristics of the system and the objectives of the study. Because dividing large

batches into smaller elements can closely approximate many continuous processes,

discrete-event simulation modeling method may be employed for many (but certainly not

all) simulation studies of continuous processes. This course primarily emphasizes

discrete, dynamic, and stochastic simulation models. It only provides limited coverage of

the static, continuous and deterministic simulation models.

TYPICAL APPLICATIONS

The application of simulation is vast. The Winter Simulation Conference (WSC) is an

excellent source o learn more about the latest in Simulation theory and applications.

Information bout upcoming WSC ca be obtained from http://www.wintersim.org. During

the early 1980s, a survey was made of major U.S. firms to learn more about their use of

simulation (Reference #2). One major finding was the identification of the functional

areas of the company where simulation was being applied. The results are shown in

Table 1 below. The survey showed that the development of simulation models has

spread beyond Operations research (or Management Science) departments. Other

functional area departments and corporate planning departments use simulation modeling

extensively. More recent reports indicate that the use of simulation continue to grow

rapidly in manufacturing, corporate planning , and finance areas. Growth in these areas

has been aided by the development of specialized programming languages for each area.

Another important recent development has been the increasing use of computer graphics

to generate animated displays of the movement of entities through the simulated system.

The computer graphics provide greater insight into the performance of the system for any

given design. They also add credibility to the results of the simulation study.

There have been numerous applications of simulation in a variety of contexts. Some of

the areas of application, are listed blow:

Manufacturing and Production

Logistic, Transportation and Distribution

Military Operations

Business Process Simulation

Construction Engineering

Health Care

Human Systems

Financial Planning

………………

For a more detailed list of application areas, see Chapter one in your textbook or visit

WSC site.

http://www.wintersim.org/
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REFERENCES

1. Banks, J., and J. S. Carson, II, Discrete-Event Simulation, Prentice-Hall, 2001

2. Christy, DS. P., and H. J. Watson, “ The Application of Simulation: A survey of

Industry Practice, ” Interfaces, 13(5): 47-52 October 1983

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Module 2: Brief Introduction to basics. Probability. Simulation, and Random numbers

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Module 2: Brief Introduction to basics. Probability. Simulation, and Random numbers

• The concept of theoretical and experimental probability

• Simulation, an example

• Random variables

• Assignments

PART I: The concept of Probability: Probability is the study of chances or the likelihood of an event

happening. Directly or indirectly, it plays a role in all of our activities.

For Example, we may say that, it will probably rain today, because most

of the day in August were rainy. However, in Science we need more

accurate way of measuring probability.

A)-Experimental Probability One way to find the probability of an event is to conduct an experiment.

EXAMPLE

A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles.

Find the experimental probability of getting a blue marble

SOLUTION

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1)- Take a marble from the bag.

2)- Record the color and return the marble.

3)- Repeat a few times (maybe 10 times).Example:

Trial # 1 2 3 4 5 6 7 8 9 10

Outcome B R R B B R B R B B

4)- Count the number of times a blue marble was picked (Suppose it is

6). The experimental probability of getting a blue marble from the bag

is 6/10 = 3/5 (Discussion: Is this correct?)

B)-Theoretical Probability We can also find the theoretical probability of an event. The equation

used to determine the theoretical probability of an event is:

Example:

A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles.

Find the theoretical probability of getting a blue marble.

Solution:

There are 8 blue marbles. Therefore, the number of favorable outcomes

= 8. There are a total of 20 marbles. Therefore, the number of total

outcomes = 20

Example:

Find the probability of rolling an even number when you roll a die

containing

the numbers 1-6. Express the probability as a fraction, decimal, ratio

and percent.

Solution:

The possible even numbers are 2, 4, 6. Number of favorable outcomes

=3.

Total number of outcomes =6

Solution:

The possible even numbers are 2, 4, 6. Number of favorable

outcomes = 3.

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Total number of outcomes = 6

The probability = (fraction) = 0.5 (decimal) = 1to 2 (ratio) = 50%

PART II: Simulation, An example

Consider the following Experiments

Class Exercise: Repeat this Experiment 12 times. Determine the experimental probability of getting an ace. What is the theoretical

probability of getting an ace 6 . compare the outcome from the two.

A Simple Experiment; Toss a die 12 times. Suppose we get the following data;

Trial# 1 2 3 4 5 6 7 8 9 10 11 12

Outcome 3 5 2 6 1 4 5 3 5 4 2 5

Computer Simulation of this experiment: Use the Excel Spreadsheet and generate the outcomes as follows A B C D E F

1 Trial

#

Outcome

2 1 = randbetween(1,6) =If(B2=1,1,0)

3 2

4 3

5 4

6 5

7 6

8 7

9 8

10 9 copy copy

11 10

12 11

13 12

Experiment 1: Tossing a die Outcome: Result of experiment (what we expect to happen)

Possible outcomes from this Experiment (Exp. #1) are the

numbers 1, 2, 3, 4, 5, and 6

Sample Space: List of all possible outcomes

Sample space, S = {1, 2, 3, 4, 5, 6}.

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14 Totals

➔

=Sum(C2:C13)

RESULTS: Value in Cell C14 C14 Experimental probability = = Total # of outcomes 12 This is the same as determining % of time we get an ace in tossing a die

12 times.

Theoretical Probability= 6 /(12*6) = 1/6 (Explain why?)

PART III: Random Variables NOTE: In order to fully understand this tutorial, you need to know what we

mean by an experiment, the outcomes of an experiment, and probability. For

a brief refresher, see the Appendix 1 attached to this module

Q)- What is a random variable? A)- In many experiments the outcomes of the experiment can be assigned

numerical values. For instance, if you roll a die, each outcome has a value

from 1 through 6. If you ascertain the midterm test score of a student in your

class, the outcome is again a number. A random variable is just a rule that

assigns a number to each outcome of an experiment. These numbers are

called the values of the random variable. We often use letters like X, Y and Z

to denote a random variable. Here are some examples

Examples

1. Experiment: Select a mutual fund; X = the number of companies in

the fund portfolio. The values of X are 2, 3, 4, …

2. Experiment: Select a soccer player; Y = the number of goals the

player has scored during the season. The values of Y are 0, 1, 2, 3, …

3. Experiment: Survey a group of 10 soccer players; Z = the average

number of goals scored by the players during the season.

The values of Z are 0, 0.1, 0.2, 0.3, …., 1.0, 1.1, …

QUESTIONS #1:

4. Experiment: Flip a coin three times. Let X= Total number of heads

you observed. In this experiment, the possible values X (a random

variable) are:

5. Experiment: Throw two dice; X = the sum of the numbers facing up. The

values of X are:

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6. Experiment: Throw one die over and over until you get a six; X = the

number of throws.

The values of X are.

Discrete and Continuous Random Variables A discrete random variable can take on only specific, isolated numerical values, like the outcome of

a roll of a die, or the number of dollars in a randomly chosen bank account. Discrete random

variables that can take on only finitely many values (like the outcome of a roll of a die) are called

finite random variables. Discrete random variables that can take on an unlimited number of values

(like the number of stars estimated to be in the universe) are infinite discrete random variables.

A continuous random variable, on the other hand, can take on any values within a continuous range

or an interval, like the temperature in Central Park, or the height of an athlete in centimeters.

Examples

Random Variable Values Type

Flip a coin three times; X = the

total number of heads.

{0, 1, 2, 3} Finite

There are only four possible

values for X.

Select a mutual fund; X = the

number of companies in the

fund portfolio.

{2, 3, 4, …} Discrete Infinite

There is no stated upper limit

to the size of the portfolio.

Measure the length of an object;

X = its length in centimeters.

Any positive real number Continuous

The set of possible

measurements can take on any

positive value.

QUESTIONS #2:

Random Variable Check the Type

Throw two dice over and over until you roll a double six;

X = the number of throws. Finite

Discrete Infinite

Continuous

Take a true-false test with 100 questions;

X = the number of questions you answered correctly. Finite

Discrete Infinite

Continuous

Invest $10,000 in stocks;

X = the value, to the nearest $1, of your investment after a

year.

Finite

Discrete Infinite

Continuous

Select a group of 50 people at random;

X = the exact average height (in m) of the group. Finite

Discrete Infinite

Continuous

Using Excel to generate Random Numbers

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Excel has two useful functions when it comes to creating random numbers.

The RAND function, and the RANDBERWEEN function .

Rand()

The RAND function creates a random decimal number between 0 and 1.

1. Select cell A1.

2. Type RAND() and press Enter. The RAND function takes no arguments.

3. To create a list of random numbers, select cell A1, click on the lower right

corner of cell A1 and drag it down.

Note that cell A1 has changed. That is because random numbers change

every time a cell on the sheet is calculated.

Randbetween (a, b)

The RANDBETWEEN function returns a random whole number

between two boundaries.

1. Select cell A1.

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2. Type RANDBETWEEN(50,75) and press Enter.

3. If you want to create random decimal numbers between 50 and 75, modify the RAND function as follows:

Assignment 1:

For each of the following experiments, simulate the situation and

1)- Answer the questions asked, in the sequence listed below.

2)- Attach a copy of your excel worksheet. At the end/bottom of each excel sheet , write

down the (excel)

equation you used to generate each column of data, and your final answers.

3)- Write your name on the answer sheets and make sure what you submit is clean and

readable.

Experiment 1 : Tossing a coin

Simulate Tossing a coin 30 times and answer these questions:

1) -Possible outcomes are: ………………………………………………

2)-Sample space, S = ………………………………………………………

3)- Experimental Probability of getting “Tail” ……………………...........

4)- Theoretical Probability of getting “Tail” ……………………………

Experiment 2: Picking a card.

In this experiment, a card is picked from a stack of six cards, which spell the word

PASCAL. (Each letter is written on one card). Simulate the experiment and answer

the following questions:

1) -Possible outcomes are: ………………………………………………

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2)-Sample space, S = ………………………………………………………

3)- Experimental Probability of getting “A”…………………………… …

4)- Theoretical Probability of getting “A” ………………………………..

Experiment 3: Suppose the ABC trucking Company delivers raw material to your factory. Each

truck carries about 18 tons of raw material. The time between subsequent arrivals of

trucks is random and changes between 30 to 75 minutes. Let us assume that the first

truck always arrives at 8:00 AM.

1)- Simulate the delivery operation for one day (8 hours /day) and

determine how many tons of raw material is delivered in a day. What

time (clock time) the last truck will arrive?

2)- Repeat the simulation in question 1 (above) 5 times and from the output data,

determine Maximum, Minimum, and average tons of raw material delivered

per day.

3)- From the 5 trials in question 2, determine Average number of trucks (arriving) per

day

APPENDIX MODULE 2

Random variables and probability

This lesson is about random variables and the basic language used to describe

populations and samples from populations.

I. Random Variable:

At the end of Chapter 2 we defined A random variable as follows: A

function that assigns a real number to each outcome in the sample space

of random experiment.

Example: X denotes the outcome of experiment, Tossing a die. Then X

can take (randomly) any of the values 1, 2, 3, 4, 5, and 6.

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Note: The outcome of an experiment need not be a number, for

example, the outcome when a coin is tossed can be 'heads' or 'tails'.

However, we often want to represent outcomes as numbers. A random

variable is a function that associates a unique numerical value with

every outcome of an experiment. The value of the random variable will

vary from trial to trial as the experiment is repeated. There are two

types of random variable - discrete and continuous

A discrete Random Variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, ... Discrete random variables are usually (but not necessarily) counts. Examples of discrete random variables include the number of students in registration office, the Friday night attendance at a cinema, the

number of cars passing a toll both. Continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, pressure, the time required to run a mile.

Examples 1. A coin is tossed ten times. The random variable X is the number

of tails that are noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

2. A light bulb is burned until it burns out. The random variable Y is its lifetime in hours. Y can take any positive real value, so Y is a continuous random variable.

Introduction to Probability The study of descriptive statistics was concerned with what has

occurred, probability is concerned with what will occur. Many of the

concepts are the same, although some of the vocabulary changes.

Descriptive statistics is concerned with (relative) frequency in the past,

probability with (relative) frequency in the future.

• Vocabulary

• Axioms

• Where do probabilities come from?

Vocabulary

Experiment:

something which generates an outcome (e.g., pick a card, roll a die,

weigh a student, look outdoors)

Outcome: (also called simple event)

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result of an experiment (e.g., jack of spades, 3 pips, 145 pounds,

partly cloudy)

Sample space: (denoted by S)

set of all possible outcomes of an experiment (e.g., for picking a card

there are 52 possible outcomes, hence 52 points in the sample space)

Event: a set of outcomes, or equivalently, a subset of the sample

space (e.g., for picking a card, events include getting a spade,

getting a deuce, getting a face card)

Note: Sometimes identifying outcomes is subtle. If you roll a pair of dice, is the total number of pips, the pair of values on the two dice, or the ordered

pair of values on the two dice the outcome?

Axioms Of Probability

A probability space entails that a probability be assigned to each outcome.

• The probability of each outcome [denoted P (xi ), where “xi ” is the

ith outcome] is always between 0 and 1.inclusive

• The probability of an event is the sum of the probabilities of the

outcomes (simple events) in the Event.

• P(S)=1; Something has to happen, the probability of the sample space

is 1.

Where do probabilities come from?

• Probabilities may be given, often in the form of a table. For example,

if an experiment has three possible outcomes: Accept , Reject, or

Compromise, one might be given the following table:

Accept Reject Compromise xi (A) ( R) (C)

P (xi ), 0.35 0.25 0.40

Note: Even if say, the 0.40 entry had been missing, you would

have been able to figure it out, since probabilities sum to 1.

• Probabilities my be historical, if it has rained during 1/3 of the days in

June during the past, one may say that the probability of rain for a day

in June is 1/3.

• Probabilities may be theoretical, if a die is fair (and there is any

justice in the universe), since there are six possible outcomes, the

probability of getting 3 pips on the top face is 1/6.

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Example: Consider the following (incomplete) table of probabilities

associated with rolling an unfair die:

xi 1 2 3 4 5 6

P (xi ) 0.2 0.1 0.2 ? 0.3 0.1

What is the probability of rolling a 5?

What is the probability of rolling an even number (2 or 4 or 6)?

Expressing Probability

A probability is usually expressed in term of a random variable. For the

die rolling example X denotes the outcome. The probability statement can be

written in either of the forms:

• Pr( x=1) = 0.2,

• Pr( x <3) = P1( x=1) + Pr(x=2) = 0.2 +0.1 =0.3

• Pr(2 ≤ x< 4) = Pr(x=2) + Pr (x=3) = 0.1+0.2 =0.30

• Pr(x ≥ 5) = Pr(x=5) + Pr(x=6) = ).3 +0.1 = 0.4 Also, the following expression may be used for the example:

• Pr(x=0) = 0

• Pr(x>6) = 0 If the set of all possible outcomes is denoted as “s” where the set

s: {1, 2, 3, 4, 5, 6], then we can use the following expressions:

• Pr(x ε s ) = 1 • if “s” is divided into a number of mutually exclusive (non-

intersecting) sun-sets s1, s2, ….sk, Then,

s = s1, + s2, + …. + sk, OR s = s1 U s2 U s3…. +U sk

Pr(x ε s ) = Pr (x ε s1 U s2 U s3…. +U sk )

21

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Module 3: Monte Carlo Simulation

• Introduction to Monte Carlo Simulation

• Random numbers from some common probability distribution

Module 3: Monte Carlo Simulation

• Introduction to Monte Carlo Simulation

• Random numbers from some common probability distribution

PART 1: Introduction to Monte Carlo simulation Computer simulation has to do with using computer models to imitate real life or make

predictions. When you create a model with a spreadsheet like Excel, you have a certain

number of input parameters and a few equations that use those inputs to give you a set of k

outputs (response variables). Figure 1 depicts such a system X1 Y1 X2 y2 ……… ………. OUTPUT INPUT ……… ………. ……… Yk

A production

System

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X n

This type of model may be deterministic, meaning that you get the same results no matter

how many times you re-calculate. For example the equation for calculating the future value

“F” of a investment of $X now with an interest rate of r% in “n” years F=P(1+r/100)n is a

deterministic model. However, in most systems, some or all input variables(Xi) are

stochastic resulting in output values (Yi) that change stochastically depending on the input

variables

Monte Carlo simulation is a method for iteratively evaluating a deterministic model using

sets of random numbers as inputs. By using random inputs, you are essentially turning the

deterministic model into a stochastic model. This method is often used when the model is

complex, nonlinear, or involves more than just a couple uncertain parameters. A simulation

can typically involve over 10,000 evaluations of the model, a task which in the past was

only practical using super computers.

Example

we used simple uniform random numbers as the inputs to the model. However, a uniform

distribution is not the only way to represent uncertainty. Before describing the steps of the

general MC simulation in detail, a little word about uncertainty propagation:

The Monte Carlo method is just one of many methods for analyzing uncertainty

propagation, where the goal is to determine how random variation, lack of knowledge, or

error affects the sensitivity, performance, or reliability of the system that is being modeled.

Monte Carlo simulation is categorized as a sampling method because the inputs are

randomly generated from probability distributions to simulate the process of sampling from

an actual population. So, we try to choose a distribution for the inputs that most closely

matches data we already have, or best represents our current state of knowledge. The data

generated from the simulation can be represented as probability distributions (or histograms)

or converted to error bars, reliability predictions, tolerance zones, and confidence intervals.

(See Figure 2).

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Random (Uncertainty) Inputs

Figure 2: the basic principal of stochastic uncertainty propagation

The steps in Monte Carlo simulation of a system shown in Figure 2 are fairly simple, and

can be easily implemented in Excel for simple models. All we need to do is follow the five

simple steps listed below:

Step 1: Create a parametric model of the system, y = f(x1, x2, ..., x n).

Step 2: Generate a set of random inputs, xi1, xi2, ..., x n. (if Xi is a random variable)

Step 3: Evaluate the model and store the results as yi.

Step 4: Repeat steps 2 and 3 for i = 1 to m.

Step 5: Analyze the results using histograms, summary statistics, confidence intervals, etc

Example 1: Monte Carlo Simulation ABC Bakery company, bakes 2500 Loaf of bread per day. Historical data shows that their