This includes solved examples on monte carlo simulation

2. System: The physical process of interest
Model: Mathematical representation of the system
◦ Models are a fundamental tool of science, engineering, business,
etc.
◦ Models always have limits of credibility
Simulation: A type of model where the computer is used to
imitate the behavior of the system
Monte Carlo simulation: Simulation that makes use of
internally generated (pseudo) random numbers
Random Number:Random numbers are numbers that occur in a
sequence such that two conditions are met: (1) the values are
uniformly distributed over a defined interval or set, and (2) it is
impossible to predict future values based on past or present
ones.
2

3. 3
Focus of course
System
Experiment w/
actual system
Experiment w/
model of system
Physical
Model
Mathematical
Model
Analytical
Model
Simulation
Model

4.  “The Monte Carlo method is a numerical solution to a problem
that models objects interacting with other objects .
 A Monte Carlo simulation is a model used to predict the
probability of different outcomes when the intervention of
random variables is present.
 Monte Carlo simulations help to explain the impact of risk and
uncertainty in prediction and forecasting models.
 A variety of fields utilize Monte Carlo simulations, including
finance, engineering, supply chain, and science.
 The basis of a Monte Carlo simulation involves assigning
multiple values to an uncertain variable to achieve multiple
results and then to average the results to obtain an estimate.
 It represents an attempt to model nature through direct
simulation of the essential dynamics of the system in question.
 In this sense the Monte Carlo method is essentially simple in its
approach.
4

5.  Business and finance are plagued by random
variables, Monte Carlo simulations have a vast
array of potential applications in these fields.
 Monte Carlo Method:
 A Monte Carlo simulation takes the variable that
has uncertainty and assigns it a random value.
 The model is then run and a result is provided.
 This process is repeated again and again while
assigning the variable in question with many
different values.
 Once the simulation is complete, the results are
averaged together to provide an estimate.
5

6. let’s consider a simple system with simple
inputs:
6
• As A, B, C and D are always the same, the output will
always be the same and it can be easily calculated
• Imagine that input A has a range of possible values –
the output will also be variable. And when there are
many more possible inputs and all of them have a range
of possible values, the output is not that simple to
calculate.
• That’s where you need to use Monte Carlo simulation.

7.  Steps in monte carlo simulation:
 Step 1:Clearly define the problem.
 Step 2:Construct the appropriate model.
 Step 3:Prepare the model for
experimentation.
 Step 4:Using step 1 to 3,experiment with the
model.
 Step 5:Summarise and examine the results
obtained in step 4.
 Step 5:Evaluate the results of the simulation.
7

8.  A manufacturing company keeps stock of a special product.
Previous experience indicates the daily demand as given
below
8
Daily
demand
5 10 15 20 25 30
probability 0.01 0.20 0.15 0.50 0.12 0.02
Simulate the demand for the next 10 days. Also find the daily
average demand for that product on the basis of simulated data.
Consider the following random numbers:
82,96,18,96,20,84,56,11,52,03

9.  Solution: Step 1:Generate tag values
9
Daily demands Probability Cumulative
probability
Tag
values(Random
num range)
5 0.01 0.01 00-00
10 0.20 0.21 01-20
15 0.15 0.36 21-35
20 0.50 0.86 36-85
25 0.12 0.98 86-97
30 0.02 1.00 98-99
Step 2: Simulate for 10 days
Days Random num Daily demand
1 82 20
2 96 25
3 18 10
4 96 25
5 20 10
6 84 20
7 56 20
8 11 10
9 52 20
10 03 10
Average demand=(20+25+10+25+10+20+20+10+20+10)/10
=170/10=17 units/day

10.  2)A tourist car operator finds that during the past few months the
cars use has varied so much that the cost of maintaining the car
varied considerably. During the past 200 days the demand for the
car fluctuated as below
10
Trips per week Frequency
0 16
1 24
2 30
3 60
4 40
5 30
Using random numbers 82,96,18,96,20,84,56,11,52,03,
simulate the demand for 10 week period

11.  Solution: Step 1:Generate tag values
11
Trips/week frequency Probability Cumulative
probability
Tag values
0 16 16/200=0.08 0.08 00-07
1 24 24/200=0.12 0.20 08-19
2 30 30/200=0.15 0.35 20-34
3 60 60/200=0.30 0.65 35-64
4 40 40/200=0.20 0.85 65-84
5 30 30/200=0.15 1.00 85-99
Frequency-Number of occurrences, Total num of occurrences(16+24+30+60+40+30)=200
Step 2: Simulation for next 10 week
Weeks Random Num Trips/week
1 82 4
2 96 5
3 18 1
4 96 5
5 20 2
6 84 4
7 56 3
8 11 1
9 52 3
10 03 0
Avg trips/week=28/10=2.8≈3 trips/week

12.  For a particular shop the daily demand of an item is given as follows, Use
random numbers 25,39,65,76,12,05,73,89,19,49.Find the average daily
demand.
 Daily demand 5 10 15 20 25 30
 Probability 0.01 0.20 0.15 0.50 0.12 0.02
Solution: Generate tag values
12
Daily demand Probability Cumulative
probability
Tag values
0 0.01 0.01 00
10 0.20 0.21 01-20
20 0.15 0.36 21-35
30 0.50 0.86 36-85
40 0.12 0.98 86-97
50 0.02 1.00 98-99

13.  Step 2: Simulation for 10 days
13
Days Random num Daily demand
1 25 20
2 39 30
3 65 30
4 76 30
5 12 10
6 05 10
7 73 30
8 89 40
9 19 10
10 49 30
Avg daily demand= 240/10=24

14.  An automobile company manufactures around 150 scooters.Daily
production varies from 146 to 154,the probability distribution is given
below.
 Step 1:Generate tag values for production/day
 Step 1:Generate tag values for production/day
14
Production
/day
146 147 148 149 150 151 152 153 154
probability 0.04 0.09 0.12 0.14 0.11 0.10 0.20 0.12 0.08
The finished scooters are transported in a lorry accomodading150 scooters. using the
following random numbers 80,81,76,75,64,43,18,26,10,12,65,68,69,61,57
simulate
1)Average number of scooters waiting in the factory
2)Average number of empty space in the lorry

15.  Step 1:Generate tag values for production/day
15
Production/day probability Cumulative probability Tag values
146 0.04 0.04 00-03
147 0.09 0.13 04-12
148 0.12 0.25 13-24
149 0.14 0.39 25-38
150 0.11 0.50 39-49
151 0.10 0.60 50-59
152 0.20 0.80 60-79
153 0.12 0.92 80-91
154 0.08 1.00 92-99
Step 2: Simulate for 15 days to get avg no of waiting scooters and empty space, lorry can accommodate 150
scooters
Days Random
num
Production/day No of scooters
waiting
No of empty space in
lorry
1 80 153 3 -
2 81 153 3 -
3 76 152 2 -
4 75 152 2 -
5 64 152 2 -
6 43 150 - -
7 18 148 - 2
8 26 149 - 1
9 10 147 - 3
10 12 147 - 3
11 65 152 2 -
12 68 152 2 -
13 69 152 2 -
14 61 152 2 -
15 57 151 1 -
Total=21 Total=9
Avg no of scooters waiting=
21/15=1.4
Avg No of space in the
lorry= 9/15=0.6

16.  An automobile production line turns out about 100 cars/day, but
deviation occur owing to many causes.Production of cars are
described by the probability distribution given below.
16
Productio
n/day
95 96 97 98 99 100 101 102 103 104 105
probabilit
y
0.0
3
0.05 0.0
7
0.10 0.1
5
0.20 0.15 0.10 0.07 0.05 0.03
Finished cars are transported across the bay at the end of each day by ferry.
If ferry has space for only 101 cars,
what will be the average number of cars waiting to be shipped and
what will be the average number of empty space on ship?
Simulate the production of cars for next 15 days,
consider the random numbers 97,02,80,66,96,55,50,29,58,51,04,86,24,39,47.

17.  Step 1:generate tag values
17
Production/d
ay
Probabilli
ty
Cumulative
probability
Tag values
95 0.03 0.03 00-02
96 0.05 0.08 03-07
97 0.07 0.15 08-14
98 0.10 0.25 15-24
99 0.15 0.40 25-39
100 0.20 0.60 40-59
101 0.15 0.75 60-74
102 0.10 0.85 75-84
103 0.07 0.92 85-91
104 0.05 0.97 92-96
105 0.03 1.00 97-99

18.  Step 2:Simulate for 15 days, ferry can transport 101 cars
18
Days Random
numbers
Productions/day No of cars
waiting
Empty space in
the ship
1 97 105 |105-101|=4 -
2 02 95 - (101-95) =6
3 80 102 1
4 66 101 - -
5 96 104 3 -
6 55 100 - 1
7 50 100 - 1
8 29 99 - 2
9 58 100 - 1
10 51 100 - 1
11 04 96 - 5
12 86 103 2 -
13 24 98 - 3
14 39 99 - 2
15 47 100 - 1
Total=10 Total=23
Avg num of cars waiting=10/15
Avg empty space in the ship=23/15

19.  Strong is a dentist who schedules all her patients for 30 minutes
appointment. Some of the patients take more or less than 30min depending
on the type of dental works to be done. The following summary shows the
various categories of work,their probability and the time actually needed to
complete the work
19
Category Filling crown cleaning extracting checkup
Time
required
45 60 15 45 15
Number of
patients
40 15 15 10 20
Simulate the dentist clinic for 4 hrs and find out the avg waiting
time for the patients as well as the idleness of doctor.Assume
that the ptients show up at the clinic at exactly scheduled time.
Arrival time starts at 8AM.Use the following random number
for handling the same 40,82,11,34,25,66,19,79

20. category Time
required
No of
patients(Frequen
cy)
probability Cumulative
probability
Tag
values
Filling 45 40 0.40 0.40 00-39
Crown 60 15 0.15 0.55 40-54
Cleaning 15 15 0.15 0.70 55-69
Extracting 45 10 0.10 0.80 70-79
Checkup 15 20 0.20 1.00 80-99
Total=100
20
Random
num
Categor
y
Time
required
(min)
Arrival
time of
patients
Service time
Start time End
time
waiting
time for
patients(mi
n)
Idleness
of
doctor
40 crown 60 8.00 8.00 9.00 0 -
82 checkup 15 8.30 9.00 9.15 30(9-8.30) -
11 Filling 45 9.00 9.15 10.00 15 -
34 Filling 45 9.30 10.00 10.45 30 -
25 Filling 45 10.00 10.45 11.30 45 -
66 Cleaning 15 10.30 11.30 11.45 60 -
19 Filling 45 11.00 11.45 12.30 45 -
79 Extractin
g
45 11.30 12.30 1.15 60 -
Step 1:find the cumulative probability and tag values
Step 2:Simulate for 4 hrs
Avg waiting time for patients=(30+15+30+45+60+45+60)/8=285/8=35.62 min≈36min
Waiting time for patients=(start time of service-arrival time)

21.  Bright Bakery keeps stock of a popular brand of
cake. Previous experience indicates the daily demand
as given below:
 Consider the following sequence of random numbers;
48, 78, 19, 51, 56, 77, 15, 14, 68,09. Using this
sequence simulate the demand for the next 10 days.
Find out the stock situation if the owner of the bakery
decides to make 30 cakes every day. Also estimate
the daily average demand for the cakes on the basis
of simulated data.
21
Daily
demand
0 10 20 30 40 50
Probability 0.01 0.20 0.15 0.50 0.12 0.02

22. Daily demand Probability Cumulative
probability
Tag values
0 0.01 0.01 00
10 0.20 0.21 01-20
20 0.15 0.36 21-35
30 0.50 0.86 36-85
40 0.12 0.98 86-97
50 0.02 1.00 98-99
22
Step 1:find the cumulative probability and tag values
Step 2:Simulate for 10 days, make 30 cakes every day
Days Random num Daily Demand Stock condition
1 48 30 -
2 78 30 -
3 19 10 20
4 51 30 -
5 56 30 -
6 77 30 -
7 15 10 20
8 14 10 20
9 68 30 -
10 09 10 20
Avg daily demand=220/10=22

23. Verification and validation are critical parts of practical
implementation
Verification pertains to whether software correctly
implements specified model
Validation pertains to whether the simulation model
(perfectly coded) is acceptable representation
◦ Are the assumptions reasonable?
Accreditation is an official determination that a
simulation is acceptable for particular purpose(s)
 Project Appraisal
 We can evaluate the likely profitability of a project
using these techniques in the light of many
uncertainties using this technique.
23

24.  RISK ANALYSIS AND MONTE CARLO SIMULATION
 Risk analysis is the systematic study of uncertainties and risks we
encounter in business, engineering, public policy, and many other areas.
 Risk analysts seek to identify the risks faced by an institution or
business unit, understand how and when they arise, and estimate the
impact (financial or otherwise) of adverse outcomes.
 Uncertainty and risk are issues that virtually every business analyst must
deal with, sooner or later.
 Monte Carlo simulation is a powerful quantitative tool often used in risk
analysis.
 Uncertainty is an intrinsic feature of some parts of nature – it is the same
for all observers. But risk is specific to a person or company – it is not
the same for all observers.
 Most business and investment decisions are choices that involve “taking
a calculated risk” – and risk analysis can give us better ways to make the
calculation.
 Risk analysis in computers are done using what-if analysis.
24