CMGT 573 Catholic University of America Simulation Modeling i attached a pic of the questionand the slides of the class you will find the example that the

CMGT 573 Catholic University of America Simulation Modeling i attached a pic of the questionand the slides of the class you will find the example that the professor mention it the question in slide #21.can i have a two version of the answers please in different files.simulation by excel or python THE CATHOLIC UNIVERSITY OF AMERICA
School of Engineering
Engineering Management Department
Lecture
10
Introduction to Systems Analysis
CMGT 575
Simulation Modeling
(Chapter-19)
Dr. José R. Febres-Andino
SECTION 01 – Monday 8:00-10:10 PM / McCort Ward Bldg., Room 209
SECTION 02 – Online / Blackboard
Monte Carlo Simulation
•
The Monte Carlo experiment is a forerunner to present-day
simulation, and is estimates stochastic or deterministic parameters
based on random sampling.
•
Examples of Monte Carlo applications include evaluation of
multiple integrals, estimation of the constant π (≈3.14159), and
matrix inversion.
•
Monte Carlo simulations are used to model the probability of
different outcomes in a process that cannot easily be predicted due
to the intervention of random variables. It is a technique used to
understand the impact of risk and uncertainty in prediction and
forecasting models.
2
Who Uses Monte Carlo Simulation?
• General Motors, Proctor and Gamble, Pfizer, Bristol-Myers Squibb, and Eli Lilly use simulation to
estimate both the average return and the risk factor of new products. At GM, this information is used by
the CEO to determine which products come to market.
• GM uses simulation for activities such as forecasting net income for the corporation, predicting
structural and purchasing costs, and determining its susceptibility to different kinds of risk (such as
interest rate changes and exchange rate fluctuations).
• Lilly uses simulation to determine the optimal plant capacity for each drug.
• Proctor and Gamble uses simulation to model and optimally hedge foreign exchange risk.
• Sears uses simulation to determine how many units of each product line should be ordered from
suppliers—for example, the number of pairs of Dockers trousers that should be ordered this year.
• Oil and drug companies use simulation to value “real options,” such as the value of an option to expand,
contract, or postpone a project.
• Financial planners use Monte Carlo simulation to determine optimal investment strategies for their
clients’ retirement.
Source: https://support.microsoft.com/en-us/office/introduction-to-monte-carlo-simulation-in-excel-64c0ba99-752a-4fa8-bbd3-4450d8db16f1?ui=en-us&rs=en-us&ad=us
3
Monte Carlo Simulation Applications
Monte Carlo (MC) simulation is a useful tool for modeling phenomena
with significant uncertainty in inputs and has numerous applications,
including (O’Connor, 2012):
•Reliability
•Random
processes simulation
(e.g., repairable systems)
•Availability
•Logistic forecasting
analysis
design
•Uncertainty propagation
•Risk analysis
•Load-strength
•Probabilistic
interference
•Geometric
dimensioning and
tolerancing
•Multiple business
applications
4
Simulation – Example 19.1-1
(x-1)2 + (y-2)2 = 25
•
A circle with r = 5 cm, and its center
(x, y) = (1, 2).
•
The estimation of area is based on a
sampling experiment that gives
equal chance to selecting any point
in the square.
x = -4 + 10R1
y = -3 + 10R2
5
Simulation Experiment –Example 19.1-1
•
•
•
The accuracy of the area estimate can be enhanced by using procedures from
ordinary statistical experiments:
1.
Increase the sample size, n.
2.
Use replications, N.
This in turn poses two questions regarding simulation experiments:
1.
How large should the sample size be?
2.
How many replications are needed?
Confidence Interval:
ҧ −

, −1
2
≤ ≤ ҧ +

, −1
, −1
2
2
ҧ −
−
− −
6
Monte Carlo Simulation Basics
Basic Steps for Performing Monte Carlo Simulation (O’Connor, 2012)
Step-1: Define the problem and the overall objectives of the study. Evaluate the available data and
outcome expectation.
Step-2: Define the system and create a parametric model, y = f (x1, x2, …xn)
Step-3: Design the simulation. Quantities of interest need t be collected, such as probability
distributions for each of the inputs. Define how many simulation runs should be used. The number
of runs, m is affected by the complexity of the model and the sought accuracy of the results.
Step-4: Generate a random set of inputs xi1, xi2, …xin.
Step-5: Run the deterministic system model with the set of random inputs. Evaluate the model and
store the results as yi.
Step-6: Repeat steps 4 and 5for i = 1 to m.
Step-7: Analyze the results statistics, confidence intervals, histograms, best fit distribution, or any
other statistical measure.
7
Types of Simulation
The execution of present-day simulation is based on the idea of sampling used
with the Monte Carlo method. It differs in that it deals with the study of the
behavior of real systems as a function of time.
1.
Continuous models – deal with systems whose behavior changes
continuously with time. These models usually use difference-differential
equations to described the interactions among the different elements of
the system. A typical example deals with the study of world population
dynamics.
2.
Discrete models – deal primarily with the study of waiting lines, with the
objective of determining such measures as the average waiting time and
length of the queue. These measures changes only when a customer
enters or leaves the system. The instants at which changes take place
occur at specific discrete points in time.
8
Elements of Discrete Event Simulation
•
The ultimate goal of simulation is to estimate some desirable measures of
performance that describe the behavior of the simulated system.
•
E.g., in a service facility, the associated measures of performance can include
the average waiting time until a customer is served, the average length of the
queue, and the average utilization of the service facility.
•
Generic Definition of Events
• All discrete event simulations described directly or indirectly, queuing
situations in which customers arrive, wait in the queue, and then receive
service before leaving the facility.
•
Sampling from Probability Distributions
• Randomness in simulation arises when the interval, t, between successive
events is probabilistic.
9
Generic Definition of Events
Discrete event simulation, regardless of the complexity of the system
that it describes, reduces to dealing with two basic events: arrivals and
departures.
• Example 19.3-1 – Describes Metalco Jobshop with rushed and regular
jobs at consecutive machines.
• Tagging a event with an attribute.
•
10
Sampling from Probability Distribution
•
•
Randomness in simulation arises when the interval, t, between
successive events is probabilistic.
1.
Inverse Method – particularly suited for analytically tractable
probability density functions, such as the exponential and uniform.
2.
Convolution Method – deals with complex cases such as normal
and Poisson.
3.
Acceptance-Rejection Method – hypothesis testing.
All three methods are rooted in the use of independent and identically
distributed (iid) uniform random numbers.
11
Inverse Method
•
To obtain a random sample x from the (continuous or discrete) probability function
f(x), the inverse method first determines a closed form expression of cumulative
density function F(x) = {y ≤ x}, where 0 ≤ F(x) ≤ 1, for all defined values of y.
•
Step 1. Generate a 0-1 random number, R.
•
Step 2. Compute the desired example = −1 .
12
Monte Carlo Inverse Method
Source: Kristov, V. 2015. Lecture-7: Review of Probabilistic Models of
Repairable Systems, lecture notes, Collection and Analysis of Reliability
Data, ENRE-640, University of Maryland, delivered April, 2015.
13
Generation of Random Numbers
Source: http://pit-claudel.fr/clement/blog/how-random-is-pseudo-randomtesting-pseudo-random-number-generators-and-measuring-randomness/
Source: O’Connor, P., & Kleyner, A. (2012). Practical reliability engineering. John Wiley & Sons.
14
Probability Distributions
•
Exponential Distribution
•

= න λ −λ = 1 − −λ , > 0
=1
1
=−
1 −
λ
1
= −
ln , = 1, 2, … ,
λ

1
=−
ෑ
λ
=1
=1
1 ≥ −λ > 1 , = 0
•
Erlang Distribution
+1
ෑ ≥ −λ > ෑ , > 0
0
•
Poisson Distribution
Normal Distribution

−
2
= +

12
In practice, take n =12
= + −6
15
Generation of Random Numbers
•
Uniform (0, 1) random numbers play a key role in sampling from distributions. True
0-1 random numbers can be generated only by electronic devices.
•
The only feasible way for generating 0-1 random numbers for use in simulation is
based on arithmetic operations. Such numbers are not truly random because the
entire sequence can be generated in advance. Thus it is more appropriate to refer
to them as pseudorandom numbers.
•
Multiplicative congruential method
= −1 + , = 1, 2, …

=
, = 1, 2, …

• The initial value, uo is usually referred to as the seed of the generator.
16
Generation of Random Variables Using Excel Functions
Source: O’Connor, P., & Kleyner, A. (2012). Practical reliability engineering. John Wiley & Sons.
17
Number of Simulation Runs and Results Accuracy
σ
=
2

μ = standard error of the mean
α = 1-C, where C is the confidence level
Zα = the standard normal statistic
2
σ = standard deviation of the output
m = number of MC runs
Source: O’Connor, P., & Kleyner, A. (2012). Practical reliability engineering. John Wiley & Sons.
18
Calculating the Probability of Exceeding Yield Strength
Consider a simple stress analysis, where a random
force F is applied to a rectangular area with
dimensions A x B. Based on previously recorded data
and goodness of fit criteria, force F can be statistically
described as a 2-parameter Weibull with β= 2.5 and
θ= 11,300 N (mean value 10,026 N). Dimension A has
a mean value of 2.0 cm with tolerance of +/- 1.0 mm
and B has a mean value of 3.0 cm with a tolerance of
+/- 1.5 mm. The structure is expected to function
properly while within the elastic strain range,
therefore the probability of exceeding the yield
strength of the material (30 x 106 N/m2) needs to be
estimated. (O’Connor, 2012)
F
A
B
Stress =

→ =

Source: O’Connor, P., & Kleyner, A. (2012). Practical reliability engineering. John Wiley & Sons.
19
Calculating the Probability of Exceeding Yield Strength (cont.)
Random variable-1: Force, F
Distribution: 2-parameter Weibull
Excel function: =(θ(-LN(RAND()))^(1/ β))
Parameters: β= 2.5 and θ= 11,300 N (mean value 10,026 N)
Random variable-2: Dimension, A
Distribution: Triangular (symmetrical) or Three point estimate
Excel function: = a+(b-a)*(RAND()+RAND())/2
Parameters: mean= 0.02 m, min= 0.019 m, max= 0.021 m
Random variable-3: Dimension, B
Distribution: Triangular (symmetrical) or Three point estimate
Excel function: = a+(b-a)*(RAND()+RAND())/2
Parameters: mean= 0.03 m, min= 0.0285 m, max= 0.0315 m
20
Calculating the Probability of Exceeding Yield Strength (cont.)
Dimension A simulation
Cell A2: = 0.019+(0.021-0.019)*(RAND()+RAND())/2
Dimension B simulation
Cell B2: = 0.0285+(0.0315-0.0285)*(RAND()+RAND())/2
Force simulation
Cell C2: =(11300*(-LN(RAND()))^(1/ 2.5))
Resulting stress simulation
Cell D2: =C2/(A2*B2)
Number of simulations (n) and replications (m)
n = 1,000, m = 1
Probability of exceeding 30 MPa (i.e., probability of failure)
=+COUNTIF(D2:D1001,”> 30000000″)/COUNT(D2:D1001)
Source: https://accendoreliability.com/reliabilityparadigm-shift-time-stress-metrics/
21
Spreadsheet Based Simulation – Example 19.5-1
•
Example 19.5-1
• Single-Server Model
• Arrival and Departure Events
• Seeking performance metrics:
• Average Utilization
• Average Number of waiting customers
• Average time a customer waits in the queue
22
Gathering Statistical Observations
•
•
A simulation experiment must satisfy three conditions:
1.
Observations are drawn from stationary distributions
2.
Observations are sampled from a normal population
3.
Observations are independent
Simulation output is a function of the length of the simulation
period. The initial period produces erratic behavior and is
usually referred to as the transient period. When the output
stabilizes it is said to operate at steady state.
23
Subinterval Method (or the Method of Batch Mean)
• This method really involves only one very long simulation run which is suitably
subdivided into an initial transient period and n batches.
• Each of these batches is then treated as an independent run of the simulation
experiment while no observations are made during the transient period which is
treated as the warm-up interval.
Warm-up
24
Replication Method
• This method is the one most popularly used.
• This was basically the suggested technique for getting n independent runs of the
simulation experiment by running the simulator n times with different initial random
seeds for the simulator’s random number generator.
25
Regenerative Method
• This method is basically intended to reduce the problem of correlation between the batches
that one may encounter in the Subinterval Method.
• A long run is used but an identified state of the system (e.g., shift, on-off cycle) is selected as
the regenerative state and the time instants when this occurs as the regenerative points.
• The batches start and end at these regenerative points once steady state has been reached.
26
Homework-6 (Due date: April 6, 2020)
1. Repeat the example “Calculating the Probability of Exceeding Yield Strength” with the
force F assuming the following distributions:
• Normal
• Lognormal
Use the following parameters:
• μ = 11,300 N
• σ = 1, 250 N
2. Run the simulation with 1,000 simulations (N) and 1 replication (m).
3. Repeat the simulation with 1,000 simulations (N) and 3 replications (m).
4. Explain the behavior of the probability of failure compared against the system material
strength, i.e., P(S>30MPa).
27
Homework-6 (Due date: April 6, 2020)
1. Repeat the example “Calculating the Probability of Exceeding Yield Strength” with the
force Fassuming the following distributions:
Normal
Lognormal
Use the following parameters:
H = 11,300 N
• 0 = 1, 250 N
2. Run the simulation with 1,000 simulations (N) and 1 replication (m).
3. Repeat the simulation with 1,000 simulations (N) and 3 replications (m).
4. Explain the behavior of the probability of failure compared against the system material
strength, i.e., P(S>30MPa).

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