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 +

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, −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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