OPERATIONS RESEARCH
MONTE CARLO SIMULATION 1
UNIVERSITY OF TECHNOLOGY, JAMAICA
FACULTY OF ENGINEERING AND COMPUTING
OPERATIONS RESEARCH
Lecturer: Junior Bennett
Project #1: Monte Carlo Simulation
Submitted By:
Jayann Walters 0804251
Ainseworth Ennis
Richard Artwell 0604152
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Table of Contents
Introduction ..................................................................................................................................... 4
What is Monte Carlo Simulation? ............................................................................................... 4
History of Monte Carlo Simulation............................................................................................. 4
How Does Monte Carlo Simulation Work? ................................................................................ 4
Applications of Monte Carlo Simulation ........................................................................................ 5
Industrial Engineering and Operations Research ........................................................................ 5
Physical Processes and Structures. .............................................................................................. 6
Random Graphs and Combinatorial Structures ........................................................................... 6
Economics and Finance ............................................................................................................... 7
Computational Statistics.............................................................................................................. 7
Simulation Optimization ................................................................................................................. 8
Monte Carlo Simulation Softwares ................................................................................................. 9
Arena Simulation Software ......................................................................................................... 9
Key Uses: ................................................................................................................................. 9
Support and Training ............................................................................................................... 9
Cost and Purchasing Options ................................................................................................... 9
FlexSim Simulation Software ................................................................................................... 10
Key Features and Benefits ..................................................................................................... 10
Support and Training ............................................................................................................. 10
Cost ........................................................................................................................................ 11
@RISK Simulation Software .................................................................................................... 11
Key Features and Benefits ..................................................................................................... 11
Support and Training ............................................................................................................. 12
Cost ........................................................................................................................................ 12
Case Study Analysis ..................................................................................................................... 13
Introduction ............................................................................................................................... 13
Description of the Problem ....................................................................................................... 13
Aim of Case Study .................................................................................................................... 13
Definition of Key Terms ........................................................................................................... 13
Data Collection .......................................................................................................................... 14
Methodology ............................................................................................................................. 14
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Findings/ Results ....................................................................................................................... 14
Optimization/Improvements...................................................................................................... 17
Limitations and Challenges ....................................................................................................... 17
Recommendations ..................................................................................................................... 17
Personal Reflections/ Observation ............................................................................................ 18
References ..................................................................................................................................... 19
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Introduction
What is Monte Carlo Simulation?
Monte Carlo Simulation refers to a computerized mathematical technique utilized in
understanding the impact of risk and uncertainty in quantitative analysis and decision making. It
is utilized in forecasting models such as project management, energy, finance, project
management, engineering, transportation, manufacturing, research and development, insurance,
oil & gas, and the environment (What is Monte Carlo Simulation?).
History of Monte Carlo Simulation
The modern Monte Carlo Method was initially developed and utilized during the Manhattan
Project, the American World War II initiative to develop nuclear weapons. Scientists John von
Neumann and Stanislaw Ulam proposed the method to examine properties of neutron travel
through radiation shielding. The method was then termed the Monte Carlo Method after the
Casino in Monaco. Neumann and Ulam established majority of the core methods of Monte Carlo
simulation (Harrison, 2010).
How Does Monte Carlo Simulation Work?
According to Harrison, Monte Carlo simulations normally follow the following the following
steps:
i.
Modelling a system as a (series of) probability density functions (PDFs);
ii.
Repetitively sample from the PDFs;
iii.
Total/compute the statistics of interest.
Risk analysis is carried out by Monte Carlo Simulation via generation of models of probable
results via substitution of a range of values (a probability distribution) for any factor that has
inherent uncertainty. These results are then repeated calculated, each calculation utilizing a
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unique set of random values from the probability functions. Depending upon the number of
uncertainties and the ranges specified for them, a Monte Carlo simulation may have thousands or
tens of thousands of recalculations before it is complete depending on the number of
uncertainties and the ranges specified for the respective uncertainties. The simulation model then
generates distributions of probable outcomes. Utilizing probability distributions is advantageous
in that variables can have different probabilities of different outcomes occurring. The use
Probability distributions offer a much more realistic manner of assessing uncertainty and risk.
In many applications of Monte Carlo Simulation, the random objects are artificially introduced
to assess purely deterministic problems and the Monte Carlo method would in this case involve
random sampling from certain probability distributions. The core concept of the Monte Carlo
techniques is to repeatedly conduct an experiment to attain many quantities of interest using the
Law of Large Numbers and other methods of statistical inference (Dirk P. Kroese, Tim Brereton,
Thomas Taimre, Zdravko I. Botev).
Applications of Monte Carlo Simulation
Numerous quantitative problems in science, engineering, and finance are solved via Monte Carlo
Simulation techniques. Some of the significant application areas are listed below:
Industrial Engineering and Operations Research
One of the main application area of simulation modelling is Industrial Engineering and
Operations Research. Typical applications of Monte Carlo Simulation in Operations Research
include simulation of inventory processes, job scheduling, queuing networks, vehicle routing and
reliability systems.
An important area of Operations Research is Mathematical Programming and Monte Carlo
Simulation has proven to be an efficient tool in Mathematical Programming, being used to
provide optimum design for scheduling and control of industrial systems. It has also proven
effective in developing new approaches to solve traditional optimization problems such as the
traveling salesman problem. The Monte Carlo Method is also utilized in the design and control
of autonomous machines and robots.
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Physical Processes and Structures.
The first application of Monte Carlo Simulation in the modern era was the direct simulation of
the process of neutron transport. Monte Carlo techniques have contributed to the simulation of
photon transport through biological tissue. Monte Carlo techniques also play a crucial role in
materials science, as they are employed in the development and analysis of new materials and
structures, such as organic, organic solar cells and Lithium-Ion batteries.
The technique is also integral in virtual materials design, where experimental data is utilized to
generate stochastic models of materials. The physical development and analysis of new materials
is generally time consuming and expensive. The advantage of the virtual materials design
approach is that it allows for generation of further data than can easily be attained from physical
experiments and allows for the virtual production and study of materials utilizing numerous
different production parameters.
Random Graphs and Combinatorial Structures
From a more calculated and probabilistic point of view, Monte Carlo techniques have proven to
be very effective in analysing the properties of random structures and graphs that arise in
statistical physics, probability theory, and computer science. The classical models of
ferromagnetism, the Ising model and the Potts model, are examples of these random structures,
where a common problem is the estimation of the partition function; see, for example, Monte
Carlo techniques also play a key role in the study of percolation (the process of a liquid slowly
passing through a filter) theory, which lies at the intersection of probability theory and statistical
physics. Monte Carlo techniques have made possible the identification of such important
quantities as the critical exponents in many percolation models long before these results have
been obtained theoretically, as an early example of work in this area.
In computer science, one issue may be to determine the number of routes in a travelling salesman
problem which have “length” less than a certain number — or else state that there are none. The
computational complexity class for such problems is known as “P”. In particular, solving a
problem in this class is at least as difficult as solving the corresponding problem. Randomized
algorithms have seen considerable success in tackling these difficult computational problems.
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Economics and Finance
The Monte Carlo Simulation is used to price financial instruments and plays a crucial role in risk
analysis in the Economics and Finance sectors.
Monte Carlo Simulation is effective in solving problems involving several different sources of
uncertainty, these include but not limited to pricing basket options. It is also employed in the
application of stochastic differential equations, which are used to model many financial time
series and subsequent papers. Monte Carlo Simulation is used in the analysis of the risk of large
portfolios of financial products (such as mortgages) and risk analysis to simulate scenario
analysis.
Computational Statistics.
Mathematical Contest in Modeling (MCM) has intensely changed the way in which Statistics is
used in today’s analysis of data. The ever increasing complexity of data (“big data”) requires
radically different statistical models and analysis techniques from those that were used 20–100
years ago. By using Monte Carlo techniques, the statistician is no longer constrained to use basic
(and often unsuitable) models to describe data. Now any probabilistic model that can be
simulated on a computer can serve as the basis for a statistical analysis. This conversion has had
the most impact in Bayesian statistics, where Monte Carlo techniques (are essential tools for
deriving the posterior distribution and related quantities. Monte Carlo techniques are also
predominant in classical statistics, where they are often referred to as resampling techniques. An
important example is the well-known bootstrap method. Numerous statistical quantities such as
p-values for statistical tests and confidence intervals can simply be determined by simulation
without the need of a sophisticated analysis of the underlying probability distributions.
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Simulation Optimization
A simulation model consists of Inputs, Mathematical calculations, and Outputs. Simulation
modelling is the method of studying mathematical models by way of simulation. The diagram
below shows a simple simulation model.
Mathematical
Inputs
Outputs
Calculations
Figure 1: Simulation Model
Simulation testing is the method of testing varying inputs in a mathematical model and
documenting the reasons for a proportionally varying output. When a model becomes too
complex, with many different possible inputs, the simulation may be difficult to compute. To
reduce this degree of difficulty, optimization is paramount. This is the process of selecting the
best input without explicitly evaluating every possibility. Below is an optimized simulation
model:
Feedback
Mathematical
Inputs
Calculations
Figure 2: Simulation Optimization Model
Optimization
Outputs
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Monte Carlo Simulation Softwares
Arena Simulation Software
Arena simulation software is used around the world by companies in a variety of industries.
Arena enables companies to address business challenges in a fast and cost-effective manner.
This software is developed by Rockwell Automation
Key Uses:
Arena is used in manufacturing, supply chain, port & terminal, oil & gas, mining, call centres,
healthcare, academics, government and military, retail, consulting and more. This software is
also used to: increased reliability, maximize cost savings and increased throughput.
Support and Training
Arena offers comprehensive support through: webinars, online, in person, phone and e-mail.
Arena is supported by a global network of partners that are fully capable of supporting your
simulation needs. These partners are from the following regions of the world.
i.
North America
ii.
Central America
iii.
South America
iv.
Africa
v.
Europe
vi.
Middle East
vii.
Australia and Asia
Cost and Purchasing Options
➢ Trial & Student Edition
Standard Edition
Full Featured. Easy Flowchart Modelling Methodology suitable for the whole enterprise.
Professional Edition
All-inclusive Features. Build higher fidelity models with finer control over the simulation
models.
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Software Editions
Trial
Standard
Professional
2D and 3D Animations
•
•
•
Model size restrictions
Limited
Unlimited
Unlimited
Flow Process
•
Packaging Template
•
Figure 3 Software Editions
FlexSim Simulation Software
FlexSim is a three-dimension (3D) simulation software that models, simulates, predicts, and
visualizes systems in manufacturing, material handling, healthcare, warehousing, mining,
logistics, etc. It is both powerful and user-friendly. FlexSim helps to optimize current and
planned processes, identify and decrease waste, reduce cost, and increase revenue.
Key Features and Benefits
This software can apply to many different processes or types of operations. Great for showing
manual and automation processes that work together. Easy to use and train new users on the
software. The flexibility of the software shows automated processes with manual processes, part
movement, and part flow all at the same time. Ability to show how things interrelate on a larger
scale.
This software offers a trial version for users who maybe curious or have limited or no knowledge
of FlexSim Software. A student version is also available but has less restriction than that of the
trial version.
Support and Training
FlexSim Software Products, Inc. is headquartered in Orem, Utah. FlexSim Products FSP has
offices in Canada, Mexico, India, Germany, and China. FlexSim has regional distributors around
the globe that provide support, training, and consulting services.
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FlexSim Products (FSP) offers training from sites in Latin America, United Kingdom and
Canada.
Cost
FlexSim Software Products cost is not provided by vendor however, unlike @Risk and Arena
FlexSim has both free and trial versions respectively.
@RISK Simulation Software
@RISK is simulation software developed Palisade by that specializes in decision tools suite and
other software for risk analysis and decision making under uncertainty. The products add-in to
Microsoft Excel and add Monte Carlo simulation, Monte Carlo for Six Sigma, decision trees and
optimization to spreadsheet models. @RISK shows you virtually all possible outcomes for any
situation and tells you how likely they are to occur. This means one should be able to judge
which risks taking on and which ones to avoid which is a critical insight in today’s uncertain
world.
Key Features and Benefits
Maintenance
➢ Maintenance is free for the first year of your software.
➢ Free upgrades when new software versions are released.
➢ Full access to Technical Support.
User Interface, Modelling
➢ 100% excel integration: -@RISK integrates seamlessly with Excel’s function set and
ribbon, letting you work in a familiar environment with results you can trust.
➢ offers a wide variety of customizable, exportable graphing and reporting options that let
you communicate risk to all stakeholders.
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➢ with a broad library of probability distributions, data fitting tools, and correlation
modelling, @RISK lets you represent any scenario in any industry with the highest level
of accuracy.
➢ by sampling different possible inputs, @RISK calculates thousands of possible future
outcomes, and the chances they will occur. This helps you avoid likely hazards—and
uncover hidden opportunities (Monte Carlo Principle).
Support and Training
@RISK Simulation Software offers comprehensive support: through webinars (scheduled and on
demand), regional training across the United States online and in person, phone and e-mail.
Arena is supported by a global network of partners that are fully capable of supporting your
simulation needs.
Technical Support is available to help with installation, operational problems, or errors.
Free with a current maintenance plan and is available nearly 24 hours.
Cost
The cost for the @RISK 7.6 Professional software is US$1,870
@RISK
7
single
user
license Price [USD]
Price [EUR]
Price [ZAR]
1st year maintenance included
Excluding VAT
Excluding VAT
Excluding VAT
@RISK Professional (Commercial)
$2,000
€1,800
R28,800
@RISK Industrial (Commercial)
$2,700
€2,400
R39,000
Figure 4
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Case Study Analysis
Introduction
To garner a better appreciation for Monte Carlo Simulations and their application to real life
industries a case study was reviewed and assessed. The case study reviewed was Monte Carlo
Simulation Based Approach to Manage Risks in Operational Networks in Green Supply Chain
by Sachin K Mangla, Pradeep Kumar and Mukesh Kumar Barua. The case study was presented
at the 12th Global Congress on Manufacturing and Management, GCMM, 2014 Conference.
Description of the Problem
Every business activity in Green Supply Chain Management is consisting of various risks and
risk factors and or drivers. These risks may have an adverse impact on the system and result in
business loss if they are accounted for. The management of an Indian poly-plastic company is
seeking to detect and assess risks related to Green Supply Chain Management at the shop floor
level to improve business profits. For a holistic approach to supply chain management, it is
crucial to have knowledge for risk and understanding how
to manage them to reduce their
consequence.
Aim of Case Study
The intent of the case study being to focus on the operational Green Supply Chain Management
risk evaluation and management by capturing of uncertainty and assessing the risks via Monte
Carlo simulation to determine the delay/disturbance consequences of the risk.
Definition of Key Terms
The following key terms are defined to better understand the scope and findings of the case
study.
➢ Supply Chain Management
According to Kenton, Supply chain management refers to the management of the flow of goods
and services and is inclusive of all processes that converts raw materials into final products.
➢ Green Supply Chain Management
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Green Supply Chain Management is defined as the integrated environmental logic into the
Supply Chain Management.
Data Collection
The data was collected by analysing past literature to identify the operational risk issues or
sources for Green Supply Chain Management. In addition to the literature an expert group was
also consulted. The expert group constituted four senior managers, three IT representatives and
three supply chain professionals.
Methodology
The methodology used to assess the risks factors in the Green Supply Chain Management the
Indian poly-plastic company consisted of:
i.
Identification of operational risks in Green Supply Chain Management
through literature and industry expert judgements
ii.
Evaluation of identified risks and risk driving factors sing industry expert
inputs
iii.
To analyse consequences (measured in time) in terms of delay/
disturbance using Monte Carlo Simulation approach through industry
experts input
iv. Discussion, Summary, Conclusions
Findings/ Results
A team of experts that was consulted determined the core risk factors and measured the
consequence of each risk in terms of Time, Brand image, Economic, Health and Safety and
Quality. The human based assessment however was not able to provide extreme scenario and so
simulation was utilized. The results of the human based assessment are summarized in the tables
below:
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Table 1: The table above shoes the operational risks description and the consequences as
determined by the team of experts.
Table 2: The table above shows the likelihood of each risk and the min, average and max
disturbance /delay time to be modelled in Monte Carlo Simulation.
The Monte Carlo Simulation approach was used to analyse the operational risk factors and their
consequences on the performance of Green Supply Chain Management. Additionally, it assists
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capturing the uncertainties in the inputs. A sensitivity analysis test was also performed to
capture the consequences of risks on the delay/disturbance profile mean.
The consequence with the main impact on the business was determined by the team of experts to
be the delay/ disturbance time cause by each risk factor. The probability of each risk factor was
considered and the factors modelled against the risk consequence (delay/disturbance time) as
probability distributions. This was accomplished via using Monte Carlo Simulation using the
@Risk software. The simulation was done using 1 simulation and 20,000 iterations where the
risk consequences were analysed at a 95 % confidence interval. An advanced sensitivity analysis
test (Sensitivity Tornado) was also been performed using @Risk to determine the effect of each
risk factor on the mean delay/disturbance time. The sensitivity analysis deduced that the ranking
of the effect of each risk on the output mean (delay/disturbance time) is given by O3 - O4 –O1 –
O2 - O5 (see table 1 above for risks code).
Figure 5: showing probability distribution form @Risk Monte Carlo Simulation
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Figure :6 Showing Sensitivity Tornado form @Risk Monte Carlo Simulation
Optimization/Improvements
The results allowed for generation of a wide variety of possible risk scenarios, as well as
associated probabilities. The simulation allowed the prediction of type of risks that enabled
anticipation of the most probable risks. This allowed for creation of a model that provided the
company with insight into the potential ecological-economic gains of a green supply chain, as
well as recommendations to optimally manage the operational risks.
Limitations and Challenges
The information used in the simulation was gathered from a group of “experts’” however there
may have been human bias in the initial assessments. It should be noted that the results of a
simulation are only as accurate as the inputs in the simulation model. Also, the results are based
on the case study conducted at one company and may hence lack external validity and
generalizability.
Recommendations
The Recommendations to minimize the limitations identified above include:
i.
Take measure to overcome human sensitivity.
ii.
Empirical research may be conducted in the future to better explore the problem and
to provide better analysis of the results to improve overall performance.
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Personal Reflections/ Observation
The team of graduate students analysing the case Monte Carlo Simulation Based Approach to
Manage Risks in Operational Networks in Green Supply Chain believes that the Monte Carlo
Simulation proved to be a useful tool in risk assessment and can contribute effectively in
managing and mitigating risks associate with Green Supply Chain Management. The team also
simulated the case study model using @RISK software and observed similar results to those
reported in the case study.
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References
1. Carson, Y., & Maria, A. (1997). Simulation Optimization: Methods And Applications.
Binghamton,
New
York,
USA.
Retrieved
from
https://www.informssim.org/wsc97papers/0118.PDF
2. Harrison, R. L. (2010, January 5). Introduction To Monte Carlo Simulation.
doi:10.1063/1.3295638
3. Kenton, W. (2019, March 13). Monte Carlo Simulation Definition. Retrieved from
Investopedia: https://www.investopedia.com/terms/m/montecarlosimulation.asp
4. KENTON, W. (2019, February 19). Supply Chain Management (SCM). Retrieved from
https://www.investopedia.com/terms/s/scm.asp
5. Kroese, D., Brereton, T., & Thomas Taimre, Z. B. (n.d.). Why Monte Carlo Method is so
important today .
6. Manglaa, S. K., Kumarb, P., & Baruac, M. K. (2014). Monte Carlo Simulation Based
Approach to Manage Risks in. 12th Global Congress On Manufacturing And
Management, GCMM 2014.
7. Monte Carlo Simulation . (n.d.). Retrieved from www.palisade.com.
8. What is Monte Carlo Simulation . (n.d.). Retrieved from www.riskamp.com:
https://www.riskamp.com/files/RiskAMP%20-%20Monte%20Carlo%20Simulation.pdf
9. Maria. A. (1997). Introduction Modelling and Simulation. Retrieved from:
acqnotes.com/Attachments/White%20Paper%20Introduction%20to%20Modeling%20an
d%20Simulation%20by%20Anu%20Maria.pdf
10. FlexSim
Simulation
Software.
https://www.flexsim.com/flexsim/
Retrieved
on
March27,
2019
from:
11. Arena
Simulation
Software.
Retrieved
on
March
26,
2018
from:
https://www.arenasimulation.com/purchasing-options/find-a-partner/north-america