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Introduction to modeling, simulation, and Optimization

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Title: Introduction to modeling, simulation, and Optimization

1
Introduction to modeling, simulation, and
Optimization
• Dr. Yan Liu
• Department of Biomedical, Industrial and Human
Factors Engineering
• Wright State University

2
Systems
• What is System
• A system is a set of components which are related
by some form of interaction and which act
together to achieve some objective or purpose
• Components are the individual parts or elements
that collectively make up the system
• Relationships are the cause-effect dependencies
between components
• Objective is the desired state or outcome which
the system is attempting to achieve

3
• Collectors
• Capture suns thermal energy
• Storage tank
• Pump
• Move the water through the tank
• Booster element
• Heat water
• Relief valve
• Cold water inlet
• Hot water outlet

Solar-Heated Water System
4
Systems
• Natural vs. Artificial Systems
• A natural system exists as a result of processes
occurring in the natural world (e.g. river,
universe)
• An artificial system owes its origin to human
activity (e.g. space shuttle, automobile)
• Static vs. Dynamic Systems
• A static system has structure but no associated
activity (e.g. bridge, building)
• A dynamic system involves time-varying behavior
(e.g. machine, U.S. economy)

5
Systems
• Open-Loop vs. Closed-Loop systems
• Inputs
• Variables that influence the behavior of the
system
• e.g. wheel, accelerator, and brake of a car
• Outputs
• Variables that are determined by the system and
may influence the surrounding environment
• e.g. direction and speed of a car
• An open-loop system cannot control or adjust its
own performance
• e.g. watch, car
• A closed-loop system controls and adjusts its own
performance in response to outputs generated by
the system through feedback
• e.g. watch with owner, car with driver
• Feedback is the system function that obtains data
on system performance (outputs), compares the
actual performance to the desired performance (a
standard or criterion), and determines the
corrective action necessary

6
(No Transcript)
7
Models
• What is Model
• A model of a system is a representation of the
construction and working of the system
• Similar to but simpler than the system it
represents
• Close approximation to the real system and
incorporate most of its salient features
• Should not be so complex that it is hard to
understand or experiment with it
• Physical Model
• A physical object that mimics some properties of
a real system
• e.g. During design of buildings, it is common to
construct small physical models with the same
shape and appearance as the real buildings to be
studied
• Through prototyping process
• Prototyping is the process of quickly putting
together a working model (a prototype) in order
to test various aspects of a design, illustrate
ideas or features and gather early user feedback

8
Models
• Mathematical Model
• A description of a system where the relationship
between variables of the system are expressed in
a mathematical form
• e.g. Ohm's law describes the relationship
between current and voltage for a resistor
Hooke's Law gives the relationship between the
force applied to an unstretched spring and the
amount the spring is stretched when the force is
applied, etc.
• Through virtual prototyping
• Deterministic vs. stochastic models
• In deterministic models, the input and output
variables are not subject to random fluctuations,
so that the system is at any time entirely
defined by the initial conditions chosen
• e.g. the return on a 5-year investment with an
annual interest rate of 7, compounded monthly
• In stochastic models, at least one of the input
or output variables is probabilistic or involves
randomness
• e.g. the number of machines that are needed to
make certain parts based on the probability of
machine failure

9
Fspring
The amount spring is stretched
spring constant
Fspring
FSpring -kx
x -FSpring/k
Hookes Law
10
Simulation
• What is Simulation
• A simulation of a system is the operation of a
model of the system, as an imitation of the real
system
• A tool to evaluate the performance of a system,
existing or proposed, under different
configurations of interest and over a long period
of time
• e.g. a simulation of an industrial process to
learn about its behavior under different
operating conditions in order to improve the
process
• Reasons for Simulation
• Experiments on real systems are too expensive,
too dangerous, or the system to be investigated
does not yet exist
• e.g. Investigating ship durability by building
ships and letting them collide is a very
expensive method of gaining information training
nuclear plant operators in handling dangerous
situations by letting the nuclear reactor enter
hazardous states is not advisable

11
Simulation
• Reasons for Simulation (Cont.)
• The time scale of the dynamics of the system is
not compatible with that of the experimenter
• e.g. It takes millions of years to observe small
changes in the development of the universe,
whereas similar changes can be quickly observed
in a computer simulation of the universe
• Easy manipulation of parameters of models (even
outside the feasible range of a particular
physical system)
• e.g. The mass of a body in a computer-based
simulation model can be increased from 40 to 500
kg at a keystroke, whereas this change might be
hard to realize in the physical system
• Suppression of disturbances
• Allow isolating particular effects and gaining a
better understanding of effects of particular
interest as a result
• e.g. simulation of free-fall objects ignores the
effect of air resistance

12
Simulation
• Dangers of Simulation
• Fall in love with a model
• Become too enthusiastic about a model and forget
about the experimental frame
• e.g. Hookes law applies only if the spring is
not stretched beyond its elastic limit
• Force reality into the constraints of a model
• e.g. Shaping of our societies after fashionable
economic theories that have a simplified view of
reality and ignoring many other important aspects
of human behavior, society, and nature
• Forget the models level of accuracy
• All models have simplifying assumptions
• e.g. Free-fall motion is a simplified model
(assuming air resistance is negligible)

13
Phases and Steps of Simulation
• Phase 1. Develop Simulation Model
• Step 1. Identify the problem
• Step 2. Formulate the problem
• Step 3. Collect and process real system data
• Step 4. Formulate and develop a model
• Step 5. Validate the model
• Step 6. Document model for future use
• Phase 2. Design and Conduct Simulation Experiment
• A test or series of tests in which meaningful
changes are made to the input variables of a
simulation model so that we may observe and
identify the reasons for changes in the
performance measures
• Step 7. Select appropriate experimental design
• Step 8. Establish experimental conditions for
runs
• Step 9. Perform simulation runs

14
Simulation
• Phase 3. Perform Simulation Analysis
• Step 10. Analyze data and present results
• Step 11. Recommend further courses of actions

15
Develop Simulation Model
• Step 1. Identify Problem
• Enumerate problems with an existing system
• Produce requirements for a proposed system
• Step 2. Formulate Problem
• Define overall objectives of the study and
specific issues to be addressed
• Define performance measures
• Quantitative criteria on the basis of which
different system configurations will be evaluated
and compared
• Develop a set of working assumptions that will
form the basis for model development
• Model boundary and scope (width of model)
• Determines what is in the model and what is out
• Level of detail (depth of model)
• Specifies how in-depth one component or entity is
modeled
• Determined by the questions being asked and data
availability
• Decide the time frame of the study
• Used for one-time or over a period of time on a
regular basis

16
Develop Simulation Model
• Step 3. Collect and Process Real System Data
• Collect data on system specifications, input
variables, performance of the existing system,
etc.
• Identify sources of randomness (stochastic input
variables) in the system
• Select an appropriate input probability
distribution for each stochastic input variable
and estimate corresponding parameters
• Standard distributions (e.g. normal, exponential,
etc.)
• Empirical distributions
• Software packages for distribution fitting (e.g.
_at_Risk, Arena, Matlab, etc.)

17
Develop Simulation Model
• Step 4. Formulate and Develop a Model
• Develop schematics and network diagrams of the
system
• How do entities flow through the system
• Translate conceptual models to simulation
software acceptable form
• Verify that the simulation model executes as
intended
• Build the model right (low-level checking)
• Traces
• Vary input parameters over their acceptable
ranges and check the output

18
Develop Simulation Model
• Step 5. Validate Model
• Check whether the model satisfies or fits the
intended usage of system (high-level checking)
• Build the right model
• Compare the model's performance under known
conditions with the performance of the real
system
• Perform statistical inference tests and get the
model examined by system experts
• Assess the confidence that the end user places on
the model and address problems if any
• Step 6. Document Model for Future Use
• Objectives, assumptions, inputs, outputs, etc.

19
Design and Conduct Simulation Experiment
• Step 7. Select Appropriate Experimental Design
• Performance measures
• Input parameters to be varied
• Ranges and legitimate combinations
• Document experiment design
• Step 8. Establish Experimental Conditions for
Runs
• Whether the system is stationary (performance
measure does not change over time) or
non-stationary (performance measure changes over
time)
• Whether a terminating or a non-terminating
simulation run is appropriate
• Starting condition
• Length of warm-up period
• Model run length
• Number of statistical replications
• Step 9. Perform Simulation Runs

20
Simulation Analysis
• Step 10. Analyze Data and Present Results
• Statistics of the performance measure for each
configuration of the model
• Mean, standard deviation, range, confidence
intervals, etc.
• Graphical displays of output data
• Histograms, scatterplot, etc.
• Document results and conclusions
• Step 11. Recommend Further Courses of Actions
• Other performance measures
• Further experiments to increase the precision and
reduce the bias of estimators
• Sensitivity analysis
• How sensitive the behavior of the model is to
changes of model parameters
• etc.

21
A machine shop contains two drills, one
straightener, and one finishing operator. Type 1
parts require drilling, straightening, and
finishing in sequence. Type 2 parts require only
drilling and finishing. The frequency of arrival
and the time to be routed to the drilling area
are deterministic for both types of parts.
22
Step 1. Identify the problem
• Assess utilization of drills, straightener, and
finishing operator
• The following modification to the original
system is of interest the frequency of arrival
of both parts is exponential with the same
respective means as in the original system

Step 2. Formulate the problem
Objectives
• Obtain the utilization of drills, straightener,
and finishing operator for the system
• Assess the modification

Performance measure
• Utilization of operations (the fraction of time
the server is busy, i.e. busy time divided by the
total time)

Assumptions
• Two drills are identical
• There is no material handling time between the
three operations
• Parts are processed on a first-come-first-serve
basis
• Parts wait in a queue till one of the two
drilling machines becomes available

23
Step 3. Collect and process real system data
• A type 1 part arrives every 30 min.
• A type 2 part arrives every 20 min.
• It takes 2 min. and 10 min. to route a type 1
part and a type 2 part to the drilling area,
• respectively
• Drilling time is normally distributed with mean
10 min. and standard deviation 1 min.
• Straightening time is exponentially distributed
with a mean of 15 min.
• Finishing requires 5 min. per part

Step 4. Formulate and develop a model
• A model of the system and the modification are
developed using a simulation package
• A trace verifies that the parts flowed through
the job shop as expected

Step 5. Validate the model
• The model of the original system is run for a
sufficiently long period, and its utilization
performance measures are judged to be reasonable
by the machine shop operators

Step 6. Document model for future use
• The models of the original system and the
modification are documented as thoroughly as
possible

24
Step 7. Select appropriate experimental design
• Performance measures are the utilization of
operations
• Vary input parameters operating times for
drilling, straightening, and arrival time of
parts (in modification)
• Document experiment design for the models of the
original and modified systems

Step 8. Establish experimental conditions for
runs
• The system is non-stationary
• There is no part in the machine shop initially
• 1000 min. warm-up period
• Each model is run three times for 4000 min.

Step 9. Perform simulation runs
• Runs are performed as specified in Steps 7 and 8

25
Step 10. Interpret and present results
Utilization Statistics of Models of Original and
Modified Systems (in parenthesis)
Drilling Straightening Finishing
Mean Run 1 0.83 (0.78) 0.51 (0.58) 0.42 (0.39)
Mean Run 2 0.82 (0.90) 0.52 (0.49) 0.41 (0.45)
Mean Run 3 0.84 (0.81) 0.42 (0.56) 0.42 (0.40)
Std. Run 1 0.69 (0.75) 0.50 (0.49) 0.49 (0.49)
Std. Run 2 0.68 (0.78) 0.50 (0.50) 0.49 (0.50)
Std. Run 3 0.69 (0.76) 0.49 (0.50) 0.49 (0.49)
• Utilization of each drill is about 80
• Utilization of straightener is about 50
• Utilization of finishing operator is about 40
• Average utilization of the original and modified
systems does not differ significantly
• The standard deviation of the drilling operation
seems to have increased because of the increased
randomness in the modification

26
Step 11. Recommend further course of action
• Other performance measures of interest may be
throughput of parts for the system, mean time in
system for both types of parts, average and
maximum queue lengths for each operation
• Other modification of interest may be the flow
of parts to the machine shop doubles

27
Simulation Tools
• General Purpose Programming Languages
• FORTRAN, PASCAL,C/C JAVA, etc.
• Little or no additional software cost
• Universally available (portable)
• No additional training
• Every model starts from scratch
• Very little reusable code
• Long development cycle for each model

28
Simulation Tools
• General Simulation Languages
• Arena, Extend, GPSS, SIMSCRIPT, SIMULINK (In
Matlab), etc.
• Standardized features in modeling
• Shorter development cycle for each model
• Very readable code
• Higher software cost (up-front)
• Additional training required
• Limited portability

29
Simulation Tools
• Special Purpose Simulation Packages
• Manufacturing (e.g. AutoMod, FACTOR/AIM, etc.),
Communications network (e.g.COMNET III, NETWORK
II.5, etc.), Business (BPIM, ProcessModel,
etc.), Health care (e.g. MedModel)
• Very quick development of complex models
• Short learning cycle
• little programming
• High cost of software
• Limited scope of applicability
• Limited flexibility

30
Optimization
• What is Optimization
• Its objective is to select the best possible
decision for a given set of circumstances without
having to enumerate all of the possibilities
• Involves maximization or minimization as desired
• How can a large manufacturing company determine
the monthly product mix at its Indianapolis plant
that maximizes corporate profitability?
• Design of civil engineering structures such as
frames, foundations, bridges, towers, chimneys
and dams for the minimum cost
• Components
• Decision variables
• Variables in the model which you have control
over
• Objective function
• A function (mathematical model) that quantifies
the quality of a solution in an optimization
problem
• Constraints
• Conditions that a solution to an optimization
problem must satisfy
• Restrict decision variables by defining
relationships among them
• Find the values of the decision variables that
maximize (minimize) the objective function value,
while staying within the constraints

31
Optimization
• Linear Programming
• The objective function and all constraints are
linear functions (e.g. no squared terms,
trigonometric functions, ratios of variables) of
the decision variables

Example Maximize z 15x110x2 subject to 0 x1
2, 0 x2 3, x1x2 4
The objective function is z 15x110x2 The
constraints are 0 x12, 0 x2 3, x1x2 4
32
x2
Feasible Region
x1
zmax 152 102 50
33
Excel Solver
• A Microsoft Excel Add-In
• Go to Tools gtgtAdd-Ins , select Solver Add-in,
click OK
• Originally designed for optimization problems but
also useful for root finding and similar
mathematical problems
• Target cell
• The objective or goal
• Maximize, minimize or set a specific value to
the target cell
• Changing cells
• Can be adjusted until the constraints in the
problem are satisfied and the cell in the Set
Target Cell box reaches its target
• Constraints
• The restrictions placed on the changing cells