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Introduction to modeling, simulation, and

Optimization

- Dr. Yan Liu
- Department of Biomedical, Industrial and Human

Factors Engineering - Wright State University

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

- 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

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)

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

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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

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

Fspring

The amount spring is stretched

spring constant

Fspring

FSpring -kx

x -FSpring/k

Hookes Law

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

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

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)

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

Simulation

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

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

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.)

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

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.

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

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.

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.

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

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

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

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

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

Simulation Tools

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

Simulation Tools

- General Simulation Languages
- Arena, Extend, GPSS, SIMSCRIPT, SIMULINK (In

Matlab), etc. - Advantages
- Standardized features in modeling
- Shorter development cycle for each model
- Very readable code
- Disadvantages
- Higher software cost (up-front)
- Additional training required
- Limited portability

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) - Advantages
- Very quick development of complex models
- Short learning cycle
- little programming
- Disadvantages
- High cost of software
- Limited scope of applicability
- Limited flexibility

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

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

x2

Feasible Region

x1

zmax 152 102 50

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