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Introduction to modeling, simulation, and Optimization

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Introduction to modeling, simulation, and Optimization Dr. Yan Liu Department of Biomedical, Industrial and Human Factors Engineering Wright State University – PowerPoint PPT presentation

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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.
  • 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

28
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

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)
  • Advantages
  • Very quick development of complex models
  • Short learning cycle
  • little programming
  • Disadvantages
  • 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
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