Introduction to modeling and simulation

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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




                                                                                            2
                            • Collectors
                              • Capture sun’s 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




                                                             3
                                      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)




                                                                                      4
                                         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
                                                                                             5
                Input       System        Output
                            Controller

                        Open-Loop System



Desired Reference
      or Input     Error     System          Output
               + Signal      Controller
                 -
                            Feedback

                        Closed-Loop System




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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

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                                           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                                                8
 Fspring              spring constant The amount spring
                                         is stretched
           Fspring
                     FSpring = -k∙x


                     x= -FSpring/k



Hooke’s Law



                                                     9
                                        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


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                                       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

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                                        Simulation

   Dangers of Simulation
       Fall in love with a model
            Become too enthusiastic about a model and forget about the experimental frame
            e.g. Hooke’s 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 model’s level of accuracy
            All models have simplifying assumptions
            e.g. Free-fall motion is a simplified model (assuming air resistance is negligible)




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                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
                                                                                 13
                                  Simulation

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




                                                        14
                        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
                                                                                            15
                       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. @Risk, Arena, Matlab, etc.)




                                                                                            16
                       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




                                                                                           17
                       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.




                                                                                  18
     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
                                                                               19
                                 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.


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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.

                                     Straightener


                  Drill #1
                                                         Finishing
                  Drill #2                               Operator

                                                              Legend:
                                                                  Type 1 parts
                                                                  Type 2 parts



                                                                                  21
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



                                                                                               22
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


                                                                                                23
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




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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



                                                                                             25
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




                                                                                        26
                                 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




                                                     27
                                 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




                                                                     28
                                 Simulation Tools

   Special Purpose Simulation Packages
       Manufacturing (e.g. AutoMod, FACTOR/AIM, etc.), Communications network
        (e.g.COMNET III, NETWORK II.5, etc.), Business (BP$IM, 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




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                                         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
                                                                                                      30
                                   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 = 15x1+10x2
        subject to
        0≤ x1 ≤2, 0 ≤ x2 ≤ 3, x1+x2 ≤4

     The objective function is z = 15x1+10x2
     The constraints are:
     0≤ x1≤2, 0 ≤ x2 ≤ 3, x1+x2 ≤4




                                                                                    31
 x
5 2                                x1=2


4




3                                                            x2=3

                                   Optimal point (x1=x2=2)
2
                                          x1+ x2 = 4
        Feasible Region

1


                                    z = 40
0
                              z = 30                                    x1
    0          1     z = 20    2              3         4           5
            z = 10

                          zmax = 15*2 + 10*2 = 50
                                                                             32
                                Excel Solver
   A Microsoft Excel Add-In
       Go to Tools >>Add-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                                 33

				
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