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					Operations Research Models
    Johannes Lüthi
    Universität der Bundeswehr München


    Visit of Slovenian Delegation
    at AStudÜbBw / IABG
        Personal Background
                 -1995: Mathematics at Univ. of Vienna, Austria:
                  game theory, optimization;
                  Masters thesis: distributed simulation

                 1995-1997: Research assistant at Dept. of Applied
                  Information Systems, Univ. Vienna, Austria:
                  performance modeling of parallel systems
                  (task graph models, queueing models)

                 1997: Dissertation: analysis of queueing networks with
                  parameter uncertainties and variabilities

                 1998- : Univ. Fed. Armed Forces, Munich:
                  performance modeling of computer systems,
                  model building and computer simulation

J.Lüthi, UniBwM                           OR - Models                      2
          Introduction
            Key aspects of OR
            Historical background
            Model types
            Application of OR Models

        Model Building
        OR Models
        Key Aspects of OR

                 Solve complex real-world problems using a
                  scientific / mathematical approach
                 Problems in systems consisting of
                   Humans
                   Machines
                   Material
                   Capital

                 Represent system in a mathematical model
                 Support people who plan and manage such
                  systems in making decisions

J.Lüthi, UniBwM                     OR - Models               4
        Historical Background
                   1938, Great Britain: “Operational Research”
                    Increase effectiveness of military operations
                    (radar, submarines) using mathematical
                   1941: Linear programming for distribution
                    problems (e.g. refinery problems)
                   1957: Dynamic programming
                   1959: Network models

                   Also   older models are used in OR

J.Lüthi, UniBwM                       OR - Models                   5
        Model Types

                    Illustrative, plastical    Abstract, symbolic          Analogy models
                            models                  models

                           Verbal                Mathematical                  Others
                           models                  models               (e.g. musical notation)

                  Models for optimization     Models for Prediction   Models for experimentation

                  E.g. linear programming,    E.g. Markov models,            Simulation
                         game theory           Forrester models

J.Lüthi, UniBwM                               OR - Models                                          6
        Application of OR Models

                  Real System                          Mathematical Model

                                                   Analysis or

             Real World Solution                         Model Solution

J.Lüthi, UniBwM                        OR - Models                          7
        Example: Continuous Model
                                                     Observed state:
                                                      volume in tank
                         V(t)                        State changes:
                                                      water flowing in and
                  a(t)                                out of the tank

                                                      Conceptual model:
                 V(t) ... Volume of water
                                                      Differential equation
                 z(t) ... Flow in tank
                  a(t) ... Flow out of tank           dV ( t )
                                                                z ( t )  a( t )


J.Lüthi, UniBwM                         OR - Models                                 8
        Example: Discrete Model
                                                   Observed state:
                                                    number of cars
                                                   State changes:
                                                    cars entering and


                  Conceptual model:             Capac ity Occ upancy
                  Petri net                            Entranc e

J.Lüthi, UniBwM                       OR - Models                       9
Model Building
         Introduction
         Model Building

             Phases of model building
             Verification & validation
             Summary

           OR Models
        Process of Building a Model

                 Problem definition
                 System analysis
                 Model formalization
                 Implementation & integration
                 Analysis / experimentation

                 During all steps: verification & validation

J.Lüthi, UniBwM                    OR - Models                  11
        Problem Definition

                 Transformation of an unsharp problem into
                  an exact problem description:

                 Which questions have to be answered?
                   E.g.:what is the optimal inspection interval for a
                    certain type of car?

                 With what accuracy?
                   E.g.:   optimal value  10%

J.Lüthi, UniBwM                     OR - Models                     12
        System Analysis

                 System decomposition
                 Interaction analysis
                 System boundaries
                 Input parameters
                 Objects to be analyzed
                 Output measures

                 Communicative conceptual model

J.Lüthi, UniBwM                  OR - Models       13
        Model Formalization

                 Choose modeling paradigm or
                  language, e.g.:
                   Linear programming model
                   Queueing model
                   Petri nets
                   Discrete event simulation model

                 Translate communicative model into
                  chosen paradigm
J.Lüthi, UniBwM                   OR - Models          14
        Model Implementation

                 Implementation
                   Writecomputer program
                   Use tool to implement formal model

                 Integration
                   Integrate  pre-built components
                   Integrate model into a federation of models
                    (e.g. via HLA)

J.Lüthi, UniBwM                   OR - Models                 15
        Analysis / Experimentation

                 Use the implemented model to produce
                  results, e.g.:

                   Obtain input values that produce optimal
                    output measures
                   Predict system behavior
                   Perform “what if” studies
                   Build meta-models

J.Lüthi, UniBwM                     OR - Models                16
        Verification & Validation

J.Lüthi, UniBwM            OR - Models   17

                 Did we build the model right?
                   Top-down    modular design
                   Structured walk-through
                   Antibugging
                   Try simplified cases
                   Continuity tests
                   Degeneracy tests
                   Consistency tests
                   Test random number generators

J.Lüthi, UniBwM                  OR - Models        18

                 Did we build the right model?
                   Validate    key aspects
                      Assumptions

                      Inputparameter values and distributions
                      Output values and conclusions

                   Use   comparison sources:
                      Expert intuition
                      Real system measurements

                      Theoretical results

J.Lüthi, UniBwM                      OR - Models                 19
                      Verification & Validation Framework
          Dirk Brade (UniBwM, brade@informatik.unibw-muenchen.de):
                                                                                  Framework of Questions for Supporting VV&A in a M&SApplication
                                                                                                             Life Cycle

                                                                                                                                                                                                                                                                                                                                                                                                                                               Experimantal Model VV&T
       for Failure

                                                                                                                                                                                                                                                                                                                                                                                             rogrammed Model VV&T
                                                                                                                                                                                                                                                                                                                                    Formal Model VV&T
                                                                                                                                                                                                                                                          Communic ative Model VV&T
     Inac curate or
       wrong data                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Model Results
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (i. e. data stream s, anim ated graphs)

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  Whic h source input data came from ?

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Is input data representative?

                                                                                                                                                                                                                                                                                                                                                                                                                              Exec utable Model                                                             Is there obviously sensless or c ryptic output?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Are c omplianc e c onditions met?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Value of Results:
                                                                                                                                                                                                                                                                                                                                                                                                                              (program on host)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Has an appropriate fram ework for the test
                                                                                                                                                                                                                                                                                                                                                                                                                                  Is the program implementation syntactic ally and
                                                                                                                                                                                                                                                                                                                                                                                                                                                           semantically correc t?                                                                                                                                                                       been c hoosen ?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (Modeling Expert)
                                                                                                                                                                                          ystem and Objectives VV&T,
                                                                                                                                                                                         Modell Qualification VV&T

   Im plemention
                                                                                                                            roblem VV&T

                                                                                                                                                                                                                                                                                                                                                                                 Exec utable Subm odels               Do the paradigms of the choosen program ming language                                                                                                                                                                             Func tionality:
                                                                                                                                                                                                                                                                                                                                                                                                                         refer to the paradigms of the form al model description?
                                                                                                                                                                                                                                                                                                                                                                                 (program on host)                  Does the program ms data structure c orrespond to the formal
                                                                                                                                                                                                                                                                                                                                                                                                                                                    m odel elements description?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Does the sim ulation program run?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (Program mer)
                                                                                                                Formulated P

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Correspond dynamic ally generated data elements to
                                                                                                                                                                                                                                                                                                                  Form al Model                                                                                                                                                                                             instanc es of model objec ts?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Does the interpretation of the formal model produc e the                                                                                                  Form alism :
                                                                                                                                                                                                                                                                                                                  (i. e. UML-desc ription, petri net)                                                                                                                                                                                      sam e results?

                                                                                                                                                                                                                                                                                                                         Has paradigm violation occ ured (syntax chec k)?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Did we hold the form al rules?
                                                                                                                                                                                                                                                                                                                                          Is it possible to proove invariants?                                                                                                                                                                                                                                                                          (Modeling Expert)

                                                                                                                                                                                                                                                                                                           HLA: There‟s no c onflic t between different desrciptions (i. E. CS
                                                                                                                                                                                                                                                                                                                                                                   and SOM?

                                                                                                                                                                                                                                                                                                                                                                                                                    Matc h the program data dependenc ies the desc ribed object          Does the desc ibed model behavior equal the observed m odel
                                                                                                                                                                                                                                    Com m unic ative Model
                                                                                                                                                                                                                                                                                                               Is this model suited to communicate the c ontents of the
                                                                                                                                                                                                                                                                                                           c ommunic ative m odel in a more acc urate, formalized way?                                                                                            interactions?                                                                   behavior?                                                                                             Transparenc y & Com pleteness:
                                                                                                                                                                                                                                                                                                                                                                                                                     Does Program structure understanding bring model c onzept                   Does the com unic ative m odel describe this behavior?
                                                                                                                                                                                                                                    (i. e. doc um entation, presenta tions)
                                                                                                                                                                                                                                                                                                                                     Is the form al desc ription c omplete?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                understanding?                       Do extreme inputs produc e the expec ted output?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Does the m odel c om munic ate its
                                                                                                                                                                                                                                   Does the system m atter expert understand the c ommunicative                                                                                                                                                                                                           Are all c hanges of inputs visible at the output?
     Conc eptual                                                                                                                                                                                                                                                                             model?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              “U welt-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       “Hauptobjekt” mit
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Modella ttributen
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        c om plete conc ept?
                                                                                                                                                                                                                                      HLA: where all form al doc umentation requirements fulfilled?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (Modeling expert)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ob je kt“
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Systemgre nze         Ob jekt

    inac c uracies                                                                                                                                                                                                                  Are the ideas on whic h the analysis of the real world system is

                                                                                                                                                                                                                                                       based reproduc ed in an appropriate way?

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ra ktio
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Ob jekt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Obje kt

       and errors                                                                                                                                                                                                                                          Is everything c arefully thought through?
                                                                                                                                                                                                                                         Have all laws and rules of the dom ain been c onsidered?

                                                                                                                                                                                                                                      Is it possible to rec onstruct the objec t hierarc hc y c ompletly          Do all objec ts appear in the formal m odel desc ription?                                          Does the c lass hierarc hy c orrespond to the objec t hierarc hy?
                                                                                                                                                                Objec t Hierarc hy, Attributes &                                                                                         and correc tly?              Can you rec onstruct the static al object hierarc hy?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Routing Space Table
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Parameter Table
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Attribute Table
                                                                                                                                                                                                                                                                   Are all defined objects integrated?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Were all com ponents used c onforming to
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Interaction Class Structure Table

                                                                                                                                                                           System Borders                                                                    Is the cause-effec t-relationship correct?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Object Class Structure Table
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Object Model Identification Table

                                                                                                                                                                                                                                     Do all objec ts support the interactions in whic h they take part
                                                                                                                                                                (i. e. dependenc y graphs, attribut tables,                                                      (do they provide the right attributes)?                                                                                                                                                                                                                                                                                                                                                their spec ific ation?
                                                                                                                                                                                                                                       Are all influenc es from outside the system border defined?
                                                                                                                                                                HLA: S  OM))
                                                                                                                                                                                                                                        Do the interac tions represent all relevant dynamic system                                                                                                                                                                                                                                                                                                                                                      (Modeling Expert & S   ME)
                                                                                                                                                                              Is there a reprensentation of the order of things                                                  behavior (acc uracy)?
                                                                                                                                                                          used for the model in the theorie of the applic ation
                                                                                                                                                                                                                     dom ain?

Com mum ication                                                                                                                                                  Is the loss of ac curacy c aused by the choswn abstraction of     Are all elements of interest explainded by the comm unic ative          Does the formal desc ription identify sim ilar models from other                                                                                                              Ist the demanded output data produced with the demanded
                                                                                            Struc tured Problem Desc ription                                                                              objects ac ceptable?                                                                model?                                                 application domains?                                                                                                                                                                ac c urac y?                                                                                                   Task:
     diffic ulties                                                                                                                                                   Did you c onsider all consequences due to non-modeling            Can you still ac c ept the results of the system analysis after
                                                                                            (i. e. detailed projec t desc ription,                                                                       spec ific c omponents?                                   having form ulated them precisely?                                                                                                                                                                                                                                                                                                                                                    Are all objects of interest and their
                                                                                                                                                                Do the c hosen attributes describe the object exac tly enough?            Is the models conc ept transparent to the sponsors SME?
                                                                                            ac c urac y Needs)
                                                                                                                                                                                                                                                Are the objec ts attributes appropriate to handle the                                                                                                                                                                                                                                                           Leistungs-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        interac tions with the environment
                                                                                                                                                                                                                                              interactions or do they need further / other attributes?
                                                                                                Do modeling experts and system m atter experts mean the
                                                                                                                                                                                                                                         Do the objects reflec t the dynamic c hange of their state                                                                                                                                                                                                                                                                                                                                                     desc ribed exactly enough?
                                                                                                                                                                                                                                                                                 accurately enough?
                                                                                                         same when reffering to the spec ified elem ents?
                                                                                                               Are the accuracy needs exactly defined?
                                                                                                                                                                                                                                      Is the system border well chosen (influenc es from outside)?                                                                                                                                                                                                                                                                                                                                                      (Modeling Expert & SME)
                                                                                                                                                                                                                                    Does the comm unicative model explain made assumptions                                                                                                                                                                                                             Is the model useful for the intended examination?
                                Sponsor Needs                                                          Are all relevant fac tors of influenc e concidered?
                                                                                                                                                                      Was it possible to integrate all demanded objec ts and
                                                                                                                                                                               attributes into the object hierarchy “naturally”?                           and the limits of the m odel to the user?                                                                                                                                                                                                                         Is m odel behavior credible?
                                                                                    Is the difference between the sponsors intention and the project              Does the (non-)consideration and c lassific ation of spec ific   Does it explain why certain fac tors of influenc e not have been
                                (i. e. c om m unic ated question)                                                                desc ription acc eptable?                                                                                                                                 modeled?
                                                                                          Is the originally intendet exam ination done in this simulation
                                                                                                                                                                                      elements make sense in the users eyes?
                                                                                                                                                                 Does the user know the statical dependenc ies from his daily      Do users believe the m odeled dependencies to make sense?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Ok, folgendes Problem ist
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      aufgetreten. Es ist die Frage
                                                 Did we recognize the true problem?                                                                 projec t?                                                                        Is the lac k of the non-modeled interac tions ac ceptable for
                                       Do we really want to exam ine T    HISproblem?HLA: Are other possible application domains regarded (S             OM:
                                                                                                                                                                                                                                                                                         model use?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     aufgekomm en, wie sich dieses
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Fahrzeug bei sc hnell
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Are the users needs satisfied?
                      Whic h other dom ains is the simulation of this problem relevant                                               Attr. Update, Ac c ept)?
                                                                                   for?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (User)

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 DB 11. 3. „99

J.Lüthi, UniBwM                                                                                                                                                                                                                                                                                                                                         OR - Models                                                                                                                                                                                                                                                                                                                       20
       Model Building - Summary
        Input                                                Phases                  Results
    Unsharp Problem                                     Problem Definition           Questions

                         Verification & Validation                                Communicative
      Observations                                       System Analysis             Model

        Modeling                                       Model Formalization         Formal Model

   Solution Technique                                     Implementation

    Input Parameters /                                                            Numbers, Graphs,
                                                     Analysis / Experimentation
      Distributions                                                                  Tables, ...

J.Lüthi, UniBwM                                               OR - Models                            21
OR Models
        Introduction
        Model Building

        OR Models

            Optimization models
            Prediction models
            Experimentation models
        OR Model Classification

                 Optimization models:
                   Deriveoptimal parameter values directly
                    from mathematical representation of the
                 Prediction models:
                   Derive predicted output (not necessarily
                    optimal) from math. Representation
                 Experimentation models:
                   Produce   output by imitating the real system

J.Lüthi, UniBwM                     OR - Models                 23
Optimization Models
      Optimization Models
       Differential Calculus
       Linear Programming

       Decision Trees (not covered)

       Game Theory

      Prediction Models
      Experimentation Models
        Differential Calculus
                 Formulate output measure of interest as
                  differentiable equation:
                          m( x1,, xn , v )  opt!
                 Differentiate equation:
                             ( x1 ,, xn , v )
                 Find extreme (optimal) values:
                             ( x1 ,, xn , v )  0
J.Lüthi, UniBwM                    OR - Models              25
        Linear Programming - Concept
                 Find extreme value of linear function:

                     c x
                      j 1
                             j    j    min! (or max!)
                 Linear boundary conditions:

                     a x
                      j 1
                             ij   j    bi   ( i  1,, k )

                     a x
                      j 1
                             ij   j    bi   ( i  k 1,, l )

                     a x
                      j 1
                             ij   j    bi   ( i  l 1,, m)

J.Lüthi, UniBwM                              OR - Models         26
        Linear Programming - Solution

                 Simplex Algorithm

                   Transform  problem into standard form
                   Linear boundary conditions describe a
                    special convex set
                   Finite number of corner points
                   Optimal solution must be at a corner
                   Simplex algorithm finds optimal point in a
                    finite number of steps

J.Lüthi, UniBwM                    OR - Models                   27
        Linear Programming - Example

                 Refinery Production
                   Various  products
                   Multistage production line
                   Capacities for production stages
                   Different qualities of products
                   Market conditions (minimum and maximum
                    production for certain products)
                   Linear relations
                   Find optimal product mix!

J.Lüthi, UniBwM                 OR - Models              28
        Linear Programming - Discussion

                 Very large systems can be solved
                  (thousands of equations)

                 Standard method for many problems in
                   Production optimization
                   Transport problems

J.Lüthi, UniBwM                   OR - Models            29
        Linear Programming - Problems

                 Nonlinear relations may exist
                   Approximation using linear equations
                   Nonlinear programming: more complex,
                    often only numerical solutions exist

                 Problems with integer variables
                   Integer   optimization

J.Lüthi, UniBwM                      OR - Models           30
        Game Theory

                 Modeling reaction in conflict situations:
                  e.g. in economy, politics
                 Conflict partners have different strategies
                  to choose from
                 For every combination of chosen strategies
                  a certain payoff is known
                 Assume rational behavior of conflict

J.Lüthi, UniBwM                   OR - Models               31
        Game Theory - The Model

                 Player A has strategies A1,...,AN
                 Player B has strategies B1,...,BM
                 If profit of A is loss of B: Zero-sum-game
                 Define payoff-matrices:

                    Fa          a1N   I F b          b1 M I
                  AG                  J BG

                                                           J
                                 
                    a         a MN
                                       J G
                                       K H b
                                                    
                                                        b K
                        M1                      N1        NM

J.Lüthi, UniBwM                   OR - Models                  32
        Game Theory - Solution

                 Consider strategy vectors x, y

                 Optimal strategies / strategy mixes can be

                 Equilibrium points

                 Evolutionary stable states

J.Lüthi, UniBwM                   OR - Models                  33
        Game Theory - Example

                 Problem: U.S. oil reserves vs. possible
                  OAPEC embargo
                 Model with various embargo strategies for
                   no embargo
                   180/270/360 days embargo with 25/50%
                    decreased oil exports
                 Various U.S. strategies for using the oil

J.Lüthi, UniBwM                    OR - Models                34
        Game Theory - Problems

                 Values in payoff matrices very difficult to
                  obtain or estimate

                 Difficult to find a set of strategies which is
                  sufficiently complete

J.Lüthi, UniBwM                     OR - Models                    35
        Game Theory - Example, continued

                 Given a fixed quantity for U.S. oil reserves,
                  optimal strategies for both conflict partners
                  can be computed
                 This was done for oil reserves from 0 to
                  1550 Mio. barrels and more than 100
                  scenario variations
                 Reasonable quantities for oil reserves
                  depending on U.S. oil imports from OAPEC
                  countries have been derived

J.Lüthi, UniBwM                   OR - Models                36
Prediction Models
        Optimization Models
        Prediction Models
         Task Graph Models
         Markov Models

         Forrester Models

        Experimentation Models
        Prediction vs. Optimization Models

                 Optimization models:
                    direct computation of optimal parameter
                 Prediction models:
                   direct computation of output measures
                  (not necessary optimal)
                 Prediction models can be used for numeric
                  optimization (simulation in the broader

J.Lüthi, UniBwM                  OR - Models               38
        Task Graph Models

                 Graphical representation of the tasks of a
                 Interdependencies of tasks
                 Timing of tasks
                           Task A
                                       tC Task C

                           Task B                 Task D
                  Start                                    End
                             tB                     tD
J.Lüthi, UniBwM                     OR - Models                  39
        Task Graphs - Analysis

                 Given a scheduled project delivery date
                 Compute:
                   earliestpossible starting time
                   latest possible starting time of tasks

                 Compute “critical path”: tasks for which
                  earliest equals latest possible time

                   delay of a task in critical path delays the
                  whole project!

J.Lüthi, UniBwM                    OR - Models                40
        Task Graphs - Example

                 Planing the production of an opera play
                 Tasks: e.g. preparation, concept, solo
                  rehearsals, ensemble rehearsals, choir
                  rehearsals, stage design, costumes
                 22 tasks were defined
                 Preparation, concept, solo rehearsals at
                  the beginning +8 more tasks at the end
                  were identified as critical

J.Lüthi, UniBwM                   OR - Models                41
        Task Graph Models - Discussion
                 Well-known and popular for planing large
                  projects such as e.g.:
                   Development    of the Alpha jet (Dornier)
                   Apollo moon missions
                   Mining projects
                   Building of the new railway track Hannover-
                 Difficult: appropriate degree of detail
                 Possible extension: associated cost

J.Lüthi, UniBwM                    OR - Models                42
        Markov Models

                 Define possible “states” S1,...,SN of a
                 Observe changes of the system in given
                  discrete time steps t
                 Consider transition probabilities pij:
                  probability that within the next time step the
                  system will change to state j, given that it is
                  currently in state i
                 Build transition probability matrix

J.Lüthi, UniBwM                    OR - Models                 43
        Markov Model - Example
                 2 machines A, B: working or out of order
                 Repair policy: if A and B out of order, A has
                  repair priority

                                  A and B       1
                  2                                         3
                    A OK,                             B OK,
                  B out of o.                       A out of o.

                                  A and B
                                  out of o. 4

J.Lüthi, UniBwM                   OR - Models                     44
        Markov Model - Example, continued

                     Transition probability matrix:
                          From/to   State 1   State 2   State 3   State 4
                          State 1   0.926     0.024     0.049     0.001
                  P=      State 2   0.190     0.760     0.010     0.040
                          State 3   0.195     0.005     0.780     0.020
                          State 4   0         0         0.200     0.800

                  P2 ... Matrix for the transition probabilities between
                         time t=0 and time t=2
                  P3 ... Matrix for the transition probabilities between
                         time t=0 and time t=3, etc.

J.Lüthi, UniBwM                         OR - Models                         45
        Markov Model - Steady State
                 Compute steady state probabilities:

          lim P    n
          n 
             From/to     State 1   State 2       State 3   State 4
             State 1     0.696     0.074         0.193     0.037
             State 2     0.696     0.074         0.193     0.037
             State 3     0.696     0.074         0.193     0.037
             State 4     0.696     0.074         0.193     0.037

                 From that, mean time in states, etc. can be
J.Lüthi, UniBwM                    OR - Models                       46
        Markov Models - Continuous Time

                 Discrete time steps Dt considered

                 Let Dt approach zero

                 Continous time Markov chains (CTMC)

                 E.g.: Queueing models

J.Lüthi, UniBwM                   OR - Models           47
        Queueing Models

                 Network of
                   Service   / Work Centers
                   Queues
                   Customer    / jobs / tasks

                 Mean service time of a job at a center

                 Open network: arrival rate of jobs
                  Closed network: number of jobs in system

J.Lüthi, UniBwM                     OR - Models              48
        Queueing Models - Examples
                 Open Model                         Closed Model


                                 CPU                          CPU

                            D1         D2                D1          D2


J.Lüthi, UniBwM                         OR - Models                       49
        Queueing Models - Analysis
                 Steady state measures for e.g.:
                   Mean queue lengths
                   Mean response times for centers
                   Mean system response time
                   Mean system throughput

                 Depending on restrictions:
                   Direct analytical solution
                   Numerical solution
                   Solution via simulation

J.Lüthi, UniBwM                    OR - Models        50
        Forrester Models
                 Based on “Control loops”
                 Symbols for:

              Varying quantities:              Quantity of interest

             „Valves“:                         Input flow or
                                               Output flow

             Influencing parameters:

             Sources, sinks:

J.Lüthi, UniBwM                        OR - Models                    51
        Forrester Models - Example

                 Extremely simplified population model
                 Assumptions
                   Onlymarried couples get children
                   Constant fertility and mortality
                   Constant “willingness” to marry

                 Quantities:
                   Population
                   Number   of married couples

J.Lüthi, UniBwM                   OR - Models             52
        Forrester Models - Example
                             to marry

                    Flow: number of                       Flow: number of
                  marriages in [t, t+1]                   births in [t, t+1]

 Number of married                         Fertility of                        Population
 couples (t=0: 250)                        married                             (t=0: 1000)

                                                          Flow: number of
                                                          deaths in [t, t+1]


J.Lüthi, UniBwM                                 OR - Models                                  53
Experimentation Models
       Optimization Models
       Prediction Models
       Experimentation Models
        Simulation in a broad sense
        Continuous Simulation

        Discrete Event Simulation
        Simulation in a Broad Sense

                 Use of optimization or prediction models for
                  experimentation, e.g.:

                   Coverage of whole parameter spaces
                   How-to-achieve models
                   What-if-models

                 Use of prediction models as objective
                  functions for numerical optimization

J.Lüthi, UniBwM                   OR - Models               55
        Actual Simulation

                 Why?

                   Optimization  and prediction models are
                    subject to certain restrictions
                    (e.g. Markov property, linearity)

                     such restrictions have to be violated (for
                   If
                    modeling reasons), the behavior of the
                    model can be simulated

J.Lüthi, UniBwM                    OR - Models                    56
        Continuous Simulation

                 Continuous models described as e.g.:
                   Systems  of equations
                   Ordinary or partial differential equations

                 If such systems cannot be solved
                  analytically, use numerical methods

                  contiuous simulation

J.Lüthi, UniBwM                    OR - Models                   57
        Discrete Simulation

                 State variables of the model are changed
                  at discrete points of time
                 Two major possibilities:
                   Time   driven simulation
                    e.g.: simulation of a discrete time Markov
                   Event driven simulation
                    e.g.: simulation of a continuous time Markov
                    model (e.g. queueing model)

J.Lüthi, UniBwM                   OR - Models                 58
        Time Driven Discrete Simulation

                 Simulation procedure:
                   Increase   virtual (simulation) time t:
                    t  t+Dt
                   Compute changes in interval [t, t+Dt]

                 Discussion:
                   Problem:   appropriate granularity of time
                   Well-suited for time-driven models

J.Lüthi, UniBwM                      OR - Models                 59
        Discrete Event Simulation - Data Structure

                 Virtual (or simulation) time VT
                 Time stamp ordered event list
                 State variables
              Virtual time    Event list           State variables

                   VT         E1   ...   t1        S1    ...   SN

                              En         tn

J.Lüthi, UniBwM                      OR - Models                     60
        Discrete Event Simulation - Algorithm

                 Simulation procedure:
                   Choose   scheduled event e with lowest time
                    stamp t
                   Set simulation time to VT = t
                   Update state variables S1,..., SN according
                    to event e
                   Insert new events in event list
                   Cancel events from event list
                   Remove event e from event list

J.Lüthi, UniBwM                    OR - Models                    61
        Deterministic vs. Stochastic Simulation

                 Deterministic simulation:
                   No uncertainty in parameters
                   No random behavior

                 Stochastic simulation:
                   Parameters  may be characterized as
                    random distributions
                   Behavior may include random choices

J.Lüthi, UniBwM                   OR - Models             62
        Stochastic Simulation - Distributions
             Characterize    model parameters (e.g. service
              time in a queueing model) as a random
             At each instant when this parameter is used in
              simulation, a sample value is drawn according
              to its distribution                  0,16



                                         density    0,1

            E.g.: normal (or Gaussian)             0,06

                  distribution                     0,04


                                                          0   2   4   6   8   10    12   14   16   18   20

J.Lüthi, UniBwM                          OR - Models                                                         63
        Stochastic Simulation - Random Numbers

                 Random variables are mathematical
                  objects, which „do not exist in computers“
                 Use random number generators
                  (deterministic series of numbers) which
                  show sufficiently „random“ behavior:
                   Uniformity
                   Independence

                 Using „bad“ random number generators
                  can make simulation results invalid!

J.Lüthi, UniBwM                    OR - Models                 64
        Stochastic Simulation - Statistics
                 Every simulation run is different !
                 Use multiple runs !
                 Results have to be analyzed using
                  statistical methods:
                   Mean   values and variances
                   Result distributions
                    (e.g. histograms for output measures)
                   Confidence intervals
                    (e.g. system throughput is in the interval
                    [10,15] with 95% certainty)
J.Lüthi, UniBwM                     OR - Models                  65
        Simulation Demonstration: Simul8

J.Lüthi, UniBwM           OR - Models      66
        Summary 1/3: Operations Research

                 Operations Research: mathematical
                  models of complex systems consisting of
                   humans
                   machines
                   material
                   capital

J.Lüthi, UniBwM                  OR - Models                67
        Summary 2/3: Model Building
                 Model building process, phases:
                   Problem  definition
                   System analysis
                   Model formalization
                   Implementation and integration
                   Analysis and/or experiments

                 During all phases:
                   Verification
                   Validation

J.Lüthi, UniBwM                    OR - Models       68
        Summary 3/3: OR Models
                 Optimization models, e.g.:
                   Differential
                   Linear Programming
                   Game theory

                 Prediction models, e.g.:
                   Task graphs (net-plan models)
                   Markov models
                   Forrester models

                 Experimentation models, i.e.: simulation

J.Lüthi, UniBwM                    OR - Models               69