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Operations Research Models Johannes Lüthi Universität der Bundeswehr München luethi@informatik.unibw-muenchen.de http://www.informatik.unibw-muenchen.de/inst4/luethi 7.4.1999 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 Overview 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 methods 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 Models 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 Abstraction Modeling Real System Mathematical Model Analysis or Application Experimentation Real World Solution Model Solution Interpretation J.Lüthi, UniBwM OR - Models 7 Example: Continuous Model z(t) 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 ) dt J.Lüthi, UniBwM OR - Models 8 Example: Discrete Model Parking Observed state: number of cars State changes: cars entering and leaving Exit 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 Verification 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 Validation 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 Reasons 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 ? P Is input data representative? Crashs? 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 errors 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) S 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? Modell- Dokumen- 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? m tation “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 Objekt Inte based reproduc ed in an appropriate way? ra ktio Ob jekt Obje kt n 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 Components: 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- beschreibung 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 Suitability: 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: business? 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 Precise Unsharp Problem Problem Definition Questions Verification & Validation Communicative Observations System Analysis Model Modeling Model Formalization Formal Model Paradigm/method Executable Solution Technique Implementation Model 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 model 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: dm ( x1 ,, xn , v ) dv Find extreme (optimal) values: dm ( x1 ,, xn , v ) 0 dv J.Lüthi, UniBwM OR - Models 25 Linear Programming - Concept Find extreme value of linear function: n c x j 1 j j min! (or max!) Linear boundary conditions: n a x j 1 ij j bi ( i 1,, k ) n a x j 1 ij j bi ( i k 1,, l ) n 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 logistics 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 partners 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 AG J BG 11 J 11 G H a a MN J G K H b b K J M1 N1 NM J.Lüthi, UniBwM OR - Models 32 Game Theory - Solution Consider strategy vectors x, y Optimal strategies / strategy mixes can be found 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 OAPEC: no embargo 180/270/360 days embargo with 25/50% decreased oil exports Various U.S. strategies for using the oil reserves 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 values Prediction models: direct computation of output measures (not necessary optimal) Prediction models can be used for numeric optimization (simulation in the broader sense) J.Lüthi, UniBwM OR - Models 38 Task Graph Models Graphical representation of the tasks of a project Interdependencies of tasks Timing of tasks Task A tA 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- Würzburg Difficult: appropriate degree of detail Possible extension: associated cost J.Lüthi, UniBwM OR - Models 42 Markov Models Define possible “states” S1,...,SN of a system 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 working 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 computed 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 Arrivals CPU CPU D1 D2 D1 D2 Departures 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 Willingness 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) women Flow: number of deaths in [t, t+1] ... Mortality 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 model 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 steps 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 variable At each instant when this parameter is used in simulation, a sample value is drawn according to its distribution 0,16 0,14 0,12 density 0,1 0,08 E.g.: normal (or Gaussian) 0,06 distribution 0,04 0,02 0 0 2 4 6 8 10 12 14 16 18 20 sec 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.: calculus 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

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