System dynamic (SD) modeling. Lisa Brouwers

Transcription

1 System dynamic (SD) modeling Lisa Brouwers

2 Agenda Modelling and simulation Different type of models System dynamics SIR in Vensim Verification and validation Modelling process (steps in a simulation project) Sid.

3 Modelling Modelling is a process of developing a representation of a system which includes only the elements and relationships that are relevant for the system behaviour. 1. Mental model model developed in human s head about how a certain system works 2. Conceptual model uses mathematical, graphical or any other means (text) for representing the mental model 3. Computer model model which can be represented using computers. It contains quantitative relations that provide basis for performing simulation. Sid 3.

4 System versus Model The model is a simplified representation of the system The simplifications made are also called abstractions Only major components and relationships are included in the model The level of abstractions should be moderate so that the system behaviour roughly corresponds to the model behaviour Modelling process Usually starts with a simple model including only the basic elements The model is expanded until it represents the system to be analyzed in a proper way Sid 4.

5 Compure simulation and analysis Simulation is a process of running a computer model to obtain one or more BOTGs and other output data Used when it is impossible, expensive or unethical to analyze the system in any other way Analysis consists of: Studying the results from the simulation Changing policies, interventions, and parameters to create multiple simulation runs (scenarios) Presenting the results obtained in a way that help decision makers understand the functioning of the system under different conditions and policies introduced Sid 5.

6 Different type of models Static or dynamic Macro Macro System dynamics Entire system Micro Discrete event models Agent-based models Microsimulation models Deterministic or stochastic Time representation Discrete or continuos Discrete (in agent models): synchronous or asynchronous update Software platforms or from scratch 6 Sid.

7 System Dynamics One way of modelling Holistic perspective Important components: Feedback Delays Stocks Flows Conceptual model: CLD Sid 7.

8 Sid 8.

9 Cause Effect Sequences A cause/effect sequence is a relation of at least two variables in the system The relationship is shown with an arrow between the variables When the increase (decrease) of one variable causes the increase (decrease) in the other one, the arrow is positive. Same direction When the increase (decrease) of one variable causes the decrease (increase) of the other one the arrow is negative. Opposit directions Motivation + Productivity Exercise - Weight Sid 9.

10 Causal Loop Diagrams (CLD) Diagrams used to model dynamic systems Use graphical way to present interactions that determine the dynamics of the system Show causal relationships and illustrate circular feedback within the system Present easily understandable conceptual model of how the system works Sid 10.

11 CLD Simple Examples Causal mapping is a tool for representing structure in complex systems Arrows indicate causal influence Immigration Population Emigration Crime rate Quality of transportations Quality of life Neighbourhood health clinics Closeness to nature Sid 11.

12 More complex example; the Growth or Decline of a Life Insurance Company Sid 12.

13 Feedback A process whereby an initial cause ripples through a chain of causation ultimately to reaffect itself Example: the shame loop (blushing + embarrassement) Embarassment + Blushing Sid 13.

14 Types of feedback loops Sid 14.

15 Positive and negative feed-back loops Sid 15.

16 Stocks - Examples Business models Debt Personnel Capital equipment Orders Customers Money Other models Population Buildings Amount of liquid (wine, water, oil,...) Inventory (of all sorts of items) Amount of work Sid 16.

17 Stock and Flow Graph Symbols Stock, level accumulation Inflow Outflow The rate of inflow or outflow Source/Sink Arrows showing causeeffect relationship Sid 17.

18 Stock and Flows Maps Stocks are presented as boxes with the name Inventory of certain items is a stock because it is accumulation of certain goods Flows are presented as valves (with source for the inflow and destination for the outflow) with a name Production is inflow for the inventory because with production the inventory increases. Shipments is outflow for the inventory because when shipping goods the inventory decreases Sid 18.

19 Units of Measurement Used for the variables in the model Help in building the models in consistent manner Make models clear to those using them Units for stocks and flows Units for the flows must be in units for the stock divided by the units of time selected Stock (liters), flow (liters/minute) or (liters/hour) Stock (SEK), flow (SEK/month) or (SEK/year) Stock (items), flow (items/month) or (items/year) Sid 19.

20 Stock and Flow Example Bank balance Sid 20.

21 21 S.I.R

22 Simple spread model Population is divided into three groups: S I R (susceptible) (infected) (recovered or removed) The flow between the groups are defined by differential equations Sid 22.

23 And fractions We have S(t), I(t) and R(t) - number of persons at each time The fraction (of total population) is also required: s(t), i(t) och r(t) Sid 23.

24 Typical graph from SIRsimulation susceptible (S) Fraction of pop (N) recovered(r) infected (I) Simulation was run for 40 days, peak after approx. 80 days Sid 24.

25 What we need to know/assume? Size of population N Initially number of infected I(0) No of days an infected is infectious recovery rate k is 1/ no of infectious days No of contacts per day b - Of these we only want to consider the susceptible contacts: b * s(t) Sid 25.

26 How no of susceptible changes: ds/dt The number of (S) decreases with those who become infected every day/time unit. The number of susceptible depends on the number of infected persons I(t), and the number of susceptible contacts b * s(t) ds/dt = -b * s(t) * I(t) Sid 26.

27 How no of recovered changes: dr/dt The no of recovered (R) increases with those who recover every day/time unit. The no depends on the no of infected persons I(t), and the recovery rate k dr/dt = k * I(t) Sid 27.

28 How no of infected changes: di/dt Increase: The no of new infections depends on number of infectious persons I(t) and number of contacts between infectious and susceptible b * s(t). Decrease: The no of recoveries every day/time unit depends on no of infected persons I(t) and the recovery rate k. di/dt = b * s(t) * I(t) k * I(t) Sid 28.

29 Not 100 % risk A contact between an infectious and a susceptible person does not always imply a transmission. Therefore, you can add an infection risk Sid 29.

30 We can test this in Vensim SIR1.mdl Sid 30.

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32 Issues in Modelling and Simulation Projects Verification and Validation Steps in a modeling and simulation project Sid 32.

33 Verification and Validation Sid 33.

34 M&S - Advantages Non-existent systems can be studied Existent systems can be studied without disrupting their operation Hypotheses about various behaviour can be tested What if analysis Creating different scenarios, policies Sensitivity analysis Inspecting how sensitive is the model to changes in parameters Sid 34.

35 M&S - Disadvantages Requires special skills and training Appropriate software If not available, programming can be difficult Good input data is often difficult to find Collecting data is tedious and expensive process Does not always provide optimal solution Sometimes is considered as a non-scientific method The results obtained from the model cannot be used unless there is credibility into the model Sid 35.

36 Are simulation models valid? A model should be developed for answering specific questions under specific conditions Limitations of the model should be known The validity of the model need to be evaluated for each question The process of validating and building the credibility in a model is costly and time consuming Sid 36.

37 What is Verification &Validation? One of the most difficult problems in simulation is to determine whether a simulation model is an accurate representation of the actual system being studied Verification and validation are two processes that help estimate the validity of the model The simulation model has credibility if the manager and the other key personnel has accepted it as enough close representation of the system Sid 37.

38 Verification Concerned with building the conceptual model correctly, translating it in the right manner into a computer model, implementing good input data Questions asked Is the model implemented correctly in the computer? Are the input data and logical structure of the model correctly represented? Sid 38.

39 How to verify the model? Have the code checked by someone other than the programmer Make a flow diagram that shows the model logic Examine the model output for a variety of input parameters Document each part of the model extensively Sid 39.

40 Validation Concerned with building a model that is accurate representation of the system. Usually includes calibration of the model through an itterative process whenever possible. Questions asked Is the model representing the behaviour of the real system? Are there discrepencies between the real system and the model? Sid 40.

41 Validation Techniques Animation The model behaviour is displayed graphically Comparison to other models The output data can be compared with those from an already validated model (analytical or simulation model) Degenerate tests Testing degenerative behavior of the model (for example testing whether the size of a single server queue increases when average service time > average inter-arrival time Sid 41.

42 Validation Techniques (cont.) Historical data validation The model is tested using historical data and the behaviour is compared with the real system behaviour Face validity Asking experts in the area about the validity of the model Confidence interval testing Check whether extra runs of the simulation produce results that are within the confidence interval Sid 42.

43 Validation Techniques (cont.) Traces Behaviour of specific entities in the model is followed and compared to the real system Turing tests Indivduals knowledgable of the system being modeleled are asked whether they can distinguish between the model and the system Many others Sid 43.

44 How to validate a Model? Four approaches depending on who does V&V: System analyst team performs verification and validation Subjective approach Users, management who ordered the project Can also be subjective since they are involved External experts Independent validation increases the credibility Apropriate for models with higher cost Scoring the model (seldom used in practice) A method for scoring different categories for a particular model is developed The model is valid if all categories have a passing score Sid 44.

45 Model Credibility Model credibility Concerned with the confidence users have to use information obtained with the model in practice All aspects of the model need not be with the same credibility It is easier to determine if the model is not accurate than prove the accuracy of the model Sid 45.

46 Model Building, Verification, Validation, Calibration, Credibility Validation Verification Real Syste m Analysis & Data Conceptual model Transfer to Computer Simulation model Make model runs, calibrate Validation Results used in decision making Establish credibility Sell model and results to policymakers Operational model Sid 46.

47 Steps in a Simulation Project Sid 47.

48 List of the Steps in a Simulation Project Sid Problem Definition 2. Project Planning 3. System Definition 4. Conceptual model formulation 5. Preliminary experimental design 6. Input data preparation 7. Model translation 8. Verification and Validation 9. Final experimental design 10. Experimentation (Sensitivity analysis) 11. Analysis and Interpretation 12. Implementation and documentation

49 Steps in a Simulation Project 1. Problem Definition Crucial for the success of the project Usually involves several parties (the manager and the analyst, the sponsor and the project team) Provides answers to the following questions: Why is the problem studied? What is the goal of the study? What questions should be answered? 2. Project Planning Plan the resources for the project (people, time, hardware, software, communication,...) Sid 49.

50 Steps in a Simulation Project (cont.) 3. System Definition Identifying elements of the system which are relevant to the problem Excluding details not interesting for the project Taking care not to oversimplify the system, so that it becomes trivial 4. Conceptual Model Formulation Dividing the system in subsystems Defining entities in the system and the material, as well as information flow Develop several models to choose from, if possible Sid 50.

51 Steps in a Simulation Project (cont.) 5. Preliminary experimental design Selecting measures of effectiveness Identifying the parameters that will vary for the What if analysis Investigate the number of samples to be carried through the experiment 6. Input data preparation Identifying whether the data needed are available Investigating whether the available data are valid Collecting data if necessary or using estimates Sid 51.

52 Steps in a Simulation Project (cont.) 7. Model translation Translating the conceptual model into a computer model Running the experiments 8. Verification and validation Checking whether the model operates as the analyst has intended Confirming that the model is representing the real-world system at a satisfactory level Calibrating the model if necessary Building credibility in the results obtained by the experiments Sid 52.

53 Steps in a Simulation Project (cont.) 9. Final Experimental Design Determining the parameters for the simulation Terminating or non-terminating system Stationary or non-stationary system Defining policies/scenarios 10. Experimentation Executing the simulation Perform sensitivity analysis Sid 53.

54 Steps in a Simulation Project (cont.) 11. Analysis and Interpretation Performing output analysis Drawing conclusions from the data generated from the simulation runs Presenting data in the way that the answers to the questions asked in problem definition can be easily seen 12. Implementation and Documentation Writing the report Presenting the work done in other format (conference paper, oral presentation) Using the results of the project Sid 54.