LOGOMaterials
Basic modeling conceptsMathematical modeling: basic
concepts Mathematical modeling: deterministic Mathematical modeling: stochasticParameter estimationVerification and validationUncertainty modeling Modeling decisionsSystems dynamicsCase studies
LOGOReferences
Murthy, D.N.P, Page, N.W, and Rodin, E.Y. (1990). Mathematical Modelling, Pergamon Press, Oxford.
Ross, S.M. Stochastic Processes, 2nd ed., John Wiley and Sons, Inc., Canada
Clemen, R.T. and Reilly, T. (2001). Making Hard Decisions with Decision Tools. California: Duxbury Thomson Learning.
LOGOWhat is a system?
Collection of one or more related objects Object: physical entity with specific characteristics
and attributes• Attributes parameters and variables• Parameters: attributes intrinsic to an object• Variables: attributes needed to describe interaction
between objects
Think system instead of single object!
LOGOSystem and its environment
The system studied is usually a subset of the bigger system Depends on the goal/objective of the study
System Variables Variables
Environment
Interaction between system and its environment is through the common variables
Similar case for interaction between objects
LOGOOpen vs closed system
Closed system: Objects within system don’t interact with other objects
of the super system
Open system: vice versa
Example: demand for soft drinks If the demand for the future only depends on the past
sales closed system If other variables such as population changes,
weather conditions, advertising are considered open system
LOGOStatic vs dynamic
Dynamic time dependent Example: rocket launch
• Rocket– Variables: position, relative velocity
• Earth • Interactions between objects: theory of dynamics
Static time independent Example: alloy selection
• Variables: coefficient of expansion for the alloy, method for production, supplier
LOGODiscrete vs continues time
A priori taken before analysis
Depends on the objective and the degree of detail required
Examples: Demand of a product is usually recorded as
discrete time River pollution (variable: pollutant concentration at
a certain point) is recorded in continuous basis
LOGOBlack box vs transparent box
Black box: inner structure of the system is ignored More interested in the interaction between system and its
environment Lack of knowledge of the inner structure Simplify the system description
Transparent box: describe all the objects within system and their attributes (variables and parameters)
Example: Manufacturer in the supply chain structure is considered as a
black box, only supply and demand are considered as entering and leaving variables
When designing a production schedule, manufacturer should be described in detail
• Need to know the inside process
LOGOWhat’s a model?
Representation of a system
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LOGOThe type of model will depend on
The question that is being asked (the problem objective)
The level of detail required The resource available (time, personnel,
computers, etc)
LOGOWhy don’t we just always build a detailed model? Models cost money
The wages of the engineer who builds the model The cost of other resources (computers, software,
company overhead)
Implication: In modeling there is always a trade-off between time
and detail
LOGOSo, we can simplify the model considerably, but …
We lose detail and accuracy The model becomes more limited in its application It may no longer be adequate for the problem
We should make our assumptions very clear to anyone who Use the model Use the result of the model
LOGOHow much detail do we need?
The purpose of modeling is to be able to answer questions and make decisions Once we have enough information to make the
decision, the model is adequate
The model is not reality We can never be 100% sure that our model gives a
perfect prediction of reality We should always attempt to indicate our confidence
in the result
LOGOAssumptions
Always try to justify assumptions With practical explanation With quick calculation to show that the neglected effects are
negligible
Only make enough assumptions to simplify the model to the level justified by the problem objectives
Too many assumptions might assume the answer as well guess work
NEVER assume data!!!
LOGOGood model?
Validate and verify Have someone else to review or check the
assumptions and results Sensitivity analysis
LOGOGood model?
Represents the actual systems Physical
• Scale down Pictorial Verbal Mathematical formulation Simulation
• Validated and verified
Adequate for the goal Focus on significant features only
LOGOSince the model is not reality….
The results are only as good as the model and data used (“garbage in garbage out”)
If the model doesn’t give a good description of reality, there is no point in optimizing a design based on it! Fix the model first
LOGOFirst questions to ask …
Have I solved this problem before? If so, do the same think again
Has someone else solved this problem? Look in textbooks, do a literature search, etc
Don’t waste time and money starting from scratch if someone has already solved the problem unless you have good reason to believe their model is not good
LOGOIf it’s a completely new problem …
Understand the system and its characteristics
Set objective Model formulation Validate Analysis
Adequate? If not revise the model
LOGOModel classification
Material or physical model
Non-material or formal model Focus on this model!
LOGOMathematical model
Symbolic representation involving an abstract mathematical model
Classification: static, dynamic, deterministic, stochastic
LOGOSimulation model
Imitation of real world system over time
Model is run instead of solved
Can be used as analysis tools for predicting the effect of change of the existing system and as a design tool to predict the performance of the new system under varying sets of circumstances
LOGOSimulation is needed when …
Dealing with complex systems
System is black box, only inputs and outputs to the system can be examined
New design or policy before implementation
Can be used to verify analytic solution