Eng345 Ch1 p

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Mathematical ModelingMathematical Modeling


� Define the problem and gather relevant data� Formulate a mathematical model to represent the

problem� Develop a procedure for driving solutions to the

problem� Test the model and refine it as needed� Prepare for the application of the model� Implementation

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Mathematical model

Numerical formulation

code verified?

Model design

Define purpose

Conceptual model

Computer program

Analyticalsolutions

Yes

No

Calibration

Verification

Prediction

Presentation of results

Comparisonwithfield data

Field data

Field data

Field data

Postaudit

includes sensitivity analyses

Mathematical ModelingMathematical ModelingDefine the Problem

� Mostly, problems described in vague and imprecise manner� It is the process of developing a well defined statement of

the problem� Defining the objective, constraints, inter relationships,

possible alternative actions� All variables should be considered� Privatizing the water network of a city, all parties should be

considered: the owner who desires profit; the employeeswho desire steady employment; the people who desire lowpriced and high quality water; the government which desirea continuous services and fair taxes

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Mathematical ModelingMathematical ModelingFormulate a Mathematical Model

� Put the problem in a form suitable for analysis� This form called models or idealized representation� Examples include: the law of motion. Chemical reactions,

etc.� Mathematical model is the set of equations that describe the

problem� The factors that affect the system output called variables, for

example, product type, price, production time, etc are thedecision variables that affect the profit of an organization


� The form that represent the relationship between thesedecision variables and the profit is called ObjectiveFunction

� Any restrictions applied to the variables called Constraints� So, mathematical model is to choose the variables that

maximize the objective function and respect the constraints� One of the most used models is the linear programming� In LP models: objective function and constraints are linear

functions� Sometimes, it is necessary to do some simplification of the

problem to make it solvable

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Let us consider construction site layout planning problem to locate the temporary facilities on site


� Constraints� Overlap constraints� Restricted area constraints� Site boundary constraints

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Solution Procedure

� Defining the steps of driving a solution� Writing a computer program� Using available software shells� Choosing among optimal solutions or applying heuristic

solutions


Model Testing

� The procedure used should be tested to make sure that it iserror-free

� Using benchmark problems� Trying problems solved previously using another procedure� This step is called model validation� Upon this validation, the model should be refined or

modified

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Procedure for Model Application

� After model testing and having an acceptable developedmodel, it is necessary to install a welldocumented systemfor how to apply the model.

� This step includes: the model, application procedure, itslimitations and any other necessary steps for implementation


Implementation

� The final step is to implement the developed system

� This is the most important step as it ensures that the modelhas been translated into an operating procedure

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ExampleExample� This model deals with the development and expansion of an

electric power system for a specific regionData Needed (gather relevant data)� The new power station will be sited not far from the grid

network of existing power lines� Demand over the next twenty years� The cost to build and operate various sizes of hydroelectric

plants, cool-fired electric plants, and nuclear power plants� Power losses along the segments of the network would

likely be important

ExampleExampleThe objective� The objective is to meet demands for power at the least total

cost where cost is the cost of building and operating theexpanded system of power plant

The constraints� Each city must be assigned sufficient power resources from

among all the plants, previously established or newly built.Decision Variables� Decision variables may be building a plant of specific type

or not. For example, decision variables may be 1 or 0. Avariable for a plant of type k at site i built to size j would be1 if such plant were established and 0 other wise

Eng345 Ch1 p

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