Define the problem and gather relevant data Formulate a mathematical model to represent the problem Develop a procedure for driving solutions to the

Transcription

1 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 1

2 Define purpose Field data Conceptual model Mathematical model Analytical solutions Numerical formulation Computer program code verified? No Yes Model design Field data Comparison with field data Calibration Verification Prediction Presentation of results Field data Postaudit includes sensitivity analyses Define 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 employees who desire steady employment; the people who desire low priced and high quality water; the government which desire a continuous services and fair taxes 2

3 Formulate 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 the decision variables that affect the profit of an organization Formulate a Mathematical Model The form that represent the relationship between these decision variables and the profit is called Objective Function 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 3

4 Let us consider construction site layout planning problem to locate the temporary facilities on site Formulate a Mathematical Model Constraints Overlap constraints Restricted area constraints Site boundary constraints 4

5 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 is error-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 5

6 Procedure for Model Application After model testing and having an acceptable developed model, it is necessary to install a welldocumented system for how to apply the model. This step includes: the model, application procedure, its limitations 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 model has been translated into an operating procedure 6

7 Example This model deals with the development and expansion of an electric power system for a specific region Data 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 Example The objective The objective is to meet demands for power at the least total cost where cost is the cost of building and operating the expanded 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. A variable for a plant of type k at site i built to size j would be 1 if such plant were established and 0 other wise 7