محاضرات الهندسة الأمثل 5 و 6

Study Notes

The lecture covers Linear Programming (LPP) and its graphical method for optimization problems.
LPP is a method to achieve the best outcome (e.g., maximum profit or minimum cost) in a mathematical model with linear relationships.
Decision Variables represent the unknown information in a problem, having domains of possible values and constraints.
Objective Function, used in optimization problems, is a linear representation of the form Z = ax + by, where a, b are constraints, and x, y are variables to be maximized or minimized.
Constraints represent real-world limits on production capacity, market demand, or available funds, etc.

Step 1: Formulate the LPP: Define the objective function and constraints in mathematical form.
Step 2: Construct the Graph and Plot Constraint Lines: Create the graph in 'n' dimensions, where 'n' is the number of decision variables. Plot the constraint lines based on the constraint equations.
Step 3: Determine the Valid Side of Each Constraint Line: Identify the correct side of each constraint line by testing the coordinates of the origin (0, 0).
Step 4: Identify the Feasible Solution Region: Find the region which is satisfied by all constraints( the intersection of the valid regions).
Step 5: Plot the Objective Function: Draw a straight line for the objective function using a random constant value for the objective function in its equation, ensuring it's distinguishable from the constraint lines.
Step 6: Find the Optimum Point: Using a ruler aligned parallel to the objective function line, slide the ruler towards the origin or away from it depending on whether maximizing or minimizing the objective function. The optimum point lies at one of the vertices of the feasible region.
Step 7: Calculate the Coordinates of the Optimum Point: Calculate the coordinates of the points representing the optimum solution. If the optimum point is at the intersection of two constraint lines, algebraically solve the equations to find the coordinates. Finally, estimate the optimized value of the objective function using the determined values of the decision variables.

Examples of LPP are provided that include formulating the problem, developing constraint lines, solving for the objective function's optimized value, and illustrating how to solve linear program graphically. These examples cover a variety of scenarios, including those related to maximizing or minimizing profit, cost, or resources.