91Ó°ÊÓ

In the world of linear programming, an "Objective Function" is what you are trying to either maximize or minimize. In simple terms, think of it as your goal. In our exercise, the goal is to maximize the function \( z = 2x + y \). Basically, the objective function translates into: how can we get the highest possible total by adjusting the values of \( x \) and \( y \)? This function is crucial because it guides where you should focus your efforts within the bounds of the given constraints.
Examining this function, we see that each unit increase in \( x \) results in a 2-unit increase in \( z \), while each unit increase in \( y \) leads to a 1-unit increase in \( z \). This difference in weight shows that adjusting \( x \) has a greater impact on increasing \( z \) than adjusting \( y \).

• Objective: Increase \( z \) as much as possible.
• Expression: \( z = 2x + y \).
• Weight: \( x \) has double the contribution compared to \( y \).

The term "Feasible Region" refers to the set of all possible points that satisfy the system of inequalities in a linear programming problem. It is a visual representation, usually depicted as a shaded area on a graph. This region outlines all plausible solutions given the problem's constraints.
For the exercise, the constraints are as follows:

• \( 3x + y \leq 15 \)
• \( 4x + 3y \leq 30 \)
• \( x \geq 0 \), \( y \geq 0 \)

Plotting these inequalities gives us a polygonal region on the graph. This region, the feasible region, is bounded by the lines of each constraint and the axes where \( x \) and \( y \) are non-negative.
Only the points inside or on the boundary of this region are potential solutions for the objective function. This is why identifying this area is crucial: it narrows the field of search for the optimal point to a finite selection of boundary points or vertices.

An "Optimization Problem" is at the heart of linear programming, where the goal is either to maximize or minimize an objective function given a set of constraints. You can think of it as finding the best "fit" or solution within a certain set of rules.
For our problem, the aim was to maximize \( z = 2x + y \) under specific conditions. Here, each constraint serves as a rule that the values of \( x \) and \( y \) must adhere to while trying to achieve the maximum possible value of \( z \).
This problem becomes a search for the point, among the feasible solutions, that yields the highest value for the objective function. The process involves evaluating the function at each vertex of the feasible region, where solutions are most likely to occur. This is rooted in the mathematical principle that for such linear problems, optimum values are found at the boundaries or corners of the feasible set. This method ensures that the best possible solution within the defined limits is found efficiently.