91Ó°ÊÓ

In linear programming, the **objective function** is the formula you want to maximize or minimize.
It represents the goal of the problem.
For example, in a business scenario, it might be profit or cost.
In the given problem, the objective function is expressed as:

• \( z = 2x + 8y \)

Here, the coefficients (2 and 8) represent how much each unit of \( x \) and \( y \) contributes to \( z \).
Solving the linear programming problem involves finding the values of \( x \) and \( y \) that either maximize or minimize \( z \) according to given constraints.
Through testing feasible solutions within the region, we find where the highest and lowest values of the objective function occur.

**Constraints** are limitations or restrictions placed on the decision variables.
They form the boundaries of the feasible region in linear programming.
Constraints can include inequalities or equations.
In our exercise, the constraints are:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( 2x + y \leq 4 \)

These inequalities restrict the region to areas where both \( x \) and \( y \) are non-negative, and their linear combination is less than or equal to four.
The constraints can be visualized as lines on a graph, and the region where all these lines overlap is the feasible region where solutions can exist.

The **feasible region** in linear programming is the set of all possible points that satisfy all the constraints.
This region contains locations where potential solutions can be found.
Graphically, the feasible region can be identified as the area of overlap when all the constraints are plotted on a coordinate plane.
In the given problem, the feasible region is defined by the intersection of the lines:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( y \leq -2x + 4 \)

The region formed in the first quadrant is the feasible region.
It includes the points

• \((0,0), (0,4), \text{ and } (2,0)\)

These corner points are significant in finding the maximum and minimum values of the objective function since these extreme values often occur at the vertices of the feasible region.

**Inequalities** form the backbone of linear programming problems.
They express the limitations and boundaries for the decision variables.
Inequalities not only tell us where the feasible region lies but also dictate its shape.
For the given exercise, the inequalities are:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( 2x + y \leq 4 \) which simplifies to \( y \leq -2x + 4 \)

The first two inequalities define that the region of interest is in the first quadrant of the coordinate plane.
The last inequality, \( y \leq -2x + 4 \), determines a boundary that slopes downward from the y-axis to the x-axis.
By shading the region that satisfies all three inequalities, we reveal the feasible region where solutions for the objective function can be found.
Understanding these inequalities is crucial, as they are the parameters within which all feasible solutions must lie.