91Ó°ÊÓ

When tackling any linear programming problem, one of the first steps is identifying the feasible region. This is where all possible solutions to the problem exist, based on the constraints given. The feasible region is defined by the area where all the constraint inequalities overlap on a graph. Think of it as a boundary within which all valid answers lie.
In our exercise, pinpointing this region requires plotting the constraints given:

• Constraint 1: \( x + 2y \leq 6 \)
• Constraint 2: \( -x + y \leq x + 4 \)
• Constraint 3: \( x + y \leq 4 \)
• Non-negativity constraints: \( x \geq 0 \), \( y \geq 0 \)

Once graphed, you'll see a shape forming on the coordinate plane, which is known in this case as a pentagon. This pentagon represents the feasible region.
Understanding the feasible region is crucial, as it narrows down the area where you have to search for the optimal solution.

The objective function is the heart of a linear programming problem. It’s the formula you want to maximize or minimize. For instance, it could represent profit, cost, or distance. In this specific problem, the objective function is \( p = 2x + y \). This function specifies what you're trying to achieve—here, maximizing the value of \( p \).
Think of the objective function as the rule or guideline that directs your decision-making within the feasible region. Once the feasible region is identified, the task is to find the point(s) within that region that give the highest (or lowest) value of the objective function.
Each vertex of the feasible region is evaluated using the objective function to determine which one optimizes (in this case, maximizes) \( p \). This ensures that resources or gains are allocated or captured in the most beneficial way possible.

Constraint inequalities play a foundational role in defining the feasible region. They are mathematical expressions that set the conditions under which the objective function must operate. Each inequality represents a limitation or requirement, such as budget limits, resource availability, or time constraints. In our example, the constraint inequalities include:

• \( x + 2y \leq 6 \)
• \( -x + y \leq x + 4 \)
• \( x + y \leq 4 \)
• \( x \geq 0 \), \( y \geq 0 \)

These constraints are visualized on a graph, helping to determine the boundaries of the feasible region.
Solving these inequalities is about understanding how available resources or situations limit possible outcomes. They ensure that the solution is realistic and applicable within the defined scope. By keeping solutions within these boundaries, we make sure that any potential solution found is practical and applicable to the real world problem at hand.

Once the feasible region is established, vertices analysis becomes the tool for finding the solution. Vertices are the corner points of the feasible region polygon, formed by the intersection of the constraint lines. These points are key, as it is often at these locations that the objective function achieves its maximum or minimum value in a linear programming problem.
In our problem, the vertices are:

• A (0, 0)
• B (0, 3)
• C (2, 2)
• D (4, 0)
• E (4, -2)

To carry out vertices analysis, you substitute these points into the objective function, \( p = 2x + y \), and calculate the value of \( p \) for each. This step tells you which vertex gives you the optimal value. In our case, vertex D (4, 0) yields the highest value for the function.
Analyzing each vertex ensures that the best possible option is selected, taking full advantage of the graph's points where constraints intersect.