Understanding LPP Feasible Regions

In Linear Programming Problems (LPP), the feasible region is the set of all points that satisfy all the given constraints. These constraints are typically linear inequalities. The question provides three linear inequalities that define the feasible region.

Analyzing the Linear Programming Constraints

Let's examine each constraint given in the problem:

• Constraint 1: \( x \geq 7 \)
  This constraint represents all points where the x-coordinate is greater than or equal to 7. Graphically, this is the region to the right of the vertical line \( x = 7 \), including the line itself.
• Constraint 2: \( y \geq 4 \)
  This constraint represents all points where the y-coordinate is greater than or equal to 4. Graphically, this is the region above the horizontal line \( y = 4 \), including the line itself.
• Constraint 3: \( x + 2y \geq 8 \)
  This constraint represents all points satisfying the inequality \( x + 2y \geq 8 \). To graph this, we first consider the line \( x + 2y = 8 \). We can find two points on this line:
  • If \( x = 0 \), then \( 2y = 8 \implies y = 4 \). Point: (0, 4)
  • If \( y = 0 \), then \( x = 8 \). Point: (8, 0)
  Now, we test a point (like the origin (0,0)) to see which side of the line satisfies the inequality. Substitute (0,0) into \( x + 2y \geq 8 \): \( 0 + 2(0) \geq 8 \implies 0 \geq 8 \). This is false. Therefore, the feasible region for this constraint is the side of the line \( x + 2y = 8 \) that does not contain the origin (0,0).

Graphing and Identifying the Feasible Region

The feasible region of the LPP is the intersection of the regions defined by all three constraints. Let's visualize this:

• The region \( x \geq 7 \) is to the right of \( x = 7 \).
• The region \( y \geq 4 \) is above \( y = 4 \).
• The region \( x + 2y \geq 8 \) is above the line \( x + 2y = 8 \) (or away from the origin).

Consider the combined region of \( x \geq 7 \) and \( y \geq 4 \). This region starts at the point (7, 4) and extends infinitely upwards and to the right. Any point in this region has \( x \geq 7 \) and \( y \geq 4 \). Let's check if such points satisfy the third constraint \( x + 2y \geq 8 \).

If \( x \geq 7 \) and \( y \geq 4 \), then \( x + 2y \geq 7 + 2(4) = 7 + 8 = 15 \). Since \( 15 \geq 8 \), any point satisfying \( x \geq 7 \) and \( y \geq 4 \) also satisfies \( x + 2y \geq 8 \). This means the region defined by \( x \geq 7 \) and \( y \geq 4 \) is entirely contained within the feasible region of \( x + 2y \geq 8 \).

Therefore, the feasible region for this LPP is simply the intersection of \( x \geq 7 \) and \( y \geq 4 \).

Determining Boundedness and Feasibility

A feasible region is considered bounded if it can be enclosed within a finite circle. It is unbounded if it extends infinitely in one or more directions.

The feasible region defined by \( x \geq 7 \) and \( y \geq 4 \) starts at the point (7, 4) and extends infinitely towards positive x and positive y values. For instance, points like (10, 4), (7, 10), (100, 100) are all in this region and it continues indefinitely. Since it extends infinitely, this region is unbounded.

A feasible region is considered feasible if there is at least one point that satisfies all constraints. In this case, the region defined by \( x \geq 7 \) and \( y \geq 4 \) exists and contains infinitely many points (e.g., (7,4), (8,5)). Thus, the feasible region is feasible.

Conclusion on the Feasible Region

Based on the analysis of the constraints \( x \geq 7 \), \( y \geq 4 \), and \( x + 2y \geq 8 \), the resulting feasible region is the set of points satisfying \( x \geq 7 \) and \( y \geq 4 \). This region is unbounded and contains points, meaning it is feasible.

Revision Table: Key Concepts in LPP Feasible Regions

Additional Information on Linear Programming

Linear Programming is a mathematical method used to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Here are some related concepts:

• Objective Function: This is the function you want to maximize or minimize (e.g., \( Z = ax + by \)). The optimal solution of an LPP is found by evaluating the objective function at the corner points (vertices) of the feasible region, provided the region is bounded.
• Corner Points (Vertices): These are the points where boundary lines of the feasible region intersect. If a bounded feasible region exists, the optimal solution occurs at one of its corner points. For an unbounded region, an optimal solution might exist at a corner point if the objective function does not increase/decrease indefinitely over the region.
• No Feasible Region: Sometimes, the constraints contradict each other, resulting in no points satisfying all constraints. In this case, the feasible region is empty, and the LPP has no solution.