Question

The solution for minimizing the function $z = x + y$ under an L.P.P. with constraints $x + y \ge 2$, $x + 2y \le 8$, $y \le 3$, $x, y \ge 0$ is ______.

• A. at the point (0, 3)
• B. at the point (8, 0)
• C. at infinite number of points but bounded set
• D. at unbounded set

Hint:

If the objective function is exactly parallel to (a scalar multiple of) one of the boundary constraint lines, the optimal solution will ALWAYS lie along the entirety of that line segment, yielding infinite solutions!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In Linear Programming, the objective function $z$ is evaluated at the corner points of the feasible region defined by the constraints.

Step 2: Formula Application:
The objective function is $z = x + y$. Notice that the constraint $x + y \ge 2$ has the same coefficients for $x$ and $y$ as the objective function.

Step 3: Explanation:
The line $x + y = 2$ represents the lower boundary of the feasible region. For any point on this line segment within the feasible region, the value of $z$ will be exactly 2. Since this is the minimum value for the region (where $x+y \ge 2$), and the line segment contains infinitely many points, the minimum occurs at an infinite number of points.

Step 4: Final Answer:
The minimum occurs at an infinite number of points within a bounded set.