Linear Programming Problem

Given, and
Let
Now, table for




x
0
2
1


y
2
0
1





At
, which is true.
So, shaded portion is towards the origin.

The corner points on shaded region are and
At point
At point
At point
Hence, maximum value of is 6 at point .

Corner Points
z = 3x + 9y


(5, 5)
z = 60


(0, 10)
z = 90


(0, 20)
z = 180


(15, 15)
z = 180




Minimum value of z occurs at (5, 5)

- Statement (I) is true because in a Linear Programming Problem (LPP), the objective function is always linear. It is of the form , where are decision variables and are constants.
- Statement (II) is also true because in an LPP, the constraints are the linear inequalities on the decision variables, such as , which limit the possible values of the decision variables.
Thus, both statements are true.

Step 1: Understanding the Concept:
This question tests a theoretical property of Linear Programming Problems (LPP). The feasible region in an LPP is a convex polygon. The objective function represents a family of parallel lines.

Step 2: Key Formula or Approach:
Recall the Multiple Optimal Solutions theorem. If an objective function reaches its maximum (or minimum) value at two distinct corner points of the feasible region, then every point on the line segment joining these two corner points will also give the same maximum (or minimum) value.

Step 3: Detailed Explanation:
Let the objective function be . Suppose the maximum value occurs at two corner points, say and . This means and . The line representing the objective function passes through both and . Since the feasible region is convex, the entire line segment connecting and lies on the boundary of the feasible region. Any point on this line segment can be represented as a convex combination of and : Let's find the value of at any such point : Since is a linear function, we can distribute it: Substitute and : This shows that every single point on the line segment joining the two corner points gives the same maximum value. Since a line segment contains an infinite number of points, the maximum value occurs at infinitely many points.

Step 4: Final Answer:
The number of points at which occurs is infinity.

Step 1: Understanding the Concept:
In linear programming, the fundamental theorem states that if an optimal value (maximum or minimum) exists, it must occur at one or more corner points of the feasible region. If the maximum value occurs at two different corner points, it means the value of the objective function evaluated at these two points must be exactly the same.

Step 2: Key Formula or Approach:
1. Evaluate the objective function at the first given point .
2. Evaluate the objective function at the second given point .
3. Set these two expressions equal to each other because they both represent the same maximum value.
4. Solve the resulting equation to find the relationship between and .

Step 3: Detailed Explanation:
The objective function is given as . Let's find the value of at the corner point : Next, find the value of at the corner point : The problem states that the maximum value of occurs at both of these points. Therefore, the value of must be equal at these two points: Substituting our expressions: Now, solve for the relation between and . Subtract from both sides: Divide both sides by 5: This can be written as .

Step 4: Final Answer:
The required relation between and is .