An objective function is $p(x,y)=3x+9y$ and constraints are,

$x+y\le \mathrm{8,}$

$x+2y\le \mathrm{4,}$

$x\ge \mathrm{0,}$

$y\ge 0$

The maximum value of the objective function is:

Concept:

Objective function: \( p(x,y) = 3x + 9y \)

Constraints:

• \( x + y \leq 8 \)
• \( x + 2y \leq 4 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

Step 1: Identify the Feasible Region

First, find the intersection points of the constraints to determine the vertices of the feasible region.

1. Intersection of x + y = 8 and x + 2y = 4 :

   Subtract the first equation from the second:

   \( (x + 2y) - (x + y) = 4 - 8 \)

   \( y = -4 \)

   Substitute y = -4 into x + y = 8 :

   \( x = 12 \)

   However, y = -4 violates y \geq 0 , so this intersection is not feasible.

2. Intersection of x + y = 8 with y = 0 :

   \( x = 8 \), y = 0 . This gives the point (8, 0) .

3. Intersection of x + 2y = 4 with x = 0 :

   \( y = 2 \), x = 0 . This gives the point (0, 2) .

4. Intersection of x + 2y = 4 with y = 0 :

   \( x = 4 \), y = 0 . This gives the point (4, 0) .

5. Origin (0, 0) :

   This point satisfies all constraints.

The feasible vertices are (0, 0) , (4, 0) , and (0, 2) . The point (8, 0) is not feasible because it violates x + 2y \leq 4 .

Step 2: Evaluate the Objective Function at Each Vertex

1. At (0, 0) :

   \( p(0, 0) = 3(0) + 9(0) = 0 \)

2. At (4, 0) :

   \( p(4, 0) = 3(4) + 9(0) = 12 \)

3. At (0, 2) :

   \( p(0, 2) = 3(0) + 9(2) = 18 \)

The maximum value of the objective function is the largest value obtained from the feasible vertices, which is 18 at the point (0, 2) .