Find the maximum value of the objective function G = 8x + 3y subject to x + y 6 3x + y 6 x 2

Solution to the Linear Programming Problem

We are given the objective function and constraints as follows:

Objective Function: $$G=8x+3y$$

Constraints:

1. $-x+y\le 6$
2. $3x+y\ge 6$
3. $x\le 2$
4. $x\ge 0,y\ge 0$ (implicit non-negativity constraints)

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Step 1: Rewrite the Constraints in Standard Form

To simplify the problem, we rewrite the inequalities in a more familiar form:

1. $-x+y\le 6$ can be rewritten as: $$y\le x+6$$

2. $3x+y\ge 6$ can be rewritten as: $$y\ge -3x+6$$

3. $x\le 2$ remains unchanged.

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Step 2: Find the Feasible Region

The feasible region is the area where all constraints are satisfied. To find the vertices of the feasible region, we solve the system of equations formed by the intersection of the boundary lines.

Intersection of $y=x+6$ and $y=-3x+6$: $$\begin{gathered} x+6=-3x+6 \\ 4x=0 \\ x=0 \end{gathered}$$ Substituting $x=0$ into $y=x+6$: $$y=0+6=6$$ So, the intersection point is $(0,6)$.

Intersection of $y=x+6$ and $x=2$: $$y=2+6=8$$ So, the intersection point is $(2,8)$.

Intersection of $y=-3x+6$ and $x=2$: $$y=-3(2)+6=-6+6=0$$ So, the intersection point is $(2,0)$.

Intersection with Axes:

• For $y=x+6$, when $x=0$, $y=6$.
• For $y=-3x+6$, when $x=0$, $y=6$.
• For $x=2$, when $y=0$, $x=2$.

The feasible region is bounded by the points:

1. $(0,0)$
2. $(0,6)$
3. $(2,0)$
4. $(2,8)$

However, we must verify which of these points satisfy all constraints. Testing $(0,0)$:

• $-0+0\le 6$ (True)
• $3(0)+0\ge 6$ (False)

Thus, $(0,0)$ is not feasible. The feasible vertices are:

• $(0,6)$
• $(2,0)$
• $(2,8)$

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Step 3: Evaluate the Objective Function at Each Vertex

We substitute the feasible vertices into the objective function $G=8x+3y$:

1. At $(0,6)$: $$G=8(0)+3(6)=0+18=18$$

2. At $(2,0)$: $$G=8(2)+3(0)=16+0=16$$

3. At $(2,8)$: $$G=8(2)+3(8)=16+24=40$$

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Step 4: Determine the Maximum Value

Comparing the values of $G$ at the feasible vertices:

• $G=18$ at $(0,6)$
• $G=16$ at $(2,0)$
• $G=40$ at $(2,8)$

The maximum value of the objective function is 40 at the point $(2,8)$.