To minimize the objective function $C = 2x + 2y$ subject to the given constraints, we'll follow these steps:

1. Identify the constraints:

• $y \leq 6$
• $4x - 2y \leq 8$
• $2x + 7 \geq 6$

2. Simplify the constraints if needed:

• The third constraint $2x + 7 \geq 6$ simplifies to $2x \geq -1$ or $x \geq -\frac{1}{2}$.

3. Write the constraints in standard form:

• $y \leq 6$
• $4x - 2y \leq 8$ or $2x - y \leq 4$
• $x \geq -\frac{1}{2}$

4. Graph the constraints and identify the feasible region:

• The inequality $y \leq 6$ is a horizontal line at $y = 6$, with the area below it included.
• The inequality $2x - y \leq 4$ can be rewritten as $y \geq 2x - 4$.
• The inequality $x \geq -\frac{1}{2}$ is a vertical line at $x = -\frac{1}{2}$, with the area to the right included.

The feasible region will be the area where all these constraints overlap.

5. Determine the vertices of the feasible region:

To find the vertices where the constraints intersect:

1. Intersection of $y = 6$ and $y = 2x - 4$: $6 = 2x - 4 \implies 2x = 10 \implies x = 5$ Vertex: $(5, 6)$

2. Intersection of $y = 2x - 4$ and $x = -\frac{1}{2}$: $y = 2\left(-\frac{1}{2}\right) - 4 = -1 - 4 = -5$ Vertex: $\left(-\frac{1}{2}, -5\right)$

3. Intersection of $y = 6$ and $x = -\frac{1}{2}$: Vertex: $\left(-\frac{1}{2}, 6\right)$

6. Evaluate the objective function at each vertex:

Substitute the vertices into the objective function $C = 2x + 2y$:

1. At $(5, 6)$: $C = 2(5) + 2(6) = 10 + 12 = 22$

2. At $\left(-\frac{1}{2}, -5\right)$: $C = 2\left(-\frac{1}{2}\right) + 2(-5) = -1 - 10 = -11$

3. At $\left(-\frac{1}{2}, 6\right)$: $C = 2\left(-\frac{1}{2}\right) + 2(6) = -1 + 12 = 11$

7. Conclusion:

The minimum value of $C$ is $-11$, which occurs at the vertex $\left(-\frac{1}{2}, -5\right)$.

Questions

1. How would the feasible region change if the constraint $y \leq 6$ was changed to $y < 6$?
2. What would happen to the optimal solution if the coefficient of $x$ in the objective function $C = 2x + 2y$ was increased?
3. How does the solution change if the constraint $2x + 7 \geq 6$ is replaced with $2x + 7 = 6$?
4. How can you verify the feasibility of a solution in a linear programming problem?
5. What would be the effect of changing the objective function to $C = 2x + 3y$?

Tip

In linear programming, always ensure that you clearly define the feasible region by graphing the constraints, as the minimum or maximum values of the objective function will always occur at one of the vertices of this region.