To solve the given linear programming problem, we will use the method of linear programming (LP) and determine the optimal values of $X_1$ and $X_2$ that maximize the objective function $Z = 3X_1 + 2X_2$ subject to the constraints:

Objective Function:

$Z = 3X_1 + 2X_2$

Constraints:

1. $2X_1 + X_2 \leq 2$
2. $3X_1 + 4X_2 \geq 12$
3. $X_1 \geq 0, X_2 \geq 0$

Step-by-Step Solution

1. Graph the Constraints: Convert each inequality to an equality and find the intercepts to graph the constraints.

2. Find Feasible Region: Identify the feasible region where all constraints are satisfied.

3. Determine Corner Points: The maximum or minimum value of the objective function in linear programming occurs at one of the corner points (vertices) of the feasible region.

4. Calculate the Objective Function at Each Corner Point: Evaluate the objective function $Z = 3X_1 + 2X_2$ at each corner point to find the maximum value.

Let's start by converting the inequalities to equations and find the points where these lines intersect the axes.

1. Graph the Constraints:

Constraint 1: $2X_1 + X_2 = 2$

• When $X_1 = 0$, $X_2 = 2$
• When $X_2 = 0$, $X_1 = 1$

Points: $(0, 2)$, $(1, 0)$

Constraint 2: $3X_1 + 4X_2 = 12$

• When $X_1 = 0$, $X_2 = 3$
• When $X_2 = 0$, $X_1 = 4$

Points: $(0, 3)$, $(4, 0)$

2. Find the Intersection of Constraints:

To find the intersection point of the two lines, solve the system of equations:

$2X_1 + X_2 = 2$ $3X_1 + 4X_2 = 12$

Solving these equations will give us the intersection point of the lines.

Let's compute this intersection point and evaluate the objective function at each corner point to find the maximum value.The intersection point of the two constraints is:

$(X_1, X_2) = \left(-\frac{4}{5}, \frac{18}{5}\right)$

Since this point includes a negative value for $X_1$, it is outside the feasible region (as $X_1 \geq 0$ and $X_2 \geq 0$).

3. Determine Feasible Region and Corner Points

Next, let's identify the feasible region by finding the corner points:

1. $(0, 2)$ from the constraint $2X_1 + X_2 = 2$
2. $(1, 0)$ from the constraint $2X_1 + X_2 = 2$
3. $(0, 3)$ from the constraint $3X_1 + 4X_2 = 12$
4. $(4, 0)$ from the constraint $3X_1 + 4X_2 = 12$

We check which of these points lie in the feasible region:

• The point must satisfy both constraints:
  • $2X_1 + X_2 \leq 2$
  • $3X_1 + 4X_2 \geq 12$

Let's evaluate these points to determine the feasible region.None of the corner points satisfy both constraints, indicating that there is no feasible region where all the constraints are simultaneously satisfied.

Conclusion

This linear programming problem has no feasible solution because there is no common region that satisfies both constraints:

• The constraint $2X_1 + X_2 \leq 2$ restricts the region to one side.
• The constraint $3X_1 + 4X_2 \geq 12$ restricts the region to another side, with no overlap between these two regions in the first quadrant.

Would you like to explore any other questions or need clarification on any steps?

Related Questions:

1. How do you solve a linear programming problem graphically?
2. What happens if there is no feasible region in a linear programming problem?
3. How can the Simplex Method be used to solve linear programming problems?
4. What are the applications of linear programming in real-life scenarios?
5. How do you handle linear programming problems with more than two variables?

Tip:

Always check the feasibility of your constraints first to determine if a solution is possible before attempting to solve the optimization problem.