Question

Solve each linear programming problem. Minimize \(z=3 x+4 y\) subject to the constraints \(x \geq 0, \quad y \geq 0, \quad 2 x+3 y \geq 6, \quad x+y \leq 8\)

Short answer

The minimum value of \( z \) is 9 at the point (3, 0).

Step-by-step solution

- Identify the Objective Function and Constraints

The objective function to minimize is given by: \[ z = 3x + 4y \]The constraints are: \[ x \, \geq \, 0 \]\[ y \, \geq \, 0 \]\[ 2x + 3y \, \geq \, 6 \]\[ x + y \, \leq \, 8 \]

- Identify the Feasible Region

The feasible region is the area on the graph where all the constraints overlap. Begin by plotting the constraints:1. Plot \( x \, \geq \, 0 \) and \( y \, \geq \, 0 \) which represent the first quadrant.2. Plot \( 2x + 3y \, = \, 6 \) by finding intercepts at \( x = 0, y = 2 \) and \( y = 0, x = 3 \).3. Plot \( x + y \, = \, 8 \) by finding intercepts at \( x = 0, y = 8 \) and \( y = 0, x = 8 \).

- Shading Feasible Regions

For each inequality:1. \( 2x + 3y \, \geq \, 6 \) - Shade the region above the line.2. \( x + y \, \leq \, 8 \) - Shade the region below the line.The feasible region is bounded by the intersection of these regions within the first quadrant.

- Identify Corner Points of the Feasible Region

Identify the corner points of the feasible region by finding intersection points between the constraints:1. Intersection of \( 2x + 3y = 6 \) and \( x + y = 8 \).2. Intersection of constraints with axes, i.e., points (0, 8) and (3, 0).

- Calculate the Objective Function at Each Corner Point

Calculate the value of the objective function at each corner point:1. At (0, 8): \( z = 3(0) + 4(8) = 32 \)2. At (3, 0): \( z = 3(3) + 4(0) = 9 \)3. At the intersection of \( 2x + 3y = 6 \) and \( x + y = 8 \): Solve these equations simultaneously to get intersection point (6, 2): \( z = 3(6) + 4(2) = 26 \)

- Identify the Minimum Value

Compare the values obtained:1. \( z = 32 \) at (0, 8)2. \( z = 9 \) at (3, 0)3. \( z = 26 \) at (6, 2)The minimum value of \( z \) is 9 at the point (3, 0).

Key concepts

Objective Function

In linear programming, every problem revolves around an objective function. This function is what you seek to either maximize or minimize in your problem. For the given exercise, the objective function is \(\text{z = 3x + 4y}\). Here, \(\text{z}\) represents the value you want to minimize, and \(\text{x}\) and \(\text{y}\) are the variables affected by constraints. Ensuring you accurately identify the objective function is crucial as it directs the outcome of the solution through calculations at various points in the feasible region.

Constraints

Constraints act as the limitations within which the objective function must be optimized. For this problem, we have four constraints:
- \(x \, \geq \, 0\)
- \(y \, \geq \, 0\)
- \(2x + 3y \, \geq \, 6\)
- \(x + y \, \leq \, 8\)
These constraints define boundaries on a graph. For example, \(x \, \geq \, 0\) and \(y \, \geq \, 0\) restrict the solution to the first quadrant. The constraints must always be satisfied simultaneously. Plotting them on a graph will reveal the area where these restrictions overlap, forming the feasible region.

Feasible Region

The feasible region is the intersection of all areas satisfying each constraint. Begin by plotting each constraint on a graph:
- The lines \(x \, \geq \, 0\) and \(y \, \geq \, 0\) form the first quadrant.
- Next, draw \(2x + 3y = 6\) by finding its intercepts at \(x = 0, y = 2\) and \(y = 0, x = 3\).
- Then, plot \(x + y = 8\) by its intercepts at \(x = 0, y = 8\) and \(y = 0, x = 8\).
Shade the regions that satisfy \(2x + 3y \, \geq \, 6\) (above the line) and \(x + y \, \leq \, 8\) (below the line). The overlapping shaded area is your feasible region. It contains all possible points that meet every constraint simultaneously.

Corner Points

Corner points of the feasible region are crucial as they potentially contain the optimal solution. These points are where the boundary lines of the constraints intersect. For the given problem, the important corner points are:
1. (0, 8) where \(x + y = 8\) meets the y-axis.
2. (3, 0) where \(2x + 3y = 6\) meets the x-axis.
3. The intersection of \(2x + 3y = 6\) and \(x + y = 8\), solved as intersection point (6, 2).
Calculate the objective function \(z = 3x + 4y\) at each corner point to identify where the minimum value occurs. This method ensures you find the most efficient solution that satisfies all constraints.