Question

Solve the following linear programming problem graphically:
Maximize \[ Z = 8000x + 12000y \] Subject to the constraints
\[ 3x + 4y \le 60 \] \[ x + 3y \le 30 \] \[ x \ge 0,\; y \ge 0 \]

Hint:

In graphical linear programming problems, the optimal value of the objective function always occurs at one of the corner points (vertices) of the feasible region.

Answer 1

Step 1: Write the given constraints.
The constraints of the linear programming problem are
\[ 3x + 4y \le 60 \] \[ x + 3y \le 30 \] \[ x \ge 0, \quad y \ge 0 \] Since \(x\) and \(y\) are non–negative, the feasible region will lie in the first quadrant.
Step 2: Convert inequalities into equations.
To draw the boundary lines, convert the inequalities into equations:
\[ 3x + 4y = 60 \] \[ x + 3y = 30 \] Step 3: Find intercepts of the first line.
For \(3x + 4y = 60\): If \(x = 0\), \[ 4y = 60 \] \[ y = 15 \] If \(y = 0\), \[ 3x = 60 \] \[ x = 20 \] Thus the line passes through points \[ (0,15) \quad \text{and} \quad (20,0) \] Step 4: Find intercepts of the second line.
For \(x + 3y = 30\): If \(x = 0\), \[ 3y = 30 \] \[ y = 10 \] If \(y = 0\), \[ x = 30 \] Thus the line passes through \[ (0,10) \quad \text{and} \quad (30,0) \] Step 5: Determine the feasible region.
The feasible region is obtained by satisfying all the inequalities simultaneously in the first quadrant.
The corner points of the feasible region are
\[ (0,0), \quad (0,10), \quad (12,6), \quad (20,0) \] Step 6: Find intersection of the two lines.
Solve \[ 3x + 4y = 60 \] \[ x + 3y = 30 \] Multiply the second equation by 3: \[ 3x + 9y = 90 \] Subtract the first equation: \[ 5y = 30 \] \[ y = 6 \] Substitute into \(x + 3y = 30\): \[ x + 18 = 30 \] \[ x = 12 \] Thus intersection point is \[ (12,6) \] Step 7: Compute the value of $Z$ at each corner point.
\[ Z = 8000x + 12000y \] At \((0,0)\): \[ Z = 0 \] At \((0,10)\): \[ Z = 12000(10) = 120000 \] At \((12,6)\): \[ Z = 8000(12) + 12000(6) \] \[ Z = 96000 + 72000 \] \[ Z = 168000 \] At \((20,0)\): \[ Z = 8000(20) = 160000 \] Step 8: Determine the maximum value.
The maximum value of \(Z\) occurs at \[ (12,6) \] Step 9: Final conclusion.
The optimal solution of the linear programming problem is \[ x = 12, \quad y = 6 \] and the maximum value of the objective function is \[ Z_{\max} = 168000 \] Final Answer:
\[ \boxed{x=12,\; y=6,\; Z_{\max}=168000} \]