Question

Solve the LPP graphically: Minimize $Z = 5x + 10y$ subject to $x + 2y \le 120$, $x + y \ge 60$, $x, y \ge 0$.

Hint:

Remember: For minimization, check all corner points of feasible region.

Answer 1

Concept: In graphical method, we:

• Convert inequalities into equations
• Plot lines and find feasible region
• Evaluate objective function at corner points

Step 1: {Convert constraints to equations.}
\[ x + 2y = 120 \quad \text{(Line 1)} \] \[ x + y = 60 \quad \text{(Line 2)} \]
Step 2: {Find intercepts.}
For $x + 2y = 120$: \[ (120,0), \quad (0,60) \] For $x + y = 60$: \[ (60,0), \quad (0,60) \]
Step 3: {Feasible region.}
Region satisfies: \[ x + 2y \le 120, \quad x + y \ge 60, \quad x,y \ge 0 \]
Step 4: {Corner points.}
\[ A(60,0), \quad B(120,0), \quad C(0,60) \]
Step 5: {Evaluate objective function.}
\[ Z = 5x + 10y \] At $A(60,0)$: \[ Z = 5(60) = 300 \] At $B(120,0)$: \[ Z = 600 \] At $C(0,60)$: \[ Z = 600 \]
Step 6: {Minimum value.}
\[ Z_{\min} = 300 \text{ at } (60,0) \]
Step 7: {Conclusion.}
The minimum value of $Z$ is 300 at $(x,y) = (60,0)$.