GK's Maths Classes

Lecture on Solving Linear Programming Problems (LPP) using GeoGebra

Introduction

• Goal: Solve three types of LPP problems using GeoGebra software.
  • Bounded and Unique Solution
  • Unbounded and Infinitely Many Solutions
  • Unbounded and No Solution
• GeoGebra: Freely downloadable software useful for solving LP problems graphically.
• For science students (maths-only): Useful for double-checking answers.
• For Applied Mathematics students: Can be used as practical exercises.

Problem 1: Bounded and Unique Solution

1. Setting Up GeoGebra

   • Open GeoGebra software.
   • Use the input window to type the inequalities and constraints.

2. Input Constraints

   • 4x + y >= 80
   • x + 5y >= 115
   • 3x + 2y <= 150
   • x >= 0
   • y >= 0

3. Typing Equations for Constraints

   • 4x + y = 80
   • x + 5y = 115
   • 3x + 2y = 150
   • x = 0
   • y = 0

4. Finding Feasible Region

   • Hide inequalities.
   • Type a && b && c && d && e to find intersection and feasible region.
   • Identify feasible region (shaded area).

5. Finding Corner Points

   • Use the intersection tool to find intersection points of lines.
   • Mark intersections as points A, B, and C.

6. Objective Function and Optimization

   • Define objective function, P(x,y) = 6x + 3y.
   • Calculate P(A), P(B), P(C).
   • Identify maximum value: 285.
   • Corresponding to x = 40 and y = 15.

Problem 2: Unbounded and Infinitely Many Solutions

1. Input Constraints

   • 3x + 4y >= 8
   • 5x + 2y >= 11
   • x >= 0
   • y >= 0

2. Typing Equations for Constraints

   • 3x + 4y = 8
   • 5x + 2y = 11
   • x = 0
   • y = 0

3. Finding Feasible Region

   • Hide inequalities.
   • Type a && b && c && d to find feasible region.
   • The region is unbounded.

4. Finding Corner Points

   • Identify intersections as points A, B, and C.

5. Objective Function and Optimization

   • Define objective function, P(x,y) = 60x + 80y.
   • Calculate P(A), P(B), P(C).
   • Minimum value: 160 (at two points: B and C).
   • Define open half-plane using constraint 60x + 80y < 160.
   • Points overlap with feasible region confirming infinitely many solutions.
   • All points between B and C form solutions.

Problem 3: Unbounded and No Solution

1. Input Constraints

   • x + y >= 5
   • x + 2y >= 6
   • x >= 0
   • y >= 0

2. Typing Equations for Constraints

   • x + y = 5
   • x + 2y = 6
   • x = 0
   • y = 0

3. Finding Feasible Region

   • Hide inequalities.
   • Type a && b && c && d && e to find feasible region.
   • The feasible region is unbounded.

4. Finding Corner Points

   • Identify intersections as points A, B, and C.

5. Objective Function and Optimization

   • Define objective function, P(x,y) = -x + 2y.
   • Calculate P(A), P(B), P(C).
   • Maximum value: 1 (at point A).
   • Define open half-plane using constraint -x + 2y > 1.
   • Overlap with feasible region confirms no maximum value.

Conclusion

• For science students: Useful for enrichment and verification.
• For Applied Mathematics students: Can be used as practical records.
• GeoGebra makes visualization and solving LP problems straightforward.

Note: For further practice, try creating and solving your own LP problems using GeoGebra.