Lecture Notes: Linear Programming with Sir Rob

Introduction to Linear Programming

• Linear programming, also known as linear optimization, is a method to achieve the best outcome in a mathematical model represented by linear relationships.
• Key goals: Maximize profit or minimize cost.
• Importance of professionalism and effective documentation in financial management.

Key Topics Discussed

• Definition of Linear Programming

  • Optimization: Making the best use of resources.
  • Applications in maximizing profits, minimizing costs, and efficient resource utilization.

• Common Terminology in Linear Programming

  • Decision Variables: Quantities to be determined for problem-solving (independent variables).
  • Objective Function: The function to be maximized or minimized (e.g., profit).
  • Constraints: Limitations or restrictions on decision variables (e.g., resource availability).
  • Non-Negativity Restriction: Decision variables should be non-negative.

Applications of Linear Programming

• Personal and professional use: route optimization, project delivery, production scheduling, inventory policies, etc.
• Financial management: Optimal portfolio mix, loan allocations.
• Human resources: Manpower optimization, patrol assignments.

Example Problems

• Delivery Route Optimization

  • JNT rider with time constraints for package deliveries.
  • Simplified linear model to reduce complexity and find efficient routes.

• Chocolate Production Problem

  • Factory manufacturing dark and light chocolates with limited ingredients.
  • Objective: Maximize profit by determining optimal production quantities.

Solving Linear Programming Problems

• Graphical Method
  • Used for two-variable problems.
  • Plot constraints on an x-y plane to find feasible regions.
  • Identify optimal solution at intersection points of constraint lines.

Example Solution using Graphical Method

• Chocolate Factory Case
  • Decision variables: x (dark chocolate units), y (light chocolate units).
  • Objective function: Max Z = 6x + 5y
  • Constraints:
    • x + y ≤ 5 (milk constraint)
    • 3x + 2y ≤ 12 (choco constraint)
    • x, y ≥ 0 (non-negativity)
  • Solution: Produce 2 million dark and 3 million light chocolate bars for max profit.

Upcoming Assignments

• Solve similar problems using graphical method.
• Future sessions: Solving linear programming problems using MS Excel.

References

• Online references and supplementary videos provided in the group chat.

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