Understanding Linear Optimization Concepts

Oct 2, 2024

Chapter 12: Linear Optimization Models

Introduction to Optimization Problems

  • Optimization problems support and improve managerial decision-making.
  • Objective is to maximize or minimize a function known as the "Objective Function."
  • Constraints are restrictions in the process and can be linear or non-linear.

Examples of Optimization Problems

  • Production Scheduling: Minimize total production and inventory costs.
  • Investment Portfolio: Maximize return on investment.
  • Advertising Budget Allocation: Maximize advertising effectiveness.
  • Warehouse Shipping: Minimize total transportation costs.

Linear Programming (LP)

  • Also known as Linear Programs, used for better decision-making.
  • Applications: GE Capital, Marathon Oil Company.
  • Initially called "programming in a linear structure."

Case Study: Par Inc. Golf Bags

  • Manufacturing golf bags with operations such as cutting, dyeing, sewing, finishing, and packaging.
  • Production times vary between standard and deluxe bags.
  • Profit contributions: $10 per standard bag, $9 per deluxe bag.

Problem Formulation

  • Translating a problem into mathematical statements.
  • Guidelines:
    • Understand the problem thoroughly.
    • Describe objectives and constraints.
    • Define decision variables.
    • Write objective and constraints in terms of decision variables.

Constraints and Objective Formulation

  • Example constraints involve available hours for operations like cutting, sewing, etc.
  • Decision variables: Number of standard bags (S) and deluxe bags (D).
  • Objective: Maximize 10S + 9D.

Solving the Optimization Problem

  • A linear programming model involves linear functions of decision variables.
  • Feasible Region: Defined by constraints; solution found at extreme points.
  • Simplex Algorithm: Developed by George Dantzig, used for finding optimal solutions.

Using Excel Solver for Linear Programming

  1. Construct a "what-if" model.
  2. Set objective box to maximize profit.
  3. Change variable cells for decision-making.
  4. Add constraints and select solving method (Simplex LP).
  5. Generate report and retain solver solution.

Conclusion

  • Linear optimization is learned best through practice and examples.
  • Upcoming: Chapter 11.
  • Note: Excel solver and tools like Wolfram Alpha are recommended for solving and visualizing optimization problems.